Finite Group Databasebeta

← #34111968

Subgroups — #34111968

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#23757961C211yesmaximal, characteristic, complemented
2#23757918C211yesmaximal, characteristic
3#23757630C211yesmaximal, characteristic, complemented
4#23757605C211yesmaximal, characteristic, complemented
5#23757600C211yesmaximal, characteristic, complemented
6#23757515C211yesmaximal, characteristic, complemented
7#23757503C211yesmaximal, characteristic, complemented
8#23757496C211yesmaximal, characteristic, complemented
9#23757436C211yesmaximal, characteristic, complemented
10#23757425C211yesmaximal, characteristic, complemented
11C4×#14578942C211yesmaximal, characteristic, complemented
12#23757338C211yesmaximal, characteristic, complemented
13#23757266C211yesmaximal, characteristic, complemented
14#102×D157C211yesmaximal, characteristic, complemented
15#478×C157C211yesmaximal, characteristic, complemented
16C2×#14578953C2211yescharacteristic
17C2×D4×D157C2211yescharacteristic
18C2×Dic628C2211yescharacteristic, complemented
19C2×#14578943C2211yescharacteristic
20C2×#14578942C2211yescharacteristic
21C2×C8×D157C2211yescharacteristic
22C2×C8×D157C2211yescharacteristic, complemented
23#19097766—21no—
24#19097762—21no—
25#19097759C2211yescharacteristic
26#19097756—21no—
27#19097755—21no—
28#19097751C2211yescharacteristic
29#19097751C2211yescharacteristic
30#19097751C2211yescharacteristic, complemented
31#19097746C2211yescharacteristic
32#19097745—21no—
33#19097741—21no—
34C4×Dic314C2211yescharacteristic, complemented
35C4×D628C2211yescharacteristic
36#19097729—21no—
37#19097729C2211yescharacteristic, complemented
38C4⋊C4×D157C2211yescharacteristic
39C2×#14578931—21no—
40#19097702C2211yescharacteristic, complemented
41#19097696C2211yescharacteristic, complemented
42#19097694C2211yescharacteristic, complemented
43#19097694C2211yescharacteristic, complemented
44#19097694C2211yescharacteristic, complemented
45#19097693C2211yescharacteristic
46#19097692C2211yescharacteristic
47#19097691C2211yescharacteristic
48#19097691C2211yescharacteristic
49#19097683C2211yescharacteristic
50#19097683C2211yescharacteristic, complemented
51C4×#11929434C2211yescharacteristic, complemented
52#125×C157C2211yescharacteristic, complemented
53C4×D4×C157C2211yescharacteristic
54#107×C157C2211yescharacteristic, complemented
55#106×C157C2211yescharacteristic, complemented
56C4⋊C8×C157C2211yescharacteristic, complemented
57#102×C157C2211yescharacteristic
58#102×C157C2211yescharacteristic
59C23×D157—21no—
60#14578953—22no—
61#14578953—41no—
62C2×Dic314C2311yescharacteristic
63C2×Dic314C2311yescharacteristic
64C2×D628—21no—
65C2×D628D411yescharacteristic
66C2×D628C2311yescharacteristic
67C2×C22⋊D157—21no—
68C2×C22⋊D157—21no—
69D4×D157—22no—
70D4×D157—41no—
71C22×Dic157—21no—
72C2×C4×D157—21no—
73C2×C4×D157C2311yescharacteristic
74C2×C4×D157C2311yescharacteristic
75Dic628—41no—
76#14578943—41no—
77#14578942—22no—
78C2×#11929434C2311yescharacteristic
79C2×#11929434C2311yescharacteristic
80C8×D157—41no—
81C8×D157—41no—
82C4⋊Dic157—21no—
83C4⋊Dic157—21no—
84C4⋊Dic157—21no—
85C4⋊Dic157D411yescharacteristic
86C4⋊Dic157C2311yescharacteristic
87#14578933—21no—
88#14578933—21no—
89#14578931—21no—
90#14578931—21no—
91#14578931—21no—
92#14578931—21no—
93#14578931—21no—
94#14578931—21no—
95#14578931—21no—
96#14578931D411yescharacteristic
97#14578931D411yescharacteristic
98C4×Dic157—21no—
99C4×Dic157—21no—
100C2×D4×C157C2311yescharacteristic
101C22×C4×C157—21no—
102C2×C8×C157C2311yescharacteristic
103C2×C8×C157C2311yescharacteristic
104C4⋊C4×C157—21no—
105C4⋊C4×C157—21no—
106C4⋊C4×C157C2311yescharacteristic
107C4⋊C4×C157C2311yescharacteristic
108C4⋊C4×C157C2311yescharacteristic
109C22⋊C4×C157—21no—
110C42×C157C2311yescharacteristic
111C22×D157—21no—
112C22×D157—42no—
113C22×D157—42no—
114C22×D157—81no—
115C22×D157C2×D411yescharacteristic
116C22×D157C2×D411yescharacteristic
117Dic314—21no—
118Dic314—21no—
119Dic314—21no—
120Dic314—21no—
121D628—22no—
122D628—21no—
123D628—21no—
124D628—81no—
125C22⋊D157—42no—
126C22⋊D157—42no—
127C2×Dic157—21no—
128C2×Dic157—21no—
129C2×Dic157—21no—
130C2×Dic157—42no—
131C2×Dic157C2×D411yescharacteristic
132C2×Dic157C2×D411yescharacteristic
133C4×D157—21no—
134C4×D157—21no—
135C4×D157—21no—
136C4×D157—21no—
137C4×D157—42no—
138#11929434—21no—
139#11929434—21no—
140C23×C157—21no—
141D4×C157—21no—
142D4×C157#4112yes—
143C2×C4×C157—21no—
