Local subgroup of finite group is contained in p-local subgroup for some prime p

Suppose $G$ is a finite group and $H$ is a local subgroup of $G$, i.e., it is the normalizer of some nontrivial solvable subgroup of $G$. Then, there exists a prime number $p$ and a p-local subgroup $K$ of $G$ such that $H \le K$.
Suppose $H = N_G(Q)$ for a solvable subgroup $Q$ of $G$. The idea is to find a prime $p$ such that $O_p(Q)$ (the p-core of $Q$) is nontrivial. Let $K = N_G(O_p(Q))$. We show that $H \le K$, completing the proof.