Direct product of quotient groups

Question

Let $ G $ is a finite solvable group, Suppose $ H $ and $ N $ are minimal normal subgroups of $ G $. Then $ G/N \times G/H \cong G/N\cap H $ ?

Answer 1

This is false.

If $N=H$ you have $$G/H \times G/H \cong G/H$$ which is not possible for groups whose order is not prime ($G/H$ is not trivial and is finite).

In the case that $N \neq H$ you have by minimality that $N \cap H = 1$. Now, it is quite easy to see that $$G/N \times G/H \cong G/1 =G$$ implies that $$|N||H|=|G|$$

Recall the lemma

If $H \subset G$ is a minimal normal subgroup, then $H$ is a $p$-group for some prime $p$.

This implies that $|G|$ has at most two prime factors. So, every solvable group whose order has at least 3 prime factors forms a counterexample to what you said.