Finite Group Databasebeta

← #34110070

Subgroups — #34110070

Order 20096.

414 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 125 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#23757785C211yesmaximal, characteristic
2C2×#19097729C211yesmaximal, characteristic, complemented
3#23757617C212yesmaximal, complemented
4#23757616C212yesmaximal, complemented
5#23757600C212yesmaximal, complemented
6#23757599C212yesmaximal, complemented
7#23757316C211yesmaximal, characteristic, complemented
8#23757275C211yesmaximal, characteristic, complemented
9#23757272C211yesmaximal, characteristic, complemented
10#23757236C211yesmaximal, characteristic, complemented
11#428×C157C211yesmaximal, characteristic, complemented
12C2×#14578954C2211yescharacteristic
13C22×C4×D157C2211yescharacteristic
14C2×#14578938C2211yescharacteristic
15#19097755—21no—
16#19097751C2211yescharacteristic
17#19097751C2211yescharacteristic
18#19097746C2211yescharacteristic
19#19097746C2211yescharacteristic
20#19097741—21no—
21C4×D628—21no—
22C4×D628—21no—
23#19097729—21no—
24#19097729—21no—
25#19097729C2212yes—
26#19097729C2212yes—
27C2×C4⋊Dic157C2211yescharacteristic
28C2×#14578931—21no—
29C2×C4×Dic157—21no—
30#19097704C2212yescomplemented
31#19097702C2212yescomplemented
32#19097696C2212yescomplemented
33#19097694C2212yescomplemented
34#19097693C2212yescomplemented
35#19097691C2212yescomplemented
36#19097683C2212yes—
37#19097683C2212yes—
38C2×M16×C157C2211yescharacteristic
39#117×C157C2211yescharacteristic
40C2×C4⋊C4×C157C2211yescharacteristic
41#107×C157C2212yescomplemented
42#106×C157C2212yescomplemented
43C23×D157D411yescharacteristic
44#14578954—22no—
45#14578954—22no—
46C2×Dic314C2311yescharacteristic
47C2×D628—21no—
48C2×D628—21no—
49C2×D628C2311yescharacteristic
50C2×C22⋊D157—21no—
51C22×Dic157—21no—
52C22×Dic157D411yescharacteristic
53C2×C4×D157—21no—
54C2×C4×D157—22no—
55C2×C4×D157—22no—
56C2×C4×D157D411yescharacteristic
57C2×C4×D157D411yescharacteristic
58C2×C4×D157C2311yescharacteristic
59C2×C4×D157C2311yescharacteristic
60#14578938—22no—
61C2×#11929434C2312yes—
62C4⋊Dic157—21no—
63C4⋊Dic157—21no—
64C4⋊Dic157—21no—
65C4⋊Dic157C2312yes—
66#14578931—22no—
67#14578931—22no—
68#14578931—21no—
69#14578931—21no—
70#14578931—21no—
71#14578931—21no—
72C4×Dic157—22no—
73C4×Dic157—22no—
74C22×C4×C157—21no—
75C22×C4×C157C2311yescharacteristic
76M16×C157—22no—
77C2×C8×C157C2312yes—
78C4⋊C4×C157—21no—
79C4⋊C4×C157C2311yescharacteristic
80C4⋊C4×C157C2312yes—
81C4⋊C4×C157C2311yescharacteristic
82C22⋊C4×C157—21no—
83C42×C157—21no—
84C22×D157—22no—
85C22×D157—21no—
86C22×D157—41no—
87C22×D157—41no—
88C22×D157C2×D411yescharacteristic
89C22×D157C2×D411yescharacteristic
90Dic314—21no—
91Dic314—21no—
92D628—21no—
93D628—21no—
94D628—42no—
95D628—42no—
96C22⋊D157—42no—
97C2×Dic157—22no—
98C2×Dic157—21no—
99C2×Dic157—22no—
100C2×Dic157—42no—
101C2×Dic157C2×D411yescharacteristic
102C2×Dic157C2×D411yescharacteristic
103C4×D157—22no—
104C4×D157—22no—
105C4×D157—21no—
106C4×D157—21no—
107C4×D157—21no—
108C4×D157—21no—
109C4×D157—42no—
110#11929434—22no—
111C23×C157C2×D411yescharacteristic
112C2×C4×C157—21no—
113C2×C4×C157—21no—
114C2×C4×C157—22no—
115C2×C4×C157—42no—
116C2×C4×C157C2×D411yescharacteristic
117C2×C4×C157#4112yes—
118C2×C4×C157#4112yes—
119C2×C4×C157C2411yescharacteristic
120C8×C157—22no—
121C22×C157—21no—
122C22×C157—21no—
123C22×C157#12312yes—
124C22×C157C22×D411yescharacteristic
125C2×D157—21no—
126C2×D157—21no—
127C2×D157—21no—
128C2×D157—21no—
129C2×D157—21no—
130C2×D157—21no—
131C2×D157—44no—
132C2×D157—41no—
133C2×D157—41no—
134C2×D157—41no—
135C4×C157—42no—
136C4×C157—41no—
137C4×C157—41no—
138C4×C157#12312yes—
139C4×C157C2×#4111yescharacteristic
140C4×C157C2×#4111yescharacteristic
141Dic157—21no—
142Dic157—21no—
143Dic157—41no—
144Dic157—42no—
145C2×C157—22no—
146C2×C157C2×#12311yescharacteristic
147C2×C157#57511yescharacteristic
