Finite Group Databasebeta

← #22578401

Subgroups — #22578401

Order 8768.

154 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 48 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
1D137⋊M16C211yesmaximal, characteristic, complemented
2#17984167C211yesmaximal, characteristic, complemented
3#17984152C214yesmaximal, complemented
4C2×#14555509C211yesmaximal, characteristic
5#14555520C2212yescomplemented
6#14555519—21no—
7#14555518—21no—
8C4⋊#10587739—21no—
9C22⋊#10587739—21no—
10C4×#10587739C2212yescomplemented
11#14555509—21no—
12#14555509—21no—
13#14555509C412yescomplemented
14#14555509C2212yes—
15C2×Dic274C411yescharacteristic, complemented
16C2×C22⋊D137—21no—
17C22×Dic137—21no—
18C2×C4×D137C2211yescharacteristic
19C2×D4×C137C411yescharacteristic, complemented
20C2×#10587739—22no—
21#11764047—22no—
22C22×D137D411yescharacteristic
23Dic274—21no—
24Dic274C2×C412yes—
25C22⋊D137—22no—
26C22⋊D137—22no—
27C2×Dic137—21no—
28C2×Dic137—22no—
29C2×Dic137—22no—
30C2×Dic137D411yescharacteristic
31C4×D137C2×C411yescharacteristic
32C4×D137D411yescharacteristic
33C4×D137D411yescharacteristic
34C4×D137C2311yescharacteristic
35C23×C137—21no—
36D4×C137—21no—
37D4×C137C2×C412yes—
38C2×C4×C137C2×C411yescharacteristic
39C22×C137—22no—
40C22×C137—42no—
41C22×C137C22⋊C411yescharacteristic
42C2×D137—41no—
43C2×D137C22⋊C411yescharacteristic
44C2×D137C2×D411yescharacteristic
45#10587739—42no—
46C4×C137C22⋊C411yescharacteristic
47C4×C137C22×C411yescharacteristic
48Dic137—22no—
49Dic137C22⋊C411yescharacteristic
50Dic137C2×D411yescharacteristic
51C2×C137—21no—
52C2×C137—42no—
53C2×C137C2×C22⋊C411yescharacteristic
54D137—21no—
55D137—21no—
56C137#42111yescharacteristic, complemented, Sylow
57#421—1371nomaximal, Sylow
58C2×#41—1371no—
59C2×M16—1371no—
60#117—1371no—
61#104—1374no—
62#41—2741no—
63#41—2741no—
64#41—1372no—
65#41—1372no—
66C2×Q8—1371no—
67C2×D4—2741no—
68C2×D4#1058773911yescharacteristic, complemented
69C22×C4—2741no—
70C22×C4—1371no—
71M16—2741no—
72M16—1372no—
73C2×C8—2741no—
74C4⋊C4—2741no—
75C22⋊C4—2741no—
76C42—1372no—
77C23—21no—
78C23—1371no—
79Q8—2741no—
80Q8—1372no—
81D4—21no—
82D4—2742no—
83D4—2742no—
84D4C2×#1058773912yes—
85C2×C4—2741no—
86C2×C4—2742no—
87C2×C4—2742no—
88C2×C4—2742no—
89C2×C4—1371no—
90C2×C4—1371no—
91C2×C4—1371no—
92C2×C4—1371no—
93C2×C4—1371no—
94C2×C4C2×#1058773911yescharacteristic
95C8—2742no—
96C22—22no—
97C22—42no—
98C22—5481no—
99C22—1371no—
100C22—1371no—
101C22C22⋊#1058773911yescharacteristic
102C4—2742no—
103C4—5482no—
104C4—1371no—
105C4—1371no—
106C4C22⋊#1058773911yescharacteristic
107C4C22×#1058773911yescharacteristic
108C2—21no—
109C2—42no—
110C2—2741no—
111C2—2741no—
112C2C2×C22⋊#1058773911yescharacteristic
subgroupG/H
centerC2C2×C22⋊#10587739
commutatorC4×C137C22×C4
FrattiniC4C22×#10587739
FittingC2×D4×C137C4
socleC2×C137C2×C22⋊C4
hypercenterC2×D4#10587739
hypocenterC137#421
LCSG ▷ C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. D137⋊M16C2semidirect
2. #17984167C2semidirect
3. #17984152C2semidirect
5. #14555520C22semidirect
10. C4×#10587739C22semidirect
13. #14555509C4semidirect
15. C2×Dic274C4semidirect
19. C2×D4×C137C4semidirect
56. C137#421semidirect
68. C2×D4#10587739semidirect