Finite Group Databasebeta

← #37543077

Subgroups — #37543077

Order 23168.

762 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C2×#19153861C211yesmaximal, characteristic, complemented
2#26042077C212yesmaximal, complemented
3#26041456C214yesmaximal, complemented
4#26041446C214yesmaximal, complemented
5#26041297C212yesmaximal, complemented
6C2×#19153733C211yesmaximal, characteristic, complemented
7#535×C181C211yesmaximal, characteristic, complemented
8#19153861—24no—
9#19153861—22no—
10#19153861C2214yes—
11C2×#15668449C2212yes—
12C2×#15668448—22no—
13C22×C22⋊D181—22no—
14C2×D4×D181—22no—
15#19153847C2214yescomplemented
16#19153846—22no—
17#19153842C2214yescomplemented
18#19153841—22no—
19#19153836—24no—
20C2×#15668433C2212yescomplemented
21#19153744C2218yescomplemented
22#19153733C2214yescomplemented
23C2×#41×C181C2211yescharacteristic, complemented
24C22×D4×C181C2211yescharacteristic
25#127×C181C2212yescomplemented
26#124×C181C2212yescomplemented
27#121×C181—22no—
28#120×C181—22no—
29#117×C181C2211yescharacteristic, complemented
30C23×D181—22no—
31#15668449—22no—
32#15668449—22no—
33#15668449D414yescomplemented
34#15668449C2314yes—
35#15668448—28no—
36#15668448—28no—
37C2×Dic362D412yescomplemented
38C2×D724D412yescomplemented
39C2×C22⋊D181—24no—
40C2×C22⋊D181—28no—
41C2×C22⋊D181—22no—
42C2×C22⋊D181—28no—
43D4×D181—28no—
44D4×D181—28no—
45C22×Dic181—22no—
46C2×C4×D181—22no—
47#15668439—24no—
48#15668437—24no—
49#15668436—24no—
50#15668434—24no—
51#15668433—22no—
52#15668433C2314yescomplemented
53C2×#13089678—22no—
54C24×C181D412yes—
55#41×C181—22no—
56#41×C181—22no—
57C2×Q8×C181C2311yescharacteristic, complemented
58C2×D4×C181—21no—
59C2×D4×C181—24no—
60C2×D4×C181—22no—
61C2×D4×C181—22no—
62C2×D4×C181D412yes—
63C2×D4×C181C2312yes—
64C2×D4×C181C2311yescharacteristic, complemented
65C22×C4×C181—21no—
66C22×C4×C181C2311yescharacteristic
67C4⋊C4×C181—21no—
68C22⋊C4×C181—21no—
69C22⋊C4×C181—24no—
70C42×C181C2312yescomplemented
71C22×D181—22no—
72C22×D181—28no—
73C22×D181—84no—
74Dic362—22no—
75Dic362C2×D414yes—
76D724—22no—
77D724C2×D414yes—
78C22⋊D181—216no—
79C22⋊D181—24no—
80C22⋊D181—24no—
81C2×Dic181—28no—
82C2×Dic181—22no—
83C4×D181—24no—
84C4×D181—24no—
85#13089678—24no—
86C23×C181—22no—
87C23×C181—22no—
88C23×C181—21no—
89C23×C181—82no—
90C23×C181C2×D414yes—
91C23×C181C2×D411yescharacteristic
92Q8×C181—22no—
93D4×C181—24no—
94D4×C181—24no—
95D4×C181—22no—
96D4×C181—41no—
97D4×C181—41no—
98D4×C181—84no—
99C2×C4×C181—22no—
100C2×C4×C181—21no—
101C2×C4×C181—42no—
102C2×C4×C181C2×D412yes—
103C2×C4×C181C2×D412yes—
104C2×C4×C181C2×D411yescharacteristic
105C2×C4×C181C2411yescharacteristic
106C22×C181—24no—
107C22×C181—41no—
108C22×C181—44no—
109C22×C181—44no—
110C22×C181—42no—
111C22×C181—41no—
112C22×C181—82no—
113C22×C181#12011yescharacteristic
114C22×C181#12011yescharacteristic
115C22×C181C22×D411yescharacteristic
116C2×D181—24no—
117C2×D181—44no—
118C2×D181—88no—
119C4×C181—42no—
120C4×C181—41no—
121C4×C181#12012yes—
122C4×C181C22×D412yes—
123Dic181—24no—
124C2×C181—21no—
125C2×C181—21no—
126C2×C181—21no—
127C2×C181—44no—
128C2×C181—81no—
129C2×C181C2×#12011yescharacteristic
130D181—44no—
131C181#301911yescharacteristic, complemented, Sylow
132#3019—1811nomaximal, Sylow
133C2×H(5,𝔽2)—1811no—
134#575—1812no—
135#535D18111yescharacteristic, complemented
136#455—1814no—
137#453—1814no—
138#421—1812no—
139C2×#100—1811no—
140H(5,𝔽2)—3624no—
141H(5,𝔽2)—3622no—
142H(5,𝔽2)—1814no—
