If $G$ is a finite group and $N_i\trianglelefteq G$ ($i=1,2$) such that $G/N_i$ is supersolvable, then, is it true that $G/(N_1\cap N_2)$ is supersolvable?
One may give only hint; I will try to prove it. But, I am not sure about supersolvability of $G/(N_1\cap N_2)$.
Hint: $G/(N_1\cap N_2)$ is isomorphic to a subgroup of $G/N_1\times G/N_2$, hence supersolvable, because subgroups of supersolvable groups are supersolvable, and the direct product of supersolvable groups is supersolvable.
References:
Prove: $G/(N_1 \cap N_2)$ is isomorphic with a subgroup of $(G/N_1) \times (G/N_2)$
How is the direct product of two supersolvable groups again supersolvable?
