Let $|G|=p^7$ and $K\le G$ with $|K|=p^3$. Then $\exists M\le G$ of order $p^4$ such that $K \trianglelefteq M$.

Question

Let $G$ be a group with order $p^7$ and $K\leq G$ with $|K|=p^3$. Then there exists a subgroup $M$ of $G$ of order $p^4$ such that $K \trianglelefteq M$.

My immediate thought is to consider the normalizer $N_G(K)$. While I can prove that $N_G(K)\neq K$, I cannot assert that it has order $p^4$. I don't need a full solution, just a hint or two.

Thanks!