Let
$G$
be a finite group and
$\unicode[STIX]{x1D70E}=\{\unicode[STIX]{x1D70E}_{i}\mid i\in I\}$
some partition of the set of all primes
$\mathbb{P}$
, that is,
$\mathbb{P}=\bigcup _{i\in I}\unicode[STIX]{x1D70E}_{i}$
and
$\unicode[STIX]{x1D70E}_{i}\cap \unicode[STIX]{x1D70E}_{j}=\emptyset$
for all
$i\neq j$
. We say that
$G$
is
$\unicode[STIX]{x1D70E}$
-primary if
$G$
is a
$\unicode[STIX]{x1D70E}_{i}$
-group for some
$i$
. A subgroup
$A$
of
$G$
is said to be:
$\unicode[STIX]{x1D70E}$
-subnormal in
$G$
if there is a subgroup chain
$A=A_{0}\leq A_{1}\leq \cdots \leq A_{n}=G$
such that either
$A_{i-1}\unlhd A_{i}$
or
$A_{i}/(A_{i-1})_{A_{i}}$
is
$\unicode[STIX]{x1D70E}$
-primary for all
$i=1,\ldots ,n$
; modular in
$G$
if the following conditions hold: (i)
$\langle X,A\cap Z\rangle =\langle X,A\rangle \cap Z$
for all
$X\leq G,Z\leq G$
such that
$X\leq Z$
and (ii)
$\langle A,Y\cap Z\rangle =\langle A,Y\rangle \cap Z$
for all
$Y\leq G,Z\leq G$
such that
$A\leq Z$
; and
$\unicode[STIX]{x1D70E}$
-quasinormal in
$G$
if
$A$
is modular and
$\unicode[STIX]{x1D70E}$
-subnormal in
$G$
. We study
$\unicode[STIX]{x1D70E}$
-quasinormal subgroups of
$G$
. In particular, we prove that if a subgroup
$H$
of
$G$
is
$\unicode[STIX]{x1D70E}$
-quasinormal in
$G$
, then every chief factor
$H/K$
of
$G$
between
$H^{G}$
and
$H_{G}$
is
$\unicode[STIX]{x1D70E}$
-central in
$G$
, that is, the semidirect product
$(H/K)\rtimes (G/C_{G}(H/K))$
is
$\unicode[STIX]{x1D70E}$
-primary.