Finite Group Databasebeta

← #42119624

Subgroups — #42119624

Order 27584.

334 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 103 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#28352185C212yesmaximal, complemented
4#28352175C212yesmaximal, complemented
5#28352162C212yesmaximal, complemented
6#28352161C212yesmaximal, complemented
7#28352159C211yesmaximal, characteristic, complemented
8#28352158C211yesmaximal, characteristic, complemented
9#28352157C211yesmaximal, characteristic, complemented
10#28352145C211yesmaximal, characteristic, complemented
11#120×C431C211yesmaximal, characteristic, complemented
12C23×D431C2211yescharacteristic
13#20293799—24no—
14C2×Dic862C2211yescharacteristic, complemented
15C2×D1724C2212yescomplemented
16C2×C22⋊D431—24no—
17C2×C22⋊D431C2211yescharacteristic
18C2×C22⋊D431C2212yes—
19C2×C22⋊D431C2212yes—
20C2×C22⋊D431C2211yescharacteristic, complemented
21C2×C22⋊D431C2212yescomplemented
22C2×C22⋊D431C2212yescomplemented
23C22×Dic431C2211yescharacteristic
24C22×Dic431C2211yescharacteristic, complemented
25C2×C4×D431C2211yescharacteristic, complemented
26Dic431⋊C4C2212yescomplemented
27C22⋊Dic431C2211yescharacteristic
28C22⋊Dic431C2212yescomplemented
29(C2×D431)⋊C4C2212yescomplemented
30(C2×D431)⋊C4C2212yescomplemented
31C4×Dic431C2212yescomplemented
32C24×C431C2211yescharacteristic
33C2×D4×C431C2211yescharacteristic
34C2×D4×C431C2212yescomplemented
35C22⋊C4×C431C2211yescharacteristic
36C22⋊C4×C431C2212yescomplemented
37C22×D431—22no—
38C22×D431—42no—
39C22×D431C2312yes—
40C22×D431C2311yescharacteristic
41Dic862—22no—
42D1724—42no—
43C22⋊D431—22no—
44C22⋊D431—24no—
45C22⋊D431—24no—
46C22⋊D431—22no—
47C22⋊D431—42no—
48C22⋊D431—42no—
49C2×Dic431—22no—
50C2×Dic431D414yescomplemented
51C2×Dic431C2312yes—
52C2×Dic431C2311yescharacteristic
53C2×Dic431C2311yescharacteristic
54C2×Dic431C2311yescharacteristic
55C4×D431—22no—
56C23×C431—22no—
57C23×C431—42no—
58C23×C431C2312yes—
59C23×C431C2311yescharacteristic
60C23×C431C2311yescharacteristic
61D4×C431—22no—
62D4×C431—42no—
63C2×C4×C431C2312yes—
64C2×C4×C431C2311yescharacteristic
65C22×C431—22no—
66C22×C431—21no—
67C22×C431—22no—
68C22×C431—22no—
69C22×C431—22no—
70C22×C431—22no—
71C22×C431—44no—
72C22×C431C2×D412yes—
73C22×C431C2411yescharacteristic
74C2×D431—21no—
75C2×D431—22no—
76C2×D431—22no—
77C2×D431—22no—
78C2×D431—22no—
79C2×D431—44no—
80C4×C431—21no—
81C4×C431—22no—
82Dic431—21no—
83Dic431—22no—
84Dic431C2×D412yes—
85Dic431C2×D412yes—
86C2×C431—22no—
87C2×C431—41no—
88C2×C431—42no—
89C2×C431C22×D411yescharacteristic
90C2×C431H(5,𝔽2)12yes—
91D431—41no—
92D431—42no—
93C431#53511yescharacteristic, complemented, Sylow
94#535—4311nomaximal, Sylow
95C2×#41—4311no—
96C22×D4—4311no—
