Finite group in which all cumulative order statistics values divide the order of the group

Definition
A finite group in which all cumulative order statistics values divide the order of the group is a finite group $$G$$with the following property: for every natural number $$d$$, the number of elements $$g$$ such that $$g^d$$ is the identity element is a divisor of the order of $$G$$.

In other words, a finite group in which all the values in the cumulative version of the order statistics divide the order of the group. Thus, to evaluate whether this property holds for a group, we simply need to know the order statistics of the group.