Finite Group Databasebeta

← #22550062

Subgroups — #22550062

Order 8128.

198 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 39 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#16861254C211yesmaximal, characteristic, complemented
2#16861178C212yesmaximal, complemented
3#16861124C211yesmaximal, characteristic, complemented
4#16861123C212yesmaximal, complemented
5#120×C127C211yesmaximal, characteristic, complemented
6#14523248—21no—
7#14523247—22no—
8C2×C22⋊D127—22no—
9C2×C22⋊D127—21no—
10C2×C22⋊D127C2212yes—
11D4×D127—22no—
12C22⋊Dic127—22no—
13C22⋊Dic127C2212yescomplemented
14C24×C127C2211yescharacteristic, complemented
15C2×D4×C127—21no—
16C2×D4×C127C2211yescharacteristic
17C22⋊C4×C127—21no—
18C22⋊C4×C127C2211yescharacteristic, complemented
19C22×D127—21no—
20C22×D127D412yescomplemented
21Dic254—21no—
22D508—21no—
23C22⋊D127—22no—
24C22⋊D127—22no—
25C22⋊D127—22no—
26C22⋊D127—44no—
27C2×Dic127—21no—
28C2×Dic127—22no—
29C2×Dic127D412yescomplemented
30C4×D127—22no—
31C23×C127—21no—
32C23×C127—22no—
33C23×C127—42no—
34C23×C127D411yescharacteristic, complemented
35C23×C127C2311yescharacteristic
36D4×C127—22no—
37D4×C127—42no—
38C2×C4×C127—21no—
39C2×C4×C127D411yescharacteristic, complemented
40C22×C127—22no—
41C22×C127—24no—
42C22×C127—21no—
43C22×C127—41no—
44C22×C127—41no—
45C22×C127—42no—
46C22×C127—41no—
47C22×C127—41no—
48C22×C127C2×D412yes—
49C22×C127C2×D411yescharacteristic
50C2×D127—22no—
51C2×D127—42no—
52C2×D127—81no—
53C4×C127—21no—
54C4×C127—41no—
55Dic127—22no—
56Dic127—42no—
57C2×C127—22no—
58C2×C127—21no—
59C2×C127—41no—
60C2×C127—42no—
61C2×C127#12011yescharacteristic
62D127—42no—
63C127B(4,𝔽2)11yescharacteristic, complemented, Sylow
64B(4,𝔽2)—1271nomaximal, Sylow
65H(5,𝔽2)—1271no—
66#120—1272no—
67#120D12711yescharacteristic, complemented
68#99—1271no—
69#99—1272no—
70C24C2×D12711yescharacteristic, complemented
71#41—2542no—
72#41—2541no—
73C2×D4—21no—
74C2×D4—1272no—
75C2×D4—2542no—
76C2×D4—2542no—
77C2×D4—2541no—
78C2×D4C2×D12711yescharacteristic
79C22⋊C4—21no—
80C22⋊C4—1272no—
81C22⋊C4—2542no—
82C22⋊C4C2×D12711yescharacteristic, complemented
83C23—22no—
84C23—21no—
85C23—42no—
86C23—1272no—
87C23—2541no—
88C23C22⋊D12711yescharacteristic, complemented
89C23C22×D12711yescharacteristic
90Q8—2541no—
91D4—22no—
92D4—42no—
93D4—5084no—
94D4—2541no—
95D4—2542no—
96D4—2542no—
97D4—2542no—
98C2×C4—21no—
99C2×C4—1272no—
100C2×C4—2542no—
101C2×C4—2542no—
102C2×C4—2541no—
103C2×C4C22⋊D12711yescharacteristic, complemented
104C22—22no—
105C22—24no—
106C22—21no—
107C22—41no—
108C22—41no—
109C22—41no—
110C22—42no—
111C22—41no—
112C22—10161no—
113C22—5082no—
114C22—2542no—
115C22D4×D12712yes—
116C22C2×C22⋊D12711yescharacteristic
117C4—21no—
118C4—41no—
119C4—5082no—
120C4—2542no—
121C2—21no—
122C2—22no—
123C2—42no—
124C2—41no—
125C2—5082no—
126C2#1686117711yescharacteristic
subgroupG/H
centerC2#16861177
commutatorC23×C127C23
FrattiniC23C22×D127
Fitting#120×C127C2
socleC2×C127#120
hypercenter#120D127
hypocenterC127B(4,𝔽2)
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C23 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #16861254C2semidirect
2. #16861178C2semidirect
3. #16861124C2semidirect
4. #16861123C2semidirect
5. #120×C127C2semidirect
13. C22⋊Dic127C22semidirect
14. C24×C127C22semidirect
18. C22⋊C4×C127C22semidirect
20. C22×D127D4semidirect
29. C2×Dic127D4semidirect
34. C23×C127D4semidirect
39. C2×C4×C127D4semidirect
63. C127B(4,𝔽2)semidirect
67. #120D127semidirect
70. C24C2×D127semidirect
82. C22⋊C4C2×D127semidirect
88. C23C22⋊D127semidirect
103. C2×C4C22⋊D127semidirect