Finite Group Databasebeta

← #22550510

Subgroups — #22550510

Order 8128.

428 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 115 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×D127C212yesmaximal, complemented
3C2×D4×D127C211yesmaximal, characteristic, complemented
4#16861205C211yesmaximal, characteristic, complemented
5#16861184C211yesmaximal, characteristic, complemented
6#16861182C212yesmaximal, complemented
7#16861179C212yesmaximal, complemented
8#16861177C212yesmaximal, complemented
9#16861171C211yesmaximal, characteristic, complemented
10C4×C22⋊D127C211yesmaximal, characteristic, complemented
11#121×C127C211yesmaximal, characteristic, complemented
12C23×D127C2212yescomplemented
13C23×D127C2212yescomplemented
14C2×D508—24no—
15C2×D508C2211yescharacteristic
16C2×D508C2211yescharacteristic, complemented
17C2×D508C2212yescomplemented
18C2×D508C2212yescomplemented
19C2×C22⋊D127C2211yescharacteristic
20C2×C22⋊D127C2212yescomplemented
21C2×C22⋊D127C2212yescomplemented
22C2×C22⋊D127C2212yescomplemented
23D4×D127—24no—
24D4×D127—28no—
25C2×C4×D127C2211yescharacteristic, complemented
26C2×C4×D127C2212yescomplemented
27Dic127⋊C4C2211yescharacteristic, complemented
28C22⋊Dic127C2211yescharacteristic, complemented
29(C2×D127)⋊C4C2212yescomplemented
30(C2×D127)⋊C4C2211yescharacteristic, complemented
31(C2×D127)⋊C4C2212yescomplemented
32C4×Dic127C2211yescharacteristic, complemented
33C2×D4×C127C2211yescharacteristic, complemented
34C2×D4×C127C2212yescomplemented
35C22×C4×C127C2211yescharacteristic, complemented
36C4⋊C4×C127C2211yescharacteristic, complemented
37C22⋊C4×C127C2212yescomplemented
38C22×D127—24no—
39C22×D127—24no—
40C22×D127—24no—
41C22×D127—24no—
42C22×D127—42no—
43C22×D127—42no—
44C22×D127C2312yes—
45C22×D127C2311yescharacteristic
46C22×D127C2312yes—
47D508—24no—
48D508—24no—
49D508—22no—
50D508D414yescomplemented
51C22⋊D127—24no—
52C22⋊D127—24no—
53C22⋊D127—42no—
54C22⋊D127D414yescomplemented
55C2×Dic127C2311yescharacteristic
56C2×Dic127C2311yescharacteristic
57C2×Dic127C2311yescharacteristic
58C4×D127—22no—
59C4×D127—24no—
60C23×C127C2312yes—
61C23×C127C2311yescharacteristic
62D4×C127—24no—
63D4×C127—22no—
64C2×C4×C127—22no—
65C2×C4×C127C2311yescharacteristic
66C2×C4×C127C2312yes—
67C2×C4×C127C2311yescharacteristic
68C22×C127—21no—
69C22×C127—21no—
70C22×C127—22no—
71C22×C127—22no—
72C22×C127—22no—
73C22×C127C2×D412yescomplemented
74C22×C127C2411yescharacteristic
75C2×D127—21no—
76C2×D127—22no—
77C2×D127—21no—
78C2×D127—22no—
79C2×D127—22no—
80C2×D127—22no—
81C2×D127—22no—
82C2×D127—48no—
83C2×D127—48no—
84C2×D127C2×D414yescomplemented
85C2×D127C2×D412yescomplemented
86C4×C127—21no—
87C4×C127—22no—
88C4×C127C2×D412yescomplemented
89Dic127—21no—
90Dic127—21no—
91Dic127C2×D412yescomplemented
92C2×C127—22no—
93C2×C127—42no—
94C2×C127C22×D411yescharacteristic
95C2×C127C22×D411yescharacteristic
96C2×C127H(5,𝔽2)11yescharacteristic
97D127—22no—
98D127—24no—
99D127—42no—
100C127D4211yescharacteristic, complemented, Sylow
101D42—1271nomaximal, Sylow
102C22×D4—1272no—
103C22×D4—1271no—
104C22×D4—1271no—
105#127—1271no—
106#121—1271no—
107#121—1272no—
108#121D12711yescharacteristic, complemented
