Finite Group Databasebeta

← #37550342

Subgroups — #37550342

Order 23168.

778 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 778 conjugacy classes of subgroups — too many to draw. Only its 200 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
1D181⋊#131C211yesmaximal, characteristic, complemented
2#26042348C211yesmaximal, characteristic, complemented
3#26042271C213yesmaximal, complemented
4#26042270C213yesmaximal, complemented
5#26042269C213yesmaximal, complemented
6C2×#19153878C213yesmaximal, complemented
7H(5,𝔽2)×D181C211yesmaximal, characteristic
8#19153904—21no—
9#19153903—23no—
10D181⋊M16C2213yescomplemented
11C2×D181⋊C8C2213yescomplemented
12Q8×#10624382—21no—
13D4×#10624382—23no—
14#19153895C2213yescomplemented
15C2×C4⋊#10624382C2213yescomplemented
16#19153879C2211yescharacteristic, complemented
17#19153879C2213yescomplemented
18#19153878C2213yescomplemented
19#19153878C2216yescomplemented
20#19153878C2213yescomplemented
21#19153862—21no—
22#19153862C411yescharacteristic, complemented
23#19153862C413yescomplemented
24#19153861—23no—
25#19153861C413yescomplemented
26#41×D181—21no—
27#41×D181C2211yescharacteristic
28#41×D181C2213yes—
29C2×D4×D181—23no—
30C2×D4×D181C2213yes—
31H(5,𝔽2)×C181C411yescharacteristic, complemented
32C22×#10624382—23no—
33#15668459—23no—
34#15668458—23no—
35D181⋊C8—23no—
36D181⋊C8C2311yescharacteristic
37D181⋊C8C2313yes—
38C2×#13089685—23no—
39C4⋊#10624382—23no—
40C4⋊#10624382C2313yes—
41C4⋊#10624382C2311yescharacteristic
42C22⋊#10624382—23no—
43C4×#10624382—23no—
44C23×D181—23no—
45#15668450—21no—
46#15668450C2×C413yescomplemented
47#15668450D411yescharacteristic
48#15668449—23no—
49#15668449—23no—
50#15668449C2×C413yes—
51#15668449C2×C413yescomplemented
52#15668448—23no—
53#15668448—23no—
54#15668448C2×C413yescomplemented
55#15668448D413yes—
56Q8×D181D411yescharacteristic
57Q8×D181C2311yescharacteristic
58C2×D724—23no—
59C2×D724C2×C413yescomplemented
60C2×C22⋊D181—26no—
61C2×C22⋊D181—23no—
62D4×D181—23no—
63D4×D181—23no—
64D4×D181—26no—
65D4×D181—23no—
66D4×D181—23no—
67D4×D181—23no—
68D4×D181C2×C416yescomplemented
69D4×D181D413yes—
70D4×D181C2313yes—
71C2×C4×D181—23no—
72C2×C4×D181C2313yes—
73#41×C181—21no—
74#41×C181C2×C411yescharacteristic
75#41×C181C2×C413yescomplemented
76C2×D4×C181—23no—
77C2×D4×C181C2×C413yescomplemented
78C2×#10624382—21no—
79C2×#10624382—23no—
80C2×#10624382—83no—
81#13089685—21no—
82#13089685—23no—
83C22×D181—23no—
84C22×D181—23no—
85C22×D181—26no—
86C22×D181—83no—
87C22×D181—83no—
88C22×D181C2×D413yes—
89Dic362—21no—
90Dic362C22⋊C413yes—
91Dic362C22×C411yescharacteristic
92D724—23no—
93D724—23no—
94D724C22⋊C413yes—
95D724C22×C413yes—
96C22⋊D181—23no—
97C22⋊D181—23no—
98C22⋊D181—26no—
99C22⋊D181—23no—
100C22⋊D181C22⋊C416yes—
101C2×Dic181—23no—
102C2×Dic181C2×D413yes—
103C4×D181—21no—
104C4×D181—23no—
105C4×D181—23no—
106C4×D181C22×C413yes—
107C4×D181C2×D413yes—
108C4×D181C2×D413yes—
109C4×D181C2411yescharacteristic
110C23×C181—23no—
111Q8×C181C22⋊C411yescharacteristic
112Q8×C181C22×C411yescharacteristic
113D4×C181—23no—
114D4×C181—23no—
115D4×C181C22⋊C413yes—
116D4×C181C22×C413yes—
117C2×C4×C181—23no—
118C2×C4×C181C22×C413yes—
119C22×C181—23no—
120C22×C181—83no—
121C22×C181C2×C22⋊C413yes—
122C2×D181—23no—
123C2×D181—23no—
124C2×D181—23no—
125C2×D181—43no—
126C2×D181—43no—
127C2×D181—86no—
128C2×D181—83no—
129C2×D181C2×C22⋊C413yes—
130C2×D181C22×D411yescharacteristic
131#10624382—43no—
132#10624382—41no—
133C4×C181—21no—
134C4×C181C2×C22⋊C413yes—
135C4×C181C23×C411yescharacteristic
136Dic181—21no—
137Dic181C2×C22⋊C413yes—
138Dic181C22×D411yescharacteristic
139C2×C181—23no—
140C2×C181—43no—
141C2×C181C22×C22⋊C411yescharacteristic
142D181—23no—
143D181—43no—
144D181#57611yescharacteristic
145D181#57711yescharacteristic
146C181#289711yescharacteristic, complemented, Sylow
147#2897—1811nomaximal, Sylow
148C2×H(5,𝔽2)—1811no—
149C2×#131—1811no—
