Steiner coset partitions for five mutually commuting subgroups

A Steiner coset partition of a group G with respect to pairwise distinct subgroups $$H_1$$ , ..., $$H_r$$ is a collection of pairwise disjoint cosets $$g_1H_1$$ , ..., $$g_rH_r$$ whose union is G. Motivated by the Herzog–Schönheim Conjecture, we have recently started exploring the groups that admit Steiner coset partitions. In previous work, we completely classified such groups for up to $$r=4$$ mutually commuting subgroups. In this paper, we extend the classification to $$r=5$$ mutually commuting subgroups.