Finite Group Databasebeta

← #22550437

Subgroups — #22550437

Order 8128.

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×D508C211yesmaximal, characteristic, complemented
2C2×D4×D127C211yesmaximal, characteristic, complemented
3#16861194C212yesmaximal, complemented
4#16861192C211yesmaximal, characteristic, complemented
5#16861184C212yesmaximal, complemented
6#16861182C212yesmaximal, complemented
7#16861179C211yesmaximal, characteristic, complemented
8#16861179C212yesmaximal, complemented
9#16861177C211yesmaximal, characteristic, complemented
10C22⋊C4×D127C211yesmaximal, characteristic, complemented
11#123×C127C211yesmaximal, characteristic, complemented
12C23×D127C2211yescharacteristic
13C23×D127C2211yescharacteristic, complemented
14C23×D127C2211yescharacteristic, complemented
15C2×D508—24no—
16C2×D508C2212yes—
17C2×D508C2211yescharacteristic, complemented
18C2×D508C2212yescomplemented
19C2×D508C2212yescomplemented
20C2×C22⋊D127C2211yescharacteristic
21C2×C22⋊D127C2211yescharacteristic, complemented
22C2×C22⋊D127C2212yescomplemented
23D4×D127—24no—
24C2×C4×D127C2211yescharacteristic, complemented
25C2×C4×D127C2212yescomplemented
26Dic127⋊C4C2212yescomplemented
27C22⋊Dic127C2211yescharacteristic, complemented
28(C2×D127)⋊C4C2212yescomplemented
29(C2×D127)⋊C4C2212yescomplemented
30(C2×D127)⋊C4C2212yescomplemented
31(C2×D127)⋊C4C2212yescomplemented
32C2×D4×C127C2211yescharacteristic
33C22×C4×C127C2211yescharacteristic, complemented
34C4⋊C4×C127C2212yescomplemented
35C22⋊C4×C127C2211yescharacteristic, complemented
36C22⋊C4×C127C2212yescomplemented
37C22×D127—22no—
38C22×D127—22no—
39C22×D127—42no—
40C22×D127—42no—
41C22×D127—42no—
42C22×D127D414yescomplemented
43C22×D127C2311yescharacteristic
44C22×D127C2312yes—
45C22×D127C2311yescharacteristic
46C22×D127C2311yescharacteristic
47D508—22no—
48D508—24no—
49D508—24no—
50D508—42no—
51C22⋊D127—22no—
52C22⋊D127—22no—
53C22⋊D127—42no—
54C2×Dic127C2312yes—
55C2×Dic127C2311yescharacteristic
56C4×D127—22no—
57C4×D127—42no—
58C23×C127C2311yescharacteristic
59C23×C127C2311yescharacteristic
60D4×C127—22no—
61C2×C4×C127—22no—
62C2×C4×C127C2312yes—
63C2×C4×C127C2312yes—
64C2×C4×C127C2311yescharacteristic
65C22×C127—22no—
66C22×C127—21no—
67C22×C127—22no—
68C22×C127C2×D412yes—
69C22×C127C2411yescharacteristic
70C2×D127—22no—
71C2×D127—22no—
72C2×D127—21no—
73C2×D127—22no—
74C2×D127—22no—
75C2×D127—22no—
76C2×D127—22no—
77C2×D127—44no—
78C2×D127—44no—
79C2×D127—44no—
80C2×D127C2×D412yes—
81C2×D127C2×D412yes—
82C4×C127—22no—
83C4×C127—21no—
84C4×C127—22no—
85Dic127—21no—
86Dic127—22no—
87C2×C127—22no—
88C2×C127—41no—
89C2×C127C22×D411yescharacteristic
90C2×C127H(5,𝔽2)12yes—
91D127—22no—
92D127—22no—
93D127—41no—
94D127—42no—
95C127#53411yescharacteristic, complemented, Sylow
96#534—1271nomaximal, Sylow
97C22×D4—1271no—
98C22×D4—1271no—
99#123—1271no—
