Finite Group Databasebeta

← #540×D163 #35227890

Subgroups — #540×D163 #35227890

Order 20864.

1260 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782271C211yesmaximal, characteristic, complemented
2#23782270C211yesmaximal, characteristic, complemented
3#23782269C211yesmaximal, characteristic, complemented
4#23782268C212yesmaximal, complemented
5#23782267C214yesmaximal, complemented
6#23782266C212yesmaximal, complemented
7#23782265C214yesmaximal, complemented
8#127×D163C211yesmaximal, characteristic, complemented
9#125×D163C211yesmaximal, characteristic, complemented
10#123×D163C214yesmaximal, complemented
11#121×D163C212yesmaximal, complemented
12#121×D163C214yesmaximal, complemented
13C4×D4×D163C212yesmaximal, complemented
14#117×D163C211yesmaximal, characteristic, complemented
15#540×C163C211yesmaximal, characteristic, complemented
16C2×D4×D163C2212yes—
17C2×D4×D163C2214yes—
18C2×D4×D163C2214yes—
19C22×C4×D163C2211yescharacteristic
20C22×C4×D163C2214yes—
21#19104732C2211yescharacteristic, complemented
22#19104731C2214yescomplemented
23#19104731C2212yescomplemented
24#19104727C2211yescharacteristic, complemented
25#19104726C2212yescomplemented
26#19104725C2214yescomplemented
27#19104720C2214yescomplemented
28#19104720C2214yescomplemented
29#19104719C2214yescomplemented
30#19104718C2214yescomplemented
31#19104717C2214yescomplemented
32#19104717C2214yescomplemented
33#19104716C2214yescomplemented
34#19104710C2212yescomplemented
35#19104710C2214yescomplemented
36#19104709C2214yescomplemented
37#19104709C2212yescomplemented
38#19104708C2214yescomplemented
39#19104708C2214yescomplemented
40#19104708C2214yescomplemented
41#19104707C2214yescomplemented
42#19104707C2214yescomplemented
43#19104707C2214yescomplemented
44#19104706C2214yescomplemented
45#19104706C2212yescomplemented
46C4×D652C2212yescomplemented
47D4×Dic163C2212yescomplemented
48#19104697C2212yescomplemented
49C4×C22⋊D163C2214yescomplemented
50(C22⋊D163)⋊C4C2214yescomplemented
51#19104694C2211yescharacteristic, complemented
52#19104693C2212yescomplemented
53#19104692C2211yescharacteristic, complemented
54#19104692C2211yescharacteristic, complemented
55#19104691C2212yescomplemented
56C4⋊C4×D163C2214yes—
57C4⋊C4×D163C2212yes—
58C4⋊C4×D163C2212yes—
59C22⋊C4×D163C2212yes—
60C22⋊C4×D163C2214yes—
61C22⋊C4×D163C2214yes—
62C42×D163C2211yescharacteristic
63C42×D163C2211yescharacteristic
64#127×C163C2211yescharacteristic, complemented
65#125×C163C2211yescharacteristic, complemented
66#123×C163C2214yescomplemented
67#121×C163C2214yescomplemented
68#121×C163C2212yescomplemented
69C4×D4×C163C2212yescomplemented
70#117×C163C2211yescharacteristic, complemented
71C23×D163C2311yescharacteristic
72C23×D163C2314yes—
73C2×D652C2312yes—
74C2×D652C2314yes—
75C2×D652C2314yes—
76C2×C22⋊D163C2314yes—
77C2×C22⋊D163C2314yes—
78C2×C22⋊D163C2314yes—
79C2×C22⋊D163C2314yes—
80C2×C22⋊D163C2314yes—
81D4×D163—28no—
82D4×D163—48no—
83D4×D163—48no—
84C22×Dic163C2311yescharacteristic
85C22×Dic163C2314yes—
86C2×C4×D163—24no—
87C2×C4×D163—48no—
88C2×C4×D163C2312yes—
89C2×C4×D163C2314yes—
90C2×C4×D163C2311yescharacteristic
91C2×C4×D163C2311yescharacteristic
92C2×C4×D163C2311yescharacteristic
93C2×C4×D163C2312yes—
94C2×C4×D163C2314yes—
95C2×C4×D163C2314yes—
