Finite Group Databasebeta

← #34111972

Subgroups — #34111972

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#23757962C211yesmaximal, characteristic, complemented
2#23757911C211yesmaximal, characteristic
3#23757629C211yesmaximal, characteristic, complemented
4#23757602C211yesmaximal, characteristic, complemented
5#23757597C211yesmaximal, characteristic, complemented
6#23757575C211yesmaximal, characteristic, complemented
7#23757513C211yesmaximal, characteristic, complemented
8#23757502C211yesmaximal, characteristic, complemented
9#23757498C211yesmaximal, characteristic, complemented
10#23757435C211yesmaximal, characteristic, complemented
11#23757425C211yesmaximal, characteristic, complemented
12#23757364C211yesmaximal, characteristic, complemented
13C8×C22⋊D157C211yesmaximal, characteristic, complemented
14#102×D157C211yesmaximal, characteristic, complemented
15#416×C157C211yesmaximal, characteristic, complemented
16C2×D4×D157C2211yescharacteristic
17C2×#14578944C2211yescharacteristic, complemented
18C2×#14578943C2211yescharacteristic
19C2×#14578942C2211yescharacteristic
20C2×C8×D157C2211yescharacteristic
21#19097764—21no—
22#19097763—21no—
23#19097762C2211yescharacteristic, complemented
24#19097760C2211yescharacteristic
25#19097754—21no—
26#19097753—21no—
27#19097751C2211yescharacteristic
28#19097750C2211yescharacteristic, complemented
29#19097747C2211yescharacteristic
30#19097745C2211yescharacteristic
31#19097743—21no—
32#19097739—21no—
33#19097736C2211yescharacteristic, complemented
34D4×Dic157C2211yescharacteristic
35C4×C22⋊D157C2211yescharacteristic
36#19097728—21no—
37C4⋊C4×D157C2211yescharacteristic
38C2×#14578931—21no—
39#19097704C2211yescharacteristic, complemented
40#19097703C2211yescharacteristic, complemented
41#19097702C2211yescharacteristic, complemented
42#19097700C2211yescharacteristic, complemented
43#19097696C2211yescharacteristic, complemented
44#19097695C2211yescharacteristic, complemented
45#19097693C2211yescharacteristic
46#19097692C2211yescharacteristic
47#19097691C2211yescharacteristic
48#19097691C2211yescharacteristic
49C22⋊#11929434C2211yescharacteristic, complemented
50#19097683C2211yescharacteristic
51C8×Dic157C2211yescharacteristic, complemented
52C2×#41×C157C2211yescharacteristic
53C22×C8×C157C2211yescharacteristic, complemented
54#117×C157C2211yescharacteristic, complemented
55#103×C157C2211yescharacteristic, complemented
56#103×C157C2211yescharacteristic, complemented
57#102×C157C2211yescharacteristic
58#102×C157C2211yescharacteristic
59C23×D157—21no—
60C2×Dic314C2311yescharacteristic
61C2×D628C2311yescharacteristic
62C2×C22⋊D157—21no—
63C2×C22⋊D157—21no—
64C2×C22⋊D157D411yescharacteristic
65D4×D157—22no—
66D4×D157—41no—
67C22×Dic157—21no—
68C2×C4×D157—21no—
69C2×C4×D157C2311yescharacteristic
70#14578944—41no—
71#14578943—41no—
72#14578942—22no—
73C2×#11929434C2311yescharacteristic
74C2×#11929434C2311yescharacteristic
75C8×D157—41no—
76C4⋊Dic157—21no—
77C4⋊Dic157—21no—
78C4⋊Dic157C2311yescharacteristic
79C4⋊Dic157C2311yescharacteristic
80#14578933—21no—
81#14578933—21no—
82#14578933—21no—
83#14578933—21no—
