Finite Group Databasebeta

← #31796020

Subgroups — #31796020

Order 17536.

264 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 90 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#22578419C212yesmaximal, complemented
2#22578248C212yesmaximal, complemented
3#22578246C212yesmaximal
4C2×#17984047C211yesmaximal, characteristic
5C4⋊#11764047—22no—
6C22⋊#11764047C2214yes—
7C4×#11764047—22no—
8#17984047—22no—
9#17984047—22no—
10C2×C4⋊Dic137C411yescharacteristic
11C2×C4⋊Dic137C2211yescharacteristic
12C2×C22⋊Dic137C411yescharacteristic
13C2×C22⋊Dic137C2211yescharacteristic
14C2×C4×Dic137C411yescharacteristic
15C2×C4×Dic137C2211yescharacteristic
16C23×C4×C137C411yescharacteristic
17C2×#11764047—24no—
18C22×Dic137C2×C411yescharacteristic
19C22×Dic137C2×C411yescharacteristic
20C22×Dic137C2×C411yescharacteristic
21C22×Dic137C2311yescharacteristic
22C4⋊Dic137—22no—
23C4⋊Dic137—22no—
24C22⋊Dic137—22no—
25C22⋊Dic137—22no—
26C4×Dic137—21no—
27C4×Dic137—21no—
28C4×Dic137D412yes—
29C4×Dic137D412yes—
30C24×C137C2×C411yescharacteristic
31C22×C4×C137—21no—
32C22×C4×C137—21no—
33C22×C4×C137—22no—
34C22×C4×C137C814yescomplemented
35C22×C4×C137C2×C411yescharacteristic
36C22×C4×C137C2×C411yescharacteristic
37#11764047—44no—
38C2×Dic137—21no—
39C2×Dic137—21no—
40C2×Dic137—41no—
41C2×Dic137—41no—
42C2×Dic137—41no—
43C2×Dic137—41no—
44C2×Dic137C22⋊C412yes—
45C2×Dic137C2×D412yes—
46C23×C137—21no—
47C23×C137—21no—
48C23×C137—41no—
49C23×C137—41no—
50C23×C137C2×C812yes—
51C23×C137C22×C411yescharacteristic
52C2×C4×C137—21no—
53C2×C4×C137—21no—
54C2×C4×C137—24no—
55C2×C4×C137—24no—
56C2×C4×C137C22⋊C411yescharacteristic
57C2×C4×C137C22⋊C411yescharacteristic
58C2×C4×C137C2×C812yes—
59C2×C4×C137C2×C812yes—
60C2×C4×C137M1612yes—
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×C137C22×C811yescharacteristic
70C22×C137C2×M1611yescharacteristic
71C4×C137—22no—
72C4×C137C22⋊C812yes—
73C4×C137C22⋊C812yes—
74Dic137—22no—
75Dic137—22no—
76Dic137—41no—
77Dic137—41no—
78C2×C137—21no—
79C2×C137—21no—
80C2×C137—42no—
81C2×C137C2×C22⋊C811yescharacteristic
82C2×C137#41011yescharacteristic
83C2×C137#41311yescharacteristic
84C137#145811yescharacteristic, complemented, Sylow
85#1458—1371nomaximal, Sylow
86C2×#117—1371no—
87#431—1372no—
88#324—1372no—
89#323—1372no—
90C23×C4#1058773911yescharacteristic
91#117—2742no—
92#117—2742no—
93C2×C4⋊C4—1371no—
94C2×C4⋊C4—1371no—
95C2×C22⋊C4—1371no—
96C2×C22⋊C4—1371no—
97C2×C42—1371no—
98C2×C42—1371no—
99C4⋊C8—2742no—
100C22⋊C8—1374no—
101C4×C8—2742no—
102C24C2×#1058773911yescharacteristic
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—22no—
106C22×C4—1371no—
107C22×C4—1371no—
108C22×C4—1371no—
109C22×C4—1371no—
110C22×C4#1176404714yescomplemented
111C22×C4C2×#1058773911yescharacteristic
112C22×C4C2×#1058773911yescharacteristic
113C2×C8—2744no—
114C4⋊C4—2742no—
115C4⋊C4—2742no—
116C22⋊C4—2742no—
117C22⋊C4—2742no—
118C42—2741no—
119C42—2741no—
120C42—1372no—
121C42—1372no—
122C23—21no—
123C23—21no—
124C23—41no—
125C23—41no—
126C23C2×#1176404712yes—
127C23C22×#1058773911yescharacteristic
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—24no—
131C2×C4—24no—
132C2×C4—2741no—
133C2×C4—2741no—
134C2×C4—5481no—
135C2×C4—5481no—
136C2×C4—5481no—
137C2×C4—5481no—
138C2×C4—1372no—
139C2×C4—1372no—
140C2×C4C22⋊#1058773911yescharacteristic
141C2×C4C22⋊#1058773911yescharacteristic
142C2×C4C2×#1176404712yes—
143C2×C4C2×#1176404712yes—
144C2×C4#1455551912yes—
145C8—5484no—
146C22—21no—
147C22—21no—
148C22—22no—
149C22—42no—
150C22—42no—
151C22—41no—
152C22—41no—
153C22C2×C22⋊#1058773911yescharacteristic
154C22C22×#1176404711yescharacteristic
155C22C2×#1455551911yescharacteristic
156C4—22no—
157C4—2742no—
158C4—2742no—
159C4—5481no—
160C4—5481no—
161C4C22⋊#1176404712yes—
162C4C22⋊#1176404712yes—
163C2—21no—
164C2—21no—
165C2—42no—
166C2C2×C22⋊#1176404711yescharacteristic
167C2#2257838211yescharacteristic
168C2#2257838911yescharacteristic
subgroupG/H
centerC2×C4C22⋊#10587739
commutatorC22×C137C22×C8
FrattiniC23C22×#10587739
FittingC23×C4×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC24×C137C2×C4
cosocleC22×Dic137C23
hypercenterC23×C4#10587739
hypocenterC137#1458
LCSG ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. #22578419C2semidirect
2. #22578248C2semidirect
34. C22×C4×C137C8semidirect
84. C137#1458semidirect
110. C22×C4#11764047semidirect

Maximal subgroups

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

subgroupindex
#225784192
#225782482
#225782462
C2×#179840472
#1458137