Finite Group Databasebeta

← #34111977

Subgroups — #34111977

Order 20096.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757965C211yesmaximal, characteristic, complemented
2#23757914C211yesmaximal, characteristic
3#23757629C211yesmaximal, characteristic, complemented
4#23757601C211yesmaximal, characteristic, complemented
5#23757600C211yesmaximal, characteristic, complemented
6#23757576C211yesmaximal, characteristic, complemented
7#23757505C211yesmaximal, characteristic, complemented
8#23757497C211yesmaximal, characteristic, complemented
9#23757434C211yesmaximal, characteristic, complemented
10#23757423C211yesmaximal, characteristic, complemented
11#23757358C211yesmaximal, characteristic, complemented
12C8×C22⋊D157C211yesmaximal, characteristic, complemented
13#23757268C211yesmaximal, characteristic, complemented
14C2×#19097693C211yesmaximal, characteristic, complemented
15#467×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic
17C22×D628C2211yescharacteristic
18C2×#14578941C2211yescharacteristic
19C2×#14578941C2211yescharacteristic
20C2×C8×D157C2211yescharacteristic, complemented
21#19097766—21no—
22#19097762—21no—
23#19097762C2211yescharacteristic, complemented
24#19097760C2211yescharacteristic
25#19097756—21no—
26#19097755—21no—
27#19097747C2211yescharacteristic, complemented
28#19097746—21no—
29#19097746C2211yescharacteristic
30#19097741—21no—
31Q8×Dic157C2211yescharacteristic, complemented
32#19097733C2211yescharacteristic
33C4×C22⋊D157C2211yescharacteristic
34#19097729C2211yescharacteristic, complemented
35#19097727—21no—
36C2×C4⋊Dic157C2211yescharacteristic
37C22⋊C4×D157—21no—
38#19097702C2211yescharacteristic, complemented
39#19097702C2211yescharacteristic, complemented
40#19097700C2211yescharacteristic, complemented
41#19097695C2211yescharacteristic, complemented
42#19097695C2211yescharacteristic, complemented
43#19097693C2211yescharacteristic
44#19097693C2211yescharacteristic
45#19097693C2211yescharacteristic
46#19097693C2211yescharacteristic
47#19097691C2211yescharacteristic
48#19097691C2211yescharacteristic
49C22⋊#11929434C2211yescharacteristic
50#19097683C2211yescharacteristic, complemented
51C8×Dic157C2211yescharacteristic, complemented
52C2×Q16×C157C2211yescharacteristic, complemented
53C22×C8×C157C2211yescharacteristic
54C22⋊Q8×C157C2211yescharacteristic
55C22⋊Q8×C157C2211yescharacteristic
56#107×C157C2211yescharacteristic, complemented
57#103×C157C2211yescharacteristic, complemented
58#103×C157C2211yescharacteristic, complemented
59C23×D157—21no—
60#14578955—22no—
61#14578955—41no—
62C2×D628—22no—
63C2×D628—21no—
64C2×D628—21no—
65C2×D628—21no—
66C2×D628—41no—
67C2×D628C2311yescharacteristic
68C2×D628C2311yescharacteristic
69C2×C22⋊D157D411yescharacteristic
70C22×Dic157—21no—
71C2×C4×D157—21no—
72C2×C4×D157—21no—
73C2×C4×D157C2311yescharacteristic
74#14578941—22no—
75#14578941—41no—
76C2×#11929434C2311yescharacteristic
77C2×#11929434C2311yescharacteristic
78C8×D157—41no—
79C4⋊Dic157—21no—
80C4⋊Dic157—21no—
81C4⋊Dic157—21no—
82C4⋊Dic157C2311yescharacteristic
83C4⋊Dic157C2311yescharacteristic
84#14578933—21no—
85#14578933—21no—
86#14578933D411yescharacteristic
87C22⋊Dic157D411yescharacteristic
88#14578931—21no—
89#14578931—21no—
90#14578931—21no—
91#14578931—21no—
92#14578931—21no—
93#14578931—21no—
94#14578931D411yescharacteristic
95C4×Dic157—21no—
96C4×Dic157—21no—
