= Subgroup correspondence theorem
{title2=$\{N\leq H\leq G\}\longleftrightarrow\{K\leq G/N\}$}
For a <normal subgroup> $N$ of $G$, projection to the <quotient group> gives a bijection between <subgroups> of $G$ containing $N$ and <subgroups> of $G/N$. The maps are $H\mapsto H/N$ and inverse image under projection. Their two composites are identities because each relevant subgroup is a union of cosets of $N$. They preserve inclusions and normality; consequently a <composition series> in a quotient lifts to a subnormal chain above $N$ with the same <composition factors>.
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