The isomorphism theorems show how quotients and images are connected. We now organize all subgroups involved in an epimorphism. The focus keyword correspondence theorem states that if $f:G\to G_1$ is onto, then subgroups of $G_1$ correspond exactly to subgroups of $G$ that contain $\ker f$. In the quotient case, subgroups of $G/N$ correspond exactly to subgroups of $G$ containing $N$. This theorem is a map of the subgroup structure after quotienting. Students often try to list subgroups of a quotient directly, but correspondence gives a systematic method.

Let $f:G\to G_1$ be an epimorphism and let $N=\ker f$. Then there is a one-to-one inclusion-preserving correspondence between $$\begin{equation*} \{H\leq G\mid N\subseteq H\} \end{equation*}$$ and $$\begin{equation*} \{K\leq G_1\}. \end{equation*}$$ The correspondence is $H\mapsto f(H)$, and its inverse is $K\mapsto f^{-1}(K)$. Moreover, $H\trianglelefteq G$ if and only if $f(H)\trianglelefteq G_1$.