Finite Group Databasebeta

← #22550520

Subgroups — #22550520

Order 8128.

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

normal subgroups only

This group has 334 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
1C2×#14523247C211yesmaximal, characteristic, complemented
2#16861196C211yesmaximal, characteristic, complemented
3#16861190C212yesmaximal, complemented
4#16861188C211yesmaximal, characteristic, complemented
5#16861181C212yesmaximal, complemented
6#16861180C211yesmaximal, characteristic, complemented
7#16861178C212yesmaximal, complemented
8D4×Dic127C211yesmaximal, characteristic, complemented
9C4×C22⋊D127C211yesmaximal, characteristic, complemented
10C2×C22⋊Dic127C212yesmaximal, complemented
11C22×D4×C127C211yesmaximal, characteristic, complemented
12#14523247—24no—
13C2×Dic254C2211yescharacteristic, complemented
14C2×C22⋊D127C2211yescharacteristic
15C2×C22⋊D127C2212yescomplemented
16C2×C22⋊D127C2212yescomplemented
17C22×Dic127C2212yescomplemented
18C22×Dic127C2212yescomplemented
19C2×C4×D127C2211yescharacteristic, complemented
20C4⋊Dic127C2211yescharacteristic, complemented
21Dic127⋊C4C2212yescomplemented
22Dic127⋊C4C2211yescharacteristic, complemented
23C22⋊Dic127C2212yes—
24C22⋊Dic127C2211yescharacteristic
25C22⋊Dic127C2212yescomplemented
26C22⋊Dic127C2212yescomplemented
27C22⋊Dic127C2212yescomplemented
28C22⋊Dic127C2212yescomplemented
29(C2×D127)⋊C4C2211yescharacteristic, complemented
30C4×Dic127C2211yescharacteristic, complemented
31C24×C127C2212yes—
32C2×D4×C127—24no—
33C2×D4×C127C2212yes—
34C2×D4×C127C2211yescharacteristic
35C2×D4×C127C2211yescharacteristic
36C22×C4×C127C2211yescharacteristic
37C22×D127C2311yescharacteristic
38Dic254—22no—
39C22⋊D127—24no—
40C22⋊D127—22no—
41C22⋊D127—42no—
42C2×Dic127—24no—
43C2×Dic127—42no—
44C2×Dic127C2311yescharacteristic
45C2×Dic127C2311yescharacteristic
46C2×Dic127C2312yes—
47C2×Dic127C2311yescharacteristic
48C2×Dic127C2312yes—
49C4×D127—22no—
50C23×C127—24no—
51C23×C127—24no—
52C23×C127—42no—
53C23×C127C2312yes—
54C23×C127C2312yes—
55C23×C127C2311yescharacteristic
56D4×C127—24no—
57D4×C127—22no—
58D4×C127D414yescomplemented
59C2×C4×C127—22no—
60C2×C4×C127C2311yescharacteristic
61C2×C4×C127C2311yescharacteristic
62C22×C127—21no—
63C22×C127—22no—
64C22×C127—21no—
65C22×C127—22no—
66C22×C127—22no—
67C22×C127—22no—
68C22×C127—22no—
69C22×C127—48no—
70C22×C127C2×D414yes—
71C22×C127#4112yescomplemented
72C22×C127C2411yescharacteristic
73C2×D127—21no—
74C2×D127—21no—
75C2×D127—21no—
76C4×C127—21no—
77C4×C127C2×D412yes—
78Dic127—21no—
79Dic127—22no—
80Dic127—21no—
81Dic127—21no—
82Dic127—22no—
83C2×C127—22no—
84C2×C127—24no—
85C2×C127—42no—
86C2×C127C22×D411yescharacteristic
87C2×C127C2×#4111yescharacteristic
88C2×C127H(5,𝔽2)11yescharacteristic
89D127—41no—
