Lattice characterizations of $p$-soluble and $p$-supersoluble finite groups

Let $G$ be a finite group, and let ${\mathcal L}(G)$ be the lattice of all subgroups of $G$. A subgroup $M$ of $G$ is called modular in $G$ if $M$ is a modular element (in the Kurosh sense) of the lattice ${ \mathcal L}(G)$, i.e., if (1) $\langle X, M \cap Z \rangle=\langle X, M \rangle \cap Z$ for all $X \leq G, Z \leq G$ such that $X \leq Z$, and (2) $\langle M, Y \cap Z \rangle=\langle M, Y \rangle \cap Z$ for all $Y \leq G, Z \leq G$ such that $M \leq Z$. If $A$ is a subgroup of $G$, then $A_{m G}$ is the subgroup of $A$ generated by all its subgroups that are modular in $G$. We say that a subgroup $A$ is $N$-modular in $G$ ($N\leq G$) if, for some modular subgroup $T$ of $G$ containing $A$, $N$ avoids the pair $(T, A_{mG})$, i.e. $N\cap T=N\cap A_{mG}$. Using these notions, we give new characterizations of $p$-soluble and $p$-supersoluble finite groups.