97C4×Dic157C2311yescharacteristic
98C2×Q8×C157C2311yescharacteristic
99C2×Q8×C157C2311yescharacteristic
100C22×C4×C157C2311yescharacteristic
101Q16×C157—41no—
102C2×C8×C157—41no—
103C2×C8×C157C2311yescharacteristic
104C2×C8×C157C2311yescharacteristic
105C4⋊C4×C157—21no—
106C4⋊C4×C157—21no—
107C4⋊C4×C157C2311yescharacteristic
108C4⋊C4×C157C2311yescharacteristic
109C22⋊C4×C157—21no—
110C22⋊C4×C157—21no—
111C22×D157—21no—
112C22×D157—21no—
113C22×D157—42no—
114C22×D157—42no—
115C22×D157—81no—
116C22×D157C2×D411yescharacteristic
117D628—21no—
118D628—21no—
119D628—21no—
120D628—42no—
121D628—42no—
122D628—81no—
123D628#4112yes—
124C22⋊D157—22no—
125C2×Dic157—21no—
126C2×Dic157—21no—
127C2×Dic157—42no—
128C2×Dic157C2×D411yescharacteristic
129C2×Dic157C2×D411yescharacteristic
130C2×Dic157C2×D411yescharacteristic
131C4×D157—21no—
132C4×D157—21no—
133C4×D157—42no—
134C4×D157—81no—
135#11929434—21no—
136#11929434—21no—
137C23×C157C2×D411yescharacteristic
138Q8×C157—21no—
139Q8×C157—21no—
140Q8×C157—21no—
141Q8×C157—21no—
142C2×C4×C157—21no—
143C2×C4×C157—21no—
144C2×C4×C157—21no—
145C2×C4×C157—21no—
146C2×C4×C157—21no—
147C2×C4×C157—21no—
148C2×C4×C157C2×D411yescharacteristic
149C2×C4×C157C2411yescharacteristic
150C8×C157—21no—
151C8×C157—21no—
152C22×C157—21no—
153C22×C157—21no—
154C22×C157#13512yes—
155C22×C157C22×D411yescharacteristic
156C2×D157—21no—
157C2×D157—22no—
158C2×D157—21no—
159C2×D157—21no—
160C2×D157—41no—
161C2×D157—41no—
162C2×D157—41no—
163C2×D157—41no—
164C2×D157—41no—
165C2×D157—84no—
166C4×C157—21no—
167C4×C157—41no—
168C4×C157—41no—
169C4×C157—41no—
170C4×C157—41no—
171C4×C157C2×#4111yescharacteristic
172C4×C157H(5,𝔽2)11yescharacteristic
173Dic157—21no—
174Dic157—21no—
175Dic157—21no—
176Dic157—41no—
177Dic157—41no—
178C2×C157—22no—
179C2×C157#54611yescharacteristic
180C2×C157C2×#13511yescharacteristic
181C2×C157#57711yescharacteristic
182D157—41no—
183D157—42no—
184D157—81no—
185C157#329611yescharacteristic, complemented, Sylow
186#3296—1571nomaximal, Sylow
187#552—1571no—
188#546—1571no—
189#488—1571no—
190#483—1571no—
191#482—1571no—
192#478—1571no—
193#467D15711yescharacteristic, complemented
194#465—1571no—
195#464—1571no—
196#452—1571no—
197#450—1571no—
198C4×SD16—1571no—
199C8×D4—1571no—
200#416—1571no—
201C2×#102—1571no—
202C2×#41—1571no—
203C22×D4—1571no—
204C2×Q16C2×D15711yescharacteristic, complemented
205C2×SD16—1571no—
206C2×SD16—1571no—
207C22×C8—1571no—
208C22×C8C2×D15711yescharacteristic
209#126—3141no—
210#125—3141no—
211#125—1571no—
212#124—1571no—
213#123—3141no—
214#123—3141no—
215C22⋊Q8—1571no—
216C22⋊Q8C2×D15711yescharacteristic
217C22⋊Q8C2×D15711yescharacteristic
218#121—3141no—
219#121—1571no—
220#120—3141no—
221C4×Q8—1571no—
222C4×D4—1571no—
223C4×D4—1571no—
224#117—3141no—
225#117—1571no—
226C2×C4⋊C4—1571no—
227C2×C22⋊C4—3141no—
228#107C2×D15711yescharacteristic, complemented
229#106—1571no—
230#106—1571no—
231C4⋊C8—1571no—
232#103—1571no—
233#103—1571no—
234#103C2×D15711yescharacteristic, complemented
235#103C2×D15711yescharacteristic, complemented
236#102—1571no—
237#102—1571no—
238#102—1571no—
239#102—1571no—
240#102—1571no—
241#102—1571no—
242C22⋊C8—1571no—
243C22⋊C8—1571no—
244C4×C8—1571no—
245C24—3141no—
246#41—3142no—
247#41—6281no—
248C2×Q8C22×D15711yescharacteristic
249C2×Q8C22×D15711yescharacteristic
250C2×D4—3142no—
251C2×D4—3141no—
252C2×D4—3141no—
