Let $G$ be a finite group, which is not necessarily solvable. I am trying to prove the following statement:
There exists a smallest normal subgroup $N(G) \lhd G$ such that $G/N(G)$ is solvable
I can prove this if I choose N(G) to be the intersection of all normal subgroups $N$ of $G$ such that $G/N$ is normal. This is because if $G/N$ and $G/M$ are solvable, then $G/(N \cap M)$ is solvable (and then proceeding with induction).
Am I always guaranteed to find at least one normal subgroup of $G$ such that this is true?