Finite Group Databasebeta

← C22×#19153789 #37546942

Subgroups — C22×#19153789 #37546942

Order 23168.

2528 conjugacy classes of subgroups; only the 645 normal subgroups are shown.

normal subgroups only

This group has 2528 conjugacy classes of subgroups — too many to draw. Only its 645 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
1C23×D724C211yesmaximal, characteristic, complemented
2C23×C22⋊D181C211yesmaximal, characteristic, complemented
3C23×C22⋊D181C211yesmaximal, characteristic, complemented
4C23×C4×D181C211yesmaximal, characteristic, complemented
5C2×#19153789C2124yesmaximal, direct product factor
6C22×#15668428C211yesmaximal, characteristic, complemented
7C22×#15668426C211yesmaximal, characteristic, complemented
8C22×C22⋊C4×C181C211yesmaximal, characteristic, complemented
9C24×D181C2211yescharacteristic
10C24×D181C2211yescharacteristic
11C22×D724—28no—
12C22×D724C22112yescomplemented
13C22×C22⋊D181—28no—
14C22×C22⋊D181—28no—
15C22×C22⋊D181C22112yescomplemented
16C22×C22⋊D181C22112yescomplemented
17C23×Dic181C2211yescharacteristic
18C23×Dic181C2211yescharacteristic
19C22×C4×D181—28no—
20C22×C4×D181C22112yescomplemented
21#19153789C22164yesdirect product factor
22C2×#15668428C22112yescomplemented
23C2×#15668426C22112yescomplemented
24C25×C181C2211yescharacteristic
25C23×C4×C181C2211yescharacteristic
26C23×C4×C181C2211yescharacteristic
27C2×C22⋊C4×C181C22112yescomplemented
28C23×D181—24no—
29C23×D181—24no—
30C23×D181—24no—
31C23×D181—24no—
32C23×D181—24no—
33C23×D181D418yescomplemented
34C23×D181C2316yes—
35C23×D181C2316yes—
36C2×D724—248no—
37C2×D724C23116yescomplemented
38C2×C22⋊D181—248no—
39C2×C22⋊D181—248no—
40C2×C22⋊D181C23116yescomplemented
41C2×C22⋊D181C23116yescomplemented
42C22×Dic181—24no—
43C22×Dic181D418yescomplemented
44C22×Dic181C2316yes—
45C22×Dic181C2316yes—
46C2×C4×D181—248no—
47C2×C4×D181C23116yescomplemented
48#15668428C23116yescomplemented
49#15668426C23116yescomplemented
50C24×C181—24no—
51C24×C181—24no—
52C24×C181—24no—
53C24×C181C2316yes—
54C24×C181C2311yescharacteristic
55C22×C4×C181—24no—
56C22×C4×C181—24no—
57C22×C4×C181C2316yes—
58C22×C4×C181C2316yes—
59C22⋊C4×C181C23116yescomplemented
60C22×D181—232no—
61C22×D181—212no—
62C22×D181—212no—
63C22×D181—212no—
64C22×D181—212no—
65C22×D181—212no—
66C22×D181—416no—
67C22×D181C2×D4124yescomplemented
68C22×D181C2414yes—
69C22×D181C2414yes—
70D724—232no—
71C22⋊D181—232no—
72C22⋊D181—232no—
73C2×Dic181—212no—
74C2×Dic181C2×D4124yescomplemented
75C2×Dic181C2414yes—
76C2×Dic181C2414yes—
77C4×D181—232no—
78C23×C181—212no—
79C23×C181—212no—
80C23×C181—212no—
81C23×C181—416no—
82C23×C181C2×D414yes—
83C23×C181C2×D414yes—
84C23×C181#4114yes—
85C23×C181C2414yes—
86C23×C181C2413yes—
87C2×C4×C181—212no—
88C2×C4×C181—212no—
89C2×C4×C181C2414yes—
90C2×C4×C181C2414yes—
91C22×C181—24no—
92C22×C181—24no—
93C22×C181—24no—
94C22×C181—424no—
95C22×C181#121116yescomplemented
96C22×C181C22×D416yes—
97C22×C181C22×D416yes—
98C22×C181C2×#4116yes—
99C22×C181C2511yescharacteristic
100C2×D181—248no—
101C2×D181—24no—
102C2×D181—24no—
103C2×D181—24no—
104C2×D181—24no—
105C2×D181—24no—
106C2×D181—424no—
107C2×D181C22×D418yescomplemented
108C4×C181—24no—
109C4×C181—24no—
110Dic181—24no—
111Dic181C22×D418yescomplemented
112C2×C181—44no—
113C2×C181C2×#121112yescomplemented
114C2×C181C23×D411yescharacteristic
115C2×C181C23×D411yescharacteristic
116C2×C181C22×#4111yescharacteristic
117D181—28no—
118D181—44no—
119C181C22×#12111yescharacteristic, complemented, Sylow
120C22×#121—1811nomaximal, Sylow
121C23×D4—1811no—
122C23×D4—1811no—
123C23×D4—1811no—
124C24×C4—1811no—
125C2×#121—18124no—
126C22×C4⋊C4—1811no—
127C22×C22⋊C4—1811no—
