On one class of finite supersoluble groups

The following theorem is proved.
Theorem. If in a non-identity finite group $G$ every primitive subgroup has a prime power index, then $G=[D]H$, where $D$ and $H$ are Hall nilpotent subgroups of $G$ and $D$ coincides with the $\mathfrak{N}$-residual $G^{\mathfrak{N}}$ of $G$.