Finite Group Databasebeta

← C2×#22578350 #31797239

Subgroups — C2×#22578350 #31797239

Order 17536.

1232 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 1232 conjugacy classes of subgroups — too many to draw. Only its 274 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
1C24×#10587739C211yesmaximal, characteristic
2C22×C4⋊#10587739C212yesmaximal, complemented
3C22×C22⋊#10587739C211yesmaximal, characteristic
4#22578350C218yesmaximal, direct product factor
5C2×(C2×#10587739)⋊C4C212yesmaximal, complemented
6C2×C22⋊C4×D137C211yesmaximal, characteristic
7C23×#10587739—28no—
8C23×#10587739—24no—
9C23×#10587739C2211yescharacteristic
10C23×#10587739C2211yescharacteristic
11C23×#10587739C2214yes—
12C23×#10587739C2212yes—
13C2×C4⋊#10587739—28no—
14C2×C4⋊#10587739C2218yescomplemented
15C2×C22⋊#10587739—24no—
16C2×C22⋊#10587739C2214yes—
17(C2×#10587739)⋊C4C2218yescomplemented
18C24×D137C2211yescharacteristic
19C22×C4×D137C2212yes—
20C2×C22⋊Dic137C411yescharacteristic, complemented
21C2×#14555487C412yescomplemented
22C22⋊C4×D137C414yescomplemented
23C22⋊C4×D137C2214yes—
24C2×C22⋊C4×C137C411yescharacteristic, complemented
25C22×#10587739—232no—
26C22×#10587739—216no—
27C22×#10587739—28no—
28C22×#10587739—24no—
29C22×#10587739—24no—
30C22×#10587739—22no—
31C22×#10587739—22no—
32C22×#10587739—24no—
33C22×#10587739—24no—
34C22×#10587739D418yes—
35C22×#10587739C2312yes—
36C22×#10587739C2312yes—
37C22×#10587739C2314yes—
38C4⋊#10587739—216no—
39C22⋊#10587739—28no—
40C23×D137—24no—
41C23×D137—24no—
42C23×D137C2×C412yes—
43C23×D137C2×C411yescharacteristic
44C23×D137D414yes—
45C23×D137Q814yes—
46C23×D137C2312yes—
47C23×D137C2311yescharacteristic
48C22×Dic137C2×C412yescomplemented
49C2×C4×D137—24no—
50C2×C4×D137—24no—
51C2×C4×D137C2×C414yescomplemented
52C2×C4×D137C2314yes—
53C22⋊Dic137C2×C414yescomplemented
54#14555487C2×C418yescomplemented
55C24×C137C2×C411yescharacteristic
56C22×C4×C137C2×C412yescomplemented
57C22⋊C4×C137C2×C414yescomplemented
58C2×#10587739—232no—
59C2×#10587739—216no—
60C2×#10587739—24no—
61C2×#10587739—24no—
62C2×#10587739—22no—
63C2×#10587739—22no—
64C2×#10587739—24no—
65C2×#10587739—24no—
66C2×#10587739—44no—
67C2×#10587739—44no—
68C2×#10587739C2×D418yes—
69C22×D137—24no—
70C22×D137—28no—
71C22×D137—28no—
72C22×D137—216no—
73C22×D137—28no—
74C22×D137—24no—
75C22×D137—24no—
76C22×D137C4⋊C418yescomplemented
77C22×D137C4⋊C414yescomplemented
78C22×D137C22×C412yes—
79C22×D137C22×C411yescharacteristic
80C22×D137C2×D414yes—
81C22×D137C2×D412yes—
82C22×D137C2×D414yes—
83C22×D137C2×Q814yes—
84C22×D137C2×Q812yes—
85C22×D137#4114yes—
