How to prove that there are not more than five different Galois groups of irreducible separable polynomial of degree 4 Let $F$ be a field and $f\in F[x]$ a polynomial that is irreducible, separable and of degree $4$.

How do I prove that there are not more than $5$ different possibilities for the Galois group Gal($f$) (not counting isomorphisms)?

I found multiple answers/sites that determine exactly what the Galois groups can be. But those were a little too advanced for me, since I just got  introduced to Galois theory. I think that's also the reason why the question says 'not more than 5' instead of being more specific.
So maybe somebody can help?
Edit: @DietrichBurde has helped a lot here below by giving a source that says that Gal($f$) must be a transitive subgroup of $S_4$, of which there are only five. However, it's not clear for me why it must be a transitive subgroup of $S_4$.

A very detailed and elementary reference are the notes of Keith Conrad on Galois groups of cubics and quartics. In Section $3$ it is proved that we have exactly $5$ possibilities, for $Gal(f)$, namely the groups $S_4,A_4,D_4,C_4$ and $C_2\times C_2$, which are just the transitive subgroups of $S_4$. In Theorem $1.1$ it is proven that $f$ is irreducible if and only if $Gal(f)$ is a transitive subgroup of $S_4$. However, it is easy to see that there are at most $5$ transitive subgroups of $S_4$, see Section $3$.
