Bound on double coset index in terms of orders of group and subgroup

For general groups
Let $$G$$ be a group, and $$H$$ be a subgroup. Then, if $$d$$ denotes the cardinality of the fact about::double coset space of $$H$$ in $$G$$, i.e., the fact about::double coset index of $$H$$ in $$G$$, we have:

$$1 + \frac{[G:H] - 1}{|H|} \le d \le [G:H]$$.

When $$H$$ is a fact about::subgroup of finite index in $$G$$, the left inequality attains equality if and only if $$H$$ is a fact about::malnormal subgroup, and the right inequality attains equality if and only if $$H$$ is a fact about::normal subgroup of $$G$$.

For finite groups
Let $$G$$ be a finite group, $$H$$ be a subgroup. Then, if $$d$$ denotes the cardinality of the fact about::double coset space of $$H$$ in $$G$$, i.e., the fact about::double coset index of $$H$$ in $$G$$, we have:

$$1 + \frac{|G|}{|H|^2} - \frac{1}{|H|} \le d \le \frac{|G|}{|H|}$$

The left inequality attains equality if and only if $$H$$ is a fact about::malnormal subgroup (i.e., it is either equal to the whole group, or is a fact about::Frobenius subgroup in the group), and the right inequality attains equality if and only if $$H$$ is a fact about::normal subgroup of $$G$$.