Finite Group Databasebeta

← #22550442

Subgroups — #22550442

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×#14523248C211yesmaximal, characteristic, complemented
2C2×D4×D127C211yesmaximal, characteristic, complemented
3#16861205C212yesmaximal, complemented
4#16861195C212yesmaximal, complemented
5#16861184C211yesmaximal, characteristic, complemented
6#16861183C211yesmaximal, characteristic, complemented
7#16861180C211yesmaximal, characteristic, complemented
8#16861179C212yesmaximal, complemented
9#16861177C212yesmaximal, complemented
10#16861167C211yesmaximal, characteristic, complemented
11#121×C127C211yesmaximal, characteristic, complemented
12C23×D127C2212yes—
13#14523248—24no—
14C2×Dic254C2211yescharacteristic, complemented
15C2×D508C2211yescharacteristic
16C2×D508C2211yescharacteristic
17C2×D508C2212yescomplemented
18C2×C22⋊D127C2212yes—
19C2×C22⋊D127C2211yescharacteristic
20C2×C22⋊D127C2211yescharacteristic, complemented
21C2×C22⋊D127C2212yescomplemented
22C2×C22⋊D127C2212yescomplemented
23D4×D127—24no—
24C2×C4×D127C2211yescharacteristic
25C2×C4×D127C2211yescharacteristic, complemented
26C4⋊Dic127C2211yescharacteristic, complemented
27C22⋊Dic127C2212yescomplemented
28(C2×D127)⋊C4C2212yes—
29(C2×D127)⋊C4C2212yescomplemented
30(C2×D127)⋊C4C2212yescomplemented
31C4×Dic127C2212yescomplemented
32C2×D4×C127C2211yescharacteristic
33C2×D4×C127C2212yescomplemented
34C22×C4×C127C2211yescharacteristic, complemented
35C4⋊C4×C127C2211yescharacteristic, complemented
36C22⋊C4×C127C2212yescomplemented
37C22×D127—24no—
38C22×D127—42no—
39C22×D127—42no—
40C22×D127C2312yes—
41C22×D127C2311yescharacteristic
42C22×D127C2311yescharacteristic
43Dic254—22no—
44D508—22no—
45D508—22no—
46D508—42no—
47C22⋊D127—22no—
48C22⋊D127—24no—
49C22⋊D127—22no—
50C22⋊D127—42no—
51C22⋊D127—42no—
52C2×Dic127C2312yes—
53C2×Dic127C2311yescharacteristic
54C2×Dic127C2311yescharacteristic
55C4×D127—22no—
56C4×D127D414yescomplemented
57C23×C127C2312yes—
58C23×C127C2311yescharacteristic
59D4×C127—22no—
60D4×C127—42no—
61C2×C4×C127—22no—
62C2×C4×C127C2311yescharacteristic
63C2×C4×C127C2312yes—
64C2×C4×C127C2311yescharacteristic
65C22×C127—21no—
66C22×C127—22no—
67C22×C127—21no—
68C22×C127—22no—
69C22×C127—21no—
70C22×C127—22no—
71C22×C127C2411yescharacteristic
72C2×D127—21no—
73C2×D127—21no—
74C2×D127—21no—
75C2×D127—22no—
76C2×D127—21no—
77C2×D127—22no—
78C2×D127—21no—
79C2×D127—22no—
80C2×D127—48no—
81C2×D127C2×D412yes—
82C4×C127—21no—
83C4×C127—22no—
84C4×C127C2×D412yes—
85Dic127—21no—
86Dic127—22no—
87Dic127C2×D412yes—
88C2×C127—42no—
89C2×C127—41no—
90C2×C127C22×D411yescharacteristic
91C2×C127H(5,𝔽2)11yescharacteristic
92C2×C127H(5,𝔽2)11yescharacteristic
93D127—22no—
94D127—41no—
95D127—42no—
96C127#53511yescharacteristic, complemented, Sylow
97#535—1271nomaximal, Sylow
98C2×#41—1271no—
