I need help with the following:

In this [paper][1] 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 group. Is it true that every [composition series][2] (or [chief series][3]) of $G$ must contain at least one solvable subgroup? 


  [1]: https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Solvable%20Generation%20of%20Finite%20Groups&btnG=
  [2]: https://en.wikipedia.org/wiki/Composition_series
  [3]: https://en.wikipedia.org/wiki/Chief_series