Suppose I have a group G , and N is a normal subgroup of G. Then, suppose we have subgroups $H_{1}$ and $H_{2}$ of G such that $H_{1}$ is a normal subgroup pf $H_2$. Also, $H_2/H_1$ is a simple sub-quotient group of G(which means $H_2/H_1$ is just a simple group).
Then,prove that either we can find a simple subquotient group $K_2/K_1$ of N($K_2$ and $K_1$ are subgroups of N, and $K_1$ is a normal subgroup pf $K_2$) such that $H_2/H_1\cong K_2/K_1$, or we can find a simple sub-quotient group $M_2/M_1$ of group $G/N$ such that $H_2/H_1\cong M_2/M_1$($M_1$ is a normal subgroup of $M_2$, and $M_1,M_2$ are sungroups of G/N)
I think, if $H_2 \subset N$, then we are done since $H_1,H_2$ are subgroups of N. Similarly, if $H_1\supset N$, then we are also done, since $H_2/H_1\cong (H_2/N)/(H_1/N)$ and ($H_2/N),(H_1/N)$ are subgroups of G/N. But, now I do not know how to contonue to prove this question. Can someone help me solve this question?