Finite Group Databasebeta

← #42119797

Subgroups — #42119797

Order 27584.

252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 91 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
1#28352185C211yesmaximal, characteristic, complemented
2#28352184C211yesmaximal, characteristic, complemented
3#28352182C211yesmaximal, characteristic, complemented
4#28352174C211yesmaximal, characteristic, complemented
5#28352173C211yesmaximal, characteristic, complemented
6#28352172C211yesmaximal, characteristic, complemented
7#28352163C211yesmaximal, characteristic, complemented
8#28352162C211yesmaximal, characteristic, complemented
9#28352161C211yesmaximal, characteristic, complemented
10#28352161C211yesmaximal, characteristic, complemented
11#28352160C211yesmaximal, characteristic, complemented
12#28352159C211yesmaximal, characteristic, complemented
13#28352157C211yesmaximal, characteristic, complemented
14#28352157C211yesmaximal, characteristic, complemented
15#121×C431C211yesmaximal, characteristic, complemented
16C23×D431C2211yescharacteristic
17C2×D1724C2211yescharacteristic
18C2×D1724C2211yescharacteristic
19C2×C22⋊D431C2211yescharacteristic
20C2×C22⋊D431C2211yescharacteristic
21C2×C22⋊D431C2211yescharacteristic
22C2×C22⋊D431C2211yescharacteristic
23C2×C22⋊D431C2211yescharacteristic
24C2×C22⋊D431C2211yescharacteristic, complemented
25C2×C22⋊D431C2211yescharacteristic, complemented
26C22×Dic431C2211yescharacteristic, complemented
27C2×C4×D431C2211yescharacteristic, complemented
28C4⋊Dic431C2211yescharacteristic, complemented
29C4⋊Dic431C2211yescharacteristic, complemented
30Dic431⋊C4C2211yescharacteristic, complemented
31Dic431⋊C4C2211yescharacteristic, complemented
32Dic431⋊C4C2211yescharacteristic, complemented
33C22⋊Dic431C2211yescharacteristic
34C22⋊Dic431C2211yescharacteristic
35C22⋊Dic431C2211yescharacteristic, complemented
36(C2×D431)⋊C4C2211yescharacteristic
37(C2×D431)⋊C4C2211yescharacteristic
38(C2×D431)⋊C4C2211yescharacteristic
39(C2×D431)⋊C4C2211yescharacteristic
40(C2×D431)⋊C4C2211yescharacteristic
41(C2×D431)⋊C4C2211yescharacteristic
42(C2×D431)⋊C4C2211yescharacteristic, complemented
43C4×Dic431C2211yescharacteristic, complemented
44C2×D4×C431C2211yescharacteristic
45C2×D4×C431C2211yescharacteristic
46C2×D4×C431C2211yescharacteristic, complemented
47C22×C4×C431C2211yescharacteristic, complemented
48C4⋊C4×C431C2211yescharacteristic, complemented
49C22⋊C4×C431C2211yescharacteristic
50C22⋊C4×C431C2211yescharacteristic, complemented
51C22×D431—41no—
52C22×D431—41no—
53C22×D431—41no—
54C22×D431C2311yescharacteristic
55C22×D431C2311yescharacteristic
56C22×D431C2311yescharacteristic
57D1724—41no—
58D1724—41no—
59C22⋊D431—41no—
60C22⋊D431—41no—
61C22⋊D431—41no—
62C22⋊D431—41no—
63C22⋊D431—41no—
64C22⋊D431—41no—
65C22⋊D431—41no—
66C2×Dic431—41no—
67C2×Dic431C2311yescharacteristic
68C2×Dic431C2311yescharacteristic
69C2×Dic431C2311yescharacteristic
70C2×Dic431C2311yescharacteristic
71C2×Dic431C2311yescharacteristic
72C4×D431—41no—
73C23×C431C2311yescharacteristic
74C23×C431C2311yescharacteristic
75C23×C431C2311yescharacteristic
76D4×C431—41no—
77D4×C431—41no—
78D4×C431—41no—
79C2×C4×C431—41no—
80C2×C4×C431C2311yescharacteristic
81C2×C4×C431C2311yescharacteristic
82C2×C4×C431C2311yescharacteristic
83C2×C4×C431C2311yescharacteristic
84C22×C431—21no—
85C22×C431—21no—
86C22×C431—21no—
87C22×C431—21no—
88C22×C431—21no—
89C22×C431—21no—
90C22×C431—21no—
91C22×C431—21no—
92C22×C431—21no—
93C22×C431C2411yescharacteristic
94C2×D431—21no—
95C2×D431—21no—
96C2×D431—21no—
97C2×D431—21no—
98C2×D431—21no—
99C2×D431—21no—
100C2×D431—21no—
101C2×D431—21no—
102C2×D431—21no—
103C2×D431—44no—
104C4×C431—21no—
105C4×C431—21no—
106C4×C431—21no—
107C4×C431—21no—
108Dic431—21no—
109Dic431—21no—
110Dic431—21no—
111Dic431—21no—
112Dic431—21no—
113C2×C431—41no—
114C2×C431—41no—
115C2×C431—41no—
116C2×C431H(5,𝔽2)11yescharacteristic
