Finite Group Databasebeta

← #34110050

Subgroups — #34110050

Order 20096.

420 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 121 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#23757786C211yesmaximal, characteristic
2C2×#19097729C211yesmaximal, characteristic, complemented
3#23757612C212yesmaximal, complemented
4#23757606C211yesmaximal, characteristic, complemented
5#23757599C212yesmaximal, complemented
6#23757592C211yesmaximal, characteristic, complemented
7#23757586C211yesmaximal, characteristic, complemented
8#23757308C212yesmaximal, complemented
9#23757268C212yesmaximal, complemented
10C22⋊C8×D157C211yesmaximal, characteristic, complemented
11#483×C157C211yesmaximal, characteristic, complemented
12C2×#14578955C2211yescharacteristic
13C22×C4×D157C2211yescharacteristic
14C2×C8×D157C2212yes—
15#19097756—21no—
16#19097747C2211yescharacteristic
17#19097746C2211yescharacteristic
18#19097745C2211yescharacteristic
19#19097741—21no—
20#19097733—21no—
21#19097731—21no—
22#19097729—21no—
23#19097729—21no—
24#19097729C2212yes—
25#19097729C2212yes—
26#19097729C2211yescharacteristic
27C2×C4⋊Dic157C2211yescharacteristic
28C2×#14578931—21no—
29C2×C4×Dic157—21no—
30#19097704C2212yescomplemented
31#19097704C2212yescomplemented
32#19097703C2212yescomplemented
33#19097695C2212yescomplemented
34#19097693C2212yescomplemented
35#19097691C2212yescomplemented
36C22⋊#11929434C2211yescharacteristic
37#19097683C2212yes—
38C22⋊Q8×C157C2211yescharacteristic
39C2×C4⋊C4×C157C2211yescharacteristic
40#106×C157C2212yescomplemented
41#103×C157C2212yescomplemented
42C22⋊C8×C157C2211yescharacteristic
43C23×D157D411yescharacteristic
44#14578955—24no—
45C2×D628—21no—
46C2×D628—21no—
47C2×D628C2311yescharacteristic
48C2×C22⋊D157—21no—
49C22×Dic157—21no—
50C22×Dic157D411yescharacteristic
51C2×C4×D157—24no—
52C2×C4×D157—21no—
53C2×C4×D157D411yescharacteristic
54C2×C4×D157D411yescharacteristic
55C2×C4×D157C2311yescharacteristic
56C2×C4×D157C2311yescharacteristic
57C2×#11929434C2312yes—
58C8×D157—24no—
59C4⋊Dic157—21no—
60C4⋊Dic157C2312yes—
61C4⋊Dic157C2311yescharacteristic
62#14578933—21no—
63#14578931—22no—
64#14578931—22no—
65#14578931—21no—
66#14578931—21no—
67#14578931—21no—
68#14578931—21no—
69C4×Dic157—22no—
70C4×Dic157—22no—
71C4×Dic157—21no—
72C2×Q8×C157C2311yescharacteristic
73C22×C4×C157—21no—
74C22×C4×C157C2311yescharacteristic
75C2×C8×C157C2312yes—
76C4⋊C4×C157—21no—
77C4⋊C4×C157—21no—
78C4⋊C4×C157C2312yes—
79C4⋊C4×C157C2311yescharacteristic
80C22⋊C4×C157—21no—
81C22×D157—22no—
82C22×D157—22no—
83C22×D157—21no—
84C22×D157—41no—
85C22×D157C2×D411yescharacteristic
86C22×D157C2×D411yescharacteristic
87D628—22no—
88D628—42no—
89D628—42no—
90C22⋊D157—42no—
91C2×Dic157—22no—
92C2×Dic157—22no—
93C2×Dic157—21no—
94C2×Dic157—42no—
95C2×Dic157C2×D411yescharacteristic
96C2×Dic157C2×D411yescharacteristic
97C4×D157—22no—
98C4×D157—22no—
99C4×D157—22no—
100C4×D157—42no—
101C4×D157#4114yes—
102#11929434—22no—
103C23×C157C2×D411yescharacteristic
104Q8×C157—22no—
105C2×C4×C157—22no—
106C2×C4×C157—21no—
107C2×C4×C157—21no—
108C2×C4×C157—22no—
109C2×C4×C157—42no—
110C2×C4×C157C2×D411yescharacteristic
111C2×C4×C157C2411yescharacteristic
112C8×C157—22no—
113C22×C157—21no—
114C22×C157—21no—
115C22×C157#13512yes—
116C22×C157C22×D411yescharacteristic
117C2×D157—21no—
118C2×D157—21no—
119C2×D157—21no—
120C2×D157—21no—
121C2×D157—21no—
122C2×D157—44no—
123C2×D157—41no—
124C2×D157—41no—
125C2×D157—41no—
126C2×D157#12312yes—
127C4×C157—21no—
128C4×C157—41no—
129C4×C157—42no—
130C4×C157—41no—
131C4×C157C2×#4112yes—
132Dic157—21no—
133Dic157—41no—
134Dic157—42no—
135Dic157#12312yes—
136C2×C157—22no—
137C2×C157C2×#12311yescharacteristic
138C2×C157C2×#13511yescharacteristic
