Let $G$ be a finite non-simple group such that all normal subgroups of $G$ are $p$-groups for a given prime $p$. Suppose also that the $p$-Sylow subgroups of $G$ are not normal. Is it true that $G$ is also a $p$-group? $G$ must not be a $p$-group if its $p$-Sylow subgroup is not normal. I am more curious if such a non-$p$-group exists. (See end of question)
So far, I have noticed that the intersection of all $p$-Sylow subgroups of $G$ is the largest normal $p$-subgroup of $G$. Hence, we know that this intersection is non-trivial. Further, I'm actually only interested in centreless groups but I can't draw any connection with $p$-groups from this other than the fact that this means that each $p$-Sylow subgroup is contained in the centralizer of some element of $g \in G$.
My updated question is does a group $G$ exist with the following properties. $G$ is non-simple, centre-less, non-$p$-group such that every normal subgroup of $N \leq G$ has order $|N|=p^k$ for $1\leq k \lneq n$ where $p^n$ is order of the Sylow $p$-subgroup of $G$.