Finite Group Databasebeta

← #42119701

Subgroups — #42119701

Order 27584.

334 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 105 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×#20293800C211yesmaximal, characteristic, complemented
2C2×D4×D431C211yesmaximal, characteristic, complemented
3#28352175C211yesmaximal, characteristic, complemented
4#28352174C211yesmaximal, characteristic, complemented
5#28352167C211yesmaximal, characteristic, complemented
6#28352164C211yesmaximal, characteristic, complemented
7#28352162C211yesmaximal, characteristic, complemented
8#28352161C211yesmaximal, characteristic, complemented
9#28352159C211yesmaximal, characteristic, complemented
10#28352159C211yesmaximal, characteristic, complemented
11#28352151C211yesmaximal, characteristic, complemented
12(C22⋊D431)⋊C4C211yesmaximal, characteristic, complemented
13C2×(C2×D431)⋊C4C211yesmaximal, characteristic, complemented
14C22⋊C4×D431C211yesmaximal, characteristic, complemented
15#123×C431C211yesmaximal, characteristic, complemented
16C23×D431C2211yescharacteristic
17C23×D431C2211yescharacteristic
18#20293800—24no—
19C2×Dic862C2211yescharacteristic, complemented
20C2×D1724C2211yescharacteristic
21C2×D1724C2211yescharacteristic
22C2×D1724C2211yescharacteristic, complemented
23C2×D1724C2211yescharacteristic, complemented
24C2×C22⋊D431C2211yescharacteristic
25C2×C22⋊D431C2211yescharacteristic
26C2×C22⋊D431C2211yescharacteristic, complemented
27C2×C22⋊D431C2211yescharacteristic, complemented
28D4×D431—24no—
29C22×Dic431C2211yescharacteristic, complemented
30C2×C4×D431C2211yescharacteristic
31C2×C4×D431C2211yescharacteristic, complemented
32C2×C4×D431C2211yescharacteristic, complemented
33C2×C4×D431C2211yescharacteristic, complemented
34Dic431⋊C4C2211yescharacteristic, complemented
35Dic431⋊C4C2211yescharacteristic, complemented
36C22⋊Dic431C2211yescharacteristic, complemented
37(C2×D431)⋊C4C2211yescharacteristic
38(C2×D431)⋊C4C2211yescharacteristic
39(C2×D431)⋊C4C2211yescharacteristic
40(C2×D431)⋊C4C2211yescharacteristic, complemented
41(C2×D431)⋊C4C2211yescharacteristic, complemented
42(C2×D431)⋊C4C2211yescharacteristic, complemented
43(C2×D431)⋊C4C2211yescharacteristic, complemented
44(C2×D431)⋊C4C2211yescharacteristic, complemented
45C4×Dic431C2211yescharacteristic, complemented
46C2×D4×C431C2211yescharacteristic
47C22×C4×C431C2211yescharacteristic, complemented
48C4⋊C4×C431C2211yescharacteristic, complemented
49C4⋊C4×C431C2211yescharacteristic, complemented
50C22⋊C4×C431C2211yescharacteristic, complemented
51C22⋊C4×C431C2211yescharacteristic, complemented
52C22⋊C4×C431C2211yescharacteristic, complemented
53C22×D431—22no—
54C22×D431—22no—
55C22×D431—22no—
56C22×D431—22no—
57C22×D431—41no—
58C22×D431—41no—
59C22×D431C2311yescharacteristic
60C22×D431C2311yescharacteristic
61C22×D431C2311yescharacteristic
62C22×D431C2311yescharacteristic
63Dic862—22no—
64D1724—22no—
65D1724—22no—
66D1724—22no—
67D1724—41no—
68C22⋊D431—22no—
69C22⋊D431—22no—
70C22⋊D431—41no—
71C22⋊D431D414yescomplemented
72C2×Dic431—41no—
73C2×Dic431C2311yescharacteristic
74C2×Dic431C2311yescharacteristic
75C2×Dic431C2311yescharacteristic
76C2×Dic431C2311yescharacteristic
77C4×D431—22no—
78C4×D431—22no—
79C4×D431—22no—
80C4×D431—41no—
81C23×C431C2311yescharacteristic
82C23×C431C2311yescharacteristic
83D4×C431—22no—
84C2×C4×C431—22no—
85C2×C4×C431C2311yescharacteristic
86C2×C4×C431C2311yescharacteristic
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×C431C2×D412yes—
96C22×C431C2411yescharacteristic
97C2×D431—21no—
98C2×D431—21no—
99C2×D431—21no—
100C2×D431—21no—
101C2×D431—21no—
102C2×D431—21no—
103C2×D431—21no—
104C2×D431—21no—
105C2×D431—21no—
106C2×D431—21no—
107C2×D431—44no—
108C2×D431—44no—
109C2×D431C2×D412yes—
110C2×D431#4112yescomplemented
111C4×C431—21no—
112C4×C431—21no—
113C4×C431—21no—
114C4×C431—21no—
115C4×C431—21no—
116Dic431—21no—
117Dic431—21no—
118Dic431—21no—
119Dic431C2×D412yes—
120C2×C431—22no—
121C2×C431—41no—
122C2×C431C22×D411yescharacteristic
123C2×C431C2×#4111yescharacteristic
124C2×C431H(5,𝔽2)11yescharacteristic
125D431—22no—
126D431—22no—
127D431—41no—
128D431—41no—