96C2×C4×D163C2311yescharacteristic
97C4⋊Dic163C2312yes—
98C4⋊Dic163C2312yes—
99C4⋊Dic163C2314yes—
100Dic163⋊C4C2312yes—
101Dic163⋊C4C2312yes—
102Dic163⋊C4C2314yes—
103Dic163⋊C4C2314yes—
104Dic163⋊C4C2314yes—
105C22⋊Dic163C2312yes—
106C22⋊Dic163C2314yes—
107C22⋊Dic163C2314yes—
108(C2×D163)⋊C4C2312yes—
109(C2×D163)⋊C4C2312yes—
110(C2×D163)⋊C4C2314yes—
111(C2×D163)⋊C4C2314yes—
112(C2×D163)⋊C4C2314yes—
113(C2×D163)⋊C4C2314yes—
114C4×Dic163C2312yes—
115C4×Dic163C2312yes—
116C4×Dic163C2311yescharacteristic
117C4×Dic163C2311yescharacteristic
118C2×D4×C163C2312yes—
119C2×D4×C163C2314yes—
120C2×D4×C163C2314yes—
121C22×C4×C163C2311yescharacteristic
122C22×C4×C163C2314yes—
123C4⋊C4×C163C2312yes—
124C4⋊C4×C163C2312yes—
125C4⋊C4×C163C2314yes—
126C22⋊C4×C163C2312yes—
127C22⋊C4×C163C2314yes—
128C22⋊C4×C163C2314yes—
129C42×C163C2311yescharacteristic
130C42×C163C2311yescharacteristic
131C22×D163—22no—
132C22×D163—28no—
133C22×D163—28no—
134C22×D163—24no—
135C22×D163—28no—
136C22×D163C2411yescharacteristic
137C22×D163C2414yes—
138C22×D163C2411yescharacteristic
139D652—24no—
140D652—44no—
141D652—44no—
142C22⋊D163—28no—
143C22⋊D163—44no—
144C22⋊D163—44no—
145C22⋊D163—44no—
146C22⋊D163—44no—
147C2×Dic163—22no—
148C2×Dic163—44no—
149C2×Dic163C2412yes—
150C2×Dic163C2412yes—
151C2×Dic163C2411yescharacteristic
152C2×Dic163C2414yes—
153C2×Dic163C2411yescharacteristic
154C4×D163—22no—
155C4×D163—24no—
156C4×D163—28no—
157C4×D163—24no—
158C4×D163—22no—
159C4×D163—22no—
160C4×D163—44no—
161C4×D163—44no—
162C4×D163#4114yes—
163C23×C163C2411yescharacteristic
164C23×C163C2414yes—
165D4×C163—24no—
166D4×C163—44no—
167D4×C163—44no—
168C2×C4×C163—22no—
169C2×C4×C163—44no—
170C2×C4×C163C2412yes—
171C2×C4×C163C2412yes—
172C2×C4×C163C2411yescharacteristic
173C2×C4×C163C2414yes—
174C2×C4×C163C2411yescharacteristic
175C22×C163—21no—
176C22×C163—24no—
177C22×C163—24no—
178C22×C163—22no—
179C22×C163—24no—
180C22×C163C2511yescharacteristic
181C2×D163—21no—
182C2×D163—24no—
183C2×D163—24no—
184C2×D163—22no—
185C2×D163—24no—
186C2×D163—44no—
187C2×D163—416no—
188C2×D163C2×#4112yes—
189C2×D163H(5,𝔽2)14yes—
190C4×C163—21no—
191C4×C163—22no—
192C4×C163—24no—
193C4×C163—22no—
194C4×C163C2×#4112yes—
195Dic163—21no—
196Dic163—22no—
197Dic163—24no—
198Dic163—22no—
199Dic163C2×#4112yes—
200C2×C163—41no—
201C2×C163—44no—
202C2×C163C22×#4111yescharacteristic
203C2×C163C2×H(5,𝔽2)12yes—
204D163—41no—
205D163—44no—
206D163#54014yesdirect product factor
207C163C2×#54011yescharacteristic, complemented, Sylow
208C2×#540—1631nomaximal, Sylow
209#540—1631no—
210#540—1631no—
211#540—1634no—
212#540—1634no—
213#540—1632no—
214#540—1632no—
215#540—1631no—
216#540D16311yescharacteristic, direct product factor
217C2×#127—1631no—
218C2×#125—1631no—
219C2×#123—1634no—
220C2×#121—1634no—
221C2×#121—1632no—
222C2×C4×D4—1632no—
223C2×#117—1631no—
224C22×D4—1632no—
225C22×D4—1634no—
226C22×D4—1634no—
227C23×C4—1631no—
228C23×C4—1634no—
229#127—1634no—
230#127—1632no—
231#127—1631no—
232#127C2×D16311yescharacteristic, complemented
233#125—1634no—
234#125—1632no—
235#125—1631no—
236#125C2×D16311yescharacteristic, complemented
237#123—1634no—
238#123—1634no—
239#123—1634no—
240#123—1634no—
241#123—1634no—
242#123—1634no—
243#123—1634no—
244#123C2×D16314yescomplemented
245#121—1634no—