100#123—1272no—
101#123D12711yescharacteristic, complemented
102#121—1272no—
103#121—1272no—
104#120—1271no—
105#120—1272no—
106#120—1271no—
107C2×C22⋊C4—1271no—
108C24—1271no—
109C24—1271no—
110C24—1271no—
111C2×D4—1272no—
112C2×D4—1271no—
113C2×D4—1271no—
114C2×D4—1272no—
115C2×D4—1272no—
116C2×D4—1271no—
117C2×D4—1272no—
118C2×D4—2544no—
119C2×D4—2544no—
120C2×D4C2×D12711yescharacteristic
121C22×C4—1272no—
122C22×C4—1271no—
123C22×C4C2×D12711yescharacteristic, complemented
124C4⋊C4—1272no—
125C4⋊C4C2×D12712yescomplemented
126C22⋊C4—1271no—
127C22⋊C4—1272no—
128C22⋊C4—1272no—
129C22⋊C4—1272no—
130C22⋊C4—1272no—
131C22⋊C4C2×D12711yescharacteristic, complemented
132C22⋊C4C2×D12712yescomplemented
133C23—1274no—
134C23—1271no—
135C23—1271no—
136C23—1272no—
137C23—1271no—
138C23—5082no—
139C23—5082no—
140C23—5082no—
141C23—2542no—
142C23—2542no—
143C23C22×D12711yescharacteristic
144C23C22×D12711yescharacteristic
145D4—22no—
146D4—5082no—
147D4—5082no—
148D4—2544no—
149D4—2544no—
150D4—2542no—
151D4—2542no—
152D4—2542no—
153C2×C4—22no—
154C2×C4—1272no—
155C2×C4—1271no—
156C2×C4—5082no—
157C2×C4—2542no—
158C2×C4C22×D12712yes—
159C2×C4C22×D12711yescharacteristic
160C2×C4C22×D12712yes—
161C22—22no—
162C22—21no—
163C22—22no—
164C22—1272no—
165C22—1272no—
166C22—5084no—
167C22—5084no—
168C22—5084no—
169C22—2542no—
170C22—2541no—
171C22—2542no—
172C22—2542no—
173C22—2542no—
174C22—2542no—
175C22—2542no—
176C22D4×D12712yes—
177C22C23×D12711yescharacteristic
178C4—22no—
179C4—21no—
180C4—22no—
181C4—2541no—
182C4—2542no—
183C2—22no—
184C2—41no—
185C2—5082no—
186C2—5081no—
187C2—2542no—
188C2—2542no—
189C2C2×D4×D12711yescharacteristic
190C2#1686125512yes—
subgroupG/H
center, FrattiniC22C23×D127
commutator, socleC22×C127C24
Fitting#123×C127C2
hypercenter#123D127
hypocenterC127#534
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C22×D508C2semidirect
2. C2×D4×D127C2semidirect
3. #16861194C2semidirect
4. #16861192C2semidirect
5. #16861184C2semidirect
6. #16861182C2semidirect
7. #16861179C2semidirect
8. #16861179C2semidirect
9. #16861177C2semidirect
10. C22⋊C4×D127C2semidirect
11. #123×C127C2semidirect
13. C23×D127C22semidirect
14. C23×D127C22semidirect
17. C2×D508C22semidirect
18. C2×D508C22semidirect
19. C2×D508C22semidirect
21. C2×C22⋊D127C22semidirect
22. C2×C22⋊D127C22semidirect
24. C2×C4×D127C22semidirect
25. C2×C4×D127C22semidirect
26. Dic127⋊C4C22semidirect
27. C22⋊Dic127C22semidirect
28. (C2×D127)⋊C4C22semidirect
29. (C2×D127)⋊C4C22semidirect
30. (C2×D127)⋊C4C22semidirect
31. (C2×D127)⋊C4C22semidirect
33. C22×C4×C127C22semidirect
34. C4⋊C4×C127C22semidirect
35. C22⋊C4×C127C22semidirect
36. C22⋊C4×C127C22semidirect
42. C22×D127D4semidirect
95. C127#534semidirect
101. #123D127semidirect
123. C22×C4C2×D127semidirect
125. C4⋊C4C2×D127semidirect
131. C22⋊C4C2×D127semidirect
132. C22⋊C4C2×D127semidirect