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A set  $  \mathfrak A $
of subgroups (cf. [[Subgroup|Subgroup]]) of a [[Group|group]]  $  G $
satisfying the following conditions: 1)  $  \mathfrak A $
contains the unit subgroup  $  1 $
and the group  $  G $
itself; and 2)  $  \mathfrak A $
is totally ordered by inclusion, i.e. for any  $  A $
and  $  B $
from  $  \mathfrak A $
either  $  A \subseteq B $
or  $  B \subseteq A $. 
One says that two subgroups  $  A $
and  $  A  ^  \prime  $
from  $  \mathfrak A $
constitute a jump if  $  A  ^  \prime  $
follows directly from  $  A $
in  $  \mathfrak A $. 
A subgroup system that is closed with respect to union and intersection is called complete. A complete subgroup system is called subnormal if for any jump  $  A $
and  $  A  ^  \prime  $
in this system,  $  A $
is a [[Normal subgroup|normal subgroup]] in  $  A  ^  \prime  $. 
The quotient group  $  A  ^  \prime  /A $
is called a factor of the system  $  \mathfrak A $. 
A subgroup system in which all members are normal subgroups of a group  $  G $
is called normal. In the case where one subnormal system contains another (in the set-theoretical sense), the first is called a refinement of the second. A normal subgroup system is called central if all its factors are central, i.e.  $  A  ^  \prime  /A $
is contained in the centre of  $  G/A $
for any jump  $  A, A  ^  \prime  $. 
A subnormal subgroup system is called solvable if all its factors are Abelian.

The presence in a group of some subgroup system enables one to distinguish various subclasses in the class of all groups, of which the ones most used are  $  RN $, 
$  \overline{RN}\; {}  ^ {*} $, 
$  \overline{RN}\; $, 
$  RI $, 
$  RI  ^ {*} $, 
$  \overline{RI}\; $, 
$  Z $, 
$  ZA $, 
$  ZD $, 
$  \overline{Z}\; $, 
$  \widetilde{N}  $, 
$  N $, 
the Kurosh–Chernikov classes of:

$  RN $-
groups: There is a solvable subnormal subgroup system;

$  \overline{RN}\; {}  ^ {*} $-
groups: There is a well-ordered ascending solvable subnormal subgroup system;

$  \overline{RN}\; $-
groups: Any subnormal subgroup system in such a group can be refined to a solvable subnormal one;

$  RI $-
groups: There is a solvable normal subgroup system;

$  RI  ^ {*} $-
groups: There is a well-ordered ascending solvable normal subgroup system;

$  \overline{RI}\; $-
groups: Any normal subgroup system in such a group can be refined to a solvable normal one;

$  Z $-
groups: There is a central subgroup system;

$  ZA $-
groups: There is a well-ordered ascending central subgroup system;

$  ZD $-
groups: There is a well-ordered descending central subgroup system;

$  \overline{Z}\; $-
groups: Any normal subgroup system of this group can be refined to a central one;

$  \widetilde{N}  $-
groups: Through any subgroup of this group there passes a subgroup system;

$  N $-
groups: Through any subgroup of this group there passes a well-ordered ascending subnormal subgroup system.

A particular case of a subgroup system is a [[Subgroup series|subgroup series]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,   "The theory of groups" , '''1–2''' , Chelsea  (1955–1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.N. Chernikov,   "Groups with given properties of subgroup systems" , Moscow  (1980)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.I. Kargapolov,   J.I. [Yu.I. Merzlyakov] Merzljakov,   "Fundamentals of the theory of groups" , Springer  (1979)  (Translated from Russian)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.J.S. Robinson,   "Finiteness condition and generalized soluble groups" , '''1–2''' , Springer  (1972)</TD></TR></table>