246#121—1632no—
247#121—1634no—
248#121—1634no—
249#121—1634no—
250#121—1634no—
251#121—1632no—
252#121—1634no—
253#121—1632no—
254#121—1634no—
255#121—1634no—
256#121—1634no—
257#121C2×D16314yescomplemented
258#121C2×D16312yescomplemented
259C4×D4—1632no—
260C4×D4—1634no—
261C4×D4—1632no—
262C4×D4—1634no—
263C4×D4—1632no—
264C4×D4C2×D16312yescomplemented
265#117—1631no—
266#117—1632no—
267#117—1632no—
268#117—1631no—
269#117—1631no—
270#117C2×D16311yescharacteristic, complemented
271C2×C4⋊C4—1632no—
272C2×C4⋊C4—1632no—
273C2×C4⋊C4—1634no—
274C2×C22⋊C4—1632no—
275C2×C22⋊C4—1634no—
276C2×C22⋊C4—1634no—
277C2×C42—1631no—
278C2×C42—1631no—
279C24—1631no—
280C24—1634no—
281C2×D4—3268no—
282C2×D4—6528no—
283C2×D4—6528no—
284C2×D4—1634no—
285C2×D4—1634no—
286C2×D4—1634no—
287C2×D4—1634no—
288C2×D4—1634no—
289C2×D4—1632no—
290C2×D4—1634no—
291C2×D4—1634no—
292C2×D4C22×D16312yes—
293C2×D4C22×D16314yes—
294C2×D4C22×D16314yes—
295C22×C4—3264no—
296C22×C4—6528no—
297C22×C4—1631no—
298C22×C4—1632no—
299C22×C4—1632no—
300C22×C4—1631no—
301C22×C4—1634no—
302C22×C4—1631no—
303C22×C4—1631no—
304C22×C4—1634no—
305C22×C4—1634no—
306C22×C4—1634no—
307C22×C4—1631no—
308C22×C4C22×D16311yescharacteristic
309C22×C4C22×D16314yes—
310C4⋊C4—1632no—
311C4⋊C4—1632no—
312C4⋊C4—1632no—
313C4⋊C4—1634no—
314C4⋊C4—1632no—
315C4⋊C4—1634no—
316C4⋊C4—1634no—
317C4⋊C4—1634no—
318C4⋊C4C22×D16312yes—
319C4⋊C4C22×D16312yes—
320C4⋊C4C22×D16314yes—
321C22⋊C4—1632no—
322C22⋊C4—1634no—
323C22⋊C4—1634no—
324C22⋊C4—1632no—
325C22⋊C4—1632no—
326C22⋊C4—1634no—
327C22⋊C4—1634no—
328C22⋊C4—1634no—
329C22⋊C4—1634no—
330C22⋊C4C22×D16312yes—
331C22⋊C4C22×D16314yes—
332C22⋊C4C22×D16314yes—
333C42—1631no—
334C42—1632no—
335C42—1631no—
336C42—1632no—
337C42C22×D16311yescharacteristic
338C42C22×D16311yescharacteristic
339C23—3262no—
340C23—3268no—
341C23—3268no—
342C23—3264no—
343C23—3268no—
344C23—1631no—
345C23—1634no—
346C23—1631no—
347C23C23×D16311yescharacteristic
348C23C23×D16314yes—
349D4—24no—
350D4—44no—
351D4—44no—
352D4—3268no—
353D4—3264no—
354D4—6524no—
355D4—6524no—
356D4—6524no—
357D4—6524no—
358D4—6524no—
359D4—6524no—
360C2×C4—22no—
361C2×C4—44no—
362C2×C4—3262no—
363C2×C4—3264no—
364C2×C4—3268no—
365C2×C4—3264no—
366C2×C4—3262no—
367C2×C4—3262no—
368C2×C4—3262no—
369C2×C4—6524no—
370C2×C4—6524no—
371C2×C4—6524no—
372C2×C4—1632no—
373C2×C4—1632no—
374C2×C4—1631no—
375C2×C4—1634no—
376C2×C4—1631no—
377C2×C4—1634no—
378C2×C4C23×D16312yes—
379C2×C4C23×D16312yes—
380C2×C4C23×D16311yescharacteristic
381C2×C4C23×D16314yes—
382C2×C4C23×D16311yescharacteristic
383C22—21no—
384C22—24no—
385C22—24no—
386C22—22no—
387C22—24no—
388C22—3261no—
389C22—3264no—
390C22—3264no—
391C22—3262no—
392C22—3264no—
393C22—6524no—
394C22—65216no—
395C22—1632no—
396C22—1634no—
397C22C24×D16311yescharacteristic
398C4—21no—
399C4—22no—
400C4—24no—
401C4—22no—
402C4—3261no—
403C4—3262no—
404C4—3264no—
405C4—3262no—
406C4—1632no—
407C4#41×D16312yes—
408C2—41no—
409C2—44no—
410C2—6521no—
411C2—6524no—
412C2—1634no—
413C2C2×#41×D16311yescharacteristic
414C2H(5,𝔽2)×D16312yes—
subgroupG/H
center, FrattiniC22C24×D163
commutator, socle, cosocleC22×C163C25
Fitting#540×C163C2
hypercenter#540D163
hypocenterC163C2×#540
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ #540

