What is the smallest set of groups $S$, such that for any group $G$ there exists either $H \in S$ or $H = H_1 \times H_2 \times \dots \times H_n$ for $H_i \in S$ such that $G$ and $H$ are isomorphic.
If anything interesting can be said about them, I'd like to hear all cases : with and without infinite groups, with and without infinite products.
Notes. I realize that there may not be a unique such set. My own only idea is the set of all non-abelian groups and cyclic groups of prime order. Also, it should be mentioned that by "small" I mean $\subset$ relation.