Finite Group Databasebeta

← #34111965

Subgroups — #34111965

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#23757919C211yesmaximal, characteristic
2#23757630C211yesmaximal, characteristic, complemented
3#23757603C211yesmaximal, characteristic, complemented
4#23757597C211yesmaximal, characteristic, complemented
5#23757570C211yesmaximal, characteristic, complemented
6#23757512C211yesmaximal, characteristic, complemented
7#23757504C211yesmaximal, characteristic, complemented
8#23757495C211yesmaximal, characteristic, complemented
9#23757433C211yesmaximal, characteristic, complemented
10#23757423C211yesmaximal, characteristic, complemented
11C4×#14578941C211yesmaximal, characteristic, complemented
12D4×#11929434C211yesmaximal, characteristic, complemented
13#23757267C211yesmaximal, characteristic, complemented
14C2×#19097691C211yesmaximal, characteristic, complemented
15#552×C157C211yesmaximal, characteristic, complemented
16C2×#14578954C2211yescharacteristic
17C22×D628C2211yescharacteristic
18C2×#14578944C2211yescharacteristic, complemented
19C2×#14578941C2211yescharacteristic
20C2×#14578941C2211yescharacteristic
21C22×#11929434C2211yescharacteristic
22C22×#11929434C2211yescharacteristic, complemented
23#19097759C2211yescharacteristic
24#19097756—21no—
25#19097750C2211yescharacteristic
26#19097745C2211yescharacteristic
27#19097744—21no—
28#19097739—21no—
29C4×D628C2211yescharacteristic
30C22⋊C4×D157—21no—
31#19097704C2211yescharacteristic, complemented
32#19097702C2211yescharacteristic, complemented
33#19097702C2211yescharacteristic, complemented
34C4⋊#11929434C2211yescharacteristic, complemented
35#19097695C2211yescharacteristic, complemented
36#19097694C2211yescharacteristic, complemented
37#19097694C2211yescharacteristic, complemented
38#19097694C2211yescharacteristic, complemented
39#19097691C2211yescharacteristic
40#19097691C2211yescharacteristic
41#19097691C2211yescharacteristic
42#19097691C2211yescharacteristic
43#19097691C2211yescharacteristic
44#19097691C2211yescharacteristic
45C22⋊#11929434C2211yescharacteristic
46C22⋊#11929434C2211yescharacteristic, complemented
47C4×#11929434C2211yescharacteristic, complemented
48#126×C157—21no—
49#125×C157—21no—
50#125×C157C2211yescharacteristic, complemented
51#123×C157—21no—
52C22⋊Q8×C157C2211yescharacteristic
53C22⋊Q8×C157C2211yescharacteristic, complemented
54C4×Q8×C157C2211yescharacteristic, complemented
55C4×D4×C157C2211yescharacteristic
56#117×C157—21no—
57#117×C157C2211yescharacteristic, complemented
58C2×C4⋊C4×C157C2211yescharacteristic
59C23×D157—21no—
60#14578954—22no—
61#14578954—41no—
62C2×Dic314C2311yescharacteristic
63C2×D628—22no—
64C2×D628—21no—
65C2×D628—41no—
66C2×D628C2311yescharacteristic
67C2×D628C2311yescharacteristic
68C2×C22⋊D157—21no—
69C2×C22⋊D157—21no—
70C2×C4×D157—21no—
71C2×C4×D157—21no—
72#14578944—41no—
73#14578941—22no—
74#14578941—41no—
75C2×#11929434—41no—
76C2×#11929434—41no—
77C2×#11929434C2311yescharacteristic
78C2×#11929434C2311yescharacteristic
79C2×#11929434C2311yescharacteristic
80C2×#11929434C2311yescharacteristic
81C4⋊Dic157C2311yescharacteristic
82#14578933—21no—
83C22⋊Dic157—21no—
84#14578931—21no—
85#14578931—21no—
86#14578931—21no—
87#14578931—21no—
88C2×Q8×C157C2311yescharacteristic