148C2×C157#57511yescharacteristic
149D157—41no—
150D157—41no—
151D157—81no—
152C157#309311yescharacteristic, complemented, Sylow
153#3093—1571nomaximal, Sylow
154#525—1571no—
155C2×#117—1571no—
156#485—1572no—
157#485—1572no—
158#482—1572no—
159#482—1572no—
160#428—1571no—
161#428D15711yescharacteristic, complemented
162#417—1571no—
163#417—1571no—
164#407—1571no—
165C2×#41—1571no—
166C23×C4—1571no—
167C2×M16—1571no—
168C2×M16C2×D15711yescharacteristic
169#123—3141no—
170C22⋊Q8—1571no—
171C22⋊Q8—1571no—
172#121—1571no—
173#121—1571no—
174#120—3141no—
175C4×D4—3141no—
176C4×D4—3141no—
177#117—3141no—
178#117—3141no—
179#117—1572no—
180#117—1572no—
181#117C2×D15711yescharacteristic
182C2×C4⋊C4—1571no—
183C2×C4⋊C4C2×D15711yescharacteristic
184C2×C22⋊C4—3141no—
185C2×C42—3141no—
186#107—1572no—
187#107C2×D15712yescomplemented
188#106—1572no—
189#106C2×D15712yescomplemented
190#103—1572no—
191#103—1572no—
192#102—1572no—
193#102—1572no—
194C22⋊C8—1572no—
195C22⋊C8—1572no—
196C24—1571no—
197#41—3142no—
198#41—3142no—
199C2×Q8—1571no—
200C2×D4—3141no—
201C2×D4—3141no—
202C2×D4—3141no—
203C2×D4—1571no—
204C22×C4—21no—
205C22×C4—3142no—
206C22×C4—3142no—
207C22×C4—3141no—
208C22×C4—3141no—
209C22×C4—1571no—
210C22×C4—1571no—
211C22×C4—1571no—
212C22×C4—1571no—
213C22×C4—1571no—
214C22×C4C22×D15711yescharacteristic
215M16—22no—
216M16—3142no—
217C2×C8—1572no—
218C2×C8C22×D15712yes—
219C4⋊C4—21no—
220C4⋊C4—3141no—
221C4⋊C4—3141no—
222C4⋊C4—3141no—
223C4⋊C4—1572no—
224C4⋊C4C22×D15712yes—
225C4⋊C4C22×D15711yescharacteristic
226C4⋊C4C22×D15711yescharacteristic
227C22⋊C4—21no—
228C22⋊C4—3142no—
229C22⋊C4—3142no—
230C22⋊C4—3141no—
231C22⋊C4—3141no—
232C22⋊C4—3141no—
233C22⋊C4—3141no—
234C42—21no—
235C42—3142no—
236C42—3142no—
237C23—3142no—
238C23—3141no—
239C23—6281no—
240C23—6281no—
241C23—1571no—
242C23—1571no—
243C23D4×D15711yescharacteristic
244Q8—3141no—
245Q8—3141no—
246D4—3141no—
247D4—3141no—
248D4—6282no—
249D4—6282no—
250D4—6282no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—42no—
255C2×C4—3142no—
256C2×C4—3142no—
257C2×C4—3142no—
258C2×C4—3141no—
259C2×C4—3141no—
260C2×C4—3141no—
261C2×C4—3141no—
262C2×C4—3142no—
263C2×C4—3141no—
264C2×C4—6282no—
265C2×C4—6282no—
266C2×C4—1571no—
267C2×C4—1571no—
268C2×C4D4×D15711yescharacteristic
269C2×C4#1457895412yes—
270C2×C4#1457895512yes—
271C2×C4C23×D15711yescharacteristic
272C8—22no—
273C8—3142no—
274C22—21no—
275C22—21no—
276C22—3141no—
277C22—3141no—
278C22—3141no—
279C22—3141no—
280C22—3141no—
281C22—3141no—
282C22—6284no—
283C22—6281no—
284C22—6281no—
285C22—6281no—
286C22#1909775612yes—
287C22C2×D4×D15711yescharacteristic
288C4—42no—
289C4—41no—
290C4—41no—
291C4—3141no—
292C4—3141no—
293C4—6281no—
294C4—6282no—
295C4#1909775612yes—
296C4C2×#1457895411yescharacteristic
297C4C2×#1457895511yescharacteristic
298C2—22no—
299C2—6281no—
300C2—6281no—
301C2—12561no—
302C2C2×#1909775611yescharacteristic
303C2#2375806911yescharacteristic
304C2#2375807211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#428×C157C2
socleC22×C157C22×D4
prime radical#23757785C2
hypercenter#428D157
hypocenterC157#3093
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
2. C2×#19097729C2semidirect
3. #23757617C2semidirect
4. #23757616C2semidirect
5. #23757600C2semidirect
6. #23757599C2semidirect
7. #23757316C2semidirect
8. #23757275C2semidirect
9. #23757272C2semidirect
10. #23757236C2semidirect
11. #428×C157C2semidirect
30. #19097704C22semidirect
31. #19097702C22semidirect
32. #19097696C22semidirect
33. #19097694C22semidirect
34. #19097693C22semidirect
35. #19097691C22semidirect
41. #107×C157C22semidirect
42. #106×C157C22semidirect
152. C157#3093semidirect
161. #428D157semidirect
187. #107C2×D157semidirect
189. #106C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237577852
C2×#190977292
#237576172
#237576162
#237576002
#237575992
#237573162
#237572752
#237572722
#237572362
#428×C1572
#3093157