Finite Group Databasebeta

← #34103451

Subgroups — #34103451

Order 20096.

1056 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 185 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
1C2×#19097739C212yesmaximal, complemented
2C22×#14578931C211yesmaximal, characteristic, complemented
3#23757184C212yesmaximal, complemented
4#23757181C212yesmaximal, complemented
5#23757153C212yesmaximal, complemented
6#23757152C214yesmaximal, complemented
7#23757136C211yesmaximal, characteristic, complemented
8C22×#41×C157C211yesmaximal, characteristic, complemented
9C24×D157C2211yescharacteristic
10C22×C22⋊D157C2214yes—
11C23×Dic157C2211yescharacteristic
12#19097739—28no—
13C2×C22⋊Dic157C2212yes—
14C2×C22⋊Dic157C2214yescomplemented
15C2×#14578931—24no—
16C2×#14578931C2212yes—
17C2×#14578931C2214yes—
18C2×#14578931C2212yes—
19#19097678C2212yescomplemented
20#19097678C2212yescomplemented
21#19097678C2214yescomplemented
22C2×#41×C157—28no—
23C2×#41×C157—24no—
24C22×Q8×C157C2211yescharacteristic, complemented
25C22×D4×C157C2212yes—
26C22×D4×C157C2211yescharacteristic, complemented
27C23×C4×C157C2211yescharacteristic
28C23×C4×C157C2212yescomplemented
29C23×D157—24no—
30C23×D157—22no—
31C23×D157—24no—
32C23×D157—42no—
33C23×D157C2312yes—
34C2×C22⋊D157—28no—
35C2×C22⋊D157—28no—
36C2×C22⋊D157—44no—
37C22×Dic157—22no—
38C22×Dic157—42no—
39C22×Dic157C2314yes—
40C22×Dic157C2312yes—
41C22⋊Dic157—24no—
42C22⋊Dic157—44no—
43#14578931—24no—
44#14578931—24no—
45#14578931—28no—
46C24×C157C2312yes—
47C24×C157C2311yescharacteristic
48#41×C157—232no—
49C2×Q8×C157—24no—
50C2×Q8×C157—22no—
51C2×D4×C157—24no—
52C2×D4×C157—24no—
53C2×D4×C157—24no—
54C2×D4×C157—22no—
55C2×D4×C157D418yescomplemented
56C22×C4×C157—24no—
57C22×C4×C157—24no—
58C22×C4×C157—24no—
59C22×C4×C157—24no—
60C22×C4×C157D414yescomplemented
61C22×C4×C157C2311yescharacteristic
62C22×C4×C157C2311yescharacteristic
63C22×C4×C157C2312yes—
64C22×D157—28no—
65C22×D157—24no—
66C22×D157—22no—
67C22×D157—22no—
68C22×D157—24no—
69C22×D157—416no—
70C22×D157—44no—
71C22×D157—44no—
72C22×D157#4114yescomplemented
73C22⋊D157—416no—
74C2×Dic157—22no—
75C2×Dic157—24no—
76C2×Dic157—24no—
77C2×Dic157—24no—
78C2×Dic157—24no—
79C2×Dic157—44no—
80C23×C157—22no—
81C23×C157—22no—
82C23×C157—22no—
83C23×C157—22no—
84C23×C157—24no—
85C23×C157—24no—
86C23×C157C2×D414yes—
87C23×C157C2×D412yes—
88C23×C157C2×D414yes—
89C23×C157C2411yescharacteristic
90Q8×C157—28no—
91D4×C157—28no—
92D4×C157—216no—
93C2×C4×C157—216no—
94C2×C4×C157—28no—
95C2×C4×C157—22no—
96C2×C4×C157—22no—
97C2×C4×C157—22no—
98C2×C4×C157—22no—
99C2×C4×C157C2×D412yes—
100C2×C4×C157C2×D414yes—
101C2×C4×C157C2×D412yes—
102C22×C157—22no—
103C22×C157—22no—
104C22×C157—28no—
105C22×C157—24no—
106C22×C157—28no—
107C22×C157—42no—
108C22×C157—42no—
109C22×C157—44no—
110C22×C157#12014yes—
111C22×C157C22×D411yescharacteristic
112C22×C157C22×D412yes—
113C22×C157C2×#4112yes—
114C22×C157H(5,𝔽2)12yes—
115C2×D157—24no—
116C2×D157—28no—
117C2×D157—416no—
118C2×D157—42no—
119C2×D157—42no—
120C2×D157—44no—
121C4×C157—22no—
122C4×C157—24no—
123C4×C157—22no—
124Dic157—42no—
125Dic157—44no—
126C2×C157—24no—
127C2×C157—44no—
128C2×C157C2×#12011yescharacteristic
129C2×C157#52511yescharacteristic
130C2×C157#53511yescharacteristic
131C2×C157#54614yes—
132D157—44no—
133C157#240411yescharacteristic, complemented, Sylow
134#2404—1571nomaximal, Sylow
135C22×#41D15711yescharacteristic, complemented
136C2×#120—1572no—
