Suppose $K/F$ is a Galois extension. Suppose $E$ and $D$ are intermediate fields of $K/F$. Let $p$ be a prime and assume that the following hold: (i) $[E:F]$ and $[D:F]$ are powers of $p$, (ii) $p$ does not divide $[K:E]$, and (iii) $K/D$ is Galois.
Show that $E$ is a subset of $D$. Show that $D$ is a subset of $E$.
I've been stuck on this problem for a few hours. Obviously, I need to use the Fundamental Theorem of Galois Theory somehow, but I can't figure out how. Any help would be appreciated.
Edit: it was pointed out that there was a mistake. It should be $D$ is a subset of $E$.