Finite Group Databasebeta

← #37532050

Subgroups — #37532050

Order 23168.

312 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#15668428C211yesmaximal, characteristic
2#26041243C211yesmaximal, characteristic
3#26041004C214yesmaximal, complemented
4#407×C181C211yesmaximal, characteristic
5C23×Dic181C411yescharacteristic
6C23×Dic181C411yescharacteristic
7C2×#15668433—21no—
8C2×#15668428—21no—
9C2×#15668428—21no—
10C2×#15668428—21no—
11C2×#15668428—21no—
12C2×#15668428C412yes—
13C2×#15668428C2212yes—
14C22⋊#13089678—21no—
15C22⋊#13089678C2212yes—
16C23×C4×C181C2211yescharacteristic
17C2×M16×C181—21no—
18C22⋊C8×C181—21no—
19C22⋊C8×C181C2212yes—
20C22×Dic181—24no—
21C22×Dic181—21no—
22C22×Dic181—21no—
23C22×Dic181—21no—
24C22×Dic181—21no—
25C22×Dic181—42no—
26C22×Dic181C2×C412yes—
27C22×Dic181C2×C412yes—
28#15668433—42no—
29C2×#13089678—22no—
30#15668428—22no—
31#15668428—22no—
32#15668428—22no—
33#15668428—22no—
34C24×C181C2×C411yescharacteristic
35C22×C4×C181—42no—
36C22×C4×C181C2×C411yescharacteristic
37C22×C4×C181D411yescharacteristic
38C22×C4×C181D411yescharacteristic
39C22×C4×C181D411yescharacteristic
40C22×C4×C181D411yescharacteristic
41C22×C4×C181C2311yescharacteristic
42M16×C181—42no—
43C2×C8×C181—22no—
44C2×Dic181—24no—
45C2×Dic181—24no—
46C2×Dic181—22no—
47C2×Dic181—22no—
48C2×Dic181—22no—
49C2×Dic181—42no—
50C2×Dic181—42no—
51C2×Dic181—42no—
52C2×Dic181M1614yescomplemented
53#13089678—42no—
54C23×C181—22no—
55C23×C181—22no—
56C23×C181—22no—
57C23×C181C22⋊C411yescharacteristic
58C23×C181C22⋊C411yescharacteristic
59C23×C181C22×C411yescharacteristic
60C2×C4×C181—22no—
61C2×C4×C181—22no—
62C2×C4×C181—41no—
63C2×C4×C181—41no—
64C2×C4×C181—41no—
65C2×C4×C181—41no—
66C2×C4×C181C22⋊C411yescharacteristic
67C2×C4×C181C22⋊C411yescharacteristic
68C2×C4×C181C2×D411yescharacteristic
69C2×C4×C181C2×D411yescharacteristic
70C8×C181—42no—
71C22×C181—28no—
72C22×C181—21no—
73C22×C181—21no—
74C22×C181—21no—
75C22×C181—21no—
76C22×C181—21no—
77C22×C181—21no—
78C22×C181#9912yes—
79C22×C181#10112yes—
80C22×C181C2×C22⋊C411yescharacteristic
81C22×C181C2×M1612yes—
82C4×C181—21no—
83C4×C181—21no—
84C4×C181—21no—
85C4×C181—21no—
86Dic181—24no—
87Dic181—42no—
88C2×C181—22no—
89C2×C181—22no—
90C2×C181—22no—
91C2×C181#40711yescharacteristic
92C2×C181C2×#9911yescharacteristic
93C2×C181C2×#10111yescharacteristic
94C181#146311yescharacteristic, complemented, Sylow
95#1463—1811nomaximal, Sylow
96C22×C4⋊C4—1811no—
97#407—1811no—
98#407D18111yescharacteristic
99#324—1814no—
100C23×C4—1811no—
101C23×C4—1811no—
102C23×C4C2×D18111yescharacteristic
103C2×M16—21no—
104C2×M16—3621no—
105C2×C4⋊C4—3621no—
106C2×C4⋊C4—3621no—
107C2×C4⋊C4—3621no—
108C2×C4⋊C4—3621no—
109C2×C4⋊C4—1812no—
110C2×C4⋊C4—1812no—
111C22⋊C8—21no—
112C22⋊C8—3621no—
113C22⋊C8—1812no—
114C22⋊C8C2×D18112yes—
115C24C4×D18111yescharacteristic
116C22×C4—42no—
117C22×C4—3621no—
118C22×C4—3621no—
119C22×C4—3624no—
120C22×C4—3621no—
121C22×C4—3621no—
122C22×C4—1812no—
123C22×C4—1812no—
124C22×C4—7242no—
125C22×C4C4×D18111yescharacteristic
126C22×C4C22⋊D18111yescharacteristic
127C22×C4C22⋊D18111yescharacteristic
128C22×C4D72411yescharacteristic
129C22×C4D72411yescharacteristic
130C22×C4C22×D18111yescharacteristic
131M16—42no—
132M16—7242no—
133C2×C8—22no—
134C2×C8—3622no—
135C4⋊C4—3622no—
136C4⋊C4—3622no—
137C4⋊C4—3622no—
138C4⋊C4—3622no—
139C23—22no—
140C23—22no—
141C23—22no—
142C23#1566842611yescharacteristic
143C23#1566842611yescharacteristic
144C23C2×C4×D18111yescharacteristic
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—3624no—
152C2×C4—3624no—
153C2×C4—3622no—
154C2×C4—3622no—
155C2×C4—3622no—
156C2×C4—1814no—
157C2×C4—7242no—
158C2×C4—7242no—
159C2×C4—7242no—
160C2×C4#1566842611yescharacteristic
161C2×C4#1566842611yescharacteristic
162C2×C4C2×C22⋊D18111yescharacteristic
163C2×C4C2×D72411yescharacteristic
164C8—42no—
165C8—7242no—
166C22—28no—
167C22—21no—
168C22—21no—
169C22—21no—
170C22—21no—
171C22—21no—
172C22—21no—
173C22#1915373012yes—
174C22#1915373412yes—
175C22C2×#1566842611yescharacteristic
176C22M16×D18112yes—
177C4—21no—
178C4—21no—
179C4—21no—
180C4—21no—
181C4—3624no—
182C4—7242no—
183C2—22no—
184C2—22no—
185C2—22no—
186C2#2604124211yescharacteristic
187C2C2×#1915373011yescharacteristic
188C2C2×#1915373411yescharacteristic
subgroupG/H
centerC22C2×#15668426
commutatorC23×C181C22×C4
FrattiniC22×C4C22×D181
Fitting#407×C181C2
socleC22×C181C2×C22⋊C4
prime radicalC24×C181C2×C4
cosocleC22×C4×C181C23
hypercenter#407D181
hypocenterC181#1463
LCSG ▷ C23×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #26041004C2semidirect
52. C2×Dic181M16semidirect
94. C181#1463semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#156684282
#260412432
#260410042
#407×C1812
#1463181