Finite Group Databasebeta

← #22550042

Subgroups — #22550042

Order 8128.

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

normal subgroups only

This group has 168 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#16861255C211yesmaximal, characteristic, complemented
2#16861235C211yesmaximal, characteristic, complemented
3#16861205C211yesmaximal, characteristic, complemented
4#16861136C211yesmaximal, characteristic, complemented
5#16861135C211yesmaximal, characteristic, complemented
6#16861125C211yesmaximal, characteristic, complemented
7AGL(1,ℤ/8)×C127C211yesmaximal, characteristic, complemented
8#14523249—21no—
9#14523248—21no—
10#14523248C2211yescharacteristic
11C2×D508—21no—
12C2×D508C2211yescharacteristic
13C2×C22⋊D127—22no—
14D4×D127—22no—
15D4×D127—21no—
16#14523236—21no—
17#14523233—21no—
18#14523232C2211yescharacteristic, complemented
19C4×Dic127C2211yescharacteristic, complemented
20#41×C127C2211yescharacteristic
21C2×D4×C127C2211yescharacteristic, complemented
22SD16×C127—21no—
23D8×C127—21no—
24M16×C127C2211yescharacteristic, complemented
25C22×D127—22no—
26C22×D127D412yes—
27Dic254D411yescharacteristic, complemented
28D508—22no—
29D508—21no—
30D508—21no—
31D508D411yescharacteristic, complemented
32C22⋊D127—22no—
33C22⋊D127—21no—
34C22⋊D127—44no—
35C2×Dic127—21no—
36C4×D127—22no—
37C4×D127—21no—
38C127⋊C8—21no—
39C23×C127—21no—
40Q8×C127D411yescharacteristic, complemented
41D4×C127—21no—
42D4×C127—21no—
43D4×C127—21no—
44D4×C127D411yescharacteristic, complemented
45C2×C4×C127—21no—
46C2×C4×C127C2311yescharacteristic
47C8×C127—21no—
48C22×C127—21no—
49C22×C127—42no—
50C22×C127—41no—
51C22×C127C2×D411yescharacteristic
52C2×D127—22no—
53C2×D127—21no—
54C2×D127—42no—
55C2×D127—82no—
56C4×C127—21no—
57C4×C127C2×D411yescharacteristic
58C4×C127C2×D411yescharacteristic
59Dic127—22no—
60Dic127—21no—
61C2×C127—21no—
62C2×C127—41no—
63C2×C127—81no—
64C2×C127#12011yescharacteristic
65D127—42no—
66D127—41no—
67C127#45311yescharacteristic, complemented, Sylow
68#453—1271nomaximal, Sylow
69H(5,𝔽2)—1271no—
70AGL(1,ℤ/8)—1271no—
71AGL(1,ℤ/8)D12711yescharacteristic, complemented
72#127—1271no—
73#104—1271no—
74#104—1271no—
75#100—1271no—
76#41—1271no—
77#41—2541no—
78#41—2541no—
79#41C2×D12711yescharacteristic
80C2×D4—1271no—
81C2×D4—2541no—
82C2×D4—2542no—
83C2×D4—2542no—
84C2×D4—2541no—
85C2×D4C2×D12711yescharacteristic, complemented
86SD16—21no—
87SD16—2541no—
88D8—21no—
89D8—2541no—
90M16—1271no—
91M16C2×D12711yescharacteristic, complemented
92C42—1271no—
93C23—21no—
94C23—1272no—
95C23—2542no—
96Q8—1271no—
97Q8C22⋊D12711yescharacteristic, complemented
98D4—21no—
99D4—21no—
100D4—21no—
101D4—1271no—
102D4—5084no—
103D4—2542no—
104D4—2542no—
105D4—2541no—
106D4—2541no—
107D4—2541no—
108D4C22⋊D12711yescharacteristic, complemented
109C2×C4—21no—
110C2×C4—2542no—
111C2×C4—2541no—
112C2×C4—2541no—
113C2×C4C22×D12711yescharacteristic
114C8—21no—
115C8—2541no—
116C22—21no—
117C22—42no—
118C22—41no—
119C22—10162no—
120C22—5082no—
121C22—2541no—
122C22—2542no—
123C22D4×D12711yescharacteristic
124C4—21no—
125C4—2542no—
126C4—2541no—
127C4D4×D12711yescharacteristic
128C4C2×C22⋊D12711yescharacteristic
129C2—21no—
130C2—41no—
131C2—81no—
132C2—5081no—
133C2—5082no—
134C2#1686117711yescharacteristic
subgroupG/H
centerC2#16861177
commutatorC2×C4×C127C23
FrattiniC2×C4C22×D127
FittingAGL(1,ℤ/8)×C127C2
socleC2×C127#120
hypercenterAGL(1,ℤ/8)D127
hypocenterC127#453
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #16861255C2semidirect
2. #16861235C2semidirect
3. #16861205C2semidirect
4. #16861136C2semidirect
5. #16861135C2semidirect
6. #16861125C2semidirect
7. AGL(1,ℤ/8)×C127C2semidirect
18. #14523232C22semidirect
19. C4×Dic127C22semidirect
21. C2×D4×C127C22semidirect
24. M16×C127C22semidirect
27. Dic254D4semidirect
31. D508D4semidirect
40. Q8×C127D4semidirect
44. D4×C127D4semidirect
67. C127#453semidirect
71. AGL(1,ℤ/8)D127semidirect
85. C2×D4C2×D127semidirect
91. M16C2×D127semidirect
97. Q8C22⋊D127semidirect
108. D4C22⋊D127semidirect