89C2×D4×C157D411yescharacteristic
90C22×C4×C157—21no—
91C22×C4×C157C2311yescharacteristic
92C22×C4×C157C2311yescharacteristic
93C4⋊C4×C157—21no—
94C4⋊C4×C157—21no—
95C4⋊C4×C157—21no—
96C4⋊C4×C157—21no—
97C4⋊C4×C157—21no—
98C4⋊C4×C157—21no—
99C4⋊C4×C157D411yescharacteristic
100C4⋊C4×C157C2311yescharacteristic
101C4⋊C4×C157C2311yescharacteristic
102C4⋊C4×C157C2311yescharacteristic
103C22⋊C4×C157—21no—
104C22⋊C4×C157—21no—
105C22⋊C4×C157—21no—
106C22⋊C4×C157D411yescharacteristic
107C22⋊C4×C157D411yescharacteristic
108C42×C157—21no—
109C42×C157—21no—
110C42×C157C2311yescharacteristic
111C22×D157—21no—
112C22×D157—21no—
113C22×D157—42no—
114C22×D157—42no—
115C22×D157—81no—
116Dic314—21no—
117Dic314—21no—
118D628—21no—
119D628—21no—
120D628—21no—
121D628—42no—
122D628#4112yes—
123C22⋊D157—42no—
124C22⋊D157—81no—
125C2×Dic157—21no—
126C2×Dic157—21no—
127C4×D157—42no—
128C4×D157—81no—
129#11929434—21no—
130#11929434—21no—
131#11929434—21no—
132#11929434—21no—
133C23×C157C2×D411yescharacteristic
134C23×C157C2×D411yescharacteristic
135Q8×C157—21no—
136Q8×C157—21no—
137D4×C157—22no—
138C2×C4×C157—21no—
139C2×C4×C157—21no—
140C2×C4×C157—21no—
141C2×C4×C157—21no—
142C2×C4×C157—21no—
143C2×C4×C157—21no—
144C2×C4×C157—21no—
145C2×C4×C157—21no—
146C2×C4×C157—42no—
147C2×C4×C157C2×D411yescharacteristic
148C2×C4×C157C2×D411yescharacteristic
149C2×C4×C157C2×D411yescharacteristic
150C2×C4×C157C2×D411yescharacteristic
151C2×C4×C157C2411yescharacteristic
152C22×C157—21no—
153C22×C157—21no—
154C22×C157—21no—
155C22×C157—21no—
156C22×C157—21no—
157C22×C157#13512yes—
158C22×C157C22×D411yescharacteristic
159C2×D157—22no—
160C2×D157—41no—
161C2×D157—41no—
162C2×D157—41no—
163C2×D157—41no—
164C2×D157—41no—
165C2×D157—84no—
166C4×C157—21no—
167C4×C157—21no—
168C4×C157—21no—
169C4×C157—21no—
170C4×C157—41no—
171C4×C157—41no—
172C4×C157—41no—
173C4×C157—41no—
174C4×C157C2×#4111yescharacteristic
175C4×C157H(5,𝔽2)11yescharacteristic
176Dic157—41no—
177Dic157—41no—
178C2×C157—22no—
179C2×C157—41no—
180C2×C157#54611yescharacteristic
181C2×C157C2×#13511yescharacteristic
182C2×C157#57711yescharacteristic
183D157—42no—
184D157—81no—
185C157#329611yescharacteristic, complemented, Sylow
186#3296—1571nomaximal, Sylow
187#552D15711yescharacteristic, complemented
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#416—1571no—
201C2×#102—1571no—
202C2×#41—1571no—
203C22×D4—1571no—
204C2×Q16—1571no—
205C2×SD16—1571no—
206C2×SD16—1571no—
207C22×C8—1571no—
208C22×C8—1571no—
209#126—21no—
210#125—21no—
211#125C2×D15711yescharacteristic, complemented
212#124—1571no—
213#123—21no—
214#123—3141no—
215C22⋊Q8—1571no—
216C22⋊Q8C2×D15711yescharacteristic
217C22⋊Q8C2×D15711yescharacteristic, complemented
218#121—3141no—
219#121—1571no—
220#120—3141no—
221C4×Q8C2×D15711yescharacteristic, complemented
222C4×D4—1571no—
223C4×D4C2×D15711yescharacteristic
224#117—21no—
225#117C2×D15711yescharacteristic, complemented
226C2×C4⋊C4C2×D15711yescharacteristic
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#102—1571no—
241#102—1571no—
242C22⋊C8—1571no—
243C22⋊C8—1571no—
244C4×C8—1571no—
245C24—3141no—
246#41—3142no—
247#41—6281no—
248C2×Q8—1571no—
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×D4C22⋊D15711yescharacteristic
