Finite Group Databasebeta

← #22550052

Subgroups — #22550052

Order 8128.

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

normal subgroups only

This group has 152 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
2#16861240C211yesmaximal, characteristic, complemented
3#16861198C211yesmaximal, characteristic, complemented
4#16861136C211yesmaximal, characteristic, complemented
5#16861135C211yesmaximal, characteristic, complemented
6#16861125C211yesmaximal, characteristic, complemented
7#137×C127C211yesmaximal, characteristic, complemented
8#14523249—21no—
9#14523248—21no—
10#14523248C2211yescharacteristic
11C2×D508—21no—
12C2×D508C2211yescharacteristic
13D4×D127—22no—
14D4×D127—21no—
15#14523238—21no—
16#14523235—21no—
17#14523232C2211yescharacteristic, complemented
18(C2×D127)⋊C4—22no—
19C4×Dic127C2211yescharacteristic, complemented
20#41×C127C2211yescharacteristic
21C2×Q8×C127C2211yescharacteristic, complemented
22Q16×C127—21no—
23SD16×C127—21no—
24M16×C127C2211yescharacteristic, complemented
25C22×D127—22no—
26C22×D127D412yes—
27Dic254D411yescharacteristic
28D508—22no—
29D508—21no—
30D508—21no—
31D508D411yescharacteristic
32C22⋊D127—22no—
33C22⋊D127—21no—
34C2×Dic127—21no—
35C4×D127—22no—
36C4×D127—21no—
37C127⋊C8—21no—
38Q8×C127—21no—
39Q8×C127—21no—
40Q8×C127D411yescharacteristic
41D4×C127—21no—
42D4×C127D411yescharacteristic
43C2×C4×C127—21no—
44C2×C4×C127—21no—
45C2×C4×C127C2311yescharacteristic
46C8×C127—21no—
47C22×C127—21no—
48C22×C127C2×D411yescharacteristic
49C2×D127—22no—
50C2×D127—21no—
51C2×D127—42no—
52C2×D127—82no—
53C4×C127—21no—
54C4×C127—41no—
55C4×C127C2×D411yescharacteristic
56C4×C127C2×D411yescharacteristic
57Dic127—21no—
58Dic127—41no—
59C2×C127—21no—
60C2×C127—41no—
61C2×C127#12011yescharacteristic
62D127—42no—
63D127—41no—
64C127#45511yescharacteristic, complemented, Sylow
65#455—1271nomaximal, Sylow
66H(5,𝔽2)—1271no—
67#137—1271no—
68#137D12711yescharacteristic, complemented
69#124—1271no—
70#104—1271no—
71#104—1271no—
72#100—1271no—
73#41—1271no—
74#41—2541no—
75#41—2541no—
76#41C2×D12711yescharacteristic
77C2×Q8C2×D12711yescharacteristic, complemented
78C2×D4—1271no—
79C2×D4—2541no—
80C2×D4—2542no—
81C2×D4—2541no—
82Q16—21no—
83Q16—2541no—
84SD16—21no—
85SD16—2541no—
86M16—1271no—
87M16C2×D12711yescharacteristic, complemented
88C22⋊C4—2542no—
89C42—1271no—
90C23—1272no—
91C23—2542no—
92Q8—21no—
93Q8—21no—
94Q8—1271no—
95Q8C22⋊D12711yescharacteristic
96D4—21no—
97D4—1271no—
98D4—2542no—
99D4—2542no—
100D4—2541no—
101D4—2541no—
102D4—2541no—
103D4C22⋊D12711yescharacteristic
104C2×C4—21no—
105C2×C4—21no—
106C2×C4—2542no—
107C2×C4—2541no—
108C2×C4—2541no—
109C2×C4C22×D12711yescharacteristic
110C8—21no—
111C8—2541no—
112C22—21no—
113C22—10162no—
114C22—5082no—
115C22—2541no—
116C22—2542no—
117C22D4×D12711yescharacteristic
118C4—21no—
119C4—41no—
120C4—5081no—
121C4—2541no—
122C4D4×D12711yescharacteristic
123C4C2×C22⋊D12711yescharacteristic
124C2—21no—
125C2—41no—
126C2—5081no—
127C2—5082no—
128C2#1686117711yescharacteristic
subgroupG/H
centerC2#16861177
commutatorC2×C4×C127C23
FrattiniC2×C4C22×D127
Fitting#137×C127C2
socleC2×C127#120
hypercenter#137D127
hypocenterC127#455
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
2. #16861240C2semidirect
3. #16861198C2semidirect
4. #16861136C2semidirect
5. #16861135C2semidirect
6. #16861125C2semidirect
7. #137×C127C2semidirect
17. #14523232C22semidirect
19. C4×Dic127C22semidirect
21. C2×Q8×C127C22semidirect
24. M16×C127C22semidirect
64. C127#455semidirect
68. #137D127semidirect
77. C2×Q8C2×D127semidirect
87. M16C2×D127semidirect