Suppose that $K$ is the splitting field of some polynomial over a field $F$ of characteristic $0$.If $[K:F]=p^{2}q$, where $p$ and $q$ are distinct primes, show that $K$ has subfields $L_{1}, L_{2}$ and $L_{3}$ such that $[K : L_{1}]=p$, $[K :L_{2}]=p^{2}$ and $[K : L3] = q$.
I know from the Fundamental Theorem of Galois Theory that $[K:F]=|Gal(K/f)|=p^{2}q$ and from Sylow's First Theorem $K$ has subfields $L_{1}, L_{2}$ and $L_{3}$ such that $|L_{1}|=p, |L_{2}|=p^{2}$ y $|L_{3}|=q$ but idk how to show that $[K:L_{1}]=|Gal(K/L_{1})|=p$