Finite Group Databasebeta

← #540×D97 #27155090

Subgroups — #540×D97 #27155090

Order 12416.

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#20261341C211yesmaximal, characteristic, complemented
2#20261340C211yesmaximal, characteristic, complemented
3#20261339C211yesmaximal, characteristic, complemented
4#20261338C212yesmaximal, complemented
5#20261337C214yesmaximal, complemented
6#20261336C212yesmaximal, complemented
7#20261335C214yesmaximal, complemented
8#127×D97C211yesmaximal, characteristic, complemented
9#125×D97C211yesmaximal, characteristic, complemented
10#123×D97C214yesmaximal, complemented
11#121×D97C214yesmaximal, complemented
12#121×D97C212yesmaximal, complemented
13C4×D4×D97C212yesmaximal, complemented
14#117×D97C211yesmaximal, characteristic, complemented
15#540×C97C211yesmaximal, characteristic, complemented
16C2×D4×D97C2212yes—
17C2×D4×D97C2214yes—
18C2×D4×D97C2214yes—
19C22×C4×D97C2211yescharacteristic
20C22×C4×D97C2214yes—
21#15690807C2211yescharacteristic, complemented
22#15690806C2214yescomplemented
23#15690806C2212yescomplemented
24#15690802C2211yescharacteristic, complemented
25#15690801C2212yescomplemented
26#15690800C2214yescomplemented
27#15690795C2214yescomplemented
28#15690795C2214yescomplemented
29#15690794C2214yescomplemented
30#15690793C2214yescomplemented
31#15690792C2214yescomplemented
32#15690792C2214yescomplemented
33#15690791C2214yescomplemented
34#15690785C2212yescomplemented
35#15690785C2214yescomplemented
36#15690784C2214yescomplemented
37#15690784C2212yescomplemented
38#15690783C2214yescomplemented
39#15690783C2214yescomplemented
40#15690783C2214yescomplemented
41#15690782C2214yescomplemented
42#15690782C2214yescomplemented
43#15690782C2214yescomplemented
44#15690781C2214yescomplemented
45#15690781C2212yescomplemented
46C4×D388C2212yescomplemented
47D4×Dic97C2212yescomplemented
48#15690772C2212yescomplemented
49C4×C22⋊D97C2214yescomplemented
50#15690770C2214yescomplemented
51#15690769C2211yescharacteristic, complemented
52#15690768C2212yescomplemented
53#15690767C2211yescharacteristic, complemented
54#15690767C2211yescharacteristic, complemented
55#15690766C2212yescomplemented
56C4⋊C4×D97C2212yes—
57C4⋊C4×D97C2212yes—
58C4⋊C4×D97C2214yes—
59C22⋊C4×D97C2212yes—
60C22⋊C4×D97C2214yes—
61C22⋊C4×D97C2214yes—
62C42×D97C2211yescharacteristic
63C42×D97C2211yescharacteristic
64#127×C97C2211yescharacteristic, complemented
65#125×C97C2211yescharacteristic, complemented
66#123×C97C2214yescomplemented
67#121×C97C2214yescomplemented
68#121×C97C2212yescomplemented
69C4×D4×C97C2212yescomplemented
70#117×C97C2211yescharacteristic, complemented
71C23×D97C2311yescharacteristic
72C23×D97C2314yes—
73C2×D388C2312yes—
74C2×D388C2314yes—
75C2×D388C2314yes—
76C2×C22⋊D97C2314yes—
77C2×C22⋊D97C2314yes—
78C2×C22⋊D97C2314yes—
79C2×C22⋊D97C2314yes—
80C2×C22⋊D97C2314yes—
81D4×D97—28no—
82D4×D97—48no—
83D4×D97—48no—
84C22×Dic97C2311yescharacteristic
85C22×Dic97C2314yes—
86C2×C4×D97—24no—
87C2×C4×D97—48no—
88C2×C4×D97C2312yes—
89C2×C4×D97C2312yes—
90C2×C4×D97C2311yescharacteristic
91C2×C4×D97C2314yes—
92C2×C4×D97C2311yescharacteristic
93C2×C4×D97C2311yescharacteristic
