Finite Group Databasebeta

← #22550441

Subgroups — #22550441

Order 8128.

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×#14523247C211yesmaximal, characteristic, complemented
2C22×C22⋊D127C211yesmaximal, characteristic, complemented
3#16861205C212yesmaximal, complemented
4#16861195C212yesmaximal, complemented
5#16861182C212yesmaximal, complemented
6#16861181C212yesmaximal, complemented
7#16861179C211yesmaximal, characteristic, complemented
8#16861178C211yesmaximal, characteristic, complemented
9#16861177C211yesmaximal, characteristic, complemented
10#16861165C211yesmaximal, characteristic, complemented
11#120×C127C211yesmaximal, characteristic, complemented
12C23×D127C2211yescharacteristic
13#14523247—24no—
14C2×Dic254C2211yescharacteristic, complemented
15C2×D508C2212yescomplemented
16C2×C22⋊D127—24no—
17C2×C22⋊D127C2212yes—
18C2×C22⋊D127C2212yes—
19C2×C22⋊D127C2211yescharacteristic
20C2×C22⋊D127C2211yescharacteristic, complemented
21C2×C22⋊D127C2212yescomplemented
22C2×C22⋊D127C2212yescomplemented
23C22×Dic127C2211yescharacteristic
24C22×Dic127C2211yescharacteristic, complemented
25C2×C4×D127C2211yescharacteristic, complemented
26Dic127⋊C4C2212yescomplemented
27C22⋊Dic127C2211yescharacteristic
28C22⋊Dic127C2212yescomplemented
29(C2×D127)⋊C4C2212yescomplemented
30(C2×D127)⋊C4C2212yescomplemented
31C4×Dic127C2212yescomplemented
32C24×C127C2211yescharacteristic
33C2×D4×C127C2211yescharacteristic
34C2×D4×C127C2212yescomplemented
35C22⋊C4×C127C2211yescharacteristic
36C22⋊C4×C127C2212yescomplemented
37C22×D127—22no—
38C22×D127—42no—
39C22×D127C2312yes—
40C22×D127C2311yescharacteristic
41Dic254—22no—
42D508—42no—
43C22⋊D127—22no—
44C22⋊D127—24no—
45C22⋊D127—24no—
46C22⋊D127—22no—
47C22⋊D127—42no—
48C22⋊D127—42no—
49C2×Dic127—22no—
50C2×Dic127D414yescomplemented
51C2×Dic127C2312yes—
52C2×Dic127C2311yescharacteristic
53C2×Dic127C2311yescharacteristic
54C2×Dic127C2311yescharacteristic
55C4×D127—22no—
56C23×C127—22no—
57C23×C127—42no—
58C23×C127C2312yes—
59C23×C127C2311yescharacteristic
60C23×C127C2311yescharacteristic
61D4×C127—22no—
62D4×C127—42no—
63C2×C4×C127C2312yes—
64C2×C4×C127C2311yescharacteristic
65C22×C127—22no—
66C22×C127—21no—
67C22×C127—22no—
68C22×C127—22no—
69C22×C127—22no—
70C22×C127—22no—
71C22×C127—44no—
72C22×C127C2×D412yes—
73C22×C127C2411yescharacteristic
74C2×D127—21no—
75C2×D127—22no—
76C2×D127—22no—
77C2×D127—22no—
78C2×D127—22no—
79C2×D127—44no—
80C4×C127—21no—
81C4×C127—22no—
82Dic127—21no—
83Dic127—22no—
84Dic127C2×D412yes—
85Dic127C2×D412yes—
86C2×C127—22no—
87C2×C127—41no—
88C2×C127—42no—
89C2×C127C22×D411yescharacteristic
90C2×C127H(5,𝔽2)12yes—
91D127—41no—
92D127—42no—
93C127#53511yescharacteristic, complemented, Sylow
