Finite Group Databasebeta

← #31796053

Subgroups — #31796053

Order 17536.

236 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 64 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#22578423C212yesmaximal
2#22578249C214yesmaximal
3C2×#17984053C211yesmaximal, characteristic
4#17984155—22no—
5C22⋊#11764047C2214yes—
6#17984143—22no—
7C22×Dic274C411yescharacteristic
8#17984053—22no—
9#17984053—22no—
10C2×#14555489C2212yes—
11C2×C4×Dic137C412yes—
12C2×C4×Dic137C2211yescharacteristic
13C2×C4⋊C4×C137C411yescharacteristic
14C2×#11764047—24no—
15C2×Dic274—24no—
16C2×Dic274—22no—
17C22×Dic137C2×C411yescharacteristic
18C22×Dic137C2×C412yes—
19C22×Dic137C2311yescharacteristic
20#14555489—24no—
21C4×Dic137—22no—
22C4×Dic137—22no—
23C4×Dic137D412yes—
24C4×Dic137D412yes—
25C22×C4×C137C2×C412yes—
26C22×C4×C137C2×C411yescharacteristic
27C4⋊C4×C137—22no—
28#11764047—44no—
29Dic274—28no—
30C2×Dic137—22no—
31C2×Dic137—22no—
32C2×Dic137—22no—
33C2×Dic137—41no—
34C2×Dic137C22⋊C411yescharacteristic
35C2×Dic137C22⋊C411yescharacteristic
36C2×Dic137M1614yes—
37C2×Dic137C2×D412yes—
38C23×C137C22×C411yescharacteristic
39C2×C4×C137—22no—
40C2×C4×C137—22no—
41C2×C4×C137—42no—
42C2×C4×C137C22⋊C412yes—
43C22×C137—21no—
44C22×C137—21no—
45C22×C137C2×C22⋊C411yescharacteristic
46C22×C137C2×M1612yes—
47C4×C137—22no—
48C4×C137—42no—
49Dic137—24no—
50Dic137—22no—
51Dic137—21no—
52Dic137#10112yes—
53C2×C137—22no—
54C2×C137#40711yescharacteristic
55C2×C137#41011yescharacteristic
56C2×C137C2×#10111yescharacteristic
57C137#146711yescharacteristic, complemented, Sylow
58#1467—1371nomaximal, Sylow
59C2×C4×Q8—1371no—
60#432—1372no—
61#324—1374no—
62C22×Q8—1371no—
63C4×Q8—2742no—
64C4×Q8—2742no—
65C2×C4⋊C4—1372no—
66C2×C4⋊C4#1058773911yescharacteristic
67C2×C42—1372no—
68C2×C42—1371no—
69C4⋊C8—2742no—
70C22⋊C8—1374no—
71#97—2742no—
72C2×Q8—2744no—
73C2×Q8—2742no—
74C22×C4—1371no—
75C22×C4—1372no—
76C22×C4—1371no—
77C22×C4C2×#1058773912yes—
78C22×C4C2×#1058773911yescharacteristic
79C2×C8—2744no—
80C4⋊C4—22no—
81C4⋊C4—2744no—
82C42—2742no—
83C42—2742no—
84C42—1372no—
85C42—1372no—
86C23C22×#1058773911yescharacteristic
87Q8—2748no—
88C2×C4—22no—
89C2×C4—22no—
90C2×C4—42no—
91C2×C4—2742no—
92C2×C4—2742no—
93C2×C4—2742no—
94C2×C4—5481no—
95C2×C4—1371no—
96C2×C4—1372no—
97C2×C4—1374no—
98C2×C4—1371no—
99C2×C4C22⋊#1058773912yes—
100C8—5484no—
101C22—21no—
102C22—21no—
103C22C2×C22⋊#1058773911yescharacteristic
104C22D137⋊M1612yes—
105C4—22no—
106C4—42no—
107C4—2744no—
108C4—2742no—
109C4—2741no—
110C4—1372no—
111C2—22no—
112C2#2257837511yescharacteristic
113C2#2257838311yescharacteristic
114C2D137⋊#10111yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutator, prime radicalC23×C137C22×C4
FrattiniC23C22×#10587739
FittingC2×C4⋊C4×C137C4
socleC22×C137C2×C22⋊C4
cosocleC22×Dic137C23
hypercenterC2×C4⋊C4#10587739
hypocenterC137#1467
LCSG ▷ C23×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
57. C137#1467semidirect

Maximal subgroups

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

subgroupindex
#225784232
#225782492
C2×#179840532
#1467137