Finite Group Databasebeta

← #540×D283 #49134615

Subgroups — #540×D283 #49134615

Order 36224.

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#31827071C211yesmaximal, characteristic, complemented
2#31827070C211yesmaximal, characteristic, complemented
3#31827069C211yesmaximal, characteristic, complemented
4#31827068C212yesmaximal, complemented
5#31827067C214yesmaximal, complemented
6#31827066C212yesmaximal, complemented
7#31827065C214yesmaximal, complemented
8#127×D283C211yesmaximal, characteristic, complemented
9#125×D283C211yesmaximal, characteristic, complemented
10#123×D283C214yesmaximal, complemented
11#121×D283C214yesmaximal, complemented
12#121×D283C212yesmaximal, complemented
13C4×D4×D283C212yesmaximal, complemented
14#117×D283C211yesmaximal, characteristic, complemented
15#540×C283C211yesmaximal, characteristic, complemented
16C2×D4×D283C2212yes—
17C2×D4×D283C2214yes—
18C2×D4×D283C2214yes—
19C22×C4×D283C2211yescharacteristic
20C22×C4×D283C2214yes—
21#22583607C2211yescharacteristic, complemented
22#22583606C2214yescomplemented
23#22583606C2212yescomplemented
24#22583602C2211yescharacteristic, complemented
25#22583601C2212yescomplemented
26#22583600C2214yescomplemented
27#22583595C2214yescomplemented
28#22583595C2214yescomplemented
29#22583594C2214yescomplemented
30#22583593C2214yescomplemented
31#22583592C2214yescomplemented
32#22583592C2214yescomplemented
33#22583591C2214yescomplemented
34#22583585C2212yescomplemented
35#22583585C2214yescomplemented
36#22583584C2214yescomplemented
37#22583584C2212yescomplemented
38#22583583C2214yescomplemented
39#22583583C2214yescomplemented
40#22583583C2214yescomplemented
41#22583582C2214yescomplemented
42#22583582C2214yescomplemented
43#22583582C2214yescomplemented
44#22583581C2214yescomplemented
45#22583581C2212yescomplemented
46C4×D1132C2212yescomplemented
47D4×Dic283C2212yescomplemented
48#22583572C2212yescomplemented
49C4×C22⋊D283C2214yescomplemented
50(C22⋊D283)⋊C4C2214yescomplemented
51#22583569C2211yescharacteristic, complemented
52#22583568C2212yescomplemented
53#22583567C2211yescharacteristic, complemented
54#22583567C2211yescharacteristic, complemented
55#22583566C2212yescomplemented
56C4⋊C4×D283C2214yes—
57C4⋊C4×D283C2212yes—
58C4⋊C4×D283C2212yes—
59C22⋊C4×D283C2212yes—
60C22⋊C4×D283C2214yes—
61C22⋊C4×D283C2214yes—
62C42×D283C2211yescharacteristic
63C42×D283C2211yescharacteristic
64#127×C283C2211yescharacteristic, complemented
65#125×C283C2211yescharacteristic, complemented
66#123×C283C2214yescomplemented
67#121×C283C2214yescomplemented
68#121×C283C2212yescomplemented
69C4×D4×C283C2212yescomplemented
70#117×C283C2211yescharacteristic, complemented
71C23×D283C2311yescharacteristic
72C23×D283C2314yes—
73C2×D1132C2312yes—
74C2×D1132C2314yes—
75C2×D1132C2314yes—
76C2×C22⋊D283C2314yes—
77C2×C22⋊D283C2314yes—
78C2×C22⋊D283C2314yes—
79C2×C22⋊D283C2314yes—
80C2×C22⋊D283C2314yes—
81D4×D283—28no—
82D4×D283—48no—
83D4×D283—48no—
84C22×Dic283C2311yescharacteristic
85C22×Dic283C2314yes—
86C2×C4×D283—24no—
87C2×C4×D283—48no—
88C2×C4×D283C2312yes—
89C2×C4×D283C2312yes—
90C2×C4×D283C2311yescharacteristic
91C2×C4×D283C2314yes—
92C2×C4×D283C2311yescharacteristic
93C2×C4×D283C2311yescharacteristic
94C2×C4×D283C2314yes—
95C2×C4×D283C2314yes—
96C2×C4×D283C2311yescharacteristic
