Finite Group Databasebeta

← #37543093

Subgroups — #37543093

Order 23168.

736 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 161 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×#19153841C212yesmaximal, complemented
2#26041843C211yesmaximal, characteristic, complemented
3#26041452C214yesmaximal, complemented
4#26041447C214yesmaximal, complemented
5#26041296C212yesmaximal, complemented
6#26041263C211yesmaximal, characteristic, complemented
7#585×C181C211yesmaximal, characteristic, complemented
8C22×D724C2211yescharacteristic, complemented
9#19153841—24no—
10#19153841C2218yescomplemented
11C2×#15668436—22no—
12C2×#15668434—22no—
13C2×#15668433C2212yescomplemented
14#19153812C2212yescomplemented
15#19153805C2212yescomplemented
16#19153790—22no—
17#19153784—22no—
18#19153775C2211yescharacteristic, complemented
19#19153743C2218yescomplemented
20#19153735C2212yescomplemented
21#19153733C2212yescomplemented
222−1+4×C181—22no—
232−1+4×C181C2212yescomplemented
24H(5,𝔽2)×C181—22no—
25H(5,𝔽2)×C181—22no—
26H(5,𝔽2)×C181C2212yescomplemented
27C2×#41×C181—24no—
28C2×#41×C181—22no—
29C2×#41×C181C2212yes—
30C2×#41×C181C2211yescharacteristic
31C23×D181—21no—
32C2×D724—22no—
33C2×D724—22no—
34C2×D724—21no—
35C2×D724C2312yes—
36C2×C22⋊D181—24no—
37#15668436—28no—
38#15668434—28no—
39#15668433—22no—
40#15668433C2314yes—
41C2×#13089678—22no—
42C4⋊Dic181—21no—
43C22⋊Dic181—21no—
44#15668426—24no—
45C4×Dic181C2312yes—
46#41×C181—24no—
47#41×C181—28no—
48#41×C181—24no—
49#41×C181—24no—
50#41×C181—24no—
51#41×C181—22no—
52#41×C181—22no—
53#41×C181—24no—
54#41×C181—22no—
55#41×C181—22no—
56#41×C181D414yescomplemented
57#41×C181C2314yes—
58C2×Q8×C181—22no—
59C2×Q8×C181—24no—
60C2×Q8×C181D412yescomplemented
61C2×Q8×C181C2311yescharacteristic
62C2×D4×C181—28no—
63C2×D4×C181—22no—
64C2×D4×C181—24no—
65C2×D4×C181—22no—
66C2×D4×C181—24no—
67C2×D4×C181D412yes—
68C2×D4×C181D412yescomplemented
69C2×D4×C181C2311yescharacteristic
70C22×C4×C181—24no—
71C22×C4×C181—22no—
72C22×C4×C181D412yes—
73C22×C4×C181C2311yescharacteristic
74C22×D181—22no—
75C22×D181—42no—
76C22×D181—82no—
77D724—24no—
78D724—41no—
79D724—41no—
80C22⋊D181—84no—
81C2×Dic181—22no—
82#13089678—24no—
83C23×C181—24no—
84C23×C181—22no—
85C23×C181C2×D412yes—
86C23×C181C2×D411yescharacteristic
87Q8×C181—24no—
88Q8×C181—22no—
89Q8×C181—22no—
90Q8×C181C2×D414yes—
91D4×C181—28no—
92D4×C181—24no—
93D4×C181—24no—
94D4×C181—24no—
95D4×C181—24no—
96D4×C181—22no—
97D4×C181—22no—
98D4×C181C2×D414yes—
99C2×C4×C181—24no—
100C2×C4×C181—28no—
101C2×C4×C181—22no—
102C2×C4×C181—24no—
103C2×C4×C181—22no—
104C2×C4×C181—22no—
105C2×C4×C181—24no—
106C2×C4×C181C2×D412yes—
107C2×C4×C181C2×D411yescharacteristic
108C2×C4×C181C2×D412yes—
109C2×C4×C181C2×D412yes—
110C2×C4×C181C2411yescharacteristic
111C22×C181—22no—
112C22×C181—24no—
113C22×C181—41no—
114C22×C181—42no—
115C22×C181—82no—
116C22×C181—84no—
117C22×C181#12012yes—
118C22×C181C22×D411yescharacteristic
119C2×D181—44no—
120C2×D181—42no—
121C2×D181—84no—
122C4×C181—24no—
123C4×C181—22no—
124C4×C181#12011yescharacteristic
125C4×C181#12011yescharacteristic
126C4×C181C22×D412yes—
127Dic181—42no—
128C2×C181—21no—
129C2×C181—22no—
130C2×C181—42no—
131C2×C181—44no—
132C2×C181C2×#12011yescharacteristic
133D181—82no—
134C181#302111yescharacteristic, complemented, Sylow
135#3021—1811nomaximal, Sylow
136#585D18111yescharacteristic, complemented
137C2×AGL(1,ℤ/8)—1812no—
138#535—1811no—
139#454—1814no—