258C22×C4—21no—
259C22×C4—3141no—
260C22×C4—3141no—
261C22×C4C22×D15711yescharacteristic
262C22×C4C22×D15711yescharacteristic
263Q16—6281no—
264SD16—3142no—
265SD16—6281no—
266C2×C8—6281no—
267C2×C8—6281no—
268C2×C8—1571no—
269C2×C8—1571no—
270C2×C8—1571no—
271C2×C8—1571no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—3141no—
279C4⋊C4—1571no—
280C4⋊C4C22⋊D15711yescharacteristic
281C4⋊C4C22×D15711yescharacteristic
282C4⋊C4C22×D15711yescharacteristic
283C4⋊C4C22×D15711yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—3141no—
288C22⋊C4—3141no—
289C22⋊C4—3141no—
290C22⋊C4—3141no—
291C22⋊C4—3141no—
292C22⋊C4C22⋊D15711yescharacteristic
293C22⋊C4C22⋊D15711yescharacteristic
294C42—21no—
295C42—21no—
296C42C22×D15711yescharacteristic
297C23—3141no—
298C23—3141no—
299C23—6282no—
300C23—6282no—
301C23—12561no—
302C23C2×C22⋊D15711yescharacteristic
303C23C2×C22⋊D15711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—3141no—
307Q8—3141no—
308D4—22no—
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—21no—
323C2×C4—21no—
324C2×C4—42no—
325C2×C4—3141no—
326C2×C4—3141no—
327C2×C4—6282no—
328C2×C4—12561no—
329C2×C4C2×C22⋊D15711yescharacteristic
330C2×C4C2×C22⋊D15711yescharacteristic
331C2×C4C2×C22⋊D15711yescharacteristic
332C2×C4C2×C22⋊D15711yescharacteristic
333C2×C4C23×D15711yescharacteristic
334C8—3141no—
335C8—3141no—
336C8—3141no—
337C8—3141no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—3142no—
344C22—6281no—
345C22—6281no—
346C22—6281no—
347C22—6281no—
348C22—6281no—
349C22—12564no—
350C22#1909779112yes—
351C22C22×C22⋊D15711yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—6281no—
361C4—6281no—
362C4#41×D15711yescharacteristic
363C4#1909781711yescharacteristic
364C2—22no—
365C2—41no—
366C2—6282no—
367C2—12561no—
368C2#2375791111yescharacteristic
369C2C2×#1909779111yescharacteristic
370C2#2375808111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#552×C157C2
socleC22×C157C22×D4
prime radical#23757919C2
hypercenter#552D157
hypocenterC157#3296
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #23757630C2semidirect
3. #23757603C2semidirect
4. #23757597C2semidirect
5. #23757570C2semidirect
6. #23757512C2semidirect
7. #23757504C2semidirect
8. #23757495C2semidirect
9. #23757433C2semidirect
10. #23757423C2semidirect
11. C4×#14578941C2semidirect
12. D4×#11929434C2semidirect
13. #23757267C2semidirect
14. C2×#19097691C2semidirect
15. #552×C157C2semidirect
18. C2×#14578944C22semidirect
22. C22×#11929434C22semidirect
31. #19097704C22semidirect
32. #19097702C22semidirect
33. #19097702C22semidirect
34. C4⋊#11929434C22semidirect
35. #19097695C22semidirect
36. #19097694C22semidirect
37. #19097694C22semidirect
38. #19097694C22semidirect
46. C22⋊#11929434C22semidirect
47. C4×#11929434C22semidirect
50. #125×C157C22semidirect
53. C22⋊Q8×C157C22semidirect
54. C4×Q8×C157C22semidirect
57. #117×C157C22semidirect
185. C157#3296semidirect
187. #552D157semidirect
211. #125C2×D157semidirect
217. C22⋊Q8C2×D157semidirect
221. C4×Q8C2×D157semidirect
225. #117C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237579192
#237576302
#237576032
#237575972
#237575702
#237575122
#237575042
#237574952
#237574332
#237574232
C4×#145789412
D4×#119294342
#237572672
C2×#190976912
#552×C1572
#3296157