Finite Group Databasebeta

← #42119696

Subgroups — #42119696

Order 27584.

334 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 107 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×#20293799C211yesmaximal, characteristic, complemented
2C22×C22⋊D431C211yesmaximal, characteristic, complemented
3#28352175C211yesmaximal, characteristic, complemented
4#28352173C211yesmaximal, characteristic, complemented
5#28352171C211yesmaximal, characteristic, complemented
6#28352166C211yesmaximal, characteristic, complemented
7#28352161C211yesmaximal, characteristic, complemented
8#28352161C211yesmaximal, characteristic, complemented
9#28352160C211yesmaximal, characteristic, complemented
10#28352157C211yesmaximal, characteristic, complemented
11D4×Dic431C211yesmaximal, characteristic, complemented
12(C22⋊D431)⋊C4C211yesmaximal, characteristic, complemented
13C2×C22⋊Dic431C211yesmaximal, characteristic, complemented
14C22⋊C4×D431C211yesmaximal, characteristic, complemented
15#120×C431C211yesmaximal, characteristic, complemented
16C23×D431C2211yescharacteristic
17#20293799—24no—
18C2×Dic862C2211yescharacteristic, complemented
19C2×C22⋊D431—24no—
20C2×C22⋊D431C2211yescharacteristic
21C2×C22⋊D431C2211yescharacteristic
22C2×C22⋊D431C2211yescharacteristic
23C2×C22⋊D431C2211yescharacteristic
24C2×C22⋊D431C2211yescharacteristic, complemented
25C2×C22⋊D431C2211yescharacteristic, complemented
26C22×Dic431C2211yescharacteristic
27C22×Dic431C2211yescharacteristic, complemented
28C22×Dic431C2211yescharacteristic, complemented
29C22×Dic431C2211yescharacteristic, complemented
30C2×C4×D431C2211yescharacteristic, complemented
31C2×C4×D431C2211yescharacteristic, complemented
32C4⋊Dic431C2211yescharacteristic, complemented
33C4⋊Dic431C2211yescharacteristic, complemented
34Dic431⋊C4C2211yescharacteristic, complemented
35Dic431⋊C4C2211yescharacteristic, complemented
36C22⋊Dic431C2211yescharacteristic
37C22⋊Dic431C2211yescharacteristic
38C22⋊Dic431C2211yescharacteristic, complemented
39C22⋊Dic431C2211yescharacteristic, complemented
40C22⋊Dic431C2211yescharacteristic, complemented
41(C2×D431)⋊C4C2211yescharacteristic, complemented
42(C2×D431)⋊C4C2211yescharacteristic, complemented
43(C2×D431)⋊C4C2211yescharacteristic, complemented
44(C2×D431)⋊C4C2211yescharacteristic, complemented
45C4×Dic431C2211yescharacteristic, complemented
46C24×C431C2211yescharacteristic
47C2×D4×C431C2211yescharacteristic
48C2×D4×C431C2211yescharacteristic, complemented
49C2×D4×C431C2211yescharacteristic, complemented
50C22⋊C4×C431C2211yescharacteristic
51C22⋊C4×C431C2211yescharacteristic, complemented
52C22⋊C4×C431C2211yescharacteristic, complemented
53C22×D431—22no—
54C22×D431—22no—
55C22×D431—41no—
56C22×D431C2311yescharacteristic
57C22×D431C2311yescharacteristic
58Dic862—22no—
59C22⋊D431—22no—
60C22⋊D431—22no—
61C22⋊D431—22no—
62C22⋊D431—22no—