150#520—1811no—
151#418—1813no—
152#417—1813no—
153#416—1813no—
154C2×#102—1813no—
155H(5,𝔽2)—3623no—
156H(5,𝔽2)—3621no—
157H(5,𝔽2)—1811no—
158H(5,𝔽2)—1813no—
159H(5,𝔽2)—1813no—
160H(5,𝔽2)#1062438211yescharacteristic, complemented
161C2×#41—3621no—
162C2×#41—1813no—
163C2×#41—1811no—
164C22×D4—3623no—
165C22×D4—1813no—
166#131—3623no—
167#131—3621no—
168C2×M16—1813no—
169C22×C8—1813no—
170C4×Q8—3621no—
171C4×D4—3623no—
172#117—1813no—
173C2×C4⋊C4—1813no—
174#103—1811no—
175#103—1813no—
176#102—1813no—
177#102—1816no—
178#102—1813no—
179C24—3623no—
180#41—21no—
181#41—3623no—
182#41—3623no—
183#41—3623no—
184#41—3621no—
185#41—3623no—
186#41—1813no—
187#41—1813no—
188#41—1813no—
189#41—1811no—
190#41—1813no—
191#41—1813no—
192#41C2×#1062438211yescharacteristic
193#41C2×#1062438213yescomplemented
194C2×Q8—1811no—
195C2×Q8—1811no—
196C2×D4—23no—
197C2×D4—3623no—
198C2×D4—3623no—
199C2×D4—3626no—
200C2×D4—3623no—
201C2×D4—3626no—
202C2×D4—3623no—
203C2×D4—3623no—
204C2×D4—3623no—
205C2×D4—3623no—
206C2×D4—1813no—
207C2×D4—1816no—
208C2×D4—1813no—
209C2×D4—1813no—
210C2×D4C2×#1062438213yescomplemented
211C22×C4—3623no—
212C22×C4—3623no—
213C22×C4—1813no—
214M16—3623no—
215M16—3623no—
216C2×C8—3623no—
217C2×C8—3623no—
218C2×C8—1811no—
219C2×C8—1813no—
220C4⋊C4—3623no—
221C4⋊C4—1813no—
222C4⋊C4—1811no—
223C22⋊C4—3623no—
224C42—3623no—
225C23—23no—
226C23—3626no—
227C23—3623no—
228C23—3623no—
229C23—14483no—
230C23—14483no—
231C23—1813no—
232Q8—3621no—
233Q8—1811no—
234Q8—1813no—
235Q8C22⋊#1062438211yescharacteristic
236Q8C22×#1062438211yescharacteristic
237D4—23no—
238D4—23no—
239D4—3623no—
240D4—3623no—
241D4—3623no—
242D4—3623no—
243D4—3626no—
244D4—3623no—
245D4—1816no—
246D4—1813no—
247D4—1813no—
248D4C22⋊#1062438213yes—
249D4C22×#1062438213yes—
250C2×C4—23no—
251C2×C4—3621no—
252C2×C4—3623no—
253C2×C4—3623no—
254C2×C4—3623no—
255C2×C4—3621no—
256C2×C4—3623no—
257C2×C4—14483no—
258C2×C4—1813no—
259C2×C4—1813no—
260C2×C4—1813no—
261C2×C4—1811no—
262C2×C4—1813no—
263C2×C4C22×#1062438213yes—
264C8—3621no—
265C8—3623no—
266C22—23no—
267C22—83no—
268C22—3623no—
269C22—3623no—
270C22—3623no—
271C22—14483no—
272C22—14486no—
273C22—1813no—
274C22—1811no—
275C22—7243no—
276C22—7243no—
277C22C2×C22⋊#1062438213yes—
278C4—21no—
279C4—3621no—
280C4—1811no—
281C4—1813no—
282C4—7243no—
283C4—7241no—
284C4C2×C22⋊#1062438213yes—
285C4C23×#1062438211yescharacteristic
286C2—23no—
287C2—43no—
288C2—3623no—
289C2—1811no—
290C2—1811no—
291C2—7243no—
292C2C22×C22⋊#1062438211yescharacteristic
subgroupG/H
centerC2C22×C22⋊#10624382
commutatorC4×C181C23×C4
FrattiniC4C23×#10624382
FittingH(5,𝔽2)×C181C4
socleC2×C181C22×C22⋊C4
prime radicalH(5,𝔽2)×D181C2
cosocleC4×D181C24
hypercenterH(5,𝔽2)#10624382
hypocenterC181#2897
LCSG ▷ C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ #41 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. D181⋊#131C2semidirect
2. #26042348C2semidirect
3. #26042271C2semidirect
4. #26042270C2semidirect
5. #26042269C2semidirect
6. C2×#19153878C2semidirect
10. D181⋊M16C22semidirect
11. C2×D181⋊C8C22semidirect
14. #19153895C22semidirect
15. C2×C4⋊#10624382C22semidirect
16. #19153879C22semidirect
17. #19153879C22semidirect
18. #19153878C22semidirect
19. #19153878C22semidirect
20. #19153878C22semidirect
22. #19153862C4semidirect
23. #19153862C4semidirect
25. #19153861C4semidirect
31. H(5,𝔽2)×C181C4semidirect
46. #15668450C2×C4semidirect
51. #15668449C2×C4semidirect
54. #15668448C2×C4semidirect
59. C2×D724C2×C4semidirect
68. D4×D181C2×C4semidirect
75. #41×C181C2×C4semidirect
77. C2×D4×C181C2×C4semidirect
146. C181#2897semidirect
160. H(5,𝔽2)#10624382semidirect
193. #41C2×#10624382semidirect
210. C2×D4C2×#10624382semidirect

Maximal subgroups

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

subgroupindex
D181⋊#1312
#260423482
#260422712
#260422702
#260422692
C2×#191538782
H(5,𝔽2)×D1812
#2897181