97C4⋊Dic283C2312yes—
98C4⋊Dic283C2312yes—
99C4⋊Dic283C2314yes—
100Dic283⋊C4C2312yes—
101Dic283⋊C4C2312yes—
102Dic283⋊C4C2314yes—
103Dic283⋊C4C2314yes—
104Dic283⋊C4C2314yes—
105C22⋊Dic283C2312yes—
106C22⋊Dic283C2314yes—
107C22⋊Dic283C2314yes—
108(C2×D283)⋊C4C2312yes—
109(C2×D283)⋊C4C2312yes—
110(C2×D283)⋊C4C2314yes—
111(C2×D283)⋊C4C2314yes—
112(C2×D283)⋊C4C2314yes—
113(C2×D283)⋊C4C2314yes—
114C4×Dic283C2312yes—
115C4×Dic283C2312yes—
116C4×Dic283C2311yescharacteristic
117C4×Dic283C2311yescharacteristic
118C2×D4×C283C2312yes—
119C2×D4×C283C2314yes—
120C2×D4×C283C2314yes—
121C22×C4×C283C2311yescharacteristic
122C22×C4×C283C2314yes—
123C4⋊C4×C283C2312yes—
124C4⋊C4×C283C2312yes—
125C4⋊C4×C283C2314yes—
126C22⋊C4×C283C2312yes—
127C22⋊C4×C283C2314yes—
128C22⋊C4×C283C2314yes—
129C42×C283C2311yescharacteristic
130C42×C283C2311yescharacteristic
131C22×D283—22no—
132C22×D283—28no—
133C22×D283—28no—
134C22×D283—24no—
135C22×D283—28no—
136C22×D283C2411yescharacteristic
137C22×D283C2414yes—
138C22×D283C2411yescharacteristic
139D1132—24no—
140D1132—44no—
141D1132—44no—
142C22⋊D283—28no—
143C22⋊D283—44no—
144C22⋊D283—44no—
145C22⋊D283—44no—
146C22⋊D283—44no—
147C2×Dic283—22no—
148C2×Dic283—44no—
149C2×Dic283C2412yes—
150C2×Dic283C2412yes—
151C2×Dic283C2411yescharacteristic
152C2×Dic283C2414yes—
153C2×Dic283C2411yescharacteristic
154C4×D283—22no—
155C4×D283—24no—
156C4×D283—28no—
157C4×D283—24no—
158C4×D283—22no—
159C4×D283—22no—
160C4×D283—44no—
161C4×D283—44no—
162C4×D283#4114yes—
163C23×C283C2411yescharacteristic
164C23×C283C2414yes—
165D4×C283—24no—
166D4×C283—44no—
167D4×C283—44no—
168C2×C4×C283—22no—
169C2×C4×C283—44no—
170C2×C4×C283C2412yes—
171C2×C4×C283C2412yes—
172C2×C4×C283C2411yescharacteristic
173C2×C4×C283C2414yes—
174C2×C4×C283C2411yescharacteristic
175C22×C283—21no—
176C22×C283—24no—
177C22×C283—24no—
178C22×C283—22no—
179C22×C283—24no—
180C22×C283C2511yescharacteristic
181C2×D283—21no—
182C2×D283—24no—
183C2×D283—24no—
184C2×D283—22no—
185C2×D283—24no—
186C2×D283—44no—
187C2×D283—416no—
188C2×D283C2×#4112yes—
189C2×D283H(5,𝔽2)14yes—
190C4×C283—21no—
191C4×C283—22no—
192C4×C283—24no—
193C4×C283—22no—
194C4×C283C2×#4112yes—
195Dic283—21no—
196Dic283—22no—
197Dic283—24no—
198Dic283—22no—
199Dic283C2×#4112yes—
200C2×C283—41no—
201C2×C283—44no—
202C2×C283C22×#4111yescharacteristic
203C2×C283C2×H(5,𝔽2)12yes—
204D283—41no—
205D283—44no—
206D283#54014yesdirect product factor
207C283C2×#54011yescharacteristic, complemented, Sylow
208C2×#540—2831nomaximal, Sylow
209#540—2831no—
210#540—2831no—
211#540—2834no—
212#540—2834no—
213#540—2832no—
214#540—2832no—
215#540—2831no—
216#540D28311yescharacteristic, direct product factor
217C2×#127—2831no—
218C2×#125—2831no—
219C2×#123—2834no—
220C2×#121—2834no—
221C2×#121—2832no—
222C2×C4×D4—2832no—
223C2×#117—2831no—
224C22×D4—2832no—
225C22×D4—2834no—
226C22×D4—2834no—
227C23×C4—2831no—
228C23×C4—2834no—
229#127—2834no—
230#127—2832no—
231#127—2831no—
232#127C2×D28311yescharacteristic, complemented
233#125—2834no—
234#125—2832no—
235#125—2831no—
236#125C2×D28311yescharacteristic, complemented
237#123—2834no—
238#123—2834no—
239#123—2834no—
240#123—2834no—
241#123—2834no—
242#123—2834no—
243#123—2834no—
244#123C2×D28314yescomplemented
245#121—2834no—
246#121—2832no—