253C2×D4—3141no—
254C2×D4—6281no—
255C2×D4—1571no—
256C2×D4—1571no—
257C2×D4—1571no—
258C22×C4—3141no—
259C22×C4—3141no—
260C22×C4—3141no—
261C22×C4—1571no—
262C22×C4C22×D15711yescharacteristic
263Q16—41no—
264SD16—3142no—
265SD16—6281no—
266C2×C8—41no—
267C2×C8—6281no—
268C2×C8—1571no—
269C2×C8—1571no—
270C2×C8C22×D15711yescharacteristic
271C2×C8C22×D15711yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—3141no—
275C4⋊C4—3141no—
276C4⋊C4—3141no—
277C4⋊C4—3141no—
278C4⋊C4—3141no—
279C4⋊C4—1571no—
280C4⋊C4—1571no—
281C4⋊C4—1571no—
282C4⋊C4C22×D15711yescharacteristic
283C4⋊C4C22×D15711yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—3141no—
287C22⋊C4—3141no—
288C22⋊C4—3141no—
289C22⋊C4—3141no—
290C22⋊C4—3141no—
291C22⋊C4—3141no—
292C22⋊C4—1571no—
293C22⋊C4—1571no—
294C42—3141no—
295C42—3141no—
296C42—1571no—
297C23—3141no—
298C23—3141no—
299C23—6282no—
300C23—6282no—
301C23—1571no—
302C23—12561no—
303C23D4×D15711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—21no—
308D4—3142no—
309D4—3141no—
310D4—3141no—
311D4—3141no—
312D4—6282no—
313D4—6282no—
314D4—1572no—
315D4—12561no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—3141no—
323C2×C4—3141no—
324C2×C4—3141no—
325C2×C4—3141no—
326C2×C4—6282no—
327C2×C4—6282no—
328C2×C4—1571no—
329C2×C4—1571no—
330C2×C4—1571no—
331C2×C4—12561no—
332C2×C4D4×D15711yescharacteristic
333C2×C4C23×D15711yescharacteristic
334C8—21no—
335C8—21no—
336C8—3141no—
337C8—3141no—
338C22—21no—
339C22—21no—
340C22—3141no—
341C22—3141no—
342C22—3142no—
343C22—3141no—
344C22—6281no—
345C22—6281no—
346C22—6281no—
347C22—6281no—
348C22—6281no—
349C22—12564no—
350C22#1909779412yes—
351C22C2×D4×D15711yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—3141no—
358C4—3141no—
359C4—3141no—
360C4—6281no—
361C4—6281no—
362C4#41×D15711yescharacteristic
363C4#1909781611yescharacteristic
364C2—22no—
365C2—6281no—
366C2—6282no—
367C2—12561no—
368C2#2375791211yescharacteristic
369C2C2×#1909779411yescharacteristic
370C2#2375808011yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#467×C157C2
socleC22×C157C22×D4
prime radical#23757914C2
hypercenter#467D157
hypocenterC157#3296
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. #23757965C2semidirect
3. #23757629C2semidirect
4. #23757601C2semidirect
5. #23757600C2semidirect
6. #23757576C2semidirect
7. #23757505C2semidirect
8. #23757497C2semidirect
9. #23757434C2semidirect
10. #23757423C2semidirect
11. #23757358C2semidirect
12. C8×C22⋊D157C2semidirect
13. #23757268C2semidirect
14. C2×#19097693C2semidirect
15. #467×C157C2semidirect
20. C2×C8×D157C22semidirect
23. #19097762C22semidirect
27. #19097747C22semidirect
31. Q8×Dic157C22semidirect
34. #19097729C22semidirect
38. #19097702C22semidirect
39. #19097702C22semidirect
40. #19097700C22semidirect
41. #19097695C22semidirect
42. #19097695C22semidirect
50. #19097683C22semidirect
51. C8×Dic157C22semidirect
52. C2×Q16×C157C22semidirect
56. #107×C157C22semidirect
57. #103×C157C22semidirect
58. #103×C157C22semidirect
185. C157#3296semidirect
193. #467D157semidirect
204. C2×Q16C2×D157semidirect
228. #107C2×D157semidirect
234. #103C2×D157semidirect
235. #103C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237579652
#237579142
#237576292
#237576012
#237576002
#237575762
#237575052
#237574972
#237574342
#237574232
#237573582
C8×C22⋊D1572
#237572682
C2×#190976932
#467×C1572
#3296157