Let $G$ be a finite group and $L$ a maximal subgroup of $G$, then all minimal normal subgroups $N$ of $G$ that satisfy $N\cap L = 1$ are isomorphic.
By the Second Isomorphism Theorem, we have $LN/N\cong L/(N\cap L)=L$, where $LN$ is precisely $G$ since $N$ is normal and $L$ is maximal.
Therefore for all minimal normal subgroups $N_1, N_2$ of $G$, $G/N_1\cong L\cong G/N_2$. But I’m stuck here, does $G/N_1\cong G/N_2$ imply $N_1\cong N_2$? I believe that the minimality of $N$ as a normal subgroup must be the key to my question, but how to do it?
Any help is sincerely appreciated! Thanks!
PS: It’s exercise 1 on page 39 of my textbook, The Theory of Finite Groups, An Introduction.