247#121—2834no—
248#121—2834no—
249#121—2834no—
250#121—2834no—
251#121—2832no—
252#121—2834no—
253#121—2832no—
254#121—2834no—
255#121—2834no—
256#121—2834no—
257#121C2×D28314yescomplemented
258#121C2×D28312yescomplemented
259C4×D4—2832no—
260C4×D4—2834no—
261C4×D4—2832no—
262C4×D4—2834no—
263C4×D4—2832no—
264C4×D4C2×D28312yescomplemented
265#117—2831no—
266#117—2832no—
267#117—2832no—
268#117—2831no—
269#117—2831no—
270#117C2×D28311yescharacteristic, complemented
271C2×C4⋊C4—2832no—
272C2×C4⋊C4—2832no—
273C2×C4⋊C4—2834no—
274C2×C22⋊C4—2832no—
275C2×C22⋊C4—2834no—
276C2×C22⋊C4—2834no—
277C2×C42—2831no—
278C2×C42—2831no—
279C24—2831no—
280C24—2834no—
281C2×D4—2834no—
282C2×D4—2834no—
283C2×D4—2834no—
284C2×D4—2834no—
285C2×D4—2834no—
286C2×D4—2832no—
287C2×D4—2834no—
288C2×D4—2834no—
289C2×D4—5668no—
290C2×D4—11328no—
291C2×D4—11328no—
292C2×D4C22×D28312yes—
293C2×D4C22×D28314yes—
294C2×D4C22×D28314yes—
295C22×C4—2831no—
296C22×C4—2831no—
297C22×C4—2834no—
298C22×C4—2832no—
299C22×C4—2832no—
300C22×C4—2831no—
301C22×C4—2834no—
302C22×C4—2831no—
303C22×C4—2831no—
304C22×C4—2834no—
305C22×C4—2834no—
306C22×C4—5664no—
307C22×C4—11328no—
308C22×C4C22×D28311yescharacteristic
309C22×C4C22×D28314yes—
310C4⋊C4—2832no—
311C4⋊C4—2832no—
312C4⋊C4—2834no—
313C4⋊C4—2834no—
314C4⋊C4—2834no—
315C4⋊C4—2832no—
316C4⋊C4—2832no—
317C4⋊C4—2834no—
318C4⋊C4C22×D28312yes—
319C4⋊C4C22×D28312yes—
320C4⋊C4C22×D28314yes—
321C22⋊C4—2832no—
322C22⋊C4—2834no—
323C22⋊C4—2834no—
324C22⋊C4—2832no—
325C22⋊C4—2832no—
326C22⋊C4—2834no—
327C22⋊C4—2834no—
328C22⋊C4—2834no—
329C22⋊C4—2834no—
330C22⋊C4C22×D28312yes—
331C22⋊C4C22×D28314yes—
332C22⋊C4C22×D28314yes—
333C42—2832no—
334C42—2831no—
335C42—2832no—
336C42—2831no—
337C42C22×D28311yescharacteristic
338C42C22×D28311yescharacteristic
339C23—2831no—
340C23—2831no—
341C23—2834no—
342C23—5662no—
343C23—5668no—
344C23—5668no—
345C23—5664no—
346C23—5668no—
347C23C23×D28311yescharacteristic
348C23C23×D28314yes—
349D4—24no—
350D4—44no—
351D4—44no—
352D4—5668no—
353D4—5664no—
354D4—11324no—
355D4—11324no—
356D4—11324no—
357D4—11324no—
358D4—11324no—
359D4—11324no—
360C2×C4—22no—
361C2×C4—44no—
362C2×C4—2831no—
363C2×C4—2834no—
364C2×C4—2832no—
365C2×C4—2832no—
366C2×C4—2831no—
367C2×C4—2834no—
368C2×C4—5662no—
369C2×C4—5662no—
370C2×C4—5664no—
371C2×C4—5668no—
372C2×C4—5664no—
373C2×C4—5662no—
374C2×C4—5662no—
375C2×C4—11324no—
376C2×C4—11324no—
377C2×C4—11324no—
378C2×C4C23×D28312yes—
379C2×C4C23×D28312yes—
380C2×C4C23×D28311yescharacteristic
381C2×C4C23×D28314yes—
382C2×C4C23×D28311yescharacteristic
383C22—21no—
384C22—24no—
385C22—24no—
386C22—22no—
387C22—24no—
388C22—2832no—
389C22—2834no—
390C22—5661no—
391C22—5664no—
392C22—5664no—
393C22—5662no—
394C22—5664no—
395C22—11324no—
396C22—113216no—
397C22C24×D28311yescharacteristic
398C4—21no—
399C4—22no—
400C4—24no—
401C4—22no—
402C4—2832no—
403C4—5661no—
404C4—5662no—
405C4—5664no—
406C4—5662no—
407C4#41×D28312yes—
408C2—41no—
409C2—44no—
410C2—2834no—
411C2—11321no—
412C2—11324no—
413C2C2×#41×D28311yescharacteristic
414C2H(5,𝔽2)×D28312yes—
subgroupG/H
center, FrattiniC22C24×D283
commutator, socle, cosocleC22×C283C25
Fitting#540×C283C2
hypercenter#540D283
hypocenterC283C2×#540
LCSG ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ #540

