The influence of non-c_p-normal maximal subgroups on the structure of the groups

We focus on the c_p -normality of maximal subgroups to classify second maximal subgroups. Moreover, the formation \widehat\mathcalF_p\text=\left\langle \left\G\left|  M\lessdot G,M\right.\right.\right.\textis p-solvable or\left| G\colon M\right| \textis \left.\left.\texta prime power\right\\right\rangle is defined and its structures is explored by considering various properties.