Finite Group Databasebeta

← #34110271

Subgroups — #34110271

Order 20096.

450 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 117 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#23757804C211yesmaximal, characteristic, complemented
2C2×#19097733C211yesmaximal, characteristic, complemented
3#23757603C211yesmaximal, characteristic, complemented
4#23757576C212yesmaximal, complemented
5#23757487C212yesmaximal, complemented
6#23757426C211yesmaximal, characteristic, complemented
7#23757406C211yesmaximal, characteristic
8#23757382C212yesmaximal, complemented
9#23757372C212yesmaximal, complemented
10#23757327C211yesmaximal, characteristic, complemented
11#480×C157C211yesmaximal, characteristic, complemented
12C22×D628C2211yescharacteristic
13C22×C4×D157—21no—
14C2×#14578943C2212yes—
15C2×#14578939C2212yes—
16#19097764—21no—
17#19097762C2211yescharacteristic, complemented
18#19097758C2211yescharacteristic, complemented
19#19097752—21no—
20#19097750C2211yescharacteristic
21#19097736C2211yescharacteristic, complemented
22D4×Dic157—21no—
23D4×Dic157C2211yescharacteristic, complemented
24#19097733—21no—
25#19097733—21no—
26#19097733C2212yes—
27#19097733C2212yes—
28#19097727—21no—
29C2×#14578931—21no—
30C2×C4×Dic157C2211yescharacteristic
31#19097702C2212yescomplemented
32#19097700C2212yescomplemented
33#19097694C2212yescomplemented
34#19097693C2212yes—
35#19097691C2212yes—
36C22⋊#11929434C2211yescharacteristic
37#19097681C2212yescomplemented
38#121×C157C2211yescharacteristic
39C2×C4⋊C4×C157C2211yescharacteristic
40#107×C157C2212yescomplemented
41#102×C157C2212yescomplemented
42C22⋊C8×C157C2211yescharacteristic
43C23×D157—21no—
44C2×Dic314C2311yescharacteristic
45C2×D628—24no—
46C2×D628—21no—
47C2×D628C2312yes—
48C22×Dic157—21no—
49C22×Dic157D411yescharacteristic
50C22×Dic157D411yescharacteristic
51C2×C4×D157—22no—
52C2×C4×D157—22no—
53C2×C4×D157—44no—
54#14578943—24no—
55#14578939—24no—
56C2×#11929434C2312yes—
57C4⋊Dic157—21no—
58C4⋊Dic157C2311yescharacteristic
59#14578933—21no—
60#14578933—21no—
61#14578933—21no—
62C22⋊Dic157—21no—
63C22⋊Dic157—21no—
64C22⋊Dic157—21no—
65C22⋊Dic157—21no—
66#14578931—22no—
67#14578931—22no—
68C4×Dic157—21no—
69C4×Dic157D411yescharacteristic
70C4×Dic157D411yescharacteristic
71C4×Dic157C2311yescharacteristic
72C4×Dic157C2311yescharacteristic
73C2×D4×C157—21no—
74C2×D4×C157C2311yescharacteristic
75C22×C4×C157—21no—
76C22×C4×C157C2311yescharacteristic
77C2×C8×C157C2312yes—
78C4⋊C4×C157—21no—
79C4⋊C4×C157C2312yes—
80C4⋊C4×C157C2311yescharacteristic
81C22⋊C4×C157—21no—
82C22×D157—24no—
83C22×D157—22no—
84C22×D157—42no—
85C22×D157—42no—
86Dic314—22no—
87D628—24no—
88D628—22no—
89D628#4114yes—
90C2×Dic157—22no—
91C2×Dic157—21no—
92C2×Dic157—21no—
93C2×Dic157—41no—
94C2×Dic157—42no—
95C2×Dic157C2×D411yescharacteristic
96C2×Dic157C2×D411yescharacteristic
97C2×Dic157C2×D411yescharacteristic
98C2×Dic157C2×D411yescharacteristic
99C4×D157—44no—
100C4×D157—44no—
101#11929434—22no—
102C23×C157—21no—
103C23×C157C2×D411yescharacteristic
104D4×C157—22no—
105D4×C157—42no—
106C2×C4×C157—22no—
107C2×C4×C157—22no—
108C2×C4×C157—21no—
109C2×C4×C157—42no—
110C2×C4×C157C2×D411yescharacteristic
111C2×C4×C157C2411yescharacteristic
112C8×C157—22no—
113C22×C157—21no—
114C22×C157—21no—
115C22×C157—41no—
116C22×C157—41no—
117C22×C157—41no—
118C22×C157AGL(1,ℤ/8)12yes—
119C22×C157C22×D411yescharacteristic
120C2×D157—24no—
121C2×D157—48no—
122C2×D157—42no—
123C2×D157—42no—
124C4×C157—21no—
125C4×C157—42no—
126C4×C157—41no—
127C4×C157C2×#4112yes—
128Dic157—21no—
129Dic157—21no—
130Dic157—21no—
131Dic157—21no—
132Dic157—41no—
133Dic157—41no—
134C2×C157—22no—
135C2×C157—81no—
136C2×C157#52511yescharacteristic