99C22×D4—1271no—
100#127—1272no—
101#124—1272no—
102#121—1271no—
103#121—1271no—
104#121—1271no—
105#121D12711yescharacteristic, complemented
106#120—1272no—
107#120—1272no—
108#117—1271no—
109C24—1272no—
110#41—2544no—
111C2×Q8—1271no—
112C2×D4—1272no—
113C2×D4—1272no—
114C2×D4—1271no—
115C2×D4—1272no—
116C2×D4—1271no—
117C2×D4—1272no—
118C2×D4—1271no—
119C2×D4—1271no—
120C2×D4—2544no—
121C2×D4C2×D12711yescharacteristic
122C2×D4C2×D12712yescomplemented
123C22×C4—1271no—
124C22×C4—1271no—
125C22×C4C2×D12711yescharacteristic, complemented
126C4⋊C4—1271no—
127C4⋊C4C2×D12711yescharacteristic, complemented
128C22⋊C4—1272no—
129C22⋊C4—1272no—
130C22⋊C4—1272no—
131C22⋊C4—1272no—
132C22⋊C4C2×D12712yescomplemented
133C42—1272no—
134C23—1271no—
135C23—1272no—
136C23—1271no—
137C23—5082no—
138C23—5082no—
139C23—2544no—
140C23C22×D12712yes—
141C23C22×D12711yescharacteristic
142Q8—2542no—
143D4—22no—
144D4—42no—
145D4—5082no—
146D4—5082no—
147D4—5082no—
148D4—2542no—
149D4—2544no—
150D4—2542no—
151D4—2542no—
152D4—2542no—
153C2×C4—22no—
154C2×C4—1271no—
155C2×C4—1274no—
156C2×C4—1272no—
157C2×C4—1271no—
158C2×C4—2542no—
159C2×C4C22×D12712yes—
160C2×C4C22×D12711yescharacteristic
161C2×C4C22×D12711yescharacteristic
162C22—21no—
163C22—22no—
164C22—21no—
165C22—22no—
166C22—21no—
167C22—22no—
168C22—1272no—
169C22—5088no—
170C22—2541no—
171C22—2542no—
172C22—2541no—
173C22—2542no—
174C22—2541no—
175C22—2542no—
176C22—2541no—
177C22—2541no—
178C22C23×D12711yescharacteristic
179C4—21no—
180C4—22no—
181C4—1272no—
182C4—2541no—
183C4—2542no—
184C4D4×D12712yes—
185C2—42no—
186C2—41no—
187C2—5082no—
188C2—5081no—
189C2—2542no—
190C2C2×D4×D12711yescharacteristic
191C2#1686125411yescharacteristic
192C2#1686125511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D127
commutator, socleC22×C127C24
Fitting#121×C127C2
hypercenter#121D127
hypocenterC127#535
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#14523248C2semidirect
2. C2×D4×D127C2semidirect
3. #16861205C2semidirect
4. #16861195C2semidirect
5. #16861184C2semidirect
6. #16861183C2semidirect
7. #16861180C2semidirect
8. #16861179C2semidirect
9. #16861177C2semidirect
10. #16861167C2semidirect
11. #121×C127C2semidirect
14. C2×Dic254C22semidirect
17. C2×D508C22semidirect
20. C2×C22⋊D127C22semidirect
21. C2×C22⋊D127C22semidirect
22. C2×C22⋊D127C22semidirect
25. C2×C4×D127C22semidirect
26. C4⋊Dic127C22semidirect
27. C22⋊Dic127C22semidirect
29. (C2×D127)⋊C4C22semidirect
30. (C2×D127)⋊C4C22semidirect
31. C4×Dic127C22semidirect
33. C2×D4×C127C22semidirect
34. C22×C4×C127C22semidirect
35. C4⋊C4×C127C22semidirect
36. C22⋊C4×C127C22semidirect
56. C4×D127D4semidirect
96. C127#535semidirect
105. #121D127semidirect
122. C2×D4C2×D127semidirect
125. C22×C4C2×D127semidirect
127. C4⋊C4C2×D127semidirect
132. C22⋊C4C2×D127semidirect