If I have a group $G$ with two two subgroups $H$ and $I$ such that $G/H \approx G/I$, what can I say about the relationship between $H$ and $I$? Are they equal; isomorphic?


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Can a Quotient of a Group by Two Different Subgroups be Isomorphic?
There exists groups $G$ with normal proper non-trivial subgroups $H$ such that $G \cong G/H$. Even finitely presented ones, as the one given by [Higman, Graham. A finitely related group with an isomorphic proper factor group. J. London Math. Soc. 26, (1951). 59--61. [MR0038347](http://www.ams.org/mathscinet-getitem?mr=MR0038347)], namely the group $G$ freely generated by $a$, $b$ and $c$ subject to $$a^{-1}ca=b^{-1}cb=c^2,$$ with $H$ normally generated by adding the relation $$aca^{-1}=bcb^{-1}.$$


