The exercise is
Assume $H$ is a proper subgroup of the finite group $G$. Prove $G\neq \bigcup\limits_{g\in G} gHg^{-1}$, i.e., $G$ is not the union of the conjugates of any proper subgroup [Hint: Put $H$ in some maximal subgroup and use the preceding exercise.]
The preceding exercise is
Recall that a proper subgroup $M$ of $G$ is called maximal if whenever $M\leq H\leq G$, either $H=M$ or $H=G$. Prove that if $M$ is a maximal subgroup then either $N_G(M)=M$ or $N_G(M)=G$. Deduce that if $M$ is a maximal subgroup of $G$ that is not normal in $G$ then the number of nonidentity elements of $G$ that are contained in conjugates of $M$ is at most $(|M|-1)|G:M|$ .
The solution to the exercise can be found here: https://math.stackexchange.com/a/121534/987127
What is the "maximal subgroup" the hint is pointing me to? I don't believe $N_G(H)$ is maximal in general, which is what the solution used.
