Finite Group Databasebeta

← #22549756

Subgroups — #22549756

Order 8128.

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

normal subgroups only

This group has 286 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
1C23×Dic127C211yesmaximal, characteristic, complemented
2C2×C22⋊Dic127C212yesmaximal, complemented
3C2×C22⋊Dic127C211yesmaximal, characteristic, complemented
4C127⋊#95C212yesmaximal, complemented
5C22×D4×C127C211yesmaximal, characteristic
6C22×Dic127—24no—
7C22×Dic127—22no—
8C22×Dic127C2212yes—
9C22×Dic127C2212yes—
10C22⋊Dic127—24no—
11C22⋊Dic127—22no—
12C24×C127C2211yescharacteristic
13C24×C127C2211yescharacteristic
14C2×D4×C127—22no—
15C2×D4×C127—22no—
16C2×D4×C127C414yescomplemented
17C22×C4×C127C2211yescharacteristic
18C2×Dic127—28no—
19C2×Dic127—22no—
20C2×Dic127—22no—
21C2×Dic127—22no—
22C2×Dic127—22no—
23C2×Dic127—22no—
24C2×Dic127D414yescomplemented
25C23×C127—22no—
26C23×C127—22no—
27C23×C127—21no—
28C23×C127—21no—
29C23×C127—22no—
30C23×C127—22no—
31C23×C127C2×C412yes—
32C23×C127C2×C412yescomplemented
33C23×C127D412yes—
34C23×C127D412yes—
35C23×C127C2311yescharacteristic
36D4×C127—28no—
37C2×C4×C127—21no—
38C2×C4×C127—21no—
39C2×C4×C127C2×C412yescomplemented
40C22×C127—24no—
41C22×C127—22no—
42C22×C127—24no—
43C22×C127—22no—
44C22×C127—22no—
45C22×C127—24no—
46C22×C127—41no—
47C22×C127—42no—
48C22×C127—41no—
49C22×C127C22⋊C414yescomplemented
50C22×C127C22×C411yescharacteristic
51C22×C127C2×D412yes—
52C22×C127C2×D411yescharacteristic
53C22×C127C2×D411yescharacteristic
54C22×C127#4112yes—
55C4×C127—22no—
56Dic127—24no—
57Dic127—42no—
58C2×C127—24no—
59C2×C127—42no—
60C2×C127C2×C22⋊C411yescharacteristic
61C2×C127C4×D412yes—
62C2×C127#12011yescharacteristic
63C2×C127#12112yes—
64C2×C127#12311yescharacteristic
65C127#38611yescharacteristic, complemented, Sylow
66#386—1271nomaximal, Sylow
67C22×D4D12711yescharacteristic
68C23×C4—1271no—
69C2×C22⋊C4—1272no—
70C2×C22⋊C4—1271no—
71#95—1272no—
72C24C2×D12711yescharacteristic
73C24C2×D12711yescharacteristic
74C2×D4—22no—
75C2×D4—22no—
76C2×D4Dic12714yescomplemented
77C22×C4—1272no—
78C22×C4—1272no—
79C22×C4—2544no—
80C22×C4—2542no—
81C22×C4C2×D12711yescharacteristic
82C22⋊C4—2544no—
83C22⋊C4—2542no—
84C23—22no—
85C23—22no—
86C23—22no—
87C23—21no—
88C23—21no—
89C23—22no—
90C23C2×Dic12712yes—
91C23C2×Dic12712yescomplemented
92C23C22⋊D12712yes—
93C23C22⋊D12712yes—
94C23C22×D12711yescharacteristic
95D4—28no—
96C2×C4—21no—
97C2×C4—21no—
98C2×C4—1274no—
99C2×C4—2542no—
100C2×C4—2542no—
101C2×C4—2542no—
102C2×C4—2548no—
103C2×C4—2542no—
104C2×C4—2542no—
105C2×C4C2×Dic12712yescomplemented
106C22—22no—
107C22—24no—
108C22—22no—
109C22—24no—
110C22—22no—
111C22—24no—
112C22—41no—
113C22—41no—
114C22—42no—
115C22C22⋊Dic12714yescomplemented
116C22C22×Dic12711yescharacteristic
117C22D4×D12712yes—
118C22C2×C22⋊D12711yescharacteristic
119C22C2×C22⋊D12711yescharacteristic
120C22#1452324712yes—
121C4—22no—
122C4—5082no—
123C4—2544no—
124C2—24no—
125C2—42no—
126C2C2×C22⋊Dic12711yescharacteristic
127C2D4×Dic12712yes—
128C2#1686117711yescharacteristic
129C2#1686118112yes—
130C2#1686119011yescharacteristic
subgroupG/H
center, FrattiniC23C22×D127
commutatorC22×C127C22×C4
FittingC22×D4×C127C2
socleC23×C127C23
hypercenterC22×D4D127
hypocenterC127#386
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C23×Dic127C2semidirect
2. C2×C22⋊Dic127C2semidirect
3. C2×C22⋊Dic127C2semidirect
4. C127⋊#95C2semidirect
16. C2×D4×C127C4semidirect
24. C2×Dic127D4semidirect
32. C23×C127C2×C4semidirect
39. C2×C4×C127C2×C4semidirect
49. C22×C127C22⋊C4semidirect
65. C127#386semidirect
76. C2×D4Dic127semidirect
91. C23C2×Dic127semidirect
105. C2×C4C2×Dic127semidirect
115. C22C22⋊Dic127semidirect