Double coset index two implies at least square root size

For finite groups
If $$G$$ is a finite group and $$H$$ is a fact about::subgroup of double coset index two, then:

$$|G| \le |H|(|H| + 1)$$.

Equivalently:

$$|H| \ge \sqrt{|G| + \frac{1}{4}} - \frac{1}{2}$$