Finite Group Databasebeta

← #31796050

Subgroups — #31796050

Order 17536.

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

normal subgroups only

This group has 300 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, complemented
2#22578247C214yesmaximal, complemented
3C2×#17984050C211yesmaximal, characteristic
4#17984155—22no—
5C22⋊#11764047C2214yes—
6#17984143—22no—
7C22×D548C411yescharacteristic
8C22×C4×D137C412yes—
9#17984050—22no—
10#17984050—22no—
11C2×#14555487C2212yes—
12C2×C4×Dic137C2211yescharacteristic
13C2×C4⋊C4×C137C411yescharacteristic
14C2×#11764047—24no—
15C23×D137C2×C412yes—
16C2×D548—24no—
17C2×D548—22no—
18C22×Dic137C2×C411yescharacteristic
19C22×Dic137C2311yescharacteristic
20C2×C4×D137—22no—
21C2×C4×D137—22no—
22C2×C4×D137—44no—
23#14555487—24no—
24C4×Dic137D412yes—
25C4×Dic137D412yes—
26C22×C4×C137C2×C412yes—
27C22×C4×C137C2×C411yescharacteristic
28C4⋊C4×C137—22no—
29#11764047—44no—
30C22×D137—22no—
31C22×D137—22no—
32C22×D137—42no—
33C22×D137—42no—
34C22×D137M1614yes—
35D548—28no—
36C2×Dic137—22no—
37C2×Dic137—41no—
38C2×Dic137C22⋊C411yescharacteristic
39C2×Dic137C22⋊C411yescharacteristic
40C2×Dic137C2×D412yes—
41C4×D137—44no—
42C4×D137—44no—
43C23×C137C22×C411yescharacteristic
44C2×C4×C137—22no—
45C2×C4×C137—22no—
46C2×C4×C137—42no—
47C2×C4×C137C22⋊C412yes—
48C22×C137—21no—
49C22×C137—21no—
50C22×C137C2×C22⋊C411yescharacteristic
51C22×C137C2×M1612yes—
52C2×D137—24no—
53C2×D137—44no—
54C2×D137—42no—
55C2×D137—42no—
56C2×D137—44no—
57C4×C137—22no—
58C4×C137—42no—
59Dic137—22no—
60Dic137—21no—
61Dic137#10012yes—
62C2×C137—22no—
63C2×C137#40711yescharacteristic
64C2×C137#41011yescharacteristic
65C2×C137C2×#10011yescharacteristic
66D137—44no—
67C137#146611yescharacteristic, complemented, Sylow
68#1466—1371nomaximal, Sylow
69C2×C4×D4—1371no—
70#432—1372no—
71#323—1374no—
72C22×D4—1371no—
73C23×C4—1372no—
74C4×D4—2742no—
75C4×D4—2742no—
76C2×C4⋊C4#1058773911yescharacteristic
77C2×C22⋊C4—1372no—
78C2×C42—1371no—
79C4⋊C8—2742no—
80C22⋊C8—1374no—
81#97—2742no—
82C24—1372no—
83C2×D4—2744no—
84C2×D4—2742no—
85C22×C4—2742no—
86C22×C4—2742no—
87C22×C4—5484no—
88C22×C4—1371no—
89C22×C4—1371no—
90C22×C4C2×#1058773912yes—
91C22×C4C2×#1058773911yescharacteristic
92C2×C8—2744no—
93C4⋊C4—22no—
94C22⋊C4—2744no—
95C42—1372no—
96C42—1372no—
97C23—2742no—
98C23—2742no—
99C23—5482no—
100C23—5482no—
101C23—1374no—
102C23C22×#1058773911yescharacteristic
103D4—2748no—
104C2×C4—22no—
105C2×C4—22no—
106C2×C4—42no—
107C2×C4—2742no—
108C2×C4—5484no—
109C2×C4—5484no—
110C2×C4—5481no—
111C2×C4—1372no—
112C2×C4—1371no—
113C2×C4—1371no—
114C2×C4C22⋊#1058773912yes—
115C8—5484no—
116C22—21no—
117C22—21no—
118C22—2744no—
119C22—5484no—
120C22—5484no—
121C22—5482no—
122C22—5482no—
123C22C2×C22⋊#1058773911yescharacteristic
124C22D137⋊M1612yes—
125C4—22no—
126C4—42no—
127C4—2742no—
128C4—2741no—
129C4—1372no—
130C2—22no—
131C2—5484no—
132C2#2257837511yescharacteristic
133C2#2257838411yescharacteristic
134C2D137⋊#10011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC23×C137C22×C4
FrattiniC23C22×#10587739
FittingC2×C4⋊C4×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC22×D548C4
cosocleC22×Dic137C23
hypercenterC2×C4⋊C4#10587739
hypocenterC137#1466
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
1. #22578423C2semidirect
2. #22578247C2semidirect
67. C137#1466semidirect

Maximal subgroups

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

subgroupindex
#225784232
#225782472
C2×#179840502
#1466137