I need help with the following:

In this paper by Michio Suzuki, it is proved that for any finite group $G$, there is a solvable subgroup $S$ in $G$ such that $G=\langle S,g^{-1}Sg \rangle$ for some $g \in G$. This theorem also implies that every finite group has a solvable subgroup.

Question: Let $G$ be a finite group such that $G$ is not a direct product of non-abelian simple groups. Is it true that every composition series (or chief series) of $G$ must contain at least one solvable subgroup?