94#535—1271nomaximal, Sylow
95C2×#41—1271no—
96C22×D4—1271no—
97#127—1272no—
98#124—1272no—
99#121—1272no—
100#121—1272no—
101#120—1271no—
102#120—1271no—
103#120—1271no—
104#120D12711yescharacteristic, complemented
105#117—1271no—
106C24—1271no—
107C24C2×D12711yescharacteristic
108#41—2544no—
109C2×Q8—1271no—
110C2×D4—1272no—
111C2×D4—1272no—
112C2×D4—1271no—
113C2×D4—1272no—
114C2×D4—1272no—
115C2×D4—1271no—
116C2×D4—1272no—
117C2×D4—2544no—
118C2×D4C2×D12711yescharacteristic
119C2×D4C2×D12712yescomplemented
120C22×C4—1271no—
121C22×C4—1271no—
122C22×C4—1271no—
123C4⋊C4—1272no—
124C22⋊C4—1271no—
125C22⋊C4—1272no—
126C22⋊C4—1272no—
127C22⋊C4—1272no—
128C22⋊C4C2×D12711yescharacteristic
129C22⋊C4C2×D12712yescomplemented
130C42—1272no—
131C23—22no—
132C23—42no—
133C23—1272no—
134C23—1271no—
135C23—5082no—
136C23—2542no—
137C23C22×D12712yes—
138C23C22×D12711yescharacteristic
139C23C22×D12711yescharacteristic
140Q8—2542no—
141D4—22no—
142D4—42no—
143D4—5082no—
144D4—5082no—
145D4—5082no—
146D4—2544no—
147D4—2544no—
148D4—2542no—
149D4—2542no—
150C2×C4—1274no—
151C2×C4—1271no—
152C2×C4—1271no—
153C2×C4—1272no—
154C2×C4—1271no—
155C2×C4—2542no—
156C2×C4—2542no—
157C2×C4C22×D12712yes—
158C2×C4C22×D12711yescharacteristic
159C22—22no—
160C22—21no—
161C22—22no—
162C22—22no—
163C22—22no—
164C22—22no—
165C22—44no—
166C22—5084no—
167C22—2541no—
168C22—2542no—
169C22—2542no—
170C22—2542no—
171C22—2542no—
172C22D4×D12712yes—
173C22C23×D12711yescharacteristic
174C4—21no—
175C4—22no—
176C4—1272no—
177C4—1272no—
178C4—2541no—
179C4—2542no—
180C2—22no—
181C2—42no—
182C2—41no—
183C2—5082no—
184C2—5081no—
185C2C2×D4×D12711yescharacteristic
186C2#1686125412yes—
subgroupG/H
center, FrattiniC22C23×D127
commutator, socleC22×C127C24
Fitting#120×C127C2
hypercenter#120D127
hypocenterC127#535
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C2×#14523247C2semidirect
2. C22×C22⋊D127C2semidirect
3. #16861205C2semidirect
4. #16861195C2semidirect
5. #16861182C2semidirect
6. #16861181C2semidirect
7. #16861179C2semidirect
8. #16861178C2semidirect
9. #16861177C2semidirect
10. #16861165C2semidirect
11. #120×C127C2semidirect
14. C2×Dic254C22semidirect
15. C2×D508C22semidirect
20. C2×C22⋊D127C22semidirect
21. C2×C22⋊D127C22semidirect
22. C2×C22⋊D127C22semidirect
24. C22×Dic127C22semidirect
25. C2×C4×D127C22semidirect
26. Dic127⋊C4C22semidirect
28. C22⋊Dic127C22semidirect
29. (C2×D127)⋊C4C22semidirect
30. (C2×D127)⋊C4C22semidirect
31. C4×Dic127C22semidirect
34. C2×D4×C127C22semidirect
36. C22⋊C4×C127C22semidirect
50. C2×Dic127D4semidirect
93. C127#535semidirect
104. #120D127semidirect
119. C2×D4C2×D127semidirect
129. C22⋊C4C2×D127semidirect