Finite Group Databasebeta

← #31797819

Subgroups — #31797819

Order 17536.

128 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 128 conjugacy classes of subgroups — too many to draw. Only its 25 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#22578501C212yesmaximal, complemented
2#22578375C211yesmaximal, characteristic
3#17984184—22no—
4C2×#14555519C411yescharacteristic
5C2×#14555519C2211yescharacteristic
6#17984145—22no—
7C22×C4×D137C411yescharacteristic
8#14555522—42no—
9#14555519—41no—
10#14555519—41no—
11C2×#11764047—21no—
12C2×#11764047D412yes—
13C23×D137C811yescharacteristic
14C22×Dic137C2×C411yescharacteristic
15C2×C4×D137—21no—
16C2×C4×D137—21no—
17C2×C4×D137—41no—
18C2×C4×D137—41no—
19C22×C4×C137C811yescharacteristic
20#11764047—22no—
21#11764047—41no—
22C22×D137—21no—
23C22×D137—42no—
24C22×D137—41no—
25C2×Dic137—21no—
26C2×Dic137—21no—
27C2×Dic137C22⋊C411yescharacteristic
28C2×Dic137M1611yescharacteristic
29C4×D137—42no—
30C4×D137—41no—
31C4×D137—41no—
32C23×C137C2×C811yescharacteristic
33C2×C4×C137—21no—
34C2×C4×C137—41no—
35C22×C137—21no—
36C22×C137—41no—
37C22×C137C22⋊C811yescharacteristic
38C2×D137—22no—
39C2×D137—41no—
40C2×D137—42no—
41C2×D137—41no—
42C2×D137—42no—
43C4×C137—41no—
44Dic137—21no—
45Dic137#9911yescharacteristic
46Dic137#10011yescharacteristic
47C2×C137—21no—
48C2×C137—41no—
49C2×C137#32311yescharacteristic
50D137—42no—
51C137#132011yescharacteristic, complemented, Sylow
52#1320—1371nomaximal, Sylow
53#407—1371no—
54#349—1372no—
55C23×C4—1371no—
56C2×M16—1371no—
57C2×M16—1371no—
58M32—2742no—
59C22⋊C8—2742no—
60C24—1371no—
61C22×C4—2741no—
62C22×C4—2741no—
63C22×C4—5481no—
64C22×C4—5481no—
65C22×C4—1371no—
66C22×C4#1176404911yescharacteristic
67M16—5481no—
68M16—5481no—
69C2×C8—2741no—
70C2×C8—1372no—
71C16—5482no—
72C23—2741no—
73C23—5481no—
74C23—5482no—
75C23C2×#1176404911yescharacteristic
76C2×C4—21no—
77C2×C4—41no—
78C2×C4—2741no—
79C2×C4—2741no—
80C2×C4—5482no—
81C2×C4—5481no—
82C2×C4—5481no—
83C2×C4—1371no—
84C2×C4—1371no—
85C8—2742no—
86C8—5481no—
87C22—21no—
88C22—41no—
89C22—2742no—
90C22—5481no—
91C22—5482no—
92C22—5482no—
93C22—5481no—
94C22C22⋊#1176404911yescharacteristic
95C4—41no—
96C4—2741no—
97C4—1371no—
98C4—1371no—
99C2—21no—
100C2—41no—
101C2—5482no—
102C2#2257848811yescharacteristic
subgroupG/H
centerC2#22578488
commutatorC23×C137C2×C8
FrattiniC23C2×#11764049
FittingC22×C4×C137C8
socleC2×C137#323
prime radicalC23×D137C8
cosocleC2×#14555519C22
hypercenterC22×C4#11764049
hypocenterC137#1320
LCSG ▷ C23×C137 ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C22 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. #22578501C2semidirect
51. C137#1320semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#225785012
#225783752
#1320137