After proving Lagrange's theorem, we now use it to compare several subgroups lying one inside another. If $K$ is a subgroup of $H$ and $H$ is a subgroup of $G$, then the index from $K$ to $G$ can be computed in two stages. First count how many cosets of $H$ are needed in $G$, and then count how many cosets of $K$ are needed in $H$. The product gives the number of cosets of $K$ in $G$. This is the index multiplication formula, and it is one of the most useful counting tools for subgroup chains.

Let $(G,\circ)$ be a group. A $\textbf{subgroup chain}$ in $G$ is a sequence of subgroups $$\begin{equation*} K\leq H\leq G \end{equation*}$$ where $K$ is a subgroup of $H$ and $H$ is a subgroup of $G$.