Finite Group Databasebeta

← #31796035

Subgroups — #31796035

Order 17536.

440 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 68 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#22578374C212yesmaximal, complemented
2#22578247C212yesmaximal, complemented
3#22578246C212yesmaximal, complemented
4C2×C22⋊C4×D137C211yesmaximal, characteristic
5C2×#14555520—22no—
6#17984145—22no—
7C22⋊#11764047C2214yescomplemented
8C24×D137C411yescharacteristic
9C22×C4×D137C411yescharacteristic
10C22×C4×D137C2211yescharacteristic
11C2×C22⋊Dic137C2211yescharacteristic
12C2×#14555487C411yescharacteristic
13C2×#14555487C2211yescharacteristic
14C22⋊C4×D137—22no—
15C22⋊C4×D137—22no—
16C2×C22⋊C4×C137C411yescharacteristic
17#14555520—44no—
18C2×#11764047—24no—
19C23×D137—21no—
20C23×D137—21no—
21C23×D137—22no—
22C23×D137—22no—
23C23×D137—42no—
24C23×D137—42no—
25C23×D137C2×C411yescharacteristic
26C23×D137C2×C411yescharacteristic
27C22×Dic137C2×C411yescharacteristic
28C22×Dic137C2311yescharacteristic
29C2×C4×D137—21no—
30C2×C4×D137—21no—
31C2×C4×D137—42no—
32C2×C4×D137—42no—
33C2×C4×D137D412yes—
34C2×C4×D137D412yes—
35C22⋊Dic137—22no—
36#14555487—22no—
37#14555487—22no—
38C24×C137C2×C411yescharacteristic
39C22×C4×C137C2×C411yescharacteristic
40C22×C4×C137C2×C411yescharacteristic
41C22⋊C4×C137—22no—
42#11764047—44no—
43C22×D137—21no—
44C22×D137—21no—
45C22×D137—22no—
46C22×D137—22no—
47C22×D137—24no—
48C22×D137—24no—
49C22×D137—21no—
50C22×D137—21no—
51C22×D137—48no—
52C22×D137—48no—
53C22×D137—42no—
54C22×D137—42no—
55C22×D137—42no—
56C22×D137—42no—
57C22×D137—41no—
58C22×D137—41no—
59C22×D137C22⋊C411yescharacteristic
60C22×D137C22⋊C411yescharacteristic
61C22×D137M1612yescomplemented
62C2×Dic137—21no—
63C2×Dic137—41no—
64C2×Dic137—41no—
65C2×Dic137C2×D412yes—
66C4×D137—24no—
67C4×D137—42no—
68C4×D137—42no—
69C4×D137—42no—
70C23×C137—21no—
71C23×C137—21no—
72C23×C137—41no—
73C23×C137—41no—
74C23×C137M1612yescomplemented
75C23×C137C22×C411yescharacteristic
76C2×C4×C137—21no—
77C2×C4×C137—41no—
78C2×C4×C137—41no—
79C2×C4×C137C22⋊C412yes—
80C22×C137—21no—
81C22×C137—21no—
82C22×C137—22no—
83C22×C137—42no—
84C22×C137—42no—
85C22×C137—41no—
86C22×C137—41no—
87C22×C137C2×C22⋊C411yescharacteristic
88C22×C137C2×M1611yescharacteristic
89C22×C137C2×M1611yescharacteristic
90C2×D137—24no—
91C2×D137—24no—
92C2×D137—21no—
93C2×D137—21no—
94C2×D137—21no—
95C2×D137—21no—
96C2×D137—22no—
97C2×D137—48no—
98C2×D137—48no—
99C2×D137—42no—
100C2×D137—41no—
101C2×D137—41no—
102C2×D137—42no—
103C2×D137#9912yes—
104C2×D137#10012yes—
105C4×C137—22no—
106C4×C137—41no—
107Dic137—22no—
108Dic137—41no—
109C2×C137—21no—
110C2×C137—21no—
111C2×C137—42no—
112C2×C137#40711yescharacteristic
113C2×C137C2×#9911yescharacteristic
114C2×C137C2×#10011yescharacteristic
115D137—22no—
116D137—22no—
117D137—42no—
