Pi-series of a finite group

Definition
Let $$G$$ be a finite group and $$\pi$$ be a set of prime numbers. A $$\pi$$-series of $$G$$ is a subgroup series:

$$\{ e \} = H_0 \le H_1 \le \dots \le H_n = G$$

with the property that the index $$[H_i:H_{i-1}]$$ is either a $$\pi$$-number (i.e., all its prime factors are in $$\pi$$ or a $$\pi'$$-number (i.e., none of its prime factors is in $$\pi$$).