Finite Group Databasebeta

← #34112155

Subgroups — #34112155

Order 20096.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757958C211yesmaximal, characteristic, complemented
2#23757907C211yesmaximal, characteristic
3#23757636C211yesmaximal, characteristic, complemented
4#23757600C211yesmaximal, characteristic, complemented
5#23757576C211yesmaximal, characteristic, complemented
6#23757520C211yesmaximal, characteristic, complemented
7#23757500C211yesmaximal, characteristic, complemented
8#23757472C211yesmaximal, characteristic, complemented
9#23757426C211yesmaximal, characteristic, complemented
10#23757425C211yesmaximal, characteristic, complemented
11#23757372C211yesmaximal, characteristic, complemented
12#23757341C211yesmaximal, characteristic, complemented
13#23757276C211yesmaximal, characteristic, complemented
14C2×#19097691C211yesmaximal, characteristic, complemented
15#480×C157C211yesmaximal, characteristic, complemented
16C22×D628C2211yescharacteristic
17C2×D4×D157—21no—
18C2×D4×D157C2211yescharacteristic
19C2×#14578943C2211yescharacteristic
20C2×#14578943C2211yescharacteristic, complemented
21C2×#14578942C2211yescharacteristic, complemented
22C2×#14578937C2211yescharacteristic, complemented
23C22×#11929434C2211yescharacteristic, complemented
24#19097767C2211yescharacteristic
25#19097766—21no—
26#19097762C2211yescharacteristic, complemented
27#19097761—21no—
28#19097754—21no—
29#19097751C2211yescharacteristic, complemented
30#19097750C2211yescharacteristic, complemented
31#19097746C2211yescharacteristic
32#19097744—21no—
33#19097741—21no—
34#19097739—21no—
35#19097736C2211yescharacteristic, complemented
36#19097733C2211yescharacteristic
37C4×C22⋊D157C2211yescharacteristic
38#19097730—21no—
39#19097729C2211yescharacteristic, complemented
40#19097702C2211yescharacteristic, complemented
41#19097700C2211yescharacteristic, complemented
42#19097694C2211yescharacteristic, complemented
43#19097693C2211yescharacteristic
44#19097693C2211yescharacteristic, complemented
45#19097692C2211yescharacteristic, complemented
46#19097691C2211yescharacteristic
47#19097691C2211yescharacteristic
48#19097691C2211yescharacteristic, complemented
49#19097691C2211yescharacteristic, complemented
50#19097683C2211yescharacteristic, complemented
51#19097681C2211yescharacteristic, complemented
52#121×C157C2211yescharacteristic
53C2×C4⋊C4×C157C2211yescharacteristic, complemented
54#107×C157C2211yescharacteristic, complemented
55#107×C157C2211yescharacteristic, complemented
56#102×C157C2211yescharacteristic, complemented
57#102×C157C2211yescharacteristic, complemented
58C22⋊C8×C157C2211yescharacteristic, complemented
59C23×D157—21no—
60C23×D157—21no—
61C2×Dic314C2311yescharacteristic
62C2×D628—22no—
63C2×D628—21no—
64C2×D628—21no—
65C2×D628—41no—
66C2×D628C2311yescharacteristic
67C2×D628C2311yescharacteristic
68C2×C22⋊D157—21no—
69C2×C22⋊D157—21no—
70C2×C22⋊D157—21no—
71C2×C22⋊D157D411yescharacteristic
72D4×D157—22no—
73D4×D157—41no—
74D4×D157—44no—
75C2×C4×D157—21no—
76C2×C4×D157C2311yescharacteristic
77#14578943—22no—
78#14578943—41no—
79#14578942—41no—
80#14578937—41no—
81C2×#11929434—41no—
82C2×#11929434C2311yescharacteristic
83C2×#11929434C2311yescharacteristic
84C4⋊Dic157—21no—
85C4⋊Dic157—21no—
86C4⋊Dic157C2311yescharacteristic
87#14578933—21no—
88#14578933—21no—
89#14578933—21no—
90#14578933D411yescharacteristic
91C22⋊Dic157—21no—
92C22⋊Dic157D411yescharacteristic
93#14578931—21no—
94#14578931—21no—
95#14578931—21no—
96#14578931—21no—
97#14578931D411yescharacteristic
98C4×Dic157—21no—
99C4×Dic157—21no—
100C4×Dic157C2311yescharacteristic
101C2×D4×C157—21no—
102C2×D4×C157C2311yescharacteristic
103C22×C4×C157—21no—
