Finite Group Databasebeta

← #22550679

Subgroups — #22550679

Order 8128.

268 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 107 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#16861255C211yesmaximal, characteristic, complemented
2#16861235C213yesmaximal, complemented
3#16861234C213yesmaximal, complemented
4#16861229C213yesmaximal, complemented
5C2×#14523233C213yesmaximal, complemented
6#16861216C211yesmaximal, characteristic, complemented
7H(5,𝔽2)×C127C211yesmaximal, characteristic, complemented
8#14523249—21no—
9#14523248C2213yescomplemented
10C2×D508C2213yescomplemented
11D4×D127—23no—
12#14523238C2211yescharacteristic, complemented
13#14523236C2213yescomplemented
14#14523235C2213yescomplemented
15#14523233C2213yescomplemented
16#14523233C2216yescomplemented
17#14523232C2213yescomplemented
18C2×C127⋊C8C2213yescomplemented
19#41×C127—21no—
20#41×C127C2211yescharacteristic
21#41×C127C2213yescomplemented
22C2×D4×C127—23no—
23C2×D4×C127C2213yescomplemented
24C22×D127—23no—
25Dic254C2311yescharacteristic
26D508—23no—
27D508C2313yes—
28C22⋊D127—23no—
29C4×D127—23no—
30C127⋊C8C2311yescharacteristic
31C127⋊C8C2313yes—
32C23×C127—23no—
33Q8×C127D411yescharacteristic
34Q8×C127C2311yescharacteristic
35D4×C127—23no—
36D4×C127—23no—
37D4×C127D413yes—
38D4×C127C2313yes—
39C2×C4×C127—23no—
40C2×C4×C127C2313yes—
41C22×C127—23no—
42C22×C127—83no—
43C22×C127C2×D413yes—
44C2×D127—23no—
45C2×D127—83no—
46C4×C127—21no—
47C4×C127C2×D413yes—
48C4×C127C2411yescharacteristic
49Dic127—21no—
50C2×C127—23no—
51C2×C127—43no—
52C2×C127C22×D411yescharacteristic
53D127—43no—
54C127#57611yescharacteristic, complemented, Sylow
55#576—1271nomaximal, Sylow
56H(5,𝔽2)—1271no—
57H(5,𝔽2)D12711yescharacteristic, complemented
58AGL(1,ℤ/8)—1273no—
59AGL(1,ℤ/8)—1273no—
60#135—1273no—
61C2×D8—1273no—
62#131—1271no—
63#41—21no—
64#41—1273no—
65#41—2541no—
66#41C2×D12711yescharacteristic
67#41C2×D12713yescomplemented
68C2×D4—23no—
69C2×D4—1273no—
70C2×D4—2543no—
71C2×D4C2×D12713yescomplemented
72Q16—1271no—
73SD16—1273no—
74SD16—1273no—
75D8—1273no—
76D8—1276no—
77M16—1273no—
78C2×C8—1273no—
79C23—23no—
80C23—2543no—
81Q8—1271no—
82Q8C22⋊D12711yescharacteristic
83Q8C22×D12711yescharacteristic
84D4—23no—
85D4—23no—
86D4—1273no—
87D4—2543no—
88D4—2543no—
89D4C22⋊D12713yes—
90D4C22×D12713yes—
91C2×C4—23no—
92C2×C4—2543no—
93C2×C4C22×D12713yes—
94C8—1271no—
95C8—1273no—
96C22—23no—
97C22—83no—
98C22—10163no—
99C22—2543no—
100C22C2×C22⋊D12713yes—
101C4—21no—
102C4—2541no—
103C4C2×C22⋊D12713yes—
104C4C23×D12711yescharacteristic
105C2—23no—
106C2—43no—
107C2—5083no—
108C2C22×C22⋊D12711yescharacteristic
subgroupG/H
centerC2C22×C22⋊D127
commutatorC4×C127C24
FrattiniC4C23×D127
FittingH(5,𝔽2)×C127C2
socleC2×C127C22×D4
hypercenterH(5,𝔽2)D127
hypocenterC127#576
LCSG ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ #41 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #16861255C2semidirect
2. #16861235C2semidirect
3. #16861234C2semidirect
4. #16861229C2semidirect
5. C2×#14523233C2semidirect
6. #16861216C2semidirect
7. H(5,𝔽2)×C127C2semidirect
9. #14523248C22semidirect
10. C2×D508C22semidirect
12. #14523238C22semidirect
13. #14523236C22semidirect
14. #14523235C22semidirect
15. #14523233C22semidirect
16. #14523233C22semidirect
17. #14523232C22semidirect
18. C2×C127⋊C8C22semidirect
21. #41×C127C22semidirect
23. C2×D4×C127C22semidirect
54. C127#576semidirect
57. H(5,𝔽2)D127semidirect
67. #41C2×D127semidirect
71. C2×D4C2×D127semidirect