Finite Group Databasebeta

← #42119625

Subgroups — #42119625

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×#20293800C211yesmaximal, characteristic, complemented
2C2×D4×D431C211yesmaximal, characteristic, complemented
3#28352185C212yesmaximal, complemented
4#28352175C212yesmaximal, complemented
5#28352164C211yesmaximal, characteristic, complemented
6#28352163C211yesmaximal, characteristic, complemented
7#28352160C211yesmaximal, characteristic, complemented
8#28352159C212yesmaximal, complemented
9#28352157C212yesmaximal, complemented
10#28352147C211yesmaximal, characteristic, complemented
11#121×C431C211yesmaximal, characteristic, complemented
12C23×D431C2212yes—
13#20293800—24no—
14C2×Dic862C2211yescharacteristic, complemented
15C2×D1724C2211yescharacteristic
16C2×D1724C2211yescharacteristic
17C2×D1724C2212yescomplemented
18C2×C22⋊D431C2211yescharacteristic
19C2×C22⋊D431C2212yes—
20C2×C22⋊D431C2211yescharacteristic, complemented
21C2×C22⋊D431C2212yescomplemented
22C2×C22⋊D431C2212yescomplemented
23D4×D431—24no—
24C2×C4×D431C2211yescharacteristic
25C2×C4×D431C2211yescharacteristic, complemented
26C4⋊Dic431C2211yescharacteristic, complemented
27C22⋊Dic431C2212yescomplemented
28(C2×D431)⋊C4C2212yes—
29(C2×D431)⋊C4C2212yescomplemented
30(C2×D431)⋊C4C2212yescomplemented
31C4×Dic431C2212yescomplemented
32C2×D4×C431C2211yescharacteristic
33C2×D4×C431C2212yescomplemented
34C22×C4×C431C2211yescharacteristic, complemented
35C4⋊C4×C431C2211yescharacteristic, complemented
36C22⋊C4×C431C2212yescomplemented
37C22×D431—24no—
38C22×D431—42no—
39C22×D431—42no—
40C22×D431C2312yes—
41C22×D431C2311yescharacteristic
42C22×D431C2311yescharacteristic
43Dic862—22no—
44D1724—22no—
45D1724—22no—
46D1724—42no—
47C22⋊D431—22no—
48C22⋊D431—24no—
49C22⋊D431—22no—
50C22⋊D431—42no—
51C22⋊D431—42no—
52C2×Dic431C2312yes—
53C2×Dic431C2311yescharacteristic
54C2×Dic431C2311yescharacteristic
55C4×D431—22no—
56C4×D431D414yescomplemented
57C23×C431C2312yes—
58C23×C431C2311yescharacteristic
59D4×C431—22no—
60D4×C431—42no—
61C2×C4×C431—22no—
62C2×C4×C431C2311yescharacteristic
63C2×C4×C431C2312yes—
64C2×C4×C431C2311yescharacteristic
65C22×C431—21no—
66C22×C431—22no—
67C22×C431—21no—
68C22×C431—22no—
69C22×C431—21no—
70C22×C431—22no—
71C22×C431C2411yescharacteristic
72C2×D431—21no—
73C2×D431—21no—
74C2×D431—21no—
75C2×D431—22no—
76C2×D431—21no—
77C2×D431—22no—
78C2×D431—21no—
79C2×D431—22no—
80C2×D431—48no—
81C2×D431C2×D412yes—
82C4×C431—21no—
83C4×C431—22no—
84C4×C431C2×D412yes—
85Dic431—21no—
86Dic431—22no—
87Dic431C2×D412yes—
88C2×C431—42no—
89C2×C431—41no—
90C2×C431C22×D411yescharacteristic
91C2×C431H(5,𝔽2)11yescharacteristic
92C2×C431H(5,𝔽2)11yescharacteristic
93D431—22no—
94D431—41no—
95D431—42no—
96C431#53511yescharacteristic, complemented, Sylow
97#535—4311nomaximal, Sylow
98C2×#41—4311no—
99C22×D4—4311no—
100#127—4312no—
101#124—4312no—
102#121—4311no—
103#121—4311no—