90C127#54611yescharacteristic, complemented, Sylow
91#546—1271nomaximal, Sylow
92C2×#41—1271no—
93C22×D4D12711yescharacteristic, complemented
94#124—1271no—
95#123—1272no—
96C22⋊Q8—1271no—
97#121—1271no—
98#121—1272no—
99#120—1272no—
100C4×D4—1271no—
101C4×D4—1271no—
102C2×C22⋊C4—1272no—
103C24C2×D12712yes—
104#41—2544no—
105C2×Q8—1271no—
106C2×D4—24no—
107C2×D4—1272no—
108C2×D4—1271no—
109C2×D4—1272no—
110C2×D4C2×D12712yes—
111C2×D4C2×D12711yescharacteristic
112C2×D4C2×D12711yescharacteristic
113C22×C4—1272no—
114C22×C4—1272no—
115C22×C4—1271no—
116C22×C4C2×D12711yescharacteristic
117C4⋊C4—1271no—
118C4⋊C4—1272no—
119C4⋊C4—1271no—
120C22⋊C4—1271no—
121C22⋊C4—1272no—
122C22⋊C4—1272no—
123C22⋊C4—1272no—
124C22⋊C4—1272no—
125C22⋊C4—1272no—
126C22⋊C4—1271no—
127C42—1271no—
128C23—24no—
129C23—24no—
130C23—42no—
131C23—1271no—
132C23C22×D12712yes—
133C23C22×D12712yes—
134C23C22×D12711yescharacteristic
135Q8—2542no—
136D4—24no—
137D4—22no—
138D4—5082no—
139D4—2544no—
140D4—2542no—
141D4C22⋊D12714yescomplemented
142C2×C4—22no—
143C2×C4—1271no—
144C2×C4—1271no—
145C2×C4—1272no—
146C2×C4—1271no—
147C2×C4—1272no—
148C2×C4—5082no—
149C2×C4—2544no—
150C2×C4—2542no—
151C2×C4C22×D12711yescharacteristic
152C2×C4C22×D12711yescharacteristic
153C22—21no—
154C22—22no—
155C22—22no—
156C22—21no—
157C22—22no—
158C22—22no—
159C22—22no—
160C22—48no—
161C22—2541no—
162C22—2541no—
163C22—2541no—
164C22C2×C22⋊D12714yes—
165C22#1452324712yescomplemented
166C22C23×D12711yescharacteristic
167C4—21no—
168C4—2541no—
169C4—2542no—
170C4—2541no—
171C4—2541no—
172C4—2542no—
173C4C2×C22⋊D12712yes—
174C2—22no—
175C2—24no—
176C2—42no—
177C2—5081no—
178C2C22×C22⋊D12711yescharacteristic
179C2C2×#1452324711yescharacteristic
180C2#1686125411yescharacteristic
subgroupG/H
center, FrattiniC22C23×D127
commutator, socleC22×C127C24
FittingC22×D4×C127C2
hypercenterC22×D4D127
hypocenterC127#546
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C2×#14523247C2semidirect
2. #16861196C2semidirect
3. #16861190C2semidirect
4. #16861188C2semidirect
5. #16861181C2semidirect
6. #16861180C2semidirect
7. #16861178C2semidirect
8. D4×Dic127C2semidirect
9. C4×C22⋊D127C2semidirect
10. C2×C22⋊Dic127C2semidirect
11. C22×D4×C127C2semidirect
13. C2×Dic254C22semidirect
15. C2×C22⋊D127C22semidirect
16. C2×C22⋊D127C22semidirect
17. C22×Dic127C22semidirect
18. C22×Dic127C22semidirect
19. C2×C4×D127C22semidirect
20. C4⋊Dic127C22semidirect
21. Dic127⋊C4C22semidirect
22. Dic127⋊C4C22semidirect
25. C22⋊Dic127C22semidirect
26. C22⋊Dic127C22semidirect
27. C22⋊Dic127C22semidirect
28. C22⋊Dic127C22semidirect
29. (C2×D127)⋊C4C22semidirect
30. C4×Dic127C22semidirect
58. D4×C127D4semidirect
71. C22×C127#41semidirect
90. C127#546semidirect
93. C22×D4D127semidirect
141. D4C22⋊D127semidirect
165. C22#14523247semidirect