Finite Group Databasebeta

← #42119619

Subgroups — #42119619

Order 27584.

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

normal subgroups only

This group has 346 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
1C22×C22⋊D431C211yesmaximal, characteristic, complemented
2C2×D4×D431C211yesmaximal, characteristic, complemented
3#28352173C211yesmaximal, characteristic, complemented
4#28352171C212yesmaximal, complemented
5#28352170C211yesmaximal, characteristic, complemented
6#28352162C212yesmaximal, complemented
7#28352160C212yesmaximal, complemented
8#28352158C211yesmaximal, characteristic, complemented
9#28352157C212yesmaximal, complemented
10C22⋊C4×D431C211yesmaximal, characteristic, complemented
11#120×C431C211yesmaximal, characteristic, complemented
12C23×D431C2211yescharacteristic
13C23×D431C2211yescharacteristic, complemented
14C2×D1724C2211yescharacteristic, complemented
15C2×C22⋊D431—24no—
16C2×C22⋊D431C2212yes—
17C2×C22⋊D431C2211yescharacteristic
18C2×C22⋊D431C2211yescharacteristic, complemented
19C2×C22⋊D431C2212yescomplemented
20C2×C22⋊D431C2212yescomplemented
21D4×D431—24no—
22C22×Dic431C2211yescharacteristic, complemented
23C2×C4×D431C2211yescharacteristic, complemented
24C2×C4×D431C2212yescomplemented
25C4⋊Dic431C2212yescomplemented
26Dic431⋊C4C2212yescomplemented
27C22⋊Dic431C2211yescharacteristic, complemented
28C22⋊Dic431C2212yescomplemented
29C22⋊Dic431C2212yescomplemented
30(C2×D431)⋊C4C2212yescomplemented
31(C2×D431)⋊C4C2212yescomplemented
32C24×C431C2211yescharacteristic, complemented
33C2×D4×C431C2211yescharacteristic
34C2×D4×C431C2212yescomplemented
35C22⋊C4×C431C2211yescharacteristic, complemented
36C22⋊C4×C431C2212yescomplemented
37C22×D431—22no—
38C22×D431—42no—
39C22×D431—42no—
40C22×D431D414yescomplemented
41C22×D431C2311yescharacteristic
42C22×D431C2311yescharacteristic
43C22×D431C2311yescharacteristic
44D1724—22no—
45C22⋊D431—22no—
46C22⋊D431—22no—
47C22⋊D431—24no—
48C22⋊D431—24no—
49C22⋊D431—42no—
50C2×Dic431—22no—
51C2×Dic431C2312yes—
52C2×Dic431C2312yes—
53C2×Dic431C2311yescharacteristic
54C4×D431—22no—
55C4×D431—42no—
56C23×C431—22no—
57C23×C431—42no—
58C23×C431C2312yes—
59C23×C431C2311yescharacteristic
60C23×C431C2311yescharacteristic
61D4×C431—22no—
62D4×C431—42no—
63C2×C4×C431C2311yescharacteristic
64C2×C4×C431C2312yes—
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—22no—
75C2×D431—22no—
76C2×D431—21no—
77C2×D431—22no—
78C2×D431—44no—
79C2×D431—44no—
80C2×D431C2×D412yes—
81C2×D431C2×D412yes—
82C4×C431—21no—
83C4×C431—22no—
84Dic431—21no—
85Dic431—22no—
86Dic431—22no—
87C2×C431—22no—
88C2×C431—41no—
89C2×C431—42no—
90C2×C431C22×D411yescharacteristic
91C2×C431H(5,𝔽2)12yes—
92D431—22no—
93D431—22no—
94D431—41no—
95C431#53411yescharacteristic, complemented, Sylow
96#534—4311nomaximal, Sylow
97C22×D4—4311no—
98C22×D4—4311no—
99#123—4311no—
100#123—4312no—
101#123—4311no—
102#121—4312no—
103#121—4312no—
104#120—4311no—
105#120—4312no—
