Finite Group Databasebeta

← #31796041

Subgroups — #31796041

Order 17536.

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

normal subgroups only

This group has 252 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#22578424C212yesmaximal, complemented
2#22578249C212yesmaximal, complemented
3#22578246C212yesmaximal
4C2×#17984044C211yesmaximal, characteristic
5C4⋊#11764047—22no—
6C22⋊#11764047C2214yes—
7#17984143—22no—
8C23×Dic137C411yescharacteristic
9#17984044—22no—
10#17984044—22no—
11C2×#14555489C411yescharacteristic
12C2×#14555489C2211yescharacteristic
13C2×C22⋊Dic137C2211yescharacteristic
14C2×C4×Dic137C411yescharacteristic
15C2×C4×Dic137C2211yescharacteristic
16C2×C22⋊C4×C137C411yescharacteristic
17C2×#11764047—24no—
18C22×Dic137—21no—
19C22×Dic137—21no—
20C22×Dic137—22no—
21C22×Dic137—22no—
22C22×Dic137C2×C411yescharacteristic
23C22×Dic137C2×C411yescharacteristic
24C22×Dic137C2×C411yescharacteristic
25C22×Dic137C2311yescharacteristic
26#14555489—22no—
27#14555489—22no—
28C22⋊Dic137—22no—
29C4×Dic137—21no—
30C4×Dic137—21no—
31C4×Dic137D412yes—
32C4×Dic137D412yes—
33C24×C137C2×C411yescharacteristic
34C22×C4×C137C2×C411yescharacteristic
35C22×C4×C137C2×C411yescharacteristic
36C22⋊C4×C137—22no—
37#11764047—44no—
38C2×Dic137—24no—
39C2×Dic137—24no—
40C2×Dic137—21no—
41C2×Dic137—21no—
42C2×Dic137—21no—
43C2×Dic137—21no—
44C2×Dic137—21no—
45C2×Dic137—41no—
46C2×Dic137—41no—
47C2×Dic137C22⋊C411yescharacteristic
48C2×Dic137C22⋊C411yescharacteristic
49C2×Dic137M1612yes—
50C2×Dic137C2×D412yes—
51C23×C137—21no—
52C23×C137—21no—
53C23×C137—41no—
54C23×C137—41no—
55C23×C137M1612yes—
56C23×C137C22×C411yescharacteristic
57C2×C4×C137—21no—
58C2×C4×C137—41no—
59C2×C4×C137—41no—
60C2×C4×C137C22⋊C412yes—
61C22×C137—21no—
62C22×C137—21no—
63C22×C137—22no—
64C22×C137—42no—
65C22×C137—42no—
66C22×C137—41no—
67C22×C137—41no—
68C22×C137C2×C22⋊C411yescharacteristic
69C22×C137C2×M1611yescharacteristic
70C22×C137C2×M1611yescharacteristic
71C4×C137—22no—
72C4×C137—41no—
73Dic137—21no—
74Dic137—21no—
75Dic137—22no—
76Dic137—22no—
77Dic137—41no—
78C2×C137—21no—
79C2×C137—21no—
80C2×C137—42no—
81C2×C137#40711yescharacteristic
82C2×C137#41011yescharacteristic
83C2×C137#41311yescharacteristic
84C137#146411yescharacteristic, complemented, Sylow
85#1464—1371nomaximal, Sylow
86C2×#117—1371no—
87#432—1372no—
88#324—1372no—
89#323—1372no—
90C23×C4—1371no—
91#117—2742no—
92#117—2742no—
93C2×C4⋊C4—1371no—
94C2×C4⋊C4—1371no—
95C2×C22⋊C4—1371no—
96C2×C22⋊C4#1058773911yescharacteristic
97C2×C42—1371no—
98C2×C42—1371no—
99C4⋊C8—2742no—
100C22⋊C8—1374no—
101#97—2742no—
102C24C2×#1058773911yescharacteristic
103C22×C4—2742no—
104C22×C4—2741no—
105C22×C4—2741no—
106C22×C4—2742no—
107C22×C4—1371no—
108C22×C4—1371no—
109C22×C4—1371no—
110C22×C4—1371no—
111C22×C4C2×#1058773911yescharacteristic
112C22×C4C2×#1058773911yescharacteristic
113C2×C8—2744no—
114C4⋊C4—2742no—
115C4⋊C4—2742no—
116C22⋊C4—22no—
117C22⋊C4—2742no—
118C42—2741no—
119C42—2741no—
120C42—1372no—
121C42—1372no—
122C23—21no—
123C23—21no—
124C23—41no—
125C23—41no—
126C23#1455552012yes—
127C23C22×#1058773911yescharacteristic
128C2×C4—21no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—2744no—
132C2×C4—2744no—
133C2×C4—2741no—
134C2×C4—2741no—
135C2×C4—2741no—
136C2×C4—2741no—
137C2×C4—2741no—
138C2×C4—5481no—
139C2×C4—5481no—
140C2×C4—1371no—
141C2×C4—1371no—
142C2×C4—1372no—
143C2×C4—1372no—
144C2×C4C22⋊#1058773912yes—
145C8—5484no—
146C22—21no—
147C22—21no—
148C22—22no—
149C22—42no—
150C22—42no—
151C22—41no—
152C22—41no—
153C22C2×C22⋊#1058773911yescharacteristic
154C22D137⋊M1611yescharacteristic
155C22C2×#1455552011yescharacteristic
156C4—22no—
157C4—41no—
158C4—2742no—
159C4—2741no—
160C4—2741no—
161C4—2742no—
162C4—5481no—
163C2—21no—
164C2—21no—
165C2—42no—
166C2#2257837411yescharacteristic
167C2#2257838311yescharacteristic
168C2#2257839011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC23×C137C22×C4
FrattiniC23C22×#10587739
FittingC2×C22⋊C4×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC24×C137C2×C4
cosocleC22×Dic137C23
hypercenterC2×C22⋊C4#10587739
hypocenterC137#1464
LCSG ▷ C23×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #22578424C2semidirect
2. #22578249C2semidirect
84. C137#1464semidirect

Maximal subgroups

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

subgroupindex
#225784242
#225782492
#225782462
C2×#179840442
#1464137