128C22×C22⋊C4D18111yescharacteristic, complemented
129C25—1811no—
130C25—1811no—
131C25C2×D18111yescharacteristic
132C22×D4—3628no—
133C22×D4—3628no—
134C22×D4—3628no—
135C22×D4—18112no—
136C22×D4—18112no—
137C22×D4—18112no—
138C23×C4—3628no—
139C23×C4—1811no—
140C23×C4—18112no—
141C23×C4—1811no—
142C23×C4C2×D18111yescharacteristic
143C23×C4C2×D18111yescharacteristic
144#121—18164no—
145C2×C4⋊C4—18112no—
146C2×C22⋊C4—18112no—
147C2×C22⋊C4C2×D181112yescomplemented
148C24—24no—
149C24—24no—
150C24—24no—
151C24—3624no—
152C24—3624no—
153C24—3624no—
154C24—3624no—
155C24—3624no—
156C24—1816no—
157C24—1818no—
158C24—1816no—
159C24C22×D18116yes—
160C24C22×D18111yescharacteristic
161C2×D4—36248no—
162C2×D4—36248no—
163C2×D4—36248no—
164C2×D4—18116no—
165C2×D4—18116no—
166C2×D4—18116no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—36248no—
170C22×C4—3624no—
171C22×C4—1816no—
172C22×C4—18116no—
173C22×C4—1818no—
174C22×C4—1816no—
175C22×C4C22×D18116yes—
176C22×C4C22×D18116yes—
177C4⋊C4—18116no—
178C22⋊C4—18116no—
179C22⋊C4C22×D181116yescomplemented
180C23—212no—
181C23—212no—
182C23—212no—
183C23—416no—
184C23—36232no—
185C23—36212no—
186C23—36212no—
187C23—36212no—
188C23—36212no—
189C23—36212no—
190C23—1814no—
191C23—18124no—
192C23—1814no—
193C23—72416no—
194C23D4×D18114yes—
195C23D4×D18114yes—
196C23#1566844914yes—
197C23C23×D18114yes—
198C23C23×D18113yes—
199D4—36232no—
200D4—36232no—
201D4—36232no—
202C2×C4—212no—
203C2×C4—212no—
204C2×C4—36232no—
205C2×C4—36212no—
206C2×C4—1814no—
207C2×C4—18124no—
208C2×C4—1814no—
209C2×C4C23×D18114yes—
210C2×C4C23×D18114yes—
211C22—24no—
212C22—24no—
213C22—24no—
214C22—424no—
215C22—36248no—
216C22—3624no—
217C22—3624no—
218C22—3624no—
219C22—3624no—
220C22—3624no—
221C22—1818no—
222C22—72424no—
223C22#19153789116yesdirect product factor
224C22C2×D4×D18116yes—
225C22C2×D4×D18116yes—
226C22C2×#1566844916yes—
227C22C24×D18111yescharacteristic
228C4—24no—
229C4—24no—
230C4—3624no—
231C4—1818no—
232C2—44no—
233C2—3628no—
234C2—7244no—
235C2C2×#19153789112yesdirect product factor
236C2C22×D4×D18111yescharacteristic
237C2C22×D4×D18111yescharacteristic
238C2C22×#1566844911yescharacteristic
subgroupG/H
centerC24C22×D181
commutator, cosocleC22×C181C25
FrattiniC22C24×D181
FittingC22×C22⋊C4×C181C2
socleC24×C181C23
hypercenterC22×C22⋊C4D181
hypocenterC181C22×#121
LCSG ▷ C22×C181 ▷ C181
UCSC1 ◁ C24 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C23×D724C2semidirect
2. C23×C22⋊D181C2semidirect
3. C23×C22⋊D181C2semidirect
4. C23×C4×D181C2semidirect
5. C2×#19153789C2direct (N × H)
6. C22×#15668428C2semidirect
7. C22×#15668426C2semidirect
8. C22×C22⋊C4×C181C2semidirect
12. C22×D724C22semidirect
15. C22×C22⋊D181C22semidirect
16. C22×C22⋊D181C22semidirect
20. C22×C4×D181C22semidirect
21. #19153789C22direct (N × H)
22. C2×#15668428C22semidirect
23. C2×#15668426C22semidirect
27. C2×C22⋊C4×C181C22semidirect
33. C23×D181D4semidirect
37. C2×D724C23semidirect
40. C2×C22⋊D181C23semidirect
41. C2×C22⋊D181C23semidirect
43. C22×Dic181D4semidirect
47. C2×C4×D181C23semidirect
48. #15668428C23semidirect
49. #15668426C23semidirect
59. C22⋊C4×C181C23semidirect
67. C22×D181C2×D4semidirect
74. C2×Dic181C2×D4semidirect
95. C22×C181#121semidirect
107. C2×D181C22×D4semidirect
111. Dic181C22×D4semidirect
113. C2×C181C2×#121semidirect
119. C181C22×#121semidirect
128. C22×C22⋊C4D181semidirect
147. C2×C22⋊C4C2×D181semidirect
179. C22⋊C4C22×D181semidirect
223. C22#19153789direct (N × H)
235. C2C2×#19153789direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D7242
C23×C22⋊D1812
C23×C4×D1812
C2×#191537892
C22×#156684282
C22×#156684262
C22×C22⋊C4×C1812
C22×#121181