97#127—4312no—
98#124—4312no—
99#121—4312no—
100#121—4312no—
101#120—4311no—
102#120—4311no—
103#120—4311no—
104#120D43111yescharacteristic, complemented
105#117—4311no—
106C24—4311no—
107C24C2×D43111yescharacteristic
108#41—8624no—
109C2×Q8—4311no—
110C2×D4—8624no—
111C2×D4—4311no—
112C2×D4—4312no—
113C2×D4—4312no—
114C2×D4—4311no—
115C2×D4—4312no—
116C2×D4—4312no—
117C2×D4—4312no—
118C2×D4C2×D43111yescharacteristic
119C2×D4C2×D43112yescomplemented
120C22×C4—4311no—
121C22×C4—4311no—
122C22×C4—4311no—
123C4⋊C4—4312no—
124C22⋊C4—4311no—
125C22⋊C4—4312no—
126C22⋊C4—4312no—
127C22⋊C4—4312no—
128C22⋊C4C2×D43111yescharacteristic
129C22⋊C4C2×D43112yescomplemented
130C42—4312no—
131C23—22no—
132C23—42no—
133C23—8622no—
134C23—4312no—
135C23—4311no—
136C23—17242no—
137C23C22×D43112yes—
138C23C22×D43111yescharacteristic
139C23C22×D43111yescharacteristic
140Q8—8622no—
141D4—22no—
142D4—42no—
143D4—8622no—
144D4—8624no—
145D4—8624no—
146D4—8622no—
147D4—17242no—
148D4—17242no—
149D4—17242no—
150C2×C4—8622no—
151C2×C4—8622no—
152C2×C4—4312no—
153C2×C4—4311no—
154C2×C4—4314no—
155C2×C4—4311no—
156C2×C4—4311no—
157C2×C4C22×D43112yes—
158C2×C4C22×D43111yescharacteristic
159C22—22no—
160C22—21no—
161C22—22no—
162C22—22no—
163C22—22no—
164C22—22no—
165C22—44no—
166C22—8621no—
167C22—8622no—
168C22—8622no—
169C22—8622no—
170C22—8622no—
171C22—17244no—
172C22D4×D43112yes—
173C22C23×D43111yescharacteristic
174C4—21no—
175C4—22no—
176C4—8621no—
177C4—8622no—
178C4—4312no—
179C4—4312no—
180C2—22no—
181C2—42no—
182C2—41no—
183C2—17242no—
184C2—17241no—
185C2C2×D4×D43111yescharacteristic
186C2#2835223412yes—
subgroupG/H
center, FrattiniC22C23×D431
commutator, socle, cosocleC22×C431C24
Fitting#120×C431C2
hypercenter#120D431
hypocenterC431#535
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. #28352185C2semidirect
4. #28352175C2semidirect
5. #28352162C2semidirect
6. #28352161C2semidirect
7. #28352159C2semidirect
8. #28352158C2semidirect
9. #28352157C2semidirect
10. #28352145C2semidirect
11. #120×C431C2semidirect
14. C2×Dic862C22semidirect
15. C2×D1724C22semidirect
20. C2×C22⋊D431C22semidirect
21. C2×C22⋊D431C22semidirect
22. C2×C22⋊D431C22semidirect
24. C22×Dic431C22semidirect
25. C2×C4×D431C22semidirect
26. Dic431⋊C4C22semidirect
28. C22⋊Dic431C22semidirect
29. (C2×D431)⋊C4C22semidirect
30. (C2×D431)⋊C4C22semidirect
31. C4×Dic431C22semidirect
34. C2×D4×C431C22semidirect
36. C22⋊C4×C431C22semidirect
50. C2×Dic431D4semidirect
93. C431#535semidirect
104. #120D431semidirect
119. C2×D4C2×D431semidirect
129. C22⋊C4C2×D431semidirect

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
#283521852
#283521752
#283521622
#283521612
#283521592
#283521582
#283521572
#283521452
#120×C4312
#535431