Finite Group Databasebeta

← #22550615

Subgroups — #22550615

Order 8128.

252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 91 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
1#16861205C211yesmaximal, characteristic, complemented
2#16861202C212yesmaximal, complemented
3#16861191C212yesmaximal, complemented
4#16861190C211yesmaximal, characteristic, complemented
5#16861182C212yesmaximal, complemented
6#16861181C212yesmaximal, complemented
7#16861180C212yesmaximal, complemented
8#16861178C212yesmaximal, complemented
9#120×C127C211yesmaximal, characteristic, complemented
10C2×D508C2211yescharacteristic, complemented
11C2×C22⋊D127C2212yes—
12C2×C22⋊D127C2212yes—
13C2×C22⋊D127C2212yes—
14C2×C22⋊D127C2212yescomplemented
15C22×Dic127C2211yescharacteristic, complemented
16C2×C4×D127C2212yescomplemented
17C4⋊Dic127C2212yescomplemented
18Dic127⋊C4C2212yescomplemented
19Dic127⋊C4C2212yescomplemented
20C22⋊Dic127C2212yes—
21C22⋊Dic127C2212yes—
22C22⋊Dic127C2212yes—
23C22⋊Dic127C2211yescharacteristic, complemented
24(C2×D127)⋊C4C2212yescomplemented
25C4×Dic127C2211yescharacteristic, complemented
26C24×C127C2211yescharacteristic
27C2×D4×C127C2211yescharacteristic
28C2×D4×C127C2212yes—
29C22⋊C4×C127C2211yescharacteristic
30C22⋊C4×C127C2212yes—
31C22×D127C2312yes—
32D508—41no—
33C22⋊D127—42no—
34C22⋊D127—42no—
35C22⋊D127—42no—
36C22⋊D127—42no—
37C2×Dic127—41no—
38C2×Dic127C2312yes—
39C2×Dic127C2312yes—
40C2×Dic127C2312yes—
41C4×D127—42no—
42C23×C127—42no—
43C23×C127—41no—
44C23×C127C2311yescharacteristic
45C23×C127C2312yes—
46C23×C127C2311yescharacteristic
47D4×C127—42no—
48D4×C127—41no—
49C2×C4×C127C2311yescharacteristic
50C2×C4×C127C2312yes—
51C22×C127—22no—
52C22×C127—21no—
53C22×C127—21no—
54C22×C127—22no—
55C22×C127—22no—
56C22×C127—22no—
57C22×C127—22no—
58C22×C127—44no—
59C22×C127C2411yescharacteristic
60C2×D127—22no—
61C2×D127—22no—
62C2×D127—22no—
63C4×C127—22no—
64C4×C127—21no—
65Dic127—22no—
66Dic127—22no—
67Dic127—22no—
68C2×C127—41no—
69C2×C127—42no—
70C2×C127—41no—
71C2×C127H(5,𝔽2)12yes—
72C2×C127H(5,𝔽2)11yescharacteristic
73D127—42no—
74C127#56011yescharacteristic, complemented, Sylow
75#560—1271nomaximal, Sylow
76#127—1271no—
77#126—1272no—
78#123—1271no—
79#123—1272no—
80#121—1272no—
81#121—1272no—
82#121—1272no—
83#120—1272no—
84#120D12711yescharacteristic, complemented
85C24C2×D12711yescharacteristic
86C2×D4—1272no—
87C2×D4—1272no—
88C2×D4—1272no—
89C2×D4—1272no—
90C2×D4—1271no—
91C2×D4C2×D12712yes—
92C2×D4C2×D12711yescharacteristic
93C22×C4—1271no—
94C22×C4—1272no—
95C4⋊C4—1272no—
96C4⋊C4—1272no—
97C4⋊C4—1272no—
98C22⋊C4—1272no—
99C22⋊C4—1272no—
100C22⋊C4—1272no—
101C22⋊C4—1271no—
102C22⋊C4—1272no—
103C22⋊C4C2×D12712yes—
104C22⋊C4C2×D12711yescharacteristic
105C42—1271no—
106C23—42no—
107C23—41no—
108C23—1272no—
109C23C22×D12712yes—
110C23C22×D12711yescharacteristic
111C23C22×D12711yescharacteristic
112D4—42no—
113D4—41no—
114D4—5081no—
115D4—5082no—
116D4—5082no—
117D4—5082no—
118D4—5082no—
119C2×C4—1272no—
120C2×C4—1272no—
121C2×C4—1272no—
122C2×C4—5082no—
123C2×C4—5081no—
124C2×C4C22×D12711yescharacteristic
125C2×C4C22×D12712yes—
126C22—21no—
127C22—22no—
128C22—22no—
129C22—21no—
130C22—22no—
131C22—22no—
132C22—22no—
133C22—44no—
134C22—2542no—
135C22—2542no—
136C22—2542no—
137C22C23×D12711yescharacteristic
138C4—22no—
139C4—21no—
140C4—2542no—
141C4—2542no—
142C4—2542no—
143C2—42no—
144C2—41no—
145C2—41no—
146C2—5082no—
147C2#1686125412yes—
148C2#1686125411yescharacteristic
subgroupG/H
center, FrattiniC22C23×D127
commutator, socleC22×C127C24
Fitting#120×C127C2
hypercenter#120D127
hypocenterC127#560
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #16861205C2semidirect
2. #16861202C2semidirect
3. #16861191C2semidirect
4. #16861190C2semidirect
5. #16861182C2semidirect
6. #16861181C2semidirect
7. #16861180C2semidirect
8. #16861178C2semidirect
9. #120×C127C2semidirect
10. C2×D508C22semidirect
14. C2×C22⋊D127C22semidirect
15. C22×Dic127C22semidirect
16. C2×C4×D127C22semidirect
17. C4⋊Dic127C22semidirect
18. Dic127⋊C4C22semidirect
19. Dic127⋊C4C22semidirect
23. C22⋊Dic127C22semidirect
24. (C2×D127)⋊C4C22semidirect
25. C4×Dic127C22semidirect
74. C127#560semidirect
84. #120D127semidirect