104#121—4311no—
105#121D43111yescharacteristic, complemented
106#120—4312no—
107#120—4312no—
108#117—4311no—
109C24—4312no—
110#41—8624no—
111C2×Q8—4311no—
112C2×D4—8624no—
113C2×D4—4311no—
114C2×D4—4312no—
115C2×D4—4311no—
116C2×D4—4312no—
117C2×D4—4312no—
118C2×D4—4311no—
119C2×D4—4311no—
120C2×D4—4312no—
121C2×D4C2×D43111yescharacteristic
122C2×D4C2×D43112yescomplemented
123C22×C4—4311no—
124C22×C4—4311no—
125C22×C4C2×D43111yescharacteristic, complemented
126C4⋊C4—4311no—
127C4⋊C4C2×D43111yescharacteristic, complemented
128C22⋊C4—4312no—
129C22⋊C4—4312no—
130C22⋊C4—4312no—
131C22⋊C4—4312no—
132C22⋊C4C2×D43112yescomplemented
133C42—4312no—
134C23—8624no—
135C23—4312no—
136C23—4311no—
137C23—4311no—
138C23—17242no—
139C23—17242no—
140C23C22×D43112yes—
141C23C22×D43111yescharacteristic
142Q8—8622no—
143D4—22no—
144D4—42no—
145D4—8622no—
146D4—8622no—
147D4—8624no—
148D4—8622no—
149D4—8622no—
150D4—17242no—
151D4—17242no—
152D4—17242no—
153C2×C4—22no—
154C2×C4—8622no—
155C2×C4—4312no—
156C2×C4—4311no—
157C2×C4—4311no—
158C2×C4—4314no—
159C2×C4C22×D43112yes—
160C2×C4C22×D43111yescharacteristic
161C2×C4C22×D43111yescharacteristic
162C22—21no—
163C22—22no—
164C22—21no—
165C22—22no—
166C22—21no—
167C22—22no—
168C22—8621no—
169C22—8621no—
170C22—8621no—
171C22—8622no—
172C22—8621no—
173C22—8622no—
174C22—8621no—
175C22—8622no—
176C22—4312no—
177C22—17248no—
178C22C23×D43111yescharacteristic
179C4—21no—
180C4—22no—
181C4—8621no—
182C4—8622no—
183C4—4312no—
184C4D4×D43112yes—
185C2—42no—
186C2—41no—
187C2—8622no—
188C2—17242no—
189C2—17241no—
190C2C2×D4×D43111yescharacteristic
191C2#2835223411yescharacteristic
192C2#2835223511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D431
commutator, socle, cosocleC22×C431C24
Fitting#121×C431C2
hypercenter#121D431
hypocenterC431#535
LCSG ▷ C22×C431 ▷ C431
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#20293800C2semidirect
2. C2×D4×D431C2semidirect
3. #28352185C2semidirect
4. #28352175C2semidirect
5. #28352164C2semidirect
6. #28352163C2semidirect
7. #28352160C2semidirect
8. #28352159C2semidirect
9. #28352157C2semidirect
10. #28352147C2semidirect
11. #121×C431C2semidirect
14. C2×Dic862C22semidirect
17. C2×D1724C22semidirect
20. C2×C22⋊D431C22semidirect
21. C2×C22⋊D431C22semidirect
22. C2×C22⋊D431C22semidirect
25. C2×C4×D431C22semidirect
26. C4⋊Dic431C22semidirect
27. C22⋊Dic431C22semidirect
29. (C2×D431)⋊C4C22semidirect
30. (C2×D431)⋊C4C22semidirect
31. C4×Dic431C22semidirect
33. C2×D4×C431C22semidirect
34. C22×C4×C431C22semidirect
35. C4⋊C4×C431C22semidirect
36. C22⋊C4×C431C22semidirect
56. C4×D431D4semidirect
96. C431#535semidirect
105. #121D431semidirect
122. C2×D4C2×D431semidirect
125. C22×C4C2×D431semidirect
127. C4⋊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
C2×#202938002
C2×D4×D4312
#283521852
#283521752
#283521642
#283521632
#283521602
#283521592
#283521572
#283521472
#121×C4312
#535431