106#120D43111yescharacteristic, complemented
107C2×C22⋊C4—4311no—
108C24—4311no—
109C24—4311no—
110C24C2×D43111yescharacteristic, complemented
111C2×D4—8624no—
112C2×D4—8624no—
113C2×D4—4312no—
114C2×D4—4311no—
115C2×D4—4311no—
116C2×D4—4312no—
117C2×D4—4312no—
118C2×D4—4311no—
119C2×D4C2×D43111yescharacteristic
120C2×D4C2×D43112yescomplemented
121C22×C4—4311no—
122C22×C4—4311no—
123C22×C4—4312no—
124C4⋊C4—4312no—
125C4⋊C4—4312no—
126C22⋊C4—4311no—
127C22⋊C4—4312no—
128C22⋊C4—4312no—
129C22⋊C4—4312no—
130C22⋊C4—4312no—
131C22⋊C4C2×D43111yescharacteristic, complemented
132C22⋊C4C2×D43112yescomplemented
133C23—22no—
134C23—42no—
135C23—8622no—
136C23—4311no—
137C23—4311no—
138C23—4311no—
139C23—4314no—
140C23—17242no—
141C23—17242no—
142C23C22×D43111yescharacteristic
143C23C22×D43112yes—
144C23C22×D43111yescharacteristic
145D4—22no—
146D4—42no—
147D4—8622no—
148D4—8622no—
149D4—8624no—
150D4—8624no—
151D4—8622no—
152D4—17242no—
153C2×C4—8622no—
154C2×C4—8622no—
155C2×C4—4312no—
156C2×C4—4311no—
157C2×C4—4312no—
158C2×C4—17242no—
159C2×C4C22×D43112yes—
160C2×C4C22×D43111yescharacteristic
161C22—22no—
162C22—21no—
163C22—22no—
164C22—22no—
165C22—22no—
166C22—22no—
167C22—44no—
168C22—8622no—
169C22—8622no—
170C22—8621no—
171C22—8622no—
172C22—4312no—
173C22—4312no—
174C22—17244no—
175C22—17244no—
176C22D4×D43112yes—
177C22C23×D43111yescharacteristic
178C4—21no—
179C4—22no—
180C4—8621no—
181C4—8622no—
182C4—8622no—
183C2—22no—
184C2—42no—
185C2—41no—
186C2—8622no—
187C2—8622no—
188C2—17241no—
189C2C2×D4×D43111yescharacteristic
190C2#2835223412yes—
subgroupG/H
center, FrattiniC22C23×D431
commutator, socle, cosocleC22×C431C24
Fitting#120×C431C2
hypercenter#120D431
hypocenterC431#534
LCSG ▷ C22×C431 ▷ C431
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊D431C2semidirect
2. C2×D4×D431C2semidirect
3. #28352173C2semidirect
4. #28352171C2semidirect
5. #28352170C2semidirect
6. #28352162C2semidirect
7. #28352160C2semidirect
8. #28352158C2semidirect
9. #28352157C2semidirect
10. C22⋊C4×D431C2semidirect
11. #120×C431C2semidirect
13. C23×D431C22semidirect
14. C2×D1724C22semidirect
18. C2×C22⋊D431C22semidirect
19. C2×C22⋊D431C22semidirect
20. C2×C22⋊D431C22semidirect
22. C22×Dic431C22semidirect
23. C2×C4×D431C22semidirect
24. C2×C4×D431C22semidirect
25. C4⋊Dic431C22semidirect
26. Dic431⋊C4C22semidirect
27. C22⋊Dic431C22semidirect
28. C22⋊Dic431C22semidirect
29. C22⋊Dic431C22semidirect
30. (C2×D431)⋊C4C22semidirect
31. (C2×D431)⋊C4C22semidirect
32. C24×C431C22semidirect
34. C2×D4×C431C22semidirect
35. C22⋊C4×C431C22semidirect
36. C22⋊C4×C431C22semidirect
40. C22×D431D4semidirect
95. C431#534semidirect
106. #120D431semidirect
110. C24C2×D431semidirect
120. C2×D4C2×D431semidirect
131. C22⋊C4C2×D431semidirect
132. C22⋊C4C2×D431semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊D4312
C2×D4×D4312
#283521732
#283521712
#283521702
#283521622
#283521602
#283521582
#283521572
C22⋊C4×D4312
#120×C4312
#534431