Finite groups with systems of generalized normal subgroups

Let $G$ be a finite group and let ${\mathscr L}_{sn}(G)$ be the lattice of all subnormal subgroups of $G$. Let $A$ and $N$ be subgroups of $G$ and let $1, G\in {\mathscr L}$ be a sublattice in ${\mathscr L}_{sn}(G)$; i.e., $B\cap C$, $\langle B, C \rangle \in {\mathscr L}$ for all $B, C \in \mathscr L$. Then: $A^{{\mathscr L}}$ is the $\mathscr L$-closure of $A$ in $G$; i.e., the intersection of all subgroups in $ {\mathscr L}$ which includes $A$ and $A_{\mathscr L}$ is the $\mathscr L$-core of $A$ in $G$, i.e., the subgroup in $A$ generated by all subgroups of $G$ belonging to $\mathscr L$. A subgroup $A$ is an $N$-${\mathscr L}$-subgroup in $G$ if either $A\in {\mathscr L}$ or $A_{{\mathscr L}} < A < A^{\mathscr L}$ and $N$ avoids each composition factor $H/K$ of $G$ between $A_{{\mathscr L}}$ and $ A^{\mathscr L}$; i.e., $N\cap H=N\cap K$. Using these notions, we give some new characterizations of soluble and supersoluble subgroups and generalize a few available results.