Must a non-simple group have a normal Sylow subgroup?

Question

In class, one way we're taught to prove a group is not simple is to exhibit a normal Sylow subgroup.

I'm wondering if the converse is true, i.e. if a group is not simple, must it have a normal Sylow subgroup? I can't seem to prove this is the case but I haven't been able to come up with a counterexample yet either.

Answer 1

For a counterexample, consider $S_n$, $n\geq 5$. No $p$-subgroup is normal as the only proper nontrivial normal subgroup is the alternating group.

Answer 2

Counterexample: if $S$ is a non-abelian simple group, then certainly $S \times S$ is not simple and has no normal Sylow subgroups.