TITLE: Give an example of a group $G$ with $N,M \triangleleft G$ such that $N \cong M$, but $G/M \not \cong G/N$.
QUESTION [1 upvotes]: I'm trying to find  an example of a group $G$ with $N$ and $M$ normal subgroups such that $N \cong M$, but $G/M \not \cong G/N$.
Can anybody help me with this ?

REPLY [4 votes]: $N=G=\mathbb Z$, $M=2\mathbb Z$.