Every subgroup is a direct factor iff direct product of elementary Abelian groups

Statement

The following are equivalent for a group:

1. Every subgroup of the group is a direct factor.
2. The group is either trivial or a direct product of elementary Abelian groups. In other words, it is a direct product of Abelian $p$-groups for possibly different primes $p$ where each $p$-group is an elementary Abelian group.

The implication (2) implies (1) requires the use of the axiom of choice.