Galois extension of degree 75 I'm studying Galois Theory and I stumbled upon this question: 

Let $E/K$ be a Galois extension of degree $75$. Show that there exists
  an intermediate subfield $K \subsetneq F \subsetneq E$ such that $F/K$
  is also a Galois extension.

I started like this: let $G = \text{gal}(E/K)$. Because $E$ is Galois over $K$, $|G| = 75$. Then, by the fundamental theorem of Galois theory, any intermediate field $F$ has the form $F = E_H$ for some subgroup $H$ of $G$ and $[E:F] = |H|$ while $[F:K] = [G:H]$. 
Any tips on how to progress further? Any help would be greatly appreciated.

Hint : It suffices to show that the Galois group has nontrivial normal subgroups. 
