Finite Group Databasebeta

← #22550057

Subgroups — #22550057

Order 8128.

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

normal subgroups only

This group has 142 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#16861256C211yesmaximal, characteristic
2#16861240C211yesmaximal, characteristic, complemented
3#16861207C211yesmaximal, characteristic, complemented
4#16861136C211yesmaximal, characteristic, complemented
5#16861135C211yesmaximal, characteristic, complemented
6#16861127C211yesmaximal, characteristic, complemented
7#137×C127C211yesmaximal, characteristic, complemented
8#14523248—21no—
9#14523248C2211yescharacteristic
10#14523247—22no—
11#14523247—21no—
12C2×Dic254—21no—
13C2×Dic254C2211yescharacteristic
14Q8×D127—21no—
15#14523238—21no—
16#14523235—21no—
17#14523232C2211yescharacteristic
18Dic127⋊C4—22no—
19C4×Dic127C2211yescharacteristic
20#41×C127C2211yescharacteristic
21C2×Q8×C127C2211yescharacteristic
22Q16×C127—21no—
23SD16×C127—21no—
24M16×C127C2211yescharacteristic
25Dic254—22no—
26Dic254—21no—
27Dic254—21no—
28Dic254D411yescharacteristic
29D508D411yescharacteristic
30C22⋊D127—22no—
31C22⋊D127—21no—
32C2×Dic127—22no—
33C2×Dic127—21no—
34C2×Dic127D412yes—
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—21no—
50C4×C127—21no—
51C4×C127—41no—
52C4×C127C2×D411yescharacteristic
53C4×C127C2×D411yescharacteristic
54Dic127—22no—
55Dic127—21no—
56Dic127—22no—
57C2×C127—21no—
58C2×C127—41no—
59C2×C127#12011yescharacteristic
60D127—41no—
61C127#45611yescharacteristic, complemented, Sylow
62#456—1271nomaximal, Sylow
632−1+4—1271no—
64#137—1271no—
65#137D12711yescharacteristic, complemented
66C4⋊Q8—1271no—
67#104—1271no—
68#104—1271no—
69#101—1271no—
70#41—1271no—
71#41—2541no—
72#41—2542no—
73#41—2541no—
74#41C2×D12711yescharacteristic
75C2×Q8—1271no—
76C2×Q8—2541no—
77C2×Q8—2541no—
78C2×Q8C2×D12711yescharacteristic
79Q16—21no—
80Q16—2541no—
81SD16—21no—
82SD16—2541no—
83M16—1271no—
84M16C2×D12711yescharacteristic
85C4⋊C4—2542no—
86C42—1271no—
87Q8—21no—
88Q8—21no—
89Q8—1271no—
90Q8—2542no—
91Q8—2541no—
92Q8—2541no—
93Q8C22⋊D12711yescharacteristic
94D4—21no—
95D4—1271no—
96D4—2542no—
97D4—2541no—
98D4C22⋊D12711yescharacteristic
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—1272no—
102C2×C4—2542no—
103C2×C4—2542no—
104C2×C4—2541no—
105C2×C4—2541no—
106C2×C4C22×D12711yescharacteristic
107C8—21no—
108C8—2541no—
109C22—21no—
110C22—2541no—
111C22D4×D12711yescharacteristic
112C4—21no—
113C4—41no—
114C4—2542no—
115C4—2542no—
116C4—2541no—
117C4D4×D12711yescharacteristic
118C4C2×C22⋊D12711yescharacteristic
119C2—21no—
120C2—41no—
121C2—5081no—
122C2#1686117711yescharacteristic
subgroupG/H
centerC2#16861177
commutatorC2×C4×C127C23
FrattiniC2×C4C22×D127
Fitting#137×C127C2
socleC2×C127#120
hypercenter#137D127
hypocenterC127#456
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. #16861207C2semidirect
4. #16861136C2semidirect
5. #16861135C2semidirect
6. #16861127C2semidirect
7. #137×C127C2semidirect
61. C127#456semidirect
65. #137D127semidirect