94C2×C4×D97C2314yes—
95C2×C4×D97C2314yes—
96C2×C4×D97C2311yescharacteristic
97C4⋊Dic97C2312yes—
98C4⋊Dic97C2312yes—
99C4⋊Dic97C2314yes—
100#13095490C2312yes—
101#13095490C2312yes—
102#13095490C2314yes—
103#13095490C2314yes—
104#13095490C2314yes—
105C22⋊Dic97C2312yes—
106C22⋊Dic97C2314yes—
107C22⋊Dic97C2314yes—
108#13095488C2312yes—
109#13095488C2312yes—
110#13095488C2314yes—
111#13095488C2314yes—
112#13095488C2314yes—
113#13095488C2314yes—
114C4×Dic97C2312yes—
115C4×Dic97C2312yes—
116C4×Dic97C2311yescharacteristic
117C4×Dic97C2311yescharacteristic
118C2×D4×C97C2312yes—
119C2×D4×C97C2314yes—
120C2×D4×C97C2314yes—
121C22×C4×C97C2311yescharacteristic
122C22×C4×C97C2314yes—
123C4⋊C4×C97C2312yes—
124C4⋊C4×C97C2312yes—
125C4⋊C4×C97C2314yes—
126C22⋊C4×C97C2312yes—
127C22⋊C4×C97C2314yes—
128C22⋊C4×C97C2314yes—
129C42×C97C2311yescharacteristic
130C42×C97C2311yescharacteristic
131C23×C97C2411yescharacteristic
132C23×C97C2414yes—
133C22×D97—22no—
134C22×D97—28no—
135C22×D97—28no—
136C22×D97—24no—
137C22×D97—28no—
138C22×D97C2411yescharacteristic
139C22×D97C2414yes—
140C22×D97C2411yescharacteristic
141D4×C97—24no—
142D4×C97—44no—
143D4×C97—44no—
144C2×C4×C97—22no—
145C2×C4×C97—44no—
146C2×C4×C97C2412yes—
147C2×C4×C97C2412yes—
148C2×C4×C97C2411yescharacteristic
149C2×C4×C97C2414yes—
150C2×C4×C97C2411yescharacteristic
151C22⋊D97—28no—
152C22⋊D97—44no—
153C22⋊D97—44no—
154C22⋊D97—44no—
155C22⋊D97—44no—
156C2×Dic97—22no—
157C2×Dic97—44no—
158C2×Dic97C2412yes—
159C2×Dic97C2412yes—
160C2×Dic97C2411yescharacteristic
161C2×Dic97C2414yes—
162C2×Dic97C2411yescharacteristic
163D388—24no—
164D388—44no—
165D388—44no—
166C4×D97—22no—
167C4×D97—24no—
168C4×D97—28no—
169C4×D97—24no—
170C4×D97—22no—
171C4×D97—22no—
172C4×D97—44no—
173C4×D97—44no—
174C4×D97#4114yes—
175C22×C97—21no—
176C22×C97—24no—
177C22×C97—24no—
178C22×C97—22no—
179C22×C97—24no—
180C22×C97C2511yescharacteristic
181C2×D97—21no—
182C2×D97—24no—
183C2×D97—24no—
184C2×D97—22no—
185C2×D97—24no—
186C2×D97—44no—
187C2×D97—416no—
188C2×D97C2×#4112yes—
189C2×D97H(5,𝔽2)14yes—
190C4×C97—21no—
191C4×C97—22no—
192C4×C97—24no—
193C4×C97—22no—
194C4×C97C2×#4112yes—
195Dic97—21no—
196Dic97—22no—
197Dic97—24no—
198Dic97—22no—
199Dic97C2×#4112yes—
200C2×C97—41no—
201C2×C97—44no—
202C2×C97C22×#4111yescharacteristic
203C2×C97C2×H(5,𝔽2)12yes—
204D97—41no—
205D97—44no—
206D97#54014yesdirect product factor
207C2×#540—971nomaximal, Sylow
208C97C2×#54011yescharacteristic, complemented, Sylow
209#540—971no—
210#540—971no—
211#540—974no—
212#540—974no—
213#540—972no—
214#540—972no—
215#540—971no—
216#540D9711yescharacteristic, direct product factor
217C2×#127—971no—
218C2×#125—971no—
219C2×#123—974no—
220C2×#121—974no—
221C2×#121—972no—
222C2×C4×D4—972no—
223C2×#117—971no—
224C22×D4—974no—
225C22×D4—974no—
226C22×D4—972no—
227C23×C4—974no—
228C23×C4—971no—
229#127—972no—
230#127—974no—
231#127—971no—
232#127C2×D9711yescharacteristic, complemented
233#125—974no—
234#125—972no—
235#125—971no—
236#125C2×D9711yescharacteristic, complemented
237#123—974no—
238#123—974no—
239#123—974no—
240#123—974no—
241#123—974no—
242#123—974no—
243#123—974no—
244#123C2×D9714yescomplemented
245#121—974no—
