Subgroup correspondence theorem ($\{N\le H\le G\}⟷\{K\le G/N\}$)

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.