Let $G$ a finite group and suppose that $S(G)$$\ne$$1$ where $S(G)$ denotes the largest normal soluble subgroup of $G$. Suppose that $N$ is non-trivial normal soluble subgroup $N$$\ne$$1$, $N$$\vartriangleleft$ $G$ such that the quotient group $G$/$N$ has no nontrivial abelian normal subgroups. Can I conclude that $S(G)$$=$$N$ ? Is it correct? Thank you very much to everyone for the help!
Take $\;G=S_3\;,\;\;N:=A_3$ . Then $\;S(G)=G\;$ and $\;G/N\cong C_3\;$ has no non-trivial subgroups whatsoever.