139C2×C157#57511yescharacteristic
140D157—22no—
141D157—41no—
142D157—81no—
143C157#309111yescharacteristic, complemented, Sylow
144#3091—1571nomaximal, Sylow
145#525—1571no—
146C2×#117—1571no—
147#485—1572no—
148#484—1571no—
149#483D15711yescharacteristic, complemented
150#482—1572no—
151#481—1571no—
152#480—1571no—
153#427—1572no—
154#416—1572no—
155C2×C22⋊C8—1571no—
156C2×#41—1571no—
157C23×C4—1571no—
158C22×C8—1572no—
159#123—3141no—
160C22⋊Q8—1571no—
161C22⋊Q8C2×D15711yescharacteristic
162#121—1571no—
163#121—1571no—
164#120—3141no—
165C4×D4—3141no—
166C4×D4—3141no—
167#117—3141no—
168#117—3141no—
169#117—1572no—
170#117—1572no—
171#117—1571no—
172C2×C4⋊C4—1571no—
173C2×C4⋊C4C2×D15711yescharacteristic
174C2×C22⋊C4—3141no—
175C2×C42—3141no—
176#107—1572no—
177#107—1572no—
178#106—1572no—
179#106C2×D15712yescomplemented
180#103—1572no—
181#103C2×D15712yescomplemented
182#102—1572no—
183#102—1572no—
184C22⋊C8—1571no—
185C22⋊C8—1572no—
186C22⋊C8C2×D15711yescharacteristic
187C24—1571no—
188#41—3144no—
189C2×Q8C22×D15711yescharacteristic
190C2×D4—3141no—
191C2×D4—3141no—
192C2×D4—3141no—
193C2×D4—1571no—
194C22×C4—21no—
195C22×C4—3144no—
196C22×C4—3141no—
197C22×C4—3141no—
198C22×C4—1571no—
199C22×C4—1571no—
200C22×C4—1571no—
201C22×C4—1571no—
202C22×C4—1571no—
203C22×C4C22×D15711yescharacteristic
204C2×C8—3144no—
205C2×C8—1572no—
206C2×C8C22×D15712yes—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—3141no—
210C4⋊C4—3141no—
211C4⋊C4—1572no—
212C4⋊C4—1571no—
213C4⋊C4C22×D15712yes—
214C4⋊C4C22×D15711yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—3142no—
217C22⋊C4—3142no—
218C22⋊C4—3141no—
219C22⋊C4—3141no—
220C22⋊C4—3141no—
221C22⋊C4—3141no—
222C42—3142no—
223C42—3142no—
224C42—3141no—
225C23—3142no—
226C23—3142no—
227C23—3141no—
228C23—6281no—
229C23—1571no—
230C23—1571no—
231C23D4×D15711yescharacteristic
232Q8—22no—
233D4—3142no—
234D4—6282no—
235D4—6282no—
236D4—6282no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—42no—
242C2×C4—3142no—
243C2×C4—3142no—
244C2×C4—3142no—
245C2×C4—3142no—
246C2×C4—3142no—
247C2×C4—3141no—
248C2×C4—6282no—
249C2×C4—6282no—
250C2×C4—1571no—
251C2×C4—1571no—
252C2×C4—1574no—
253C2×C4D4×D15711yescharacteristic
254C2×C4C23×D15711yescharacteristic
255C8—22no—
256C8—3142no—
257C22—21no—
258C22—21no—
259C22—3141no—
260C22—3141no—
261C22—3141no—
262C22—3141no—
263C22—3141no—
264C22—6284no—
265C22—6281no—
266C22—6281no—
267C22—6281no—
268C22—1572no—
269C22#1909779212yes—
270C22C2×D4×D15711yescharacteristic
271C4—21no—
272C4—41no—
273C4—42no—
274C4—41no—
275C4—3141no—
276C4—6281no—
277C4—6282no—
278C4—1572no—
279C4#41×D15712yes—
280C2—22no—
281C2—3142no—
282C2—6281no—
283C2—12561no—
284C2#123×D15711yescharacteristic
285C2C2×#1909779211yescharacteristic
286C2#2375807211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#483×C157C2
socleC22×C157C22×D4
prime radical#23757786C2
hypercenter#483D157
hypocenterC157#3091
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
2. C2×#19097729C2semidirect
3. #23757612C2semidirect
4. #23757606C2semidirect
5. #23757599C2semidirect
6. #23757592C2semidirect
7. #23757586C2semidirect
8. #23757308C2semidirect
9. #23757268C2semidirect
10. C22⋊C8×D157C2semidirect
11. #483×C157C2semidirect
30. #19097704C22semidirect
31. #19097704C22semidirect
32. #19097703C22semidirect
33. #19097695C22semidirect
34. #19097693C22semidirect
35. #19097691C22semidirect
40. #106×C157C22semidirect
41. #103×C157C22semidirect
143. C157#3091semidirect
149. #483D157semidirect
179. #106C2×D157semidirect
181. #103C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237577862
C2×#190977292
#237576122
#237576062
#237575992
#237575922
#237575862
#237573082
#237572682
C22⋊C8×D1572
#483×C1572
#3091157