Complements

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

NH ≅ G/Ntype
1. #23782271C2semidirect
2. #23782270C2semidirect
3. #23782269C2semidirect
4. #23782268C2semidirect
5. #23782267C2semidirect
6. #23782266C2semidirect
7. #23782265C2semidirect
8. #127×D163C2semidirect
9. #125×D163C2semidirect
10. #123×D163C2semidirect
11. #121×D163C2semidirect
12. #121×D163C2semidirect
13. C4×D4×D163C2semidirect
14. #117×D163C2semidirect
15. #540×C163C2semidirect
21. #19104732C22semidirect
22. #19104731C22semidirect
23. #19104731C22semidirect
24. #19104727C22semidirect
25. #19104726C22semidirect
26. #19104725C22semidirect
27. #19104720C22semidirect
28. #19104720C22semidirect
29. #19104719C22semidirect
30. #19104718C22semidirect
31. #19104717C22semidirect
32. #19104717C22semidirect
33. #19104716C22semidirect
34. #19104710C22semidirect
35. #19104710C22semidirect
36. #19104709C22semidirect
37. #19104709C22semidirect
38. #19104708C22semidirect
39. #19104708C22semidirect
40. #19104708C22semidirect
41. #19104707C22semidirect
42. #19104707C22semidirect
43. #19104707C22semidirect
44. #19104706C22semidirect
45. #19104706C22semidirect
46. C4×D652C22semidirect
47. D4×Dic163C22semidirect
48. #19104697C22semidirect
49. C4×C22⋊D163C22semidirect
50. (C22⋊D163)⋊C4C22semidirect
51. #19104694C22semidirect
52. #19104693C22semidirect
53. #19104692C22semidirect
54. #19104692C22semidirect
55. #19104691C22semidirect
64. #127×C163C22semidirect
65. #125×C163C22semidirect
66. #123×C163C22semidirect
67. #121×C163C22semidirect
68. #121×C163C22semidirect
69. C4×D4×C163C22semidirect
70. #117×C163C22semidirect
206. D163#540direct (N × H)
207. C163C2×#540semidirect
216. #540D163direct (N × H)
232. #127C2×D163semidirect
236. #125C2×D163semidirect
244. #123C2×D163semidirect
257. #121C2×D163semidirect
258. #121C2×D163semidirect
264. C4×D4C2×D163semidirect
270. #117C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237822712
#237822702
#237822692
#237822682
#237822672
#237822662
#237822652
#127×D1632
#125×D1632
#123×D1632
#121×D1632
C4×D4×D1632
#117×D1632
#540×C1632
C2×#540163