Finite Group Databasebeta

← #34111975

Subgroups — #34111975

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#23757917C211yesmaximal, characteristic
3#23757626C211yesmaximal, characteristic, complemented
4#23757604C211yesmaximal, characteristic, complemented
5#23757596C211yesmaximal, characteristic, complemented
6#23757575C211yesmaximal, characteristic, complemented
7#23757512C211yesmaximal, characteristic, complemented
8#23757499C211yesmaximal, characteristic, complemented
9#23757493C211yesmaximal, characteristic, complemented
10#23757432C211yesmaximal, characteristic, complemented
11SD16×Dic157C211yesmaximal, characteristic, complemented
12#23757336C211yesmaximal, characteristic, complemented
13#23757266C211yesmaximal, characteristic, complemented
14C2×#19097692C211yesmaximal, characteristic, complemented
15#450×C157C211yesmaximal, characteristic, complemented
16C2×#14578953C2211yescharacteristic
17C2×#14578944C2211yescharacteristic, complemented
18C22×#11929434C2211yescharacteristic
19C2×C8×D157C2211yescharacteristic, complemented
20#19097766—21no—
21#19097762—21no—
22#19097762C2211yescharacteristic, complemented
23#19097758C2211yescharacteristic
24#19097755—21no—
25#19097752—21no—
26#19097750C2211yescharacteristic
27#19097747C2211yescharacteristic, complemented
28#19097743—21no—
29#19097742C2211yescharacteristic
30#19097740—21no—
31Q8×Dic157C2211yescharacteristic, complemented
32D4×Dic157C2211yescharacteristic
33C4×C22⋊D157C2211yescharacteristic
34#19097729C2211yescharacteristic, complemented
35#19097727—21no—
36C2×C4⋊Dic157C2211yescharacteristic
37C2×C22⋊Dic157—21no—
38#19097704C2211yescharacteristic, complemented
39#19097703C2211yescharacteristic, complemented
40#19097703C2211yescharacteristic, complemented
41#19097700C2211yescharacteristic, complemented
42#19097696C2211yescharacteristic, complemented
43#19097696C2211yescharacteristic, complemented
44#19097695C2211yescharacteristic, complemented
45#19097694C2211yescharacteristic, complemented
46#19097692C2211yescharacteristic
47#19097692C2211yescharacteristic
48#19097692C2211yescharacteristic
49#19097692C2211yescharacteristic
50#19097683C2211yescharacteristic, complemented
51C8×Dic157C2211yescharacteristic, complemented
52C22×D4×C157C2211yescharacteristic
53C2×SD16×C157C2211yescharacteristic
54C2×SD16×C157C2211yescharacteristic
55C22⋊Q8×C157C2211yescharacteristic
56#102×C157C2211yescharacteristic
57#102×C157C2211yescharacteristic
58C22⋊C8×C157C2211yescharacteristic
59#14578953—22no—
60#14578953—41no—
61C2×Dic314C2311yescharacteristic
62C2×C22⋊D157—21no—
63C2×C22⋊D157—21no—
64C2×C22⋊D157D411yescharacteristic
65C22×Dic157—21no—
66C22×Dic157—21no—
67C22×Dic157—21no—
68C2×C4×D157C2311yescharacteristic
69#14578944—41no—
70C2×#11929434—41no—
71C2×#11929434C2311yescharacteristic
72C2×#11929434C2311yescharacteristic
73C8×D157—41no—
74C4⋊Dic157—21no—
75C4⋊Dic157—21no—
76C4⋊Dic157—21no—
77C4⋊Dic157C2311yescharacteristic
78C4⋊Dic157C2311yescharacteristic
79C4⋊Dic157C2311yescharacteristic
80#14578933—21no—
81#14578933—21no—
82#14578933—21no—
83#14578933D411yescharacteristic
84C22⋊Dic157—21no—
85C22⋊Dic157—21no—
86C22⋊Dic157—21no—
87C22⋊Dic157—21no—
88C22⋊Dic157—21no—
89C22⋊Dic157D411yescharacteristic
90#14578931—21no—
91#14578931—21no—
92#14578931D411yescharacteristic
93C4×Dic157—21no—
94C4×Dic157—21no—
95C4×Dic157C2311yescharacteristic
96C24×C157—21no—
97C2×Q8×C157C2311yescharacteristic
98C2×D4×C157—22no—
99C2×D4×C157—21no—
100C2×D4×C157—41no—
101C2×D4×C157C2311yescharacteristic
102C2×D4×C157C2311yescharacteristic
103C22×C4×C157C2311yescharacteristic
104SD16×C157—22no—
105SD16×C157—41no—
106C2×C8×C157C2311yescharacteristic
107C2×C8×C157C2311yescharacteristic
108C4⋊C4×C157—21no—
109C4⋊C4×C157C2311yescharacteristic
110C22⋊C4×C157—21no—
111C22×D157C2×D411yescharacteristic
112Dic314—21no—
113Dic314—21no—
114C22⋊D157—22no—
115C22⋊D157—42no—
116C22⋊D157—81no—
117C2×Dic157—21no—
118C2×Dic157—21no—
119C2×Dic157—21no—
120C2×Dic157—21no—
121C2×Dic157—42no—
122C2×Dic157—42no—
123C2×Dic157—81no—
124C2×Dic157C2×D411yescharacteristic
125C2×Dic157C2×D411yescharacteristic
126C2×Dic157C2×D411yescharacteristic
127C4×D157—21no—
128C4×D157—21no—
129#11929434—21no—
130#11929434—21no—
131C23×C157—21no—
132C23×C157—21no—
133C23×C157—42no—
134C23×C157—42no—
135C23×C157—81no—
136C23×C157C2×D411yescharacteristic
137Q8×C157—21no—
138Q8×C157—21no—
139D4×C157—21no—
140D4×C157—21no—
141D4×C157—21no—
142D4×C157—42no—
143D4×C157#4112yes—
