about the Galois extension

Question

Suppose that $E_2/E_1$ and $E_3/E_2$ are two Galois extension, with the expension times m and n. i.e $[E_2: E_1]=m, [E_3:E_2]=n$. As is known to all, $E_3/E_1$ may not be a Galois extension. So, we consider the Galois closure $F$ of the extension field $E_3$. There exist an extension times $t=[F:E_1]$. My question is that if $p$ is a prime, and $p\nmid m, p\nmid n$, can we assert that $p\nmid t$?