Correspondence theorem

Statement

Let $G$ be a group and let $N$ be a normal subgroup of $G$. Then the map $K \mapsto K/N$ is a bijection from the set of subgroups $K$ of $G$ with $N \subseteq K$ to the set of all subgroups of $G/N$. Here $G/N$ is the quotient group. Let $K$ and $L$ be subgroups of $G$ that contain $N$. Then $K \subseteq L$ if and only if $K/N \subseteq L/N$. Also, $K$ is normal in $G$ if and only if $K/N$ is normal in $G/N$.

Remarks

The inverse map sends a subgroup $\bar{K}$ of $G/N$ to its preimage $\{g \in G : gN \in \bar{K}\}$. The bijection is an isomorphism of lattices of subgroups, and it keeps indices: $[L : K] = [L/N : K/N]$ when $N \subseteq K \subseteq L$. For example, the subgroups of $\mathbb{Z}/12\mathbb{Z}$ correspond to the subgroups $d\mathbb{Z}$ of $\mathbb{Z}$ with $d$ dividing $12$.

Explanations

L1 human

The subgroups of a quotient group $G/N$ are the same as the subgroups of $G$ that contain $N$. Passing to the quotient does not create new subgroups or lose old ones above $N$. So the picture of the subgroups above $N$ in $G$ is the full picture of the subgroups of $G/N$.