144C2×C4×C157—21no—
145C2×C4×C157—21no—
146C2×C4×C157—81no—
147C2×C4×C157C2×D411yescharacteristic
148C2×C4×C157C2×D411yescharacteristic
149C2×C4×C157C2411yescharacteristic
150C8×C157—21no—
151C8×C157—21no—
152C22×C157—22no—
153C22×C157—41no—
154C22×C157—41no—
155C22×C157C22×D411yescharacteristic
156C2×D157—21no—
157C2×D157—21no—
158C2×D157—21no—
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—21no—
168C4×C157—41no—
169C4×C157—41no—
170C4×C157—41no—
171C4×C157C2×#4111yescharacteristic
172C4×C157H(5,𝔽2)11yescharacteristic
173Dic157—21no—
174Dic157—21no—
175Dic157—41no—
176Dic157—41no—
177Dic157—41no—
178C2×C157—42no—
179C2×C157#54611yescharacteristic
180C2×C157C2×#13511yescharacteristic
181C2×C157#57711yescharacteristic
182D157—22no—
183D157—41no—
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#478D15711yescharacteristic, complemented
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—3141no—
210#125—3141no—
211#125C2×D15711yescharacteristic, complemented
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×D4C2×D15711yescharacteristic
224#117—3141no—
225#117—1571no—
226C2×C4⋊C4—1571no—
227C2×C22⋊C4—3141no—
228#107C2×D15711yescharacteristic, complemented
229#106—1571no—
230#106C2×D15711yescharacteristic, complemented
231C4⋊C8C2×D15711yescharacteristic, complemented
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⋊C8—1571no—
244C4×C8—1571no—
245C24—3141no—
246#41—3142no—
247#41—6281no—
248C2×Q8—1571no—
249C2×Q8—1571no—
250C2×D4—3141no—
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×C4—1571no—
263Q16—6281no—
264SD16—3142no—
265SD16—6281no—
266C2×C8—6281no—
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⋊C4C22×D15711yescharacteristic
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—3141no—
295C42—3141no—
296C42C22×D15711yescharacteristic
297C23—21no—
298C23—3141no—
299C23—6282no—
300C23—6282no—
301C23—1571no—
302C23—1571no—
303C23—12561no—
304Q8—3141no—
305Q8—3141no—
306Q8—3141no—
307Q8—3141no—
308D4—21no—
309D4—3142no—
310D4—3141no—
311D4—3141no—
312D4—6282no—
313D4—6282no—
314D4—12561no—
315D4#1457895512yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—81no—
320C2×C4—3141no—
321C2×C4—3141no—
322C2×C4—3141no—
323C2×C4—3141no—
324C2×C4—3141no—
325C2×C4—3141no—
326C2×C4—3141no—
327C2×C4—6282no—
328C2×C4—6282no—
329C2×C4—1571no—
330C2×C4—1571no—
331C2×C4D4×D15711yescharacteristic
332C2×C4D4×D15711yescharacteristic
333C2×C4C23×D15711yescharacteristic
334C8—21no—
335C8—21no—
336C8—3141no—
337C8—3141no—
338C22—22no—
339C22—41no—
340C22—41no—
341C22—3141no—
342C22—3141no—
343C22—3141no—
344C22—3141no—
345C22—3141no—
346C22—6281no—
347C22—6281no—
348C22—6281no—
349C22—1572no—
350C22—12564no—
351C22C2×D4×D15711yescharacteristic
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—3141no—
358C4—3141no—
359C4—6281no—
360C4—6281no—
361C4—6281no—
362C4C2×#1457895511yescharacteristic
363C4#1909781711yescharacteristic
364C2—42no—
365C2—3142no—
366C2—6281no—
367C2—12561no—
368C2#2375791411yescharacteristic
369C2#135×D15711yescharacteristic
370C2#2375808311yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#478×C157C2
socleC22×C157C22×D4
prime radical#23757918C2
hypercenter#478D157
hypocenterC157#3296
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #23757961C2semidirect
3. #23757630C2semidirect
4. #23757605C2semidirect
5. #23757600C2semidirect
6. #23757515C2semidirect
7. #23757503C2semidirect
8. #23757496C2semidirect
9. #23757436C2semidirect
10. #23757425C2semidirect
11. C4×#14578942C2semidirect
12. #23757338C2semidirect
13. #23757266C2semidirect
14. #102×D157C2semidirect
15. #478×C157C2semidirect
18. C2×Dic628C22semidirect
22. C2×C8×D157C22semidirect
30. #19097751C22semidirect
34. C4×Dic314C22semidirect
37. #19097729C22semidirect
40. #19097702C22semidirect
41. #19097696C22semidirect
42. #19097694C22semidirect
43. #19097694C22semidirect
44. #19097694C22semidirect
50. #19097683C22semidirect
51. C4×#11929434C22semidirect
52. #125×C157C22semidirect
54. #107×C157C22semidirect
55. #106×C157C22semidirect
56. C4⋊C8×C157C22semidirect
185. C157#3296semidirect
192. #478D157semidirect
211. #125C2×D157semidirect
228. #107C2×D157semidirect
230. #106C2×D157semidirect
231. C4⋊C8C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237579612
#237579182
#237576302
#237576052
#237576002
#237575152
#237575032
#237574962
#237574362
#237574252
C4×#145789422
#237573382
#237572662
#102×D1572
#478×C1572
#3296157