$G/(M \cap N) \cong G/M \times G/N$ where $G=MN$ and $M, N$ normal in $G$.

Question

This is exercise 3.3.7 in Dummit and Foote. I was able to finish this exercise by defining a map $\phi: G \to G/M \times G/N $ by sending $g$ to $(gM, gN)$ and showing the kernel is $M \cap N$ and then conclude the result by the first isomorphism theorem. However, the hint given for the question is "Draw the lattice." so I was wondering if there is another way to do this exercise using the 2nd, 3rd or 4th isomorphism theorems.