84#14578933—21no—
85#14578933D411yescharacteristic
86C22⋊Dic157—21no—
87C22⋊Dic157—21no—
88C22⋊Dic157D411yescharacteristic
89#14578931—21no—
90#14578931—21no—
91#14578931—21no—
92#14578931—21no—
93#14578931—21no—
94#14578931D411yescharacteristic
95C4×Dic157—21no—
96C4×Dic157C2311yescharacteristic
97#41×C157—22no—
98#41×C157—41no—
99C2×Q8×C157C2311yescharacteristic
100C2×D4×C157—21no—
101C2×D4×C157C2311yescharacteristic
102C22×C4×C157—21no—
103C22×C4×C157C2311yescharacteristic
104C2×C8×C157—41no—
105C2×C8×C157C2311yescharacteristic
106C2×C8×C157C2311yescharacteristic
107C4⋊C4×C157C2311yescharacteristic
108C4⋊C4×C157C2311yescharacteristic
109C22⋊C4×C157—21no—
110C42×C157—21no—
111C22×D157—21no—
112C22×D157—42no—
113C22×D157—42no—
114C22×D157—81no—
115C22×D157C2×D411yescharacteristic
116Dic314—21no—
117Dic314—21no—
118D628—21no—
119D628—21no—
120C22⋊D157—22no—
121C22⋊D157—42no—
122C22⋊D157—81no—
123C2×Dic157—21no—
124C2×Dic157—21no—
125C2×Dic157—21no—
126C2×Dic157—81no—
127C2×Dic157C2×D411yescharacteristic
128C2×Dic157C2×D411yescharacteristic
129C2×Dic157C2×D411yescharacteristic
130C4×D157—21no—
131C4×D157—21no—
132C4×D157—42no—
133#11929434—21no—
134#11929434—21no—
135C23×C157—21no—
136C23×C157C2×D411yescharacteristic
137Q8×C157—21no—
138Q8×C157—21no—
139D4×C157—21no—
140D4×C157—42no—
141D4×C157#4112yes—
142C2×C4×C157—21no—
143C2×C4×C157—21no—
144C2×C4×C157—21no—
145C2×C4×C157—21no—
146C2×C4×C157—21no—
147C2×C4×C157—42no—
148C2×C4×C157C2×D411yescharacteristic
149C2×C4×C157C2411yescharacteristic
150C8×C157—21no—
151C8×C157—21no—
152C22×C157—21no—
153C22×C157—22no—
154C22×C157—21no—
155C22×C157—21no—
156C22×C157—41no—
157C22×C157—41no—
158C22×C157C22×D411yescharacteristic
159C2×D157—21no—
160C2×D157—21no—
161C2×D157—41no—
162C2×D157—41no—
163C2×D157—41no—
164C2×D157—84no—
165C2×D157#13512yes—
166C4×C157—21no—
167C4×C157—41no—
168C4×C157—41no—
169C4×C157—41no—
170C4×C157C2×#4111yescharacteristic
171C4×C157H(5,𝔽2)11yescharacteristic
172Dic157—21no—
173Dic157—21no—
174Dic157—21no—
175Dic157—41no—
176Dic157—41no—
177Dic157—41no—
178C2×C157—42no—
179C2×C157—41no—
180C2×C157#54611yescharacteristic
181C2×C157C2×#13511yescharacteristic
182C2×C157#57711yescharacteristic
183D157—22no—
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#467—1571no—
194#465—1571no—
195#464—1571no—
196#452—1571no—
197#450—1571no—
198C4×SD16—1571no—
199C8×D4—1571no—
200#416D15711yescharacteristic, complemented
201C2×#102—1571no—
202C2×#41C2×D15711yescharacteristic
203C22×D4—1571no—
204C2×Q16—1571no—
205C2×SD16—1571no—
206C2×SD16—1571no—
207C22×C8—1571no—
208C22×C8C2×D15711yescharacteristic, complemented
209#126—3141no—
210#125—3141no—
211#125—1571no—
212#124—1571no—
213#123—3141no—
214#123—3141no—
215C22⋊Q8—1571no—
216C22⋊Q8—1571no—
217C22⋊Q8—1571no—
218#121—3141no—
219#121—1571no—
220#120—3141no—
221C4×Q8—1571no—
222C4×D4—1571no—
223C4×D4—1571no—
224#117—3141no—
225#117C2×D15711yescharacteristic, complemented
226C2×C4⋊C4—1571no—
227C2×C22⋊C4—3141no—
228#107—1571no—
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#102C2×D15711yescharacteristic
241#102C2×D15711yescharacteristic