Complements

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

NH ≅ G/Ntype
1. #31827071C2semidirect
2. #31827070C2semidirect
3. #31827069C2semidirect
4. #31827068C2semidirect
5. #31827067C2semidirect
6. #31827066C2semidirect
7. #31827065C2semidirect
8. #127×D283C2semidirect
9. #125×D283C2semidirect
10. #123×D283C2semidirect
11. #121×D283C2semidirect
12. #121×D283C2semidirect
13. C4×D4×D283C2semidirect
14. #117×D283C2semidirect
15. #540×C283C2semidirect
21. #22583607C22semidirect
22. #22583606C22semidirect
23. #22583606C22semidirect
24. #22583602C22semidirect
25. #22583601C22semidirect
26. #22583600C22semidirect
27. #22583595C22semidirect
28. #22583595C22semidirect
29. #22583594C22semidirect
30. #22583593C22semidirect
31. #22583592C22semidirect
32. #22583592C22semidirect
33. #22583591C22semidirect
34. #22583585C22semidirect
35. #22583585C22semidirect
36. #22583584C22semidirect
37. #22583584C22semidirect
38. #22583583C22semidirect
39. #22583583C22semidirect
40. #22583583C22semidirect
41. #22583582C22semidirect
42. #22583582C22semidirect
43. #22583582C22semidirect
44. #22583581C22semidirect
45. #22583581C22semidirect
46. C4×D1132C22semidirect
47. D4×Dic283C22semidirect
48. #22583572C22semidirect
49. C4×C22⋊D283C22semidirect
50. (C22⋊D283)⋊C4C22semidirect
51. #22583569C22semidirect
52. #22583568C22semidirect
53. #22583567C22semidirect
54. #22583567C22semidirect
55. #22583566C22semidirect
64. #127×C283C22semidirect
65. #125×C283C22semidirect
66. #123×C283C22semidirect
67. #121×C283C22semidirect
68. #121×C283C22semidirect
69. C4×D4×C283C22semidirect
70. #117×C283C22semidirect
206. D283#540direct (N × H)
207. C283C2×#540semidirect
216. #540D283direct (N × H)
232. #127C2×D283semidirect
236. #125C2×D283semidirect
244. #123C2×D283semidirect
257. #121C2×D283semidirect
258. #121C2×D283semidirect
264. C4×D4C2×D283semidirect
270. #117C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318270712
#318270702
#318270692
#318270682
#318270672
#318270662
#318270652
#127×D2832
#125×D2832
#123×D2832
#121×D2832
C4×D4×D2832
#117×D2832
#540×C2832
C2×#540283