Finite Group Databasebeta

← #22550127

Subgroups — #22550127

Order 8128.

130 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 130 conjugacy classes of subgroups — too many to draw. Only its 41 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×#14523233C211yesmaximal, characteristic, complemented
2C2×#14523232C211yesmaximal, characteristic, complemented
3#16861183C211yesmaximal, characteristic, complemented
4#16861140C211yesmaximal, characteristic, complemented
5#16861130C211yesmaximal, characteristic, complemented
6#16861129C211yesmaximal, characteristic, complemented
7#121×C127C211yesmaximal, characteristic, complemented
8C2×D508C2211yescharacteristic
9C2×C22⋊D127—21no—
10#14523233—22no—
11#14523232—22no—
12C2×C127⋊C8C2211yescharacteristic
13C2×C127⋊C8C2211yescharacteristic
14C4⋊Dic127C2211yescharacteristic
15(C2×D127)⋊C4—21no—
16C2×D4×C127—21no—
17C2×D4×C127C2211yescharacteristic
18C22×C4×C127C2211yescharacteristic
19C4⋊C4×C127C2211yescharacteristic
20C22⋊C4×C127—21no—
21C22×D127—21no—
22D508—21no—
23D508—21no—
24C22⋊D127—42no—
25C2×Dic127—21no—
26C127⋊C8—21no—
27C127⋊C8D412yescomplemented
28C23×C127—21no—
29C23×C127D411yescharacteristic
30D4×C127—21no—
31D4×C127—21no—
32D4×C127—42no—
33C2×C4×C127—21no—
34C2×C4×C127—21no—
35C2×C4×C127—21no—
36C2×C4×C127D411yescharacteristic
37C2×C4×C127C2311yescharacteristic
38C22×C127—21no—
39C22×C127—21no—
40C22×C127—21no—
41C22×C127—41no—
42C22×C127—41no—
43C22×C127—41no—
44C22×C127C2×D411yescharacteristic
45C2×D127—41no—
46C2×D127—41no—
47C2×D127—41no—
48C4×C127—21no—
49C4×C127—41no—
50C4×C127C2×D411yescharacteristic
51C4×C127#4111yescharacteristic
52Dic127—41no—
53C2×C127—41no—
54C2×C127—81no—
55C2×C127#12111yescharacteristic
56C2×C127AGL(1,ℤ/8)11yescharacteristic
57C2×C127AGL(1,ℤ/8)11yescharacteristic
58D127—81no—
59C127#46911yescharacteristic, complemented, Sylow
60#469—1271nomaximal, Sylow
61C2×D8—1271no—
62C2×M16—1271no—
63#121—1271no—
64#121D12711yescharacteristic, complemented
65#106—1271no—
66#102—1271no—
67#102—1271no—
68C2×D4—21no—
69C2×D4—1271no—
70C2×D4—2541no—
71C2×D4C2×D12711yescharacteristic
72C22×C4C2×D12711yescharacteristic
73D8—2542no—
74M16—2542no—
75C2×C8—1271no—
76C2×C8—1271no—
77C4⋊C4—1271no—
78C4⋊C4C2×D12711yescharacteristic
79C22⋊C4—21no—
80C22⋊C4—2541no—
81C23—21no—
82C23—2541no—
83C23C22⋊D12711yescharacteristic
84D4—21no—
85D4—21no—
86D4—42no—
87D4—5082no—
88D4—2541no—
89D4—2541no—
90C2×C4—21no—
91C2×C4—21no—
92C2×C4—21no—
93C2×C4—2541no—
94C2×C4C22⋊D12711yescharacteristic
95C2×C4C22×D12711yescharacteristic
96C8—1272no—
97C8—2541no—
98C22—21no—
99C22—21no—
100C22—21no—
101C22—41no—
102C22—41no—
103C22—41no—
104C22—5081no—
105C22—5081no—
106C22—5081no—
107C22C2×C22⋊D12711yescharacteristic
108C4—21no—
109C4—41no—
110C4—5081no—
111C4D4×D12711yescharacteristic
112C4#1452324711yescharacteristic
113C2—41no—
114C2—81no—
115C2—10161no—
116C2#1686118111yescharacteristic
117C2#1686123411yescharacteristic
118C2#1686123511yescharacteristic
subgroupG/H
centerC22C2×C22⋊D127
commutatorC2×C4×C127C23
FrattiniC2×C4C22×D127
Fitting#121×C127C2
socleC22×C127C2×D4
hypercenter#121D127
hypocenterC127#469
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#14523233C2semidirect
2. C2×#14523232C2semidirect
3. #16861183C2semidirect
4. #16861140C2semidirect
5. #16861130C2semidirect
6. #16861129C2semidirect
7. #121×C127C2semidirect
27. C127⋊C8D4semidirect
59. C127#469semidirect
64. #121D127semidirect