I was remeditating Sylow subgroups recently, after reading somewhere that it served as a partial converse to Lagrange's theorem. After a bit more pondering I started wondering if we can find the groups such that a "perfect" converse to Lagrange's theorem holds, i.e those groups such that if $n\mid |G|$ then $G$ has a subgroup of order $n$. Call such groups Lagrange groups.
Clearly $p$-groups and abelian groups are Lagrange groups. I am almost certain a product of two Lagrange groups is a Lagrange group. Beyond this, I did not know how to further my search for Lagrange groups, and if a complete classification is even doable.
Any help would be greatly appreciated, even just ideas to pursue, thanks in advance.