63C22⋊D431—41no—
64C22⋊D431D414yescomplemented
65C2×Dic431—22no—
66C2×Dic431—22no—
67C2×Dic431—22no—
68C2×Dic431—41no—
69C2×Dic431C2311yescharacteristic
70C2×Dic431C2311yescharacteristic
71C2×Dic431C2311yescharacteristic
72C2×Dic431C2311yescharacteristic
73C2×Dic431C2311yescharacteristic
74C2×Dic431C2311yescharacteristic
75C4×D431—22no—
76C4×D431—41no—
77C23×C431—22no—
78C23×C431—22no—
79C23×C431—41no—
80C23×C431C2311yescharacteristic
81C23×C431C2311yescharacteristic
82C23×C431C2311yescharacteristic
83C23×C431C2311yescharacteristic
84D4×C431—22no—
85D4×C431—22no—
86D4×C431—41no—
87C2×C4×C431C2311yescharacteristic
88C2×C4×C431C2311yescharacteristic
89C2×C4×C431C2311yescharacteristic
90C22×C431—21no—
91C22×C431—21no—
92C22×C431—21no—
93C22×C431—21no—
94C22×C431—21no—
95C22×C431—21no—
96C22×C431—21no—
97C22×C431—21no—
98C22×C431—21no—
99C22×C431—21no—
100C22×C431—44no—
101C22×C431C2×D412yes—
102C22×C431#4112yescomplemented
103C22×C431C2411yescharacteristic
104C2×D431—21no—
105C2×D431—21no—
106C2×D431—21no—
107C2×D431—21no—
108C2×D431—21no—
109C2×D431—44no—
110C2×D431C2×D412yes—
111C4×C431—21no—
112C4×C431—21no—
113C4×C431—21no—
114Dic431—21no—
115Dic431—21no—
116Dic431—21no—
117Dic431—21no—
118Dic431—21no—
119Dic431C2×D412yes—
120C2×C431—22no—
121C2×C431—22no—
122C2×C431—41no—
123C2×C431—41no—
124C2×C431C22×D411yescharacteristic
125C2×C431C2×#4111yescharacteristic
126C2×C431H(5,𝔽2)11yescharacteristic
127D431—22no—
128D431—41no—
129C431#54611yescharacteristic, complemented, Sylow
130#546—4311nomaximal, Sylow
131C2×#41—4311no—
132C22×D4—4311no—
133#124—4311no—
134#123—4311no—
135#123—4311no—
136C22⋊Q8—4311no—
137#121—4311no—
138#121—4311no—
139#121—4311no—
140#120—4311no—
141#120D43111yescharacteristic, complemented
142C4×D4—4311no—
143C4×D4—4311no—
144C2×C22⋊C4—4311no—
145C2×C22⋊C4—4311no—
146C24—4311no—
147C24C2×D43111yescharacteristic
148#41—8624no—
149C2×Q8—4311no—
150C2×D4—8624no—
151C2×D4—4311no—
152C2×D4—4311no—
153C2×D4—4311no—
154C2×D4—4311no—
155C2×D4—4311no—
156C2×D4—4311no—
157C2×D4C2×D43111yescharacteristic
158C2×D4C2×D43111yescharacteristic, complemented
159C2×D4C2×D43111yescharacteristic, complemented
160C22×C4—4311no—
161C22×C4—4311no—
162C22×C4—4311no—
163C22×C4—4311no—
164C22×C4—4311no—
165C22×C4—4311no—
166C4⋊C4—4311no—
167C4⋊C4—4311no—
168C4⋊C4—4311no—
169C4⋊C4—4311no—
170C22⋊C4—4311no—
171C22⋊C4—4311no—
172C22⋊C4—4311no—
173C22⋊C4—4311no—
174C22⋊C4—4311no—
175C22⋊C4—4311no—
176C22⋊C4—4311no—
177C22⋊C4—4311no—
178C22⋊C4—4311no—
179C22⋊C4C2×D43111yescharacteristic
180C22⋊C4C2×D43111yescharacteristic, complemented
181C22⋊C4C2×D43111yescharacteristic, complemented
182C42—4311no—
183C23—22no—
184C23—22no—
185C23—41no—
186C23—8622no—
187C23—8622no—
188C23—4311no—
189C23—4311no—
190C23—17241no—
191C23C22×D43111yescharacteristic
192C23C22×D43111yescharacteristic
193C23C22×D43111yescharacteristic
