Finite Group Databasebeta

← #22550067

Subgroups — #22550067

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#16861193C212yesmaximal, complemented
3#16861190C211yesmaximal, characteristic, complemented
4#16861123C212yesmaximal, complemented
5#99×C127C211yesmaximal, characteristic, complemented
6#14523248—21no—
7#14523247—22no—
8C2×C22⋊D127—21no—
9C2×C22⋊D127C2212yes—
10D4×D127—22no—
11C22×Dic127C2211yescharacteristic, complemented
12C4⋊Dic127—22no—
13Dic127⋊C4—21no—
14C22⋊Dic127—21no—
15C22⋊Dic127C2211yescharacteristic, complemented
16(C2×D127)⋊C4—22no—
17C2×D4×C127C2211yescharacteristic
18C22⋊C4×C127C2212yescomplemented
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×Dic127—21no—
29C2×Dic127—21no—
30C2×Dic127D412yes—
31C4×D127—22no—
32C23×C127D411yescharacteristic
33C23×C127C2311yescharacteristic
34D4×C127—22no—
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—42no—
47Dic127—22no—
48Dic127—41no—
49Dic127—41no—
50C2×C127—21no—
51C2×C127—22no—
52C2×C127—41no—
53C2×C127#12011yescharacteristic
54D127—42no—
55C127#45811yescharacteristic, complemented, Sylow
56#458—1271nomaximal, Sylow
57H(5,𝔽2)—1271no—
58#123—1271no—
59#123—1272no—
60#99—1272no—
61#99D12711yescharacteristic, complemented
62#41—2542no—
63#41—2541no—
64C2×D4—1272no—
65C2×D4—2542no—
66C2×D4—2541no—
67C2×D4C2×D12711yescharacteristic
68C22×C4—1271no—
69C4⋊C4—2542no—
70C4⋊C4—2541no—
71C22⋊C4—1271no—
72C22⋊C4—2542no—
73C22⋊C4—2541no—
74C22⋊C4C2×D12712yescomplemented
75C23—1272no—
76C23—2541no—
77C23D50811yescharacteristic
78C23C22×D12711yescharacteristic
79Q8—2541no—
80D4—22no—
81D4—2541no—
82D4—2542no—
83D4—2542no—
84D4—2542no—
85C2×C4—22no—
86C2×C4—1272no—
87C2×C4—2542no—
88C2×C4—2542no—
89C2×C4—2541no—
90C2×C4—2541no—
91C2×C4—2541no—
92C2×C4D50811yescharacteristic
93C22—21no—
94C22—41no—
95C22—41no—
96C22—10161no—
97C22—5082no—
98C22—2542no—
99C22D4×D12712yes—
100C22C2×D50811yescharacteristic
101C4—21no—
102C4—42no—
103C4—5081no—
104C4—5081no—
105C4—2542no—
106C2—22no—
107C2—21no—
108C2—41no—
109C2—5082no—
110C2#1686117911yescharacteristic
subgroupG/H
centerC2#16861179
commutatorC23×C127C23
FrattiniC23C22×D127
Fitting#99×C127C2
socleC2×C127#120
hypercenter#99D127
hypocenterC127#458
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C23 ◁ #99

Complements

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

NH ≅ G/Ntype
2. #16861193C2semidirect
3. #16861190C2semidirect
4. #16861123C2semidirect
5. #99×C127C2semidirect
11. C22×Dic127C22semidirect
15. C22⋊Dic127C22semidirect
18. C22⋊C4×C127C22semidirect
55. C127#458semidirect
61. #99D127semidirect
74. C22⋊C4C2×D127semidirect