143C2×#41—3622no—
144C2×#41—1812no—
145C2×#41C2×D18111yescharacteristic, complemented
146C22×D4—3622no—
147C22×D4—3622no—
148C22×D4C2×D18111yescharacteristic
149#137—3622no—
150#137—1814no—
151AGL(1,ℤ/8)—3622no—
152AGL(1,ℤ/8)—1814no—
153#135—3624no—
154C2×M16—1812no—
155#127C2×D18112yescomplemented
156#124C2×D18112yescomplemented
157#121—22no—
158#120—22no—
159#117C2×D18111yescharacteristic, complemented
160#104—1818no—
161#100—1814no—
162C24—3622no—
163C24C22⋊D18112yes—
164#41—22no—
165#41—22no—
166#41—3628no—
167#41—3628no—
168#41—3622no—
169#41—3622no—
170#41—1814no—
171#41—1814no—
172C2×Q8—1812no—
173C2×Q8C22×D18111yescharacteristic, complemented
174C2×D4—21no—
175C2×D4—24no—
176C2×D4—22no—
177C2×D4—22no—
178C2×D4—3628no—
179C2×D4—3628no—
180C2×D4—3624no—
181C2×D4—3628no—
182C2×D4—3622no—
183C2×D4—3628no—
184C2×D4—1812no—
185C2×D4C22⋊D18112yes—
186C2×D4C22×D18112yes—
187C2×D4C22×D18111yescharacteristic, complemented
188C22×C4—21no—
189C22×C4—3622no—
190C22×C4—3622no—
191C22×C4C22×D18111yescharacteristic
192Q16—3624no—
193SD16—3624no—
194SD16—3624no—
195D8—3624no—
196M16—3622no—
197M16—1814no—
198C2×C8—3622no—
199C4⋊C4—21no—
200C22⋊C4—21no—
201C22⋊C4—24no—
202C42C22×D18112yescomplemented
203C23—22no—
204C23—22no—
205C23—21no—
206C23—82no—
207C23—3628no—
208C23—3622no—
209C23—14484no—
210C23C2×C22⋊D18114yes—
211C23C2×C22⋊D18111yescharacteristic
212Q8—22no—
213Q8—3622no—
214Q8—1814no—
215D4—24no—
216D4—24no—
217D4—22no—
218D4—41no—
219D4—41no—
220D4—84no—
221D4—36216no—
222D4—3624no—
223D4—3624no—
224D4—3622no—
225D4—1814no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—42no—
229C2×C4—3628no—
230C2×C4—3622no—
231C2×C4—3624no—
232C2×C4—3624no—
233C2×C4D4×D18112yes—
234C2×C4D4×D18112yes—
235C2×C4C2×C22⋊D18111yescharacteristic
236C2×C4C23×D18111yescharacteristic
237C8—3624no—
238C22—24no—
239C22—41no—
240C22—44no—
241C22—44no—
242C22—42no—
243C22—41no—
244C22—82no—
245C22—3624no—
246C22—14488no—
247C22—7244no—
248C22#1915378411yescharacteristic
249C22#1915378411yescharacteristic
250C22C22×C22⋊D18111yescharacteristic
251C4—42no—
252C4—41no—
253C4—3624no—
254C4#1915378412yes—
255C4C2×D4×D18112yes—
256C2—21no—
257C2—21no—
258C2—21no—
259C2—44no—
260C2—81no—
261C2—7244no—
262C2C2×#1915378411yescharacteristic
subgroupG/H
centerC2C2×#19153784
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#535×C181C2
socleC2×C181C2×#120
hypercenter#535D181
hypocenterC181#3019
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#19153861C2semidirect
2. #26042077C2semidirect
3. #26041456C2semidirect
4. #26041446C2semidirect
5. #26041297C2semidirect
6. C2×#19153733C2semidirect
7. #535×C181C2semidirect
15. #19153847C22semidirect
17. #19153842C22semidirect
20. C2×#15668433C22semidirect
21. #19153744C22semidirect
22. #19153733C22semidirect
23. C2×#41×C181C22semidirect
25. #127×C181C22semidirect
26. #124×C181C22semidirect
29. #117×C181C22semidirect
33. #15668449D4semidirect
37. C2×Dic362D4semidirect
38. C2×D724D4semidirect
52. #15668433C23semidirect
57. C2×Q8×C181C23semidirect
64. C2×D4×C181C23semidirect
70. C42×C181C23semidirect
131. C181#3019semidirect
135. #535D181semidirect
145. C2×#41C2×D181semidirect
155. #127C2×D181semidirect
156. #124C2×D181semidirect
159. #117C2×D181semidirect
173. C2×Q8C22×D181semidirect
187. C2×D4C22×D181semidirect
202. C42C22×D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538612
#260420772
#260414562
#260414462
#260412972
C2×#191537332
#535×C1812
#3019181