Finite Group Databasebeta

← #34110044

Subgroups — #34110044

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#23757783C211yesmaximal, characteristic
2C2×#19097729C211yesmaximal, characteristic, complemented
3#23757616C212yesmaximal, complemented
4#23757609C211yesmaximal, characteristic, complemented
5#23757603C211yesmaximal, characteristic, complemented
6#23757595C212yesmaximal, complemented
7#23757589C211yesmaximal, characteristic, complemented
8#23757310C212yesmaximal, complemented
9#23757266C212yesmaximal, complemented
10C22⋊C8×D157C211yesmaximal, characteristic, complemented
11#480×C157C211yesmaximal, characteristic, complemented
12C2×#14578953C2211yescharacteristic
13C22×C4×D157C2211yescharacteristic
14C2×C8×D157C2212yes—
15#19097753—21no—
16#19097751C2211yescharacteristic
17#19097750C2211yescharacteristic
18#19097742C2211yescharacteristic
19#19097739—21no—
20D4×Dic157—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#19097705C2212yescomplemented
31#19097702C2212yescomplemented
32#19097702C2212yescomplemented
33#19097696C2212yescomplemented
34#19097694C2212yescomplemented
35#19097692C2212yescomplemented
36C22⋊#11929434C2211yescharacteristic
37#19097683C2212yes—
38#121×C157C2211yescharacteristic
39C2×C4⋊C4×C157C2211yescharacteristic
40#107×C157C2212yescomplemented
41#102×C157C2212yescomplemented
42C22⋊C8×C157C2211yescharacteristic
43C23×D157D411yescharacteristic
44#14578953—24no—
45C2×Dic314C2311yescharacteristic
46C2×C22⋊D157—21no—
47C2×C22⋊D157—21no—
48C22×Dic157—21no—
49C22×Dic157—21no—
50C22×Dic157D411yescharacteristic
51C2×C4×D157—24no—
52C2×C4×D157D411yescharacteristic
53C2×C4×D157D411yescharacteristic
54C2×C4×D157C2311yescharacteristic
55C2×C4×D157C2311yescharacteristic
56C2×#11929434C2312yes—
57C8×D157—24no—
58C4⋊Dic157—21no—
59C4⋊Dic157—21no—
60C4⋊Dic157C2312yes—
61C4⋊Dic157C2311yescharacteristic
62#14578933—21no—
63C22⋊Dic157—21no—
64C22⋊Dic157—21no—
65#14578931—22no—
66#14578931—22no—
67#14578931—21no—
68#14578931—21no—
69C4×Dic157—22no—
70C4×Dic157—22no—
71C4×Dic157—21no—
72C2×D4×C157—21no—
73C2×D4×C157C2311yescharacteristic
74C22×C4×C157—21no—
75C22×C4×C157C2311yescharacteristic
76C2×C8×C157C2312yes—
77C4⋊C4×C157—21no—
78C4⋊C4×C157C2312yes—
79C4⋊C4×C157C2311yescharacteristic
80C22⋊C4×C157—21no—
81C22×D157—22no—
82C22×D157—22no—
83C22×D157—41no—
84C22×D157C2×D411yescharacteristic
85C22×D157C2×D411yescharacteristic
86Dic314—22no—
87C22⋊D157—42no—
88C22⋊D157—42no—
89C2×Dic157—22no—
90C2×Dic157—21no—
91C2×Dic157—22no—
92C2×Dic157—21no—
93C2×Dic157—42no—
94C2×Dic157—42no—
95C2×Dic157C2×D411yescharacteristic
96C2×Dic157C2×D411yescharacteristic
97C4×D157—22no—
98C4×D157—22no—
99C4×D157—22no—
100C4×D157#4114yes—
101#11929434—22no—
102C23×C157—21no—
103C23×C157C2×D411yescharacteristic
104D4×C157—22no—
105D4×C157—42no—
106C2×C4×C157—22no—
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—41no—
116C22×C157—41no—
117C22×C157—41no—
118C22×C157#13512yes—
119C22×C157C22×D411yescharacteristic
120C2×D157—21no—
121C2×D157—21no—
122C2×D157—21no—
123C2×D157—21no—
124C2×D157—21no—