86C22×D137C2411yescharacteristic
87C2×Dic137—24no—
88C2×Dic137C22×C414yescomplemented
89C4×D137—28no—
90C4×D137—24no—
91C4×D137—24no—
92C23×C137—24no—
93C23×C137C4⋊C414yescomplemented
94C23×C137C22×C412yes—
95C23×C137C22×C411yescharacteristic
96C2×C4×C137—24no—
97C2×C4×C137C22×C414yescomplemented
98C22×C137—24no—
99C22×C137—28no—
100C22×C137C2×C4⋊C412yes—
101C22×C137C2×C4⋊C414yescomplemented
102C22×C137C4×D414yes—
103C22×C137C23×C411yescharacteristic
104C2×D137—24no—
105C2×D137—28no—
106C2×D137—24no—
107C2×D137—28no—
108C2×D137—24no—
109C2×D137—216no—
110C2×D137C2×C4⋊C412yes—
111C2×D137C2×C4⋊C414yescomplemented
112C2×D137C4×D414yes—
113C2×D137#12014yes—
114C2×D137C22⋊Q818yes—
115C2×D137#12314yes—
116C2×D137C22×D412yes—
117C2×D137C22×D411yescharacteristic
118C2×D137C22×Q811yescharacteristic
119C2×D137C2×#4112yes—
120#10587739—28no—
121#10587739—42no—
122#10587739—42no—
123C4×C137—24no—
124Dic137—24no—
125C2×C137—24no—
126C2×C137#38514yescomplemented
127C2×C137C22×C4⋊C411yescharacteristic
128C2×C137C2×C4×D412yes—
129D137—24no—
130D137#38514yescomplemented
131D137C2×#12011yescharacteristic
132D137C2×C22⋊Q812yes—
133D137C2×#12311yescharacteristic
134C137C2×#38511yescharacteristic, complemented, Sylow
135C2×#385—1371nomaximal, Sylow
136C24×C4—1371no—
137C22×C4⋊C4—1372no—
138C22×C22⋊C4—1371no—
139C22×C22⋊C4—1371no—
140#385—1378no—
141C2×#95—1372no—
142C25—1371no—
143C23×C4—2744no—
144C23×C4—2748no—
145C23×C4—1371no—
146C23×C4—1371no—
147C23×C4—1374no—
148C23×C4—1372no—
149C23×C4—1372no—
150C2×C4⋊C4—2748no—
151C2×C4⋊C4—1378no—
152C2×C22⋊C4—2744no—
153C2×C22⋊C4—1374no—
154C2×C22⋊C4—1371no—
155C2×C22⋊C4—1372no—
156C2×C22⋊C4—1374no—
157C2×C22⋊C4—1374no—
158C2×C22⋊C4#1058773911yescharacteristic, complemented
159#95—1378no—
160C24—2744no—
161C24—2744no—
162C24—1374no—
163C24—1374no—
164C24—1372no—
165C24—1372no—
166C24—1371no—
167C24—1371no—
168C24C2×#1058773911yescharacteristic
169C22×C4—2744no—
170C22×C4—2744no—
171C22×C4—27432no—
172C22×C4—2748no—
173C22×C4—2744no—
174C22×C4—2744no—
175C22×C4—27416no—
176C22×C4—2742no—
177C22×C4—2742no—
178C22×C4—2744no—
179C22×C4—2744no—
180C22×C4—1372no—
181C22×C4—1372no—
182C22×C4—1374no—
183C22×C4—1378no—
184C22×C4—1372no—
185C22×C4—1374no—
186C22×C4—1374no—
187C22×C4C2×#1058773912yescomplemented
188C4⋊C4—27416no—
189C22⋊C4—2748no—
190C22⋊C4—1374no—
191C22⋊C4—1378no—
192C22⋊C4C2×#1058773914yescomplemented
193C23—24no—
194C23—2744no—
195C23—2748no—
196C23—2744no—
197C23—2748no—
198C23—2748no—
199C23—27416no—
200C23—2744no—
201C23—1374no—
202C23—1374no—
203C23—1378no—
204C23—1374no—
205C23—1372no—
206C23—1372no—
207C23—1372no—
