Suppose G is a finite group, $\mathfrak{M}_G$ is the set of all its maximal proper subgroups. Is there any way to describe $\mathfrak{M}_{G \times G}$ - the set of all maximal proper subgroups of $G \times G$?
It is quite easy to show that for each $A \in \mathfrak{M}_G$ the subgroups $\{(x, y) \in G \times G | x \in A, y \in G\}$ and $\{(x, y) \in G \times G| x \in G, y \in A\}$ are maximal proper. However not all maximal proper subgroups are of those types: the subgroup $\{(g, g) \in G \times G | g \in G\}$ does not lie in any of the aforementioned subgroups, but a maximal proper subgroup, that contains it, definitely does exist. So the question remains unanswered...
Any help will be appreciated.
The remaining maximal subgroups of $G \times G$ have the form $ \{ (g,h) : \phi(gN_1)=hN_2 \}$, where $N_1$ and $N_2$ maximal normal subgroups of $G$, such that $\phi:G/N_1 \to G/N_2$ is an isomorphism (so $G/N_1$ and $G/N_2$ are isomorphic simple groups).
There is no need to assume that $G$ is finite.