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—22no—
155C22×C157—41no—
156C22×C157—41no—
157C22×C157—41no—
158C22×C157—41no—
159C22×C157—41no—
160C22×C157—84no—
161C22×C157#13512yes—
162C22×C157C22×D411yescharacteristic
163C2×D157—21no—
164C2×D157—21no—
165C2×D157—21no—
166C4×C157—21no—
167C4×C157—41no—
168C4×C157—41no—
169C4×C157C2×#4111yescharacteristic
170C4×C157H(5,𝔽2)11yescharacteristic
171Dic157—21no—
172Dic157—21no—
173Dic157—21no—
174Dic157—41no—
175Dic157—41no—
176Dic157—41no—
177Dic157—41no—
178C2×C157—22no—
179C2×C157—42no—
180C2×C157—81no—
181C2×C157#54611yescharacteristic
182C2×C157C2×#13511yescharacteristic
183C2×C157#57711yescharacteristic
184D157—41no—
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#450D15711yescharacteristic, complemented
198C4×SD16—1571no—
199C8×D4—1571no—
200#416—1571no—
201C2×#102—1571no—
202C2×#41—1571no—
203C22×D4C2×D15711yescharacteristic
204C2×Q16—1571no—
205C2×SD16C2×D15711yescharacteristic
206C2×SD16C2×D15711yescharacteristic
207C22×C8—1571no—
208C22×C8—1571no—
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⋊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#107—1571no—
229#106—1571no—
230#106—1571no—
231C4⋊C8—1571no—
232#103—1571no—
233#103—1571no—
234#103—1571no—
235#103—1571no—
236#102—1571no—
237#102—1571no—
238#102—1571no—
239#102—1571no—
240#102C2×D15711yescharacteristic
241#102C2×D15711yescharacteristic
242C22⋊C8—1571no—
243C22⋊C8C2×D15711yescharacteristic
244C4×C8—1571no—
245C24—21no—
246#41—3142no—
247#41—6281no—
248C2×Q8—1571no—
249C2×Q8C22×D15711yescharacteristic
250C2×D4—22no—
251C2×D4—21no—
252C2×D4—41no—
253C2×D4—3141no—
254C2×D4—3141no—
255C2×D4—1571no—
256C2×D4C22×D15711yescharacteristic
257C2×D4C22×D15711yescharacteristic
258C22×C4—3141no—
259C22×C4—3141no—
260C22×C4—3141no—
261C22×C4—1571no—
262C22×C4C22×D15711yescharacteristic
263Q16—6281no—
264SD16—22no—
265SD16—41no—
266C2×C8—6281no—
267C2×C8—6281no—
268C2×C8—1571no—
269C2×C8—1571no—
270C2×C8C22×D15711yescharacteristic
271C2×C8C22×D15711yescharacteristic
272C4⋊C4—21no—
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⋊C4—1571no—
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—3141no—
295C42—3141no—
296C42—1571no—
297C23—21no—
298C23—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23—1571no—
303C23D4×D15711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—3141no—
307Q8—3141no—
308D4—21no—
309D4—21no—
310D4—21no—
311D4—42no—
312D4—3142no—
313D4—6282no—
314D4—12561no—
315D4#1457895312yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—3141no—
321C2×C4—3141no—
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—22no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—84no—
347C22—3141no—
348C22—3141no—
349C22—3141no—
350C22#1909779212yes—
351C22C2×D4×D15711yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—3141no—
356C4—3141no—
357C4—3141no—
358C4—6281no—
359C4—6281no—
360C4—6281no—
361C4—6281no—
362C4C2×#1457895311yescharacteristic
363C4#1909781611yescharacteristic
364C2—22no—
365C2—42no—
366C2—81no—
367C2—6281no—
368C2#2375791011yescharacteristic
369C2C2×#1909779211yescharacteristic
370C2#2375808311yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#450×C157C2
socleC22×C157C22×D4
prime radical#23757917C2
hypercenter#450D157
hypocenterC157#3296
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #23757965C2semidirect
3. #23757626C2semidirect
4. #23757604C2semidirect
5. #23757596C2semidirect
6. #23757575C2semidirect
7. #23757512C2semidirect
8. #23757499C2semidirect
9. #23757493C2semidirect
10. #23757432C2semidirect
11. SD16×Dic157C2semidirect
12. #23757336C2semidirect
13. #23757266C2semidirect
14. C2×#19097692C2semidirect
15. #450×C157C2semidirect
17. C2×#14578944C22semidirect
19. C2×C8×D157C22semidirect
22. #19097762C22semidirect
27. #19097747C22semidirect
31. Q8×Dic157C22semidirect
34. #19097729C22semidirect
38. #19097704C22semidirect
39. #19097703C22semidirect
40. #19097703C22semidirect
41. #19097700C22semidirect
42. #19097696C22semidirect
43. #19097696C22semidirect
44. #19097695C22semidirect
45. #19097694C22semidirect
50. #19097683C22semidirect
51. C8×Dic157C22semidirect
185. C157#3296semidirect
197. #450D157semidirect

Maximal subgroups

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

subgroupindex
#237579652
#237579172
#237576262
#237576042
#237575962
#237575752
#237575122
#237574992
#237574932
#237574322
SD16×Dic1572
#237573362
#237572662
C2×#190976922
#450×C1572
#3296157