Finite Group Databasebeta

← #22549784

Subgroups — #22549784

Order 8128.

322 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 69 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
1C22×C22⋊D127C212yesmaximal, complemented
2C2×C22⋊Dic127C212yesmaximal, complemented
3C2×(C2×D127)⋊C4C211yesmaximal, characteristic, complemented
4C127⋊#95C211yesmaximal, characteristic, complemented
5C22×D4×C127C211yesmaximal, characteristic, complemented
6C23×D127C2211yescharacteristic
7C2×C22⋊D127—24no—
8C2×C22⋊D127—24no—
9C2×C22⋊D127—24no—
10C22×Dic127C2211yescharacteristic
11C22×Dic127C2212yes—
12C22⋊Dic127—24no—
13(C2×D127)⋊C4—22no—
14C24×C127C2212yes—
15C2×D4×C127—24no—
16C2×D4×C127—22no—
17C22×C4×C127C2211yescharacteristic
18C22×D127—21no—
19C22×D127—22no—
20C22×D127—22no—
21C22×D127—21no—
22C22×D127D412yescomplemented
23C22⋊D127—216no—
24C2×Dic127—22no—
25C2×Dic127—22no—
26C2×Dic127—21no—
27C2×Dic127—22no—
28C2×Dic127D414yescomplemented
29C23×C127—22no—
30C23×C127—22no—
31C23×C127—22no—
32C23×C127—22no—
33C23×C127—22no—
34C23×C127—22no—
35C23×C127D414yescomplemented
36C23×C127C2311yescharacteristic
37D4×C127—28no—
38C2×C4×C127—22no—
39C2×C4×C127D412yescomplemented
40C22×C127—24no—
41C22×C127—24no—
42C22×C127—24no—
43C22×C127—42no—
44C22×C127—42no—
45C22×C127—42no—
46C22×C127—42no—
47C22×C127C2×D411yescharacteristic
48C22×C127C2×D412yes—
49C22×C127C2×D412yes—
50C22×C127C2×D411yescharacteristic
51C22×C127#4111yescharacteristic
52C2×D127—24no—
53C2×D127—22no—
54C2×D127—41no—
55C2×D127—42no—
56C2×D127—41no—
57C4×C127—22no—
58Dic127—24no—
59Dic127—41no—
60C2×C127—44no—
61C2×C127#12012yes—
62C2×C127#12011yescharacteristic
63C2×C127#12111yescharacteristic
64C2×C127#12112yes—
65C2×C127#12711yescharacteristic
66D127—42no—
67C127#39211yescharacteristic, complemented, Sylow
68#392—1271nomaximal, Sylow
69C22×D4—1272no—
70C22×D4D12711yescharacteristic, complemented
71C2×C22⋊C4—1272no—
72C2×C22⋊C4—1271no—
73#95—1271no—
74C24—1271no—
75C24C2×D12712yes—
76C2×D4—24no—
77C2×D4—22no—
78C2×D4—2544no—
79C2×D4—2544no—
80C2×D4—2544no—
81C22×C4—1272no—
82C22×C4—1271no—
83C22×C4C2×D12711yescharacteristic
84C22⋊C4—2544no—
85C22⋊C4—2542no—
86C23—22no—
87C23—22no—
88C23—22no—
89C23—22no—
90C23—22no—
91C23—22no—
92C23—1272no—
93C23—2542no—
94C23—2541no—
95C23—2542no—
96C23—2541no—
97C23C22⋊D12714yescomplemented
98C23C22×D12711yescharacteristic
99D4—28no—
100D4—25416no—
101C2×C4—22no—
102C2×C4—1274no—
103C2×C4—2542no—
104C2×C4—2541no—
105C2×C4—2542no—
106C2×C4—2542no—
107C2×C4C22⋊D12712yescomplemented
108C22—24no—
109C22—24no—
110C22—24no—
111C22—42no—
112C22—42no—
113C22—42no—
114C22—42no—
115C22—5081no—
116C22—5081no—
117C22—5082no—
118C22—2544no—
119C22—2542no—
120C22D4×D12712yes—
121C22D4×D12711yescharacteristic
122C22C2×C22⋊D12711yescharacteristic
123C22C2×C22⋊D12712yes—
124C22#1452324711yescharacteristic
125C4—22no—
126C4—5081no—
127C4—2544no—
128C2—44no—
129C2—5082no—
130C2#1686117712yes—
131C2#1686117811yescharacteristic
132C2#1686118011yescharacteristic
133C2#1686118112yes—
134C2#1686120511yescharacteristic
subgroupG/H
center, FrattiniC23C22×D127
commutator, socleC23×C127C23
FittingC22×D4×C127C2
hypercenterC22×D4D127
hypocenterC127#392
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊D127C2semidirect
2. C2×C22⋊Dic127C2semidirect
3. C2×(C2×D127)⋊C4C2semidirect
4. C127⋊#95C2semidirect
5. C22×D4×C127C2semidirect
22. C22×D127D4semidirect
28. C2×Dic127D4semidirect
35. C23×C127D4semidirect
39. C2×C4×C127D4semidirect
67. C127#392semidirect
70. C22×D4D127semidirect
97. C23C22⋊D127semidirect
107. C2×C4C22⋊D127semidirect