Sylow subgroup of a subgroup from double cosets

If $P$ is a Sylow subgroup of a finite group $G$ and $H\le G$, some intersection $H\cap xP{x}^{-1}$ is a Sylow subgroup of $H$. Otherwise every double-coset sizewould be divisible by a larger power of $p$ than $|G|$, as would their sum $|G|$.