104C22×C4×C157C2311yescharacteristic
105C2×C8×C157C2311yescharacteristic
106C2×C8×C157C2311yescharacteristic
107C4⋊C4×C157—21no—
108C4⋊C4×C157C2311yescharacteristic
109C4⋊C4×C157C2311yescharacteristic
110C4⋊C4×C157C2311yescharacteristic
111C22⋊C4×C157—21no—
112C22×D157—21no—
113C22×D157—21no—
114C22×D157—42no—
115C22×D157—42no—
116C22×D157—42no—
117C22×D157—42no—
118C22×D157—81no—
119C22×D157—81no—
120C22×D157C2×D411yescharacteristic
121Dic314—21no—
122Dic314—21no—
123D628—21no—
124D628—21no—
125D628—21no—
126D628—42no—
127D628—42no—
128D628#4112yes—
129C22⋊D157—22no—
130C22⋊D157—42no—
131C22⋊D157—42no—
132C22⋊D157—81no—
133C2×Dic157—21no—
134C2×Dic157—21no—
135C2×Dic157C2×D411yescharacteristic
136C2×Dic157C2×D411yescharacteristic
137C2×Dic157C2×D411yescharacteristic
138C4×D157—21no—
139C4×D157—21no—
140C4×D157—42no—
141#11929434—21no—
142#11929434—21no—
143C23×C157—21no—
144C23×C157C2×D411yescharacteristic
145D4×C157—21no—
146D4×C157—21no—
147D4×C157—42no—
148C2×C4×C157—21no—
149C2×C4×C157—21no—
150C2×C4×C157—21no—
151C2×C4×C157—21no—
152C2×C4×C157—21no—
153C2×C4×C157—42no—
154C2×C4×C157C2×D411yescharacteristic
155C2×C4×C157C2411yescharacteristic
156C8×C157—21no—
157C8×C157—21no—
158C22×C157—21no—
159C22×C157—21no—
160C22×C157—41no—
161C22×C157—41no—
162C22×C157—41no—
163C22×C157AGL(1,ℤ/8)12yes—
164C22×C157C22×D411yescharacteristic
165C2×D157—21no—
166C2×D157—22no—
167C2×D157—21no—
168C2×D157—21no—
169C2×D157—41no—
170C2×D157—41no—
171C2×D157—41no—
172C2×D157—41no—
173C2×D157—41no—
174C2×D157—84no—
175C2×D157—84no—
176C4×C157—21no—
177C4×C157—41no—
178C4×C157—41no—
179C4×C157—41no—
180C4×C157C2×#4111yescharacteristic
181C4×C157H(5,𝔽2)11yescharacteristic
182Dic157—21no—
183Dic157—21no—
184Dic157—21no—
185Dic157—41no—
186Dic157—41no—
187C2×C157—22no—
188C2×C157—81no—
189C2×C157#54611yescharacteristic
190C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C157#57711yescharacteristic
192D157—41no—
193D157—42no—
194D157—81no—
195C157#331011yescharacteristic, complemented, Sylow
196#3310—1571nomaximal, Sylow
197#552—1571no—
198D42—1571no—
199#490—1571no—
200#482—1571no—
201#480D15711yescharacteristic, complemented
202#478—1571no—
203#468—1571no—
204#465—1571no—
205#460—1571no—
206#450—1571no—
207#450—1571no—
208#440—1571no—
209#435—1571no—
210#418—1571no—
211C2×#102—1571no—
212C22×D4—3141no—
213C22×D4—1571no—
214C22×D4—1571no—
215C2×SD16—1571no—
216C2×SD16—1571no—
217C2×SD16—1571no—
218C2×M16—1571no—
219C22×C8—1571no—
220#127—1571no—
221#126—3141no—
222#125—3141no—
223#125—1571no—
224#123—3141no—
225C22⋊Q8—1571no—
226C22⋊Q8—1571no—
227#121—3141no—
228#121—1571no—
229#121C2×D15711yescharacteristic
230#120—3141no—
231#120—3141no—
232C4×Q8—1571no—
233C4×D4—1571no—
234C4×D4—1571no—
235#117—3141no—
236#117—1571no—
237C2×C4⋊C4C2×D15711yescharacteristic, complemented
238#107C2×D15711yescharacteristic, complemented
239#107C2×D15711yescharacteristic, complemented
240#106—1571no—
241C4⋊C8—1571no—
242#103—1571no—
243#102—1571no—
244#102—1571no—
245#102—1571no—
246#102—1571no—
247#102—1571no—
248#102—1571no—
249#102—1571no—
250#102C2×D15711yescharacteristic, complemented
251#102C2×D15711yescharacteristic, complemented
252C22⋊C8—1571no—
253C22⋊C8C2×D15711yescharacteristic, complemented
254#97—1571no—
255C24—3141no—
256C24—3141no—
257C2×Q8—1571no—
258C2×D4—21no—
259C2×D4—3142no—
260C2×D4—3141no—
261C2×D4—3142no—
262C2×D4—3141no—
263C2×D4—3141no—
264C2×D4—3141no—
265C2×D4—3141no—
266C2×D4—6281no—
267C2×D4—6281no—
268C2×D4—6284no—
269C2×D4—1571no—
270C2×D4—1571no—
271C2×D4—1571no—
272C2×D4C22×D15711yescharacteristic
273C22×C4—21no—
274C22×C4—3141no—
275C22×C4—1571no—
276C22×C4C22×D15711yescharacteristic
