non-abelian simple group

Question

Let G be a non-solvable group and $(\frac{G}{Z(G)})^{'}=\frac{G}{Z(G)}$. Can we say that there is a normal subgroup $N$ of $G$ with property $Z(G)\leq N$ such that $\frac{G}{N}$ is a non-abelian simple group ?

Answer 1

Choose $N \supseteq Z(G)$ maximal normal. Then obviously $G/N$ is simple. Assume $G/N$ is abelian, then $G' \subseteq N$. Since $(G/Z(G))'=G/Z(G)$, it follows that $G=G'Z(G)$. But then $G=N$, since $N$ contains $Z(G)$ and $G'$. $N$ was maximal and hence proper in $G$, a contradiction, so $G/N$ is non-abelian.