125C2×D157—44no—
126C2×D157#12312yes—
127C4×C157—21no—
128C4×C157—42no—
129C4×C157—41no—
130C4×C157C2×#4112yes—
131Dic157—21no—
132Dic157—41no—
133Dic157—42no—
134Dic157—41no—
135Dic157#12312yes—
136C2×C157—22no—
137C2×C157—81no—
138C2×C157C2×#12311yescharacteristic
139C2×C157C2×#13511yescharacteristic
140C2×C157#57511yescharacteristic
141D157—22no—
142D157—41no—
143C157#309111yescharacteristic, complemented, Sylow
144#3091—1571nomaximal, Sylow
145#525—1571no—
146C2×#117—1571no—
147#485—1572no—
148#484—1571no—
149#483—1571no—
150#482—1572no—
151#481—1571no—
152#480D15711yescharacteristic, complemented
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⋊Q8—1571no—
162#121—1571no—
163#121C2×D15711yescharacteristic
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#107C2×D15712yescomplemented
178#106—1572no—
179#106—1572no—
180#103—1572no—
181#103—1572no—
182#102—1572no—
183#102C2×D15712yescomplemented
184C22⋊C8—1571no—
185C22⋊C8—1572no—
186C22⋊C8C2×D15711yescharacteristic
187C24—1571no—
188#41—3144no—
189C2×Q8—1571no—
190C2×D4—21no—
191C2×D4—3141no—
192C2×D4—3141no—
193C2×D4C22×D15711yescharacteristic
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—3141no—
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—21no—
226C23—3142no—
227C23—3142no—
228C23—6281no—
229C23—1571no—
230C23—1571no—
231C23D4×D15711yescharacteristic
232Q8—3142no—
233D4—22no—
234D4—42no—
235D4—6282no—
236D4—6282no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—42no—
241C2×C4—3142no—
242C2×C4—3142no—
243C2×C4—3142no—
244C2×C4—3142no—
245C2×C4—3142no—
246C2×C4—3141no—
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—41no—
260C22—41no—
261C22—41no—
262C22—3141no—
263C22—3141no—
264C22—3141no—
265C22—3141no—
266C22—3141no—
267C22—6284no—
268C22—1572no—
269C22#1909779012yes—
270C22C2×D4×D15711yescharacteristic
271C4—21no—
272C4—42no—
273C4—41no—
274C4—3141no—
275C4—6281no—
276C4—6282no—
277C4—6281no—
278C4—1572no—
279C4#41×D15712yes—
280C2—22no—
281C2—81no—
282C2—3142no—
283C2—6281no—
284C2#123×D15711yescharacteristic
285C2C2×#1909779011yescharacteristic
286C2#2375807211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#480×C157C2
socleC22×C157C22×D4
prime radical#23757783C2
hypercenter#480D157
hypocenterC157#3091
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
2. C2×#19097729C2semidirect
3. #23757616C2semidirect
4. #23757609C2semidirect
5. #23757603C2semidirect
6. #23757595C2semidirect
7. #23757589C2semidirect
8. #23757310C2semidirect
9. #23757266C2semidirect
10. C22⋊C8×D157C2semidirect
11. #480×C157C2semidirect
30. #19097705C22semidirect
31. #19097702C22semidirect
32. #19097702C22semidirect
33. #19097696C22semidirect
34. #19097694C22semidirect
35. #19097692C22semidirect
40. #107×C157C22semidirect
41. #102×C157C22semidirect
143. C157#3091semidirect
152. #480D157semidirect
177. #107C2×D157semidirect
183. #102C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237577832
C2×#190977292
#237576162
#237576092
#237576032
#237575952
#237575892
#237573102
#237572662
C22⋊C8×D1572
#480×C1572
#3091157