109#120—1272no—
110#120—1272no—
111C4×D4—1271no—
112C4×D4—1271no—
113C24—1272no—
114C24—1272no—
115C2×D4—1271no—
116C2×D4—1272no—
117C2×D4—1272no—
118C2×D4—1272no—
119C2×D4—1271no—
120C2×D4—1272no—
121C2×D4—1272no—
122C2×D4—1271no—
123C2×D4—2544no—
124C2×D4—2548no—
125C2×D4—2544no—
126C2×D4C2×D12712yescomplemented
127C2×D4C2×D12711yescharacteristic, complemented
128C22×C4—1272no—
129C22×C4—1271no—
130C22×C4C2×D12711yescharacteristic, complemented
131C4⋊C4—1271no—
132C4⋊C4C2×D12711yescharacteristic, complemented
133C22⋊C4—1271no—
134C22⋊C4—1272no—
135C22⋊C4—1271no—
136C22⋊C4—1272no—
137C22⋊C4C2×D12712yescomplemented
138C42—1271no—
139C23—1271no—
140C23—1272no—
141C23—1272no—
142C23—5082no—
143C23—5082no—
144C23—2544no—
145C23—2544no—
146C23—2544no—
147C23—2544no—
148C23C22×D12712yes—
149C23C22×D12711yescharacteristic
150D4—24no—
151D4—22no—
152D4—1274no—
153D4—1274no—
154D4—5082no—
155D4—2544no—
156D4—2544no—
157D4—2544no—
158D4—2544no—
159D4—2542no—
160C2×C4—22no—
161C2×C4—1271no—
162C2×C4—1271no—
163C2×C4—1271no—
164C2×C4—2544no—
165C2×C4—2542no—
166C2×C4C22×D12712yes—
167C2×C4C22×D12711yescharacteristic
168C2×C4C22×D12711yescharacteristic
169C22—21no—
170C22—21no—
171C22—22no—
172C22—22no—
173C22—22no—
174C22—1272no—
175C22—1274no—
176C22—5088no—
177C22—5088no—
178C22—2542no—
179C22—2542no—
180C22—2542no—
181C22—2542no—
182C22—2541no—
183C22—2542no—
184C22—2541no—
185C22D4×D12712yescomplemented
186C22C23×D12711yescharacteristic
187C4—21no—
188C4—22no—
189C4—1272no—
190C4—2541no—
191C4—2541no—
192C4D4×D12712yescomplemented
193C2—22no—
194C2—42no—
195C2—5082no—
196C2—2544no—
197C2—2542no—
198C2C2×D4×D12711yescharacteristic
199C2C2×D4×D12711yescharacteristic
200C2#1686125511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D127
commutator, socleC22×C127C24
Fitting#121×C127C2
hypercenter#121D127
hypocenterC127D42
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C22×D508C2semidirect
2. C2×D4×D127C2semidirect
3. C2×D4×D127C2semidirect
4. #16861205C2semidirect
5. #16861184C2semidirect
6. #16861182C2semidirect
7. #16861179C2semidirect
8. #16861177C2semidirect
9. #16861171C2semidirect
10. C4×C22⋊D127C2semidirect
11. #121×C127C2semidirect
12. C23×D127C22semidirect
13. C23×D127C22semidirect
16. C2×D508C22semidirect
17. C2×D508C22semidirect
18. C2×D508C22semidirect
20. C2×C22⋊D127C22semidirect
21. C2×C22⋊D127C22semidirect
22. C2×C22⋊D127C22semidirect
25. C2×C4×D127C22semidirect
26. C2×C4×D127C22semidirect
27. Dic127⋊C4C22semidirect
28. C22⋊Dic127C22semidirect
29. (C2×D127)⋊C4C22semidirect
30. (C2×D127)⋊C4C22semidirect
31. (C2×D127)⋊C4C22semidirect
32. C4×Dic127C22semidirect
33. C2×D4×C127C22semidirect
34. C2×D4×C127C22semidirect
35. C22×C4×C127C22semidirect
36. C4⋊C4×C127C22semidirect
37. C22⋊C4×C127C22semidirect
50. D508D4semidirect
54. C22⋊D127D4semidirect
73. C22×C127C2×D4semidirect
84. C2×D127C2×D4semidirect
85. C2×D127C2×D4semidirect
88. C4×C127C2×D4semidirect
91. Dic127C2×D4semidirect
100. C127D42semidirect
108. #121D127semidirect
126. C2×D4C2×D127semidirect
127. C2×D4C2×D127semidirect
130. C22×C4C2×D127semidirect
132. C4⋊C4C2×D127semidirect
137. C22⋊C4C2×D127semidirect
185. C22D4×D127semidirect
192. C4D4×D127semidirect