  In this paper we consider the Fitting subgroup $F(G)$ of a finite group $G$
and its generalizations: the quasinilpotent radical $F^*(G)$ and the
generalized Fitting subgroup $\tilde{F}(G)$ defined by $\tilde{F}(G)\supseteq
\Phi(G)$ and $\tilde{F}(G)/\Phi(G)=Soc(G/\Phi(G))$. We sum up known properties
of $\tilde{F}(G)$ and suggest some new ones. Let $R$ be a subgroup of a group
$G$. We shall call a subgroup $H$ of $G$ the $R$-subnormal subgroup if $H$ is
subnormal in $ \langle H,R\rangle$. In this work the influence of $R$-subnormal
subgroups (maximal, Sylow, cyclic primary) on the structure of finite groups
are studied in the case when $R\in\{F(G), F^*(G),\tilde{F}(G)\}$.