137C22×C22⋊C4—1571no—
138#393—1572no—
139#392—1572no—
140#386—1572no—
141#386—1574no—
142#381—1571no—
143C25—1571no—
144C2×#41—28no—
145C2×#41—24no—
146C22×Q8C2×D15711yescharacteristic, complemented
147C22×D4—1574no—
148C22×D4C2×D15712yes—
149C22×D4C2×D15711yescharacteristic, complemented
150C23×C4—1571no—
151C23×C4C2×D15711yescharacteristic
152C23×C4C2×D15712yescomplemented
153#120—3148no—
154C2×C22⋊C4—3144no—
155C2×C22⋊C4—1572no—
156C2×C22⋊C4—1572no—
157C2×C22⋊C4—1572no—
158C2×C22⋊C4—1574no—
159C2×C22⋊C4—1574no—
160#95—1572no—
161#95—1572no—
162#95—1574no—
163C24—3144no—
164C24—3142no—
165C24—3144no—
166C24—6282no—
167C24—1572no—
168C24C22×D15712yes—
169C24C22×D15711yescharacteristic
170#41—232no—
171C2×Q8—24no—
172C2×Q8—22no—
173C2×D4—24no—
174C2×D4—24no—
175C2×D4—24no—
176C2×D4—22no—
177C2×D4—3148no—
178C2×D4—3148no—
179C2×D4—6284no—
180C2×D4C22⋊D15718yescomplemented
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—3142no—
186C22×C4—6282no—
187C22×C4—1574no—
188C22×C4—1572no—
189C22×C4C22⋊D15714yescomplemented
190C22×C4C22×D15711yescharacteristic
191C22×C4C22×D15711yescharacteristic
192C22×C4C22×D15712yes—
193C22⋊C4—3144no—
194C22⋊C4—3144no—
195C22⋊C4—3144no—
196C22⋊C4—3148no—
197C22⋊C4—6284no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23—24no—
203C23—24no—
204C23—3148no—
205C23—3142no—
206C23—3144no—
207C23—3142no—
208C23—3144no—
209C23—62816no—
210C23—6284no—
211C23—6284no—
212C23—1574no—
213C23C2×C22⋊D15714yes—
214C23C2×C22⋊D15714yes—
215C23C2×C22⋊D15712yes—
216C23C23×D15711yescharacteristic
217Q8—28no—
218D4—28no—
219D4—216no—
220D4—62816no—
221C2×C4—216no—
222C2×C4—28no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—3142no—
228C2×C4—3144no—
229C2×C4—3144no—
230C2×C4—3144no—
231C2×C4—3144no—
232C2×C4—6284no—
233C2×C4C2×C22⋊D15712yes—
234C2×C4C2×C22⋊D15714yes—
235C2×C4C2×C22⋊D15712yes—
236C22—22no—
237C22—28no—
238C22—22no—
239C22—28no—
240C22—24no—
241C22—42no—
242C22—44no—
243C22—42no—
244C22—3148no—
245C22—3144no—
246C22—62816no—
247C22—6282no—
248C22—6284no—
249C22—6282no—
250C22#1909774014yes—
251C22C22×C22⋊D15711yescharacteristic
252C22C22×C22⋊D15712yes—
253C22#41×D15712yes—
254C22#1909781712yes—
255C4—22no—
256C4—24no—
257C4—22no—
258C4—6282no—
259C4—6284no—
260C2—24no—
261C2—44no—
262C2—6284no—
263C2C2×#1909774011yescharacteristic
264C2#2375778011yescharacteristic
265C2#2375783711yescharacteristic
266C2#2375791114yes—
subgroupG/H
center, FrattiniC23C23×D157
commutator, socle, cosocleC23×C157C24
FittingC22×#41×C157C2
hypercenterC22×#41D157
hypocenterC157#2404
LCSG ▷ C23×C157 ▷ C157
UCSC1 ◁ C23 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#19097739C2semidirect
2. C22×#14578931C2semidirect
3. #23757184C2semidirect
4. #23757181C2semidirect
5. #23757153C2semidirect
6. #23757152C2semidirect
7. #23757136C2semidirect
8. C22×#41×C157C2semidirect
14. C2×C22⋊Dic157C22semidirect
19. #19097678C22semidirect
20. #19097678C22semidirect
21. #19097678C22semidirect
24. C22×Q8×C157C22semidirect
26. C22×D4×C157C22semidirect
28. C23×C4×C157C22semidirect
55. C2×D4×C157D4semidirect
60. C22×C4×C157D4semidirect
72. C22×D157#41semidirect
133. C157#2404semidirect
135. C22×#41D157semidirect
146. C22×Q8C2×D157semidirect
149. C22×D4C2×D157semidirect
152. C23×C4C2×D157semidirect
180. C2×D4C22⋊D157semidirect
189. C22×C4C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977392
C22×#145789312
#237571842
#237571812
#237571532
#237571522
#237571362
C22×#41×C1572
#2404157