208C23—1374no—
209C23—1374no—
210C23—1371no—
211C23—1371no—
212C23C4⋊#1058773914yescomplemented
213C23C22×#1058773912yes—
214C23C22×#1058773911yescharacteristic
215C2×C4—24no—
216C2×C4—2744no—
217C2×C4—2744no—
218C2×C4—2748no—
219C2×C4—27432no—
220C2×C4—2744no—
221C2×C4—2744no—
222C2×C4—27416no—
223C2×C4—2744no—
224C2×C4—2742no—
225C2×C4—2742no—
226C2×C4—2744no—
227C2×C4—2744no—
228C2×C4—5484no—
229C2×C4—5484no—
230C2×C4—1378no—
231C2×C4—1374no—
232C2×C4C22×#1058773914yescomplemented
233C22—24no—
234C22—28no—
235C22—2744no—
236C22—2744no—
237C22—2748no—
238C22—27416no—
239C22—2744no—
240C22—2748no—
241C22—1374no—
242C22—1371no—
243C22—1372no—
244C22—1371no—
245C22—1372no—
246C22—1372no—
247C22—1374no—
248C22—1374no—
249C22—1374no—
250C22—1378no—
251C22C2×C4⋊#1058773912yes—
252C22C2×C4⋊#1058773914yescomplemented
253C22D4×#1058773914yes—
254C22C23×#1058773911yescharacteristic
255C4—24no—
256C4—2748no—
257C4—2744no—
258C4—5482no—
259C4—5482no—
260C2—24no—
261C2—2744no—
262C2—1371no—
263C2—1371no—
264C2—1372no—
265C2—1374no—
266C2#2257835014yesdirect product factor
267C2C22×C4⋊#1058773911yescharacteristic
268C2C2×D4×#1058773912yes—
subgroupG/H
centerC23C22×#10587739
commutatorC22×C137C23×C4
FrattiniC22C23×#10587739
FittingC2×C22⋊C4×C137C4
socleC23×C137C22×C4
prime radicalC24×D137C22
cosocleC22×D137C24
hypercenterC2×C22⋊C4#10587739
hypocenterC137C2×#385
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊#10587739C2semidirect
4. #22578350C2direct (N × H)
5. C2×(C2×#10587739)⋊C4C2semidirect
14. C2×C4⋊#10587739C22semidirect
17. (C2×#10587739)⋊C4C22semidirect
20. C2×C22⋊Dic137C4semidirect
21. C2×#14555487C4semidirect
22. C22⋊C4×D137C4semidirect
24. C2×C22⋊C4×C137C4semidirect
48. C22×Dic137C2×C4semidirect
51. C2×C4×D137C2×C4semidirect
53. C22⋊Dic137C2×C4semidirect
54. #14555487C2×C4semidirect
56. C22×C4×C137C2×C4semidirect
57. C22⋊C4×C137C2×C4semidirect
76. C22×D137C4⋊C4semidirect
77. C22×D137C4⋊C4semidirect
88. C2×Dic137C22×C4semidirect
93. C23×C137C4⋊C4semidirect
97. C2×C4×C137C22×C4semidirect
101. C22×C137C2×C4⋊C4semidirect
111. C2×D137C2×C4⋊C4semidirect
126. C2×C137#385semidirect
130. D137#385semidirect
134. C137C2×#385semidirect
158. C2×C22⋊C4#10587739semidirect
187. C22×C4C2×#10587739semidirect
192. C22⋊C4C2×#10587739semidirect
212. C23C4⋊#10587739semidirect
232. C2×C4C22×#10587739semidirect
252. C22C2×C4⋊#10587739semidirect
266. C2#22578350direct (N × H)

Maximal subgroups

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

subgroupindex
C24×#105877392
C22×C4⋊#105877392
C22×C22⋊#105877392
#225783502
C2×(C2×#10587739)⋊C42
C2×C22⋊C4×D1372
C2×#385137