I'm trying to prove the following claims are equivalent:

1. Every simple group of odd order is of the type $\mathbb{Z}\_{p}$
for prime $p$
2. Every group of odd order is solvable.

Getting from 2 to 1 was easy but I'm having problem with the other direction. Obviously I only need to show that given 1 every non-simple group of odd order is solvable. So if I assume $G$ is a non-simple group of odd order then it has a non-trivial normal subgroup and this is where I get stuck. I'd appreciate a hint that will lead me towards the solution without giving it up completely :)

Thanks in advance!