277SD16—3142no—
278SD16—6281no—
279SD16—6281no—
280M16—6281no—
281C2×C8—6281no—
282C2×C8—1571no—
283C2×C8—1571no—
284C2×C8C22×D15711yescharacteristic
285C2×C8C22×D15711yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—3141no—
288C4⋊C4—3141no—
289C4⋊C4—3141no—
290C4⋊C4—3141no—
291C4⋊C4—3141no—
292C4⋊C4—1571no—
293C4⋊C4—1571no—
294C4⋊C4C22×D15711yescharacteristic
295C4⋊C4C22×D15711yescharacteristic
296C4⋊C4C22×D15711yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—3141no—
299C22⋊C4—3141no—
300C22⋊C4—3141no—
301C22⋊C4—3141no—
302C22⋊C4—3141no—
303C22⋊C4—1571no—
304C22⋊C4—1571no—
305C42—3141no—
306C42—3141no—
307C42—1571no—
308C23—21no—
309C23—3141no—
310C23—3141no—
311C23—6282no—
312C23—6282no—
313C23—6282no—
314C23—6282no—
315C23—1571no—
316C23—12561no—
317C23—12561no—
318C23D4×D15711yescharacteristic
319Q8—3141no—
320Q8—3141no—
321D4—21no—
322D4—21no—
323D4—42no—
324D4—3142no—
325D4—3141no—
326D4—3141no—
327D4—3141no—
328D4—6282no—
329D4—6282no—
330D4—6282no—
331D4—6282no—
332D4—1572no—
333D4—12561no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—42no—
340C2×C4—3141no—
341C2×C4—3141no—
342C2×C4—3141no—
343C2×C4—3141no—
344C2×C4—6282no—
345C2×C4—1571no—
346C2×C4—1571no—
347C2×C4—1571no—
348C2×C4D4×D15711yescharacteristic
349C2×C4C23×D15711yescharacteristic
350C8—21no—
351C8—21no—
352C8—3141no—
353C8—3141no—
354C22—21no—
355C22—21no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—3141no—
360C22—3141no—
361C22—3142no—
362C22—3141no—
363C22—6281no—
364C22—6281no—
365C22—6281no—
366C22—6281no—
367C22—6281no—
368C22—12564no—
369C22—12564no—
370C22#1909779512yes—
371C22C2×D4×D15711yescharacteristic
372C4—21no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—3141no—
377C4—3141no—
378C4—3141no—
379C4—6281no—
380C4—6281no—
381C4#41×D15711yescharacteristic
382C4#1909781711yescharacteristic
383C2—22no—
384C2—81no—
385C2—6281no—
386C2—6282no—
387C2—12561no—
388C2#2375791511yescharacteristic
389C2C2×#1909779511yescharacteristic
390C2#2375808211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#480×C157C2
socleC22×C157C22×D4
prime radical#23757907C2
hypercenter#480D157
hypocenterC157#3310
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. #23757958C2semidirect
3. #23757636C2semidirect
4. #23757600C2semidirect
5. #23757576C2semidirect
6. #23757520C2semidirect
7. #23757500C2semidirect
8. #23757472C2semidirect
9. #23757426C2semidirect
10. #23757425C2semidirect
11. #23757372C2semidirect
12. #23757341C2semidirect
13. #23757276C2semidirect
14. C2×#19097691C2semidirect
15. #480×C157C2semidirect
20. C2×#14578943C22semidirect
21. C2×#14578942C22semidirect
22. C2×#14578937C22semidirect
23. C22×#11929434C22semidirect
26. #19097762C22semidirect
29. #19097751C22semidirect
30. #19097750C22semidirect
35. #19097736C22semidirect
39. #19097729C22semidirect
40. #19097702C22semidirect
41. #19097700C22semidirect
42. #19097694C22semidirect
44. #19097693C22semidirect
45. #19097692C22semidirect
48. #19097691C22semidirect
49. #19097691C22semidirect
50. #19097683C22semidirect
51. #19097681C22semidirect
53. C2×C4⋊C4×C157C22semidirect
54. #107×C157C22semidirect
55. #107×C157C22semidirect
56. #102×C157C22semidirect
57. #102×C157C22semidirect
58. C22⋊C8×C157C22semidirect
195. C157#3310semidirect
201. #480D157semidirect
237. C2×C4⋊C4C2×D157semidirect
238. #107C2×D157semidirect
239. #107C2×D157semidirect
250. #102C2×D157semidirect
251. #102C2×D157semidirect
253. C22⋊C8C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237579582
#237579072
#237576362
#237576002
#237575762
#237575202
#237575002
#237574722
#237574262
#237574252
#237573722
#237573412
#237572762
C2×#190976912
#480×C1572
#3310157