129C431#54611yescharacteristic, complemented, Sylow
130#546—4311nomaximal, Sylow
131C2×#41—4311no—
132C22×D4—4311no—
133#124—4311no—
134#123—4311no—
135#123D43111yescharacteristic, complemented
136C22⋊Q8—4311no—
137#121—4311no—
138#121—4311no—
139#121—4311no—
140#120—4311no—
141#120—4311no—
142C4×D4—4311no—
143C4×D4—4311no—
144C2×C22⋊C4—4311no—
145C2×C22⋊C4—4311no—
146C24—4311no—
147C24—4311no—
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×D4—4311no—
158C2×D4—4311no—
159C2×D4C2×D43111yescharacteristic
160C22×C4—4311no—
161C22×C4—4311no—
162C22×C4—4311no—
163C22×C4—4311no—
164C22×C4—4311no—
165C22×C4C2×D43111yescharacteristic, complemented
166C4⋊C4—4311no—
167C4⋊C4—4311no—
168C4⋊C4C2×D43111yescharacteristic, complemented
169C4⋊C4C2×D43111yescharacteristic, complemented
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, complemented
180C22⋊C4C2×D43111yescharacteristic, complemented
181C22⋊C4C2×D43111yescharacteristic, complemented
182C42—4311no—
183C23—8622no—
184C23—8622no—
185C23—8622no—
186C23—8622no—
187C23—4311no—
188C23—4311no—
189C23—4311no—
190C23—4311no—
191C23—17241no—
192C23—17241no—
193C23C22×D43111yescharacteristic
194C23C22×D43111yescharacteristic
195Q8—8622no—
196D4—22no—
197D4—8622no—
198D4—8622no—
199D4—8622no—
200D4—8622no—
201D4—8622no—
202D4—4314no—
203D4—17241no—
204D4—17241no—
205C2×C4—22no—
206C2×C4—8622no—
207C2×C4—8622no—
208C2×C4—8622no—
209C2×C4—4311no—
210C2×C4—4311no—
211C2×C4—4311no—
212C2×C4—4311no—
213C2×C4—17241no—
214C2×C4—17241no—
215C2×C4C22×D43111yescharacteristic
216C2×C4C22×D43111yescharacteristic
217C2×C4C22×D43111yescharacteristic
218C2×C4C22×D43111yescharacteristic
219C2×C4C22×D43111yescharacteristic
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—8621no—
226C22—8621no—
227C22—8621no—
228C22—8621no—
229C22—8621no—
230C22—8621no—
231C22—8621no—
232C22—8621no—
233C22—8621no—
234C22—8621no—
235C22—4312no—
236C22—4312no—
237C22—17244no—
238C22—17244no—
239C22D4×D43112yes—
240C22C23×D43111yescharacteristic
241C4—21no—
242C4—21no—
243C4—21no—
244C4—21no—
245C4—21no—
246C4—8621no—
247C4—8621no—
248C4—8621no—
249C4—4312no—
250C2—22no—
251C2—41no—
252C2—8622no—
253C2—8622no—
254C2—17241no—
255C2—17241no—
256C2C2×D4×D43111yescharacteristic
257C2#41×D43111yescharacteristic
258C2#2835223511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D431
commutator, socle, cosocleC22×C431C24
Fitting#123×C431C2
hypercenter#123D431
hypocenterC431#546
LCSG ▷ C22×C431 ▷ C431
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C2×#20293800C2semidirect
2. C2×D4×D431C2semidirect
3. #28352175C2semidirect
4. #28352174C2semidirect
5. #28352167C2semidirect
6. #28352164C2semidirect
7. #28352162C2semidirect
8. #28352161C2semidirect
9. #28352159C2semidirect
10. #28352159C2semidirect
11. #28352151C2semidirect
12. (C22⋊D431)⋊C4C2semidirect
13. C2×(C2×D431)⋊C4C2semidirect
14. C22⋊C4×D431C2semidirect
15. #123×C431C2semidirect
19. C2×Dic862C22semidirect
22. C2×D1724C22semidirect
23. C2×D1724C22semidirect
26. C2×C22⋊D431C22semidirect
27. C2×C22⋊D431C22semidirect
29. C22×Dic431C22semidirect
31. C2×C4×D431C22semidirect
32. C2×C4×D431C22semidirect
33. C2×C4×D431C22semidirect
34. Dic431⋊C4C22semidirect
35. Dic431⋊C4C22semidirect
36. C22⋊Dic431C22semidirect
40. (C2×D431)⋊C4C22semidirect
41. (C2×D431)⋊C4C22semidirect
42. (C2×D431)⋊C4C22semidirect
43. (C2×D431)⋊C4C22semidirect
44. (C2×D431)⋊C4C22semidirect
45. C4×Dic431C22semidirect
47. C22×C4×C431C22semidirect
48. C4⋊C4×C431C22semidirect
49. C4⋊C4×C431C22semidirect
50. C22⋊C4×C431C22semidirect
51. C22⋊C4×C431C22semidirect
52. C22⋊C4×C431C22semidirect
71. C22⋊D431D4semidirect
110. C2×D431#41semidirect
129. C431#546semidirect
135. #123D431semidirect
165. C22×C4C2×D431semidirect
168. C4⋊C4C2×D431semidirect
169. C4⋊C4C2×D431semidirect
179. C22⋊C4C2×D431semidirect
180. C22⋊C4C2×D431semidirect
181. C22⋊C4C2×D431semidirect

Maximal subgroups

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

subgroupindex
C2×#202938002
C2×D4×D4312
#283521752
#283521742
#283521672
#283521642
#283521622
#283521612
#283521592
#283521512
(C22⋊D431)⋊C42
C2×(C2×D431)⋊C42
C22⋊C4×D4312
#123×C4312
#546431