Finite groups in which every maximal subgroup is nilpotent or normal or has $p^{\mathrm{\prime }}$-order

Let $G$ be a finite group and $p$ a fixed prime divisor of $\mathrm{\mid }G\mathrm{\mid }$. Combining the nilpotence, the normality and the order of groups together, we prove that if every maximal subgroup of $G$ is nilpotent or normal or has $p^{\mathrm{\prime }}$-order, then (1) $G$ is solvable; (2) $G$ has a Sylow tower; (3) There exists at most one prime divisor $q$ of $\mathrm{\mid }G\mathrm{\mid }$ such that $G$ is neither $q$-nilpotent nor $q$-closed, where $q\mathrm{\ne }p$.