If $H$ is a maximal subgroup of $G$, then the action of $G$ on the cosets of $H$ is primitive.

Question

If $H$ is a maximal subgroup of $G$, then the action of $G$ on the cosets of $H$ is primitive.

This is a corollary in my group theory lecture notes with no proof. My thought is the following (I don't know if my proof is correct):

If the action is not primitive, say $B$ is a proper block, let $Hg'\in B$, the stabilizer is $G(Hg')=\{g\in G : Hg'g=Hg'\}=g'Hg'^{-1}$.

Then $G(Hg')<G(B)<G$.

So $H<g'^{-1}G(B)g'<G$, which contradicts the maximality assumption.