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 $H$.
People write $(G:H)$, $[G:H]$ or $\lvert G:H\rvert$ to mean $\lvert G/H\rvert$. Why not just $\lvert G/H\rvert$?
If $H$ is a subgroup of $G$ and $K$ is a subgroup of $H$, then
\[\lvert G/K\rvert = \lvert G/H\rvert\cdot\lvert H/K\rvert.\]