140#453—1814no—
141#421—1812no—
142#413—1811no—
1432−1+4—22no—
1442−1+4C2×D18112yescomplemented
145H(5,𝔽2)—22no—
146H(5,𝔽2)—22no—
147H(5,𝔽2)C2×D18112yescomplemented
148C2×#41—24no—
149C2×#41—22no—
150C2×#41C2×D18112yes—
151C2×#41C2×D18111yescharacteristic
152C22×D4—1811no—
153AGL(1,ℤ/8)—3624no—
154AGL(1,ℤ/8)—1818no—
155C2×SD16—3622no—
156C2×D8—3622no—
157C2×M16—1812no—
158#127—1812no—
159#124—1812no—
160#121—3622no—
161#120—3622no—
162#117—1811no—
163#104—1818no—
164#101—1812no—
165#100—1812no—
166C24—3621no—
167#41—24no—
168#41—28no—
169#41—24no—
170#41—24no—
171#41—24no—
172#41—22no—
173#41—22no—
174#41—24no—
175#41—22no—
176#41—22no—
177#41C22⋊D18114yescomplemented
178#41C22×D18114yes—
179C2×Q8—22no—
180C2×Q8—24no—
181C2×Q8C22⋊D18112yescomplemented
182C2×Q8C22×D18111yescharacteristic
183C2×D4—28no—
184C2×D4—22no—
185C2×D4—24no—
186C2×D4—22no—
187C2×D4—24no—
188C2×D4—3624no—
189C2×D4—3622no—
190C2×D4—3622no—
191C2×D4—3621no—
192C2×D4—1812no—
193C2×D4C22⋊D18112yes—
194C2×D4C22⋊D18112yescomplemented
195C2×D4C22×D18111yescharacteristic
196C22×C4—24no—
197C22×C4—22no—
198C22×C4C22⋊D18112yes—
199C22×C4C22×D18111yescharacteristic
200SD16—3628no—
201D8—3628no—
202M16—3622no—
203M16—1814no—
204C2×C8—3622no—
205C4⋊C4—3621no—
206C22⋊C4—3621no—
207C22⋊C4—3624no—
208C42—1812no—
209C23—24no—
210C23—22no—
211C23—3622no—
212C23—14482no—
213C23—7242no—
214C23C2×C22⋊D18112yes—
215C23C2×C22⋊D18111yescharacteristic
216Q8—24no—
217Q8—22no—
218Q8—22no—
219Q8C2×C22⋊D18114yes—
220D4—28no—
221D4—24no—
222D4—24no—
223D4—24no—
224D4—24no—
225D4—22no—
226D4—22no—
227D4—3624no—
228D4—14484no—
229D4—7241no—
230D4—7241no—
231D4C2×C22⋊D18114yes—
232C2×C4—24no—
233C2×C4—28no—
234C2×C4—22no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—24no—
239C2×C4—3622no—
240C2×C4C2×C22⋊D18112yes—
241C2×C4C2×C22⋊D18111yescharacteristic
242C2×C4C2×C22⋊D18112yes—
243C2×C4C2×C22⋊D18112yes—
244C2×C4C23×D18111yescharacteristic
245C8—3624no—
246C22—22no—
247C22—24no—
248C22—41no—
249C22—42no—
250C22—82no—
251C22—84no—
252C22—14484no—
253C22—7244no—
254C22—7242no—
255C22#1915378512yes—
256C22C22×C22⋊D18111yescharacteristic
257C4—24no—
258C4—22no—
259C4—7242no—
260C4#1915378511yescharacteristic
261C4#1915378511yescharacteristic
262C4C22×C22⋊D18112yes—
263C2—21no—
264C2—22no—
265C2—42no—
266C2—44no—
267C2—14482no—
268C2C2×#1915378511yescharacteristic
subgroupG/H
centerC2C2×#19153785
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#585×C181C2
socleC2×C181C2×#120
hypercenter#585D181
hypocenterC181#3021
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. C2×#19153841C2semidirect
2. #26041843C2semidirect
3. #26041452C2semidirect
4. #26041447C2semidirect
5. #26041296C2semidirect
6. #26041263C2semidirect
7. #585×C181C2semidirect
8. C22×D724C22semidirect
10. #19153841C22semidirect
13. C2×#15668433C22semidirect
14. #19153812C22semidirect
15. #19153805C22semidirect
18. #19153775C22semidirect
19. #19153743C22semidirect
20. #19153735C22semidirect
21. #19153733C22semidirect
23. 2−1+4×C181C22semidirect
26. H(5,𝔽2)×C181C22semidirect
56. #41×C181D4semidirect
60. C2×Q8×C181D4semidirect
68. C2×D4×C181D4semidirect
134. C181#3021semidirect
136. #585D181semidirect
144. 2−1+4C2×D181semidirect
147. H(5,𝔽2)C2×D181semidirect
177. #41C22⋊D181semidirect
181. C2×Q8C22⋊D181semidirect
194. C2×D4C22⋊D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538412
#260418432
#260414522
#260414472
#260412962
#260412632
#585×C1812
#3021181