246#121—972no—
247#121—974no—
248#121—974no—
249#121—974no—
250#121—974no—
251#121—972no—
252#121—974no—
253#121—972no—
254#121—974no—
255#121—974no—
256#121—974no—
257#121C2×D9714yescomplemented
258#121C2×D9712yescomplemented
259C4×D4—972no—
260C4×D4—974no—
261C4×D4—972no—
262C4×D4—974no—
263C4×D4—972no—
264C4×D4C2×D9712yescomplemented
265#117—971no—
266#117—972no—
267#117—972no—
268#117—971no—
269#117—971no—
270#117C2×D9711yescharacteristic, complemented
271C2×C4⋊C4—972no—
272C2×C4⋊C4—972no—
273C2×C4⋊C4—974no—
274C2×C22⋊C4—972no—
275C2×C22⋊C4—974no—
276C2×C22⋊C4—974no—
277C2×C42—971no—
278C2×C42—971no—
279C24—974no—
280C24—971no—
281C2×D4—974no—
282C2×D4—974no—
283C2×D4—974no—
284C2×D4—974no—
285C2×D4—974no—
286C2×D4—974no—
287C2×D4—974no—
288C2×D4—972no—
289C2×D4—3888no—
290C2×D4—3888no—
291C2×D4—1948no—
292C2×D4C22×D9712yes—
293C2×D4C22×D9714yes—
294C2×D4C22×D9714yes—
295C22×C4—971no—
296C22×C4—974no—
297C22×C4—971no—
298C22×C4—974no—
299C22×C4—974no—
300C22×C4—972no—
301C22×C4—972no—
302C22×C4—971no—
303C22×C4—974no—
304C22×C4—971no—
305C22×C4—971no—
306C22×C4—3888no—
307C22×C4—1944no—
308C22×C4C22×D9711yescharacteristic
309C22×C4C22×D9714yes—
310C4⋊C4—972no—
311C4⋊C4—972no—
312C4⋊C4—974no—
313C4⋊C4—974no—
314C4⋊C4—974no—
315C4⋊C4—972no—
316C4⋊C4—972no—
317C4⋊C4—974no—
318C4⋊C4C22×D9712yes—
319C4⋊C4C22×D9712yes—
320C4⋊C4C22×D9714yes—
321C22⋊C4—972no—
322C22⋊C4—974no—
323C22⋊C4—974no—
324C22⋊C4—972no—
325C22⋊C4—972no—
326C22⋊C4—974no—
327C22⋊C4—974no—
328C22⋊C4—974no—
329C22⋊C4—974no—
330C22⋊C4C22×D9712yes—
331C22⋊C4C22×D9714yes—
332C22⋊C4C22×D9714yes—
333C42—972no—
334C42—971no—
335C42—972no—
336C42—971no—
337C42C22×D9711yescharacteristic
338C42C22×D9711yescharacteristic
339C23—971no—
340C23—971no—
341C23—974no—
342C23—1948no—
343C23—1942no—
344C23—1948no—
345C23—1944no—
346C23—1948no—
347C23C23×D9711yescharacteristic
348C23C23×D9714yes—
349D4—24no—
350D4—44no—
351D4—44no—
352D4—3884no—
353D4—3884no—
354D4—3884no—
355D4—3884no—
356D4—3884no—
357D4—3884no—
358D4—1948no—
359D4—1944no—
360C2×C4—22no—
361C2×C4—44no—
362C2×C4—971no—
363C2×C4—974no—
364C2×C4—972no—
365C2×C4—972no—
366C2×C4—971no—
367C2×C4—974no—
368C2×C4—3884no—
369C2×C4—3884no—
370C2×C4—3884no—
371C2×C4—1942no—
372C2×C4—1944no—
373C2×C4—1948no—
374C2×C4—1942no—
375C2×C4—1944no—
376C2×C4—1942no—
377C2×C4—1942no—
378C2×C4C23×D9712yes—
379C2×C4C23×D9712yes—
380C2×C4C23×D9711yescharacteristic
381C2×C4C23×D9714yes—
382C2×C4C23×D9711yescharacteristic
383C22—21no—
384C22—24no—
385C22—24no—
386C22—22no—
387C22—24no—
388C22—972no—
389C22—974no—
390C22—38816no—
391C22—3884no—
392C22—1941no—
393C22—1944no—
394C22—1944no—
395C22—1942no—
396C22—1944no—
397C22C24×D9711yescharacteristic
398C4—21no—
399C4—22no—
400C4—24no—
401C4—22no—
402C4—972no—
403C4—1941no—
404C4—1942no—
405C4—1944no—
406C4—1942no—
407C4#41×D9712yes—
408C2—41no—
409C2—44no—
410C2—974no—
411C2—3881no—
412C2—3884no—
413C2C2×#41×D9711yescharacteristic
414C2H(5,𝔽2)×D9712yes—
subgroupG/H
center, FrattiniC22C24×D97
commutator, socle, cosocleC22×C97C25
Fitting#540×C97C2
hypercenter#540D97
hypocenterC97C2×#540
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ #540

