Zassenhaus butterfly lemma

Let $A^{\prime }⊴A$ and $B^{\prime }⊴B$ be subgroups of one group. Put $C=A\cap B$ and $D=(A^{\prime }\cap B)(A\cap B^{\prime })$. Then $D⊴C$ and the first isomorphism theorem identifies bothwith $C/D$. Indeed $C$ maps onto either displayed quotient group, and its kernel in either case is $D$. This gives the matching factors used in the Schreier refinement theorem.