Finite Group Databasebeta

← #22550065

Subgroups — #22550065

Order 8128.

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

normal subgroups only

This group has 160 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
2#16861192C212yesmaximal, complemented
3#16861124C211yesmaximal, characteristic, complemented
4#16861123C212yesmaximal, complemented
5#123×C127C211yesmaximal, characteristic, complemented
6#14523248—21no—
7#14523247—22no—
8C2×C22⋊D127—21no—
9C2×C22⋊D127C2212yes—
10D4×D127—22no—
11Dic127⋊C4—22no—
12C22⋊Dic127C2212yescomplemented
13(C2×D127)⋊C4—22no—
14C2×D4×C127C2211yescharacteristic
15C22×C4×C127C2211yescharacteristic, complemented
16C4⋊C4×C127—21no—
17C22⋊C4×C127—21no—
18C22⋊C4×C127C2211yescharacteristic, complemented
19C22×D127—21no—
20C22×D127D412yes—
21Dic254—21no—
22D508—21no—
23C22⋊D127—22no—
24C22⋊D127—22no—
25C22⋊D127—22no—
26C2×Dic127—21no—
27C2×Dic127—22no—
28C2×Dic127D412yes—
29C4×D127—22no—
30C23×C127D411yescharacteristic
31C23×C127C2311yescharacteristic
32D4×C127—22no—
33C2×C4×C127—21no—
34C2×C4×C127—21no—
35C2×C4×C127—22no—
36C2×C4×C127D411yescharacteristic
37C22×C127—21no—
38C22×C127—41no—
39C22×C127—41no—
40C22×C127C2×D412yes—
41C22×C127C2×D411yescharacteristic
42C2×D127—22no—
43C2×D127—42no—
44C2×D127—81no—
45C4×C127—21no—
46C4×C127—41no—
47C4×C127—41no—
48Dic127—22no—
49Dic127—42no—
50C2×C127—22no—
51C2×C127—21no—
52C2×C127—41no—
53C2×C127#12011yescharacteristic
54D127—42no—
55C127#45811yescharacteristic, complemented, Sylow
56#458—1271nomaximal, Sylow
57H(5,𝔽2)—1271no—
58#123—1272no—
59#123D12711yescharacteristic, complemented
60#99—1271no—
61#99—1272no—
62#41—2542no—
63#41—2541no—
64C2×D4—1272no—
65C2×D4—2542no—
66C2×D4—2541no—
67C2×D4C2×D12711yescharacteristic
68C22×C4C2×D12711yescharacteristic, complemented
69C4⋊C4—21no—
70C4⋊C4—2542no—
71C22⋊C4—21no—
72C22⋊C4—1272no—
73C22⋊C4—2542no—
74C22⋊C4C2×D12711yescharacteristic, complemented
75C23—1272no—
76C23—2541no—
77C23C22⋊D12711yescharacteristic
78C23C22×D12711yescharacteristic
79Q8—2541no—
80D4—22no—
81D4—2541no—
82D4—2542no—
83D4—2542no—
84D4—2542no—
85C2×C4—22no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—1272no—
89C2×C4—2542no—
90C2×C4—2542no—
91C2×C4—2541no—
92C2×C4C22⋊D12711yescharacteristic
93C22—21no—
94C22—41no—
95C22—41no—
96C22—10161no—
97C22—5082no—
98C22—2542no—
99C22D4×D12712yes—
100C22C2×C22⋊D12711yescharacteristic
101C4—21no—
102C4—41no—
103C4—41no—
104C4—5082no—
105C4—2542no—
106C2—21no—
107C2—22no—
108C2—41no—
109C2—5082no—
110C2#1686117711yescharacteristic
subgroupG/H
centerC2#16861177
commutatorC23×C127C23
FrattiniC23C22×D127
Fitting#123×C127C2
socleC2×C127#120
hypercenter#123D127
hypocenterC127#458
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C23 ◁ #123

Complements

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

NH ≅ G/Ntype
2. #16861192C2semidirect
3. #16861124C2semidirect
4. #16861123C2semidirect
5. #123×C127C2semidirect
12. C22⋊Dic127C22semidirect
15. C22×C4×C127C22semidirect
18. C22⋊C4×C127C22semidirect
55. C127#458semidirect
59. #123D127semidirect
68. C22×C4C2×D127semidirect
74. C22⋊C4C2×D127semidirect