242C22⋊C8—1571no—
243C22⋊C8—1571no—
244C4×C8—1571no—
245C24—3141no—
246#41—22no—
247#41—41no—
248C2×Q8—1571no—
249C2×Q8C22×D15711yescharacteristic
250C2×D4—21no—
251C2×D4—3142no—
252C2×D4—3141no—
253C2×D4—3141no—
254C2×D4—6281no—
255C2×D4—1571no—
256C2×D4—1571no—
257C2×D4C22×D15711yescharacteristic
258C22×C4—21no—
259C22×C4—3141no—
260C22×C4—3141no—
261C22×C4—1571no—
262C22×C4C22×D15711yescharacteristic
263Q16—6281no—
264SD16—3142no—
265SD16—6281no—
266C2×C8—41no—
267C2×C8—6281no—
268C2×C8—1571no—
269C2×C8—1571no—
270C2×C8C22×D15711yescharacteristic
271C2×C8C22×D15711yescharacteristic
272C4⋊C4—3141no—
273C4⋊C4—3141no—
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—3141no—
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—21no—
295C42—3141no—
296C42—1571no—
297C23—21no—
298C23—3141no—
299C23—6282no—
300C23—6282no—
301C23—1571no—
302C23—12561no—
303C23D4×D15711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—3141no—
307Q8—3141no—
308D4—21no—
309D4—42no—
310D4—3142no—
311D4—3141no—
312D4—3141no—
313D4—6282no—
314D4—12561no—
315D4#1457895412yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—3141no—
323C2×C4—3141no—
324C2×C4—3141no—
325C2×C4—3141no—
326C2×C4—3141no—
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—22no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—3141no—
345C22—3141no—
346C22—6281no—
347C22—6281no—
348C22—6281no—
349C22—1572no—
350C22—12564no—
351C22C2×D4×D15711yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—3141no—
357C4—3141no—
358C4—3141no—
359C4—6281no—
360C4—6281no—
361C4—6281no—
362C4C2×#1457895411yescharacteristic
363C4#1909781611yescharacteristic
364C2—41no—
365C2—42no—
366C2—3142no—
367C2—12561no—
368C2#2375791311yescharacteristic
369C2#135×D15711yescharacteristic
370C2#2375808011yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#416×C157C2
socleC22×C157C22×D4
prime radical#23757911C2
hypercenter#416D157
hypocenterC157#3296
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #23757962C2semidirect
3. #23757629C2semidirect
4. #23757602C2semidirect
5. #23757597C2semidirect
6. #23757575C2semidirect
7. #23757513C2semidirect
8. #23757502C2semidirect
9. #23757498C2semidirect
10. #23757435C2semidirect
11. #23757425C2semidirect
12. #23757364C2semidirect
13. C8×C22⋊D157C2semidirect
14. #102×D157C2semidirect
15. #416×C157C2semidirect
17. C2×#14578944C22semidirect
23. #19097762C22semidirect
28. #19097750C22semidirect
33. #19097736C22semidirect
39. #19097704C22semidirect
40. #19097703C22semidirect
41. #19097702C22semidirect
42. #19097700C22semidirect
43. #19097696C22semidirect
44. #19097695C22semidirect
49. C22⋊#11929434C22semidirect
51. C8×Dic157C22semidirect
53. C22×C8×C157C22semidirect
54. #117×C157C22semidirect
55. #103×C157C22semidirect
56. #103×C157C22semidirect
185. C157#3296semidirect
200. #416D157semidirect
208. C22×C8C2×D157semidirect
225. #117C2×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
#237579622
#237579112
#237576292
#237576022
#237575972
#237575752
#237575132
#237575022
#237574982
#237574352
#237574252
#237573642
C8×C22⋊D1572
#102×D1572
#416×C1572
#3296157