Let $G$ be a group and $H$ be a subgroup. The index of $H$ in $G$ is
\[\lvert G / H \rvert,\]the cardinality of the set of all left cosets of $H$.
For any three groups $K \leq H \leq G$,
\[\lvert G / K\rvert = \lvert G / H \rvert \cdot \lvert H / K \rvert.\]