137C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C157#57511yescharacteristic
139D157—44no—
140C157#311411yescharacteristic, complemented, Sylow
141#3114—1571nomaximal, Sylow
142#529—1571no—
143C2×C4×D4—1571no—
144#483—1571no—
145#480D15711yescharacteristic, complemented
146#478—1572no—
147#463—1572no—
148#450—1571no—
149#447—1571no—
150#442—1572no—
151#440—1572no—
152#432—1571no—
153C22×D4—1571no—
154C23×C4—3141no—
155C2×SD16—1572no—
156C2×D8—1572no—
157#126—3141no—
158#125—1571no—
159#124—1571no—
160#123—3141no—
161C22⋊Q8—1571no—
162#121C2×D15711yescharacteristic
163C4×Q8—1571no—
164C4×D4—3141no—
165C4×D4—3141no—
166C4×D4—3141no—
167C4×D4—1571no—
168C4×D4—1572no—
169C4×D4—1572no—
170#117—3141no—
171C2×C4⋊C4C2×D15711yescharacteristic
172C2×C22⋊C4—3141no—
173C2×C42—1571no—
174#107C2×D15712yescomplemented
175#106—1572no—
176C4⋊C8—1572no—
177#103—1572no—
178#102—1572no—
179#102—1572no—
180#102C2×D15712yescomplemented
181C22⋊C8—1571no—
182C22⋊C8C2×D15711yescharacteristic
183#97—1572no—
184C24—3141no—
185C2×Q8—1571no—
186C2×D4—21no—
187C2×D4—3144no—
188C2×D4—3141no—
189C2×D4—1572no—
190C2×D4C22×D15711yescharacteristic
191C22×C4—21no—
192C22×C4—3142no—
193C22×C4—3142no—
194C22×C4—3141no—
195C22×C4—6284no—
196C22×C4—1571no—
197C22×C4—1571no—
198C22×C4C22×D15711yescharacteristic
199SD16—3144no—
200D8—3144no—
201C2×C8—1572no—
202C2×C8C22×D15712yes—
203C4⋊C4—21no—
204C4⋊C4—3141no—
205C4⋊C4—3141no—
206C4⋊C4—3141no—
207C4⋊C4—3141no—
208C4⋊C4—1571no—
209C4⋊C4C22×D15711yescharacteristic
210C4⋊C4C22×D15712yes—
211C22⋊C4—21no—
212C22⋊C4—3142no—
213C22⋊C4—3142no—
214C22⋊C4—3141no—
215C22⋊C4—3141no—
216C22⋊C4—3141no—
217C22⋊C4—3141no—
218C42—3141no—
219C42—1571no—
220C42—1571no—
221C42—1571no—
222C42—1571no—
223C23—21no—
224C23—3144no—
225C23—3142no—
226C23—6282no—
227C23—6282no—
228C23D4×D15711yescharacteristic
229Q8—3142no—
230D4—22no—
231D4—42no—
232D4—3144no—
233D4—3142no—
234D4—1574no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—42no—
239C2×C4—3142no—
240C2×C4—3141no—
241C2×C4—3141no—
242C2×C4—6284no—
243C2×C4—6284no—
244C2×C4—6281no—
245C2×C4—6282no—
246C2×C4—1571no—
247C2×C4—1571no—
248C2×C4—1571no—
249C2×C4—1571no—
250C2×C4D4×D15711yescharacteristic
251C2×C4C23×D15711yescharacteristic
252C8—22no—
253C8—3142no—
254C22—21no—
255C22—21no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—3144no—
260C22—6288no—
261C22—6282no—
262C22—6282no—
263C22#1909779512yes—
264C22C2×D4×D15711yescharacteristic
265C4—21no—
266C4—42no—
267C4—41no—
268C4—3141no—
269C4—3141no—
270C4—3141no—
271C4—3141no—
272C4—6281no—
273C4—6281no—
274C4#41×D15712yes—
275C2—22no—
276C2—81no—
277C2—6284no—
278C2#2375778111yescharacteristic
279C2C2×#1909779511yescharacteristic
280C2#2375807211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#480×C157C2
socleC22×C157C22×D4
prime radical#23757406C2
hypercenter#480D157
hypocenterC157#3114
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. #23757804C2semidirect
2. C2×#19097733C2semidirect
3. #23757603C2semidirect
4. #23757576C2semidirect
5. #23757487C2semidirect
6. #23757426C2semidirect
8. #23757382C2semidirect
9. #23757372C2semidirect
10. #23757327C2semidirect
11. #480×C157C2semidirect
17. #19097762C22semidirect
18. #19097758C22semidirect
21. #19097736C22semidirect
23. D4×Dic157C22semidirect
31. #19097702C22semidirect
32. #19097700C22semidirect
33. #19097694C22semidirect
37. #19097681C22semidirect
40. #107×C157C22semidirect
41. #102×C157C22semidirect
140. C157#3114semidirect
145. #480D157semidirect
174. #107C2×D157semidirect
180. #102C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237578042
C2×#190977332
#237576032
#237575762
#237574872
#237574262
#237574062
#237573822
#237573722
#237573272
#480×C1572
#3114157