194C23C22×D43111yescharacteristic
195Q8—8622no—
196D4—22no—
197D4—22no—
198D4—41no—
199D4—8622no—
200D4—8622no—
201D4—8622no—
202D4—8622no—
203D4—4314no—
204D4—17241no—
205C2×C4—8622no—
206C2×C4—8622no—
207C2×C4—8622no—
208C2×C4—8622no—
209C2×C4—4311no—
210C2×C4—4311no—
211C2×C4—4311no—
212C2×C4—4311no—
213C2×C4—4311no—
214C2×C4—4311no—
215C2×C4—17241no—
216C2×C4—17241no—
217C2×C4C22×D43111yescharacteristic
218C2×C4C22×D43111yescharacteristic
219C2×C4C22×D43111yescharacteristic
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—21no—
230C22—44no—
231C22—8621no—
232C22—8621no—
233C22—8621no—
234C22—8621no—
235C22—8621no—
236C22—4312no—
237C22—17244no—
238C22D4×D43112yes—
239C22#2029379912yescomplemented
240C22C23×D43111yescharacteristic
241C4—21no—
242C4—21no—
243C4—21no—
244C4—8621no—
245C4—8621no—
246C4—8621no—
247C4—8621no—
248C4—8621no—
249C4—4312no—
250C2—22no—
251C2—22no—
252C2—41no—
253C2—41no—
254C2—8622no—
255C2—17241no—
256C2C2×D4×D43111yescharacteristic
257C2C2×#2029379911yescharacteristic
258C2#2835223411yescharacteristic
subgroupG/H
center, FrattiniC22C23×D431
commutator, socle, cosocleC22×C431C24
Fitting#120×C431C2
hypercenter#120D431
hypocenterC431#546
LCSG ▷ C22×C431 ▷ C431
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C2×#20293799C2semidirect
2. C22×C22⋊D431C2semidirect
3. #28352175C2semidirect
4. #28352173C2semidirect
5. #28352171C2semidirect
6. #28352166C2semidirect
7. #28352161C2semidirect
8. #28352161C2semidirect
9. #28352160C2semidirect
10. #28352157C2semidirect
11. D4×Dic431C2semidirect
12. (C22⋊D431)⋊C4C2semidirect
13. C2×C22⋊Dic431C2semidirect
14. C22⋊C4×D431C2semidirect
15. #120×C431C2semidirect
18. C2×Dic862C22semidirect
24. C2×C22⋊D431C22semidirect
25. C2×C22⋊D431C22semidirect
27. C22×Dic431C22semidirect
28. C22×Dic431C22semidirect
29. C22×Dic431C22semidirect
30. C2×C4×D431C22semidirect
31. C2×C4×D431C22semidirect
32. C4⋊Dic431C22semidirect
33. C4⋊Dic431C22semidirect
34. Dic431⋊C4C22semidirect
35. Dic431⋊C4C22semidirect
38. C22⋊Dic431C22semidirect
39. C22⋊Dic431C22semidirect
40. C22⋊Dic431C22semidirect
41. (C2×D431)⋊C4C22semidirect
42. (C2×D431)⋊C4C22semidirect
43. (C2×D431)⋊C4C22semidirect
44. (C2×D431)⋊C4C22semidirect
45. C4×Dic431C22semidirect
48. C2×D4×C431C22semidirect
49. C2×D4×C431C22semidirect
51. C22⋊C4×C431C22semidirect
52. C22⋊C4×C431C22semidirect
64. C22⋊D431D4semidirect
102. C22×C431#41semidirect
129. C431#546semidirect
141. #120D431semidirect
158. C2×D4C2×D431semidirect
159. C2×D4C2×D431semidirect
180. C22⋊C4C2×D431semidirect
181. C22⋊C4C2×D431semidirect
239. C22#20293799semidirect

Maximal subgroups

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

subgroupindex
C2×#202937992
C22×C22⋊D4312
#283521752
#283521732
#283521712
#283521662
#283521612
#283521602
#283521572
D4×Dic4312
(C22⋊D431)⋊C42
C2×C22⋊Dic4312
C22⋊C4×D4312
#120×C4312
#546431