117C2×C431H(5,𝔽2)11yescharacteristic
118C2×C431H(5,𝔽2)11yescharacteristic
119D431—41no—
120D431—41no—
121D431—41no—
122C431#56011yescharacteristic, complemented, Sylow
123#560—4311nomaximal, Sylow
124#127—4311no—
125#126—4311no—
126#126—4311no—
127#123—4311no—
128#123—4311no—
129#123—4311no—
130#121—4311no—
131#121—4311no—
132#121—4311no—
133#121—4311no—
134#121—4311no—
135#121D43111yescharacteristic, complemented
136#120—4311no—
137#120—4311no—
138#120—4311no—
139C24—4311no—
140C2×D4—4311no—
141C2×D4—4311no—
142C2×D4—4311no—
143C2×D4—4311no—
144C2×D4—4311no—
145C2×D4—4311no—
146C2×D4—4311no—
147C2×D4—4311no—
148C2×D4—4311no—
149C2×D4C2×D43111yescharacteristic
150C2×D4C2×D43111yescharacteristic
151C2×D4C2×D43111yescharacteristic, complemented
152C22×C4—4311no—
153C22×C4—4311no—
154C22×C4C2×D43111yescharacteristic, complemented
155C4⋊C4—4311no—
156C4⋊C4—4311no—
157C4⋊C4—4311no—
158C4⋊C4—4311no—
159C4⋊C4—4311no—
160C4⋊C4C2×D43111yescharacteristic, complemented
161C22⋊C4—4311no—
162C22⋊C4—4311no—
163C22⋊C4—4311no—
164C22⋊C4—4311no—
165C22⋊C4—4311no—
166C22⋊C4—4311no—
167C22⋊C4—4311no—
168C22⋊C4—4311no—
169C22⋊C4—4311no—
170C22⋊C4—4311no—
171C22⋊C4C2×D43111yescharacteristic
172C22⋊C4C2×D43111yescharacteristic, complemented
173C42—4311no—
174C23—4311no—
175C23—4311no—
176C23—4311no—
177C23—17241no—
178C23—17241no—
179C23—17241no—
180C23C22×D43111yescharacteristic
181C23C22×D43111yescharacteristic
182C23C22×D43111yescharacteristic
183D4—41no—
184D4—41no—
185D4—41no—
186D4—17241no—
187D4—17241no—
188D4—17241no—
189D4—17241no—
190D4—17241no—
191D4—17241no—
192D4—17241no—
193D4—17241no—
194D4—17241no—
195C2×C4—41no—
196C2×C4—4311no—
197C2×C4—4311no—
198C2×C4—4311no—
199C2×C4—4311no—
200C2×C4—4311no—
201C2×C4—17241no—
202C2×C4—17241no—
203C2×C4C22×D43111yescharacteristic
204C2×C4C22×D43111yescharacteristic
205C2×C4C22×D43111yescharacteristic
206C2×C4C22×D43111yescharacteristic
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—8621no—
217C22—8621no—
218C22—8621no—
219C22—8621no—
220C22—8621no—
221C22—8621no—
222C22—8621no—
223C22—8621no—
224C22—8621no—
225C22—17244no—
226C22C23×D43111yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—21no—
231C4—8621no—
232C4—8621no—
233C4—8621no—
234C4—8621no—
235C4—8621no—
236C2—41no—
237C2—41no—
238C2—41no—
239C2—17241no—
240C2—17241no—
241C2—17241no—
242C2#2835223411yescharacteristic
243C2#2835223411yescharacteristic
244C2#2835223511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D431
commutator, socle, cosocleC22×C431C24
Fitting#121×C431C2
hypercenter#121D431
hypocenterC431#560
LCSG ▷ C22×C431 ▷ C431
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #28352185C2semidirect
2. #28352184C2semidirect
3. #28352182C2semidirect
4. #28352174C2semidirect
5. #28352173C2semidirect
6. #28352172C2semidirect
7. #28352163C2semidirect
8. #28352162C2semidirect
9. #28352161C2semidirect
10. #28352161C2semidirect
11. #28352160C2semidirect
12. #28352159C2semidirect
13. #28352157C2semidirect
14. #28352157C2semidirect
15. #121×C431C2semidirect
24. C2×C22⋊D431C22semidirect
25. C2×C22⋊D431C22semidirect
26. C22×Dic431C22semidirect
27. C2×C4×D431C22semidirect
28. C4⋊Dic431C22semidirect
29. C4⋊Dic431C22semidirect
30. Dic431⋊C4C22semidirect
31. Dic431⋊C4C22semidirect
32. Dic431⋊C4C22semidirect
35. C22⋊Dic431C22semidirect
42. (C2×D431)⋊C4C22semidirect
43. C4×Dic431C22semidirect
46. C2×D4×C431C22semidirect
47. C22×C4×C431C22semidirect
48. C4⋊C4×C431C22semidirect
50. C22⋊C4×C431C22semidirect
122. C431#560semidirect
135. #121D431semidirect
151. C2×D4C2×D431semidirect
154. C22×C4C2×D431semidirect
160. C4⋊C4C2×D431semidirect
172. C22⋊C4C2×D431semidirect

Maximal subgroups

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

subgroupindex
#283521852
#283521842
#283521822
#283521742
#283521732
#283521722
#283521632
#283521622
#283521612
#283521602
#283521592
#283521572
#121×C4312
#560431