Finiteness is extension-closed

Statement
Suppose $$G$$ is a group with a normal subgroup $$H$$ having quotient group $$G/H$$. Then, if both $$H$$ and $$G/H$$ are finite groups, $$G$$ is also a finite group.

Quantitative version

 * Order of extension group is product of order of normal subgroup and quotient group

Facts used

 * 1) uses::Lagrange's theorem