118C137#146211yescharacteristic, complemented, Sylow
119#1462—1371nomaximal, Sylow
120C22×C22⋊C4—1371no—
121#407—1372no—
122#323—1372no—
123#323—1372no—
124C25—1371no—
125C23×C4—1371no—
126C23×C4—1371no—
127C2×M16—2742no—
128C2×C22⋊C4—2742no—
129C2×C22⋊C4—2742no—
130C2×C22⋊C4—1371no—
131C2×C22⋊C4—1371no—
132C2×C22⋊C4—1371no—
133C2×C22⋊C4#1058773911yescharacteristic
134C22⋊C8—2742no—
135C22⋊C8—1374no—
136C24—2741no—
137C24—2741no—
138C24—2742no—
139C24—2742no—
140C24—5482no—
141C24—5482no—
142C24—1371no—
143C24—1371no—
144C24C2×#1058773911yescharacteristic
145C22×C4—2741no—
146C22×C4—2741no—
147C22×C4—5482no—
148C22×C4—5482no—
149C22×C4—1372no—
150C22×C4—1371no—
151C22×C4—1371no—
152C22×C4—1372no—
153C22×C4C2×#1058773911yescharacteristic
154C22×C4C2×#1058773911yescharacteristic
155M16—5484no—
156C2×C8—2744no—
157C22⋊C4—22no—
158C22⋊C4—2742no—
159C22⋊C4—2742no—
160C22⋊C4—2742no—
161C23—21no—
162C23—21no—
163C23—41no—
164C23—41no—
165C23—2742no—
166C23—2742no—
167C23—2744no—
168C23—2744no—
169C23—2741no—
170C23—2741no—
171C23—2741no—
172C23—2741no—
173C23—5481no—
174C23—5481no—
175C23—5488no—
176C23—5488no—
177C23—5482no—
178C23—5482no—
179C23—5482no—
180C23—5482no—
181C23—1371no—
182C23—1371no—
183C23—1372no—
184C23#1455552012yescomplemented
185C23C22×#1058773911yescharacteristic
186C2×C4—21no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—2744no—
190C2×C4—2741no—
191C2×C4—5482no—
192C2×C4—5482no—
193C2×C4—5482no—
194C2×C4—5481no—
195C2×C4—5481no—
196C2×C4—1372no—
197C2×C4C22⋊#1058773912yes—
198C8—5484no—
199C22—21no—
200C22—21no—
201C22—22no—
202C22—42no—
203C22—42no—
204C22—41no—
205C22—41no—
206C22—2744no—
207C22—2744no—
208C22—2741no—
209C22—2741no—
210C22—2741no—
211C22—2741no—
212C22—2742no—
213C22—5482no—
214C22—5482no—
215C22—5488no—
216C22—5488no—
217C22—5481no—
218C22—5481no—
219C22—1372no—
220C22—1372no—
221C22C2×C22⋊#1058773911yescharacteristic
222C22D137⋊M1611yescharacteristic
223C22C2×#1455552011yescharacteristic
224C4—22no—
225C4—41no—
226C4—2742no—
227C4—5481no—
228C2—21no—
229C2—21no—
230C2—42no—
231C2—2742no—
232C2—2742no—
233C2—5482no—
234C2#2257837411yescharacteristic
235C2D137⋊#9911yescharacteristic
236C2D137⋊#10011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC23×C137C22×C4
FrattiniC23C22×#10587739
FittingC2×C22⋊C4×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC24×D137C4
cosocleC22×Dic137C23
hypercenterC2×C22⋊C4#10587739
hypocenterC137#1462
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. #22578374C2semidirect
2. #22578247C2semidirect
3. #22578246C2semidirect
7. C22⋊#11764047C22semidirect
61. C22×D137M16semidirect
74. C23×C137M16semidirect
118. C137#1462semidirect
184. C23#14555520semidirect

Maximal subgroups

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

subgroupindex
#225783742
#225782472
#225782462
C2×C22⋊C4×D1372
#1462137