Complements

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

NH ≅ G/Ntype
1. #20261341C2semidirect
2. #20261340C2semidirect
3. #20261339C2semidirect
4. #20261338C2semidirect
5. #20261337C2semidirect
6. #20261336C2semidirect
7. #20261335C2semidirect
8. #127×D97C2semidirect
9. #125×D97C2semidirect
10. #123×D97C2semidirect
11. #121×D97C2semidirect
12. #121×D97C2semidirect
13. C4×D4×D97C2semidirect
14. #117×D97C2semidirect
15. #540×C97C2semidirect
21. #15690807C22semidirect
22. #15690806C22semidirect
23. #15690806C22semidirect
24. #15690802C22semidirect
25. #15690801C22semidirect
26. #15690800C22semidirect
27. #15690795C22semidirect
28. #15690795C22semidirect
29. #15690794C22semidirect
30. #15690793C22semidirect
31. #15690792C22semidirect
32. #15690792C22semidirect
33. #15690791C22semidirect
34. #15690785C22semidirect
35. #15690785C22semidirect
36. #15690784C22semidirect
37. #15690784C22semidirect
38. #15690783C22semidirect
39. #15690783C22semidirect
40. #15690783C22semidirect
41. #15690782C22semidirect
42. #15690782C22semidirect
43. #15690782C22semidirect
44. #15690781C22semidirect
45. #15690781C22semidirect
46. C4×D388C22semidirect
47. D4×Dic97C22semidirect
48. #15690772C22semidirect
49. C4×C22⋊D97C22semidirect
50. #15690770C22semidirect
51. #15690769C22semidirect
52. #15690768C22semidirect
53. #15690767C22semidirect
54. #15690767C22semidirect
55. #15690766C22semidirect
64. #127×C97C22semidirect
65. #125×C97C22semidirect
66. #123×C97C22semidirect
67. #121×C97C22semidirect
68. #121×C97C22semidirect
69. C4×D4×C97C22semidirect
70. #117×C97C22semidirect
206. D97#540direct (N × H)
208. C97C2×#540semidirect
216. #540D97direct (N × H)
232. #127C2×D97semidirect
236. #125C2×D97semidirect
244. #123C2×D97semidirect
257. #121C2×D97semidirect
258. #121C2×D97semidirect
264. C4×D4C2×D97semidirect
270. #117C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202613412
#202613402
#202613392
#202613382
#202613372
#202613362
#202613352
#127×D972
#125×D972
#123×D972
#121×D972
C4×D4×D972
#117×D972
#540×C972
C2×#54097