Finite Group Databasebeta

← #12256145

Subgroups — #12256145

Order 1280.

568 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3280C211yesmaximal, characteristic
2#10611144C211yesmaximal, characteristic, complemented
3#10602876C211yesmaximal, characteristic, complemented
4#10602870C211yesmaximal, characteristic, complemented
5#10602829C211yesmaximal, characteristic, complemented
6C2×#64782C211yesmaximal, characteristic, complemented
7#10602117C211yesmaximal, characteristic, complemented
8#10602111C211yesmaximal, characteristic, complemented
9#10601874C211yesmaximal, characteristic, complemented
10#10601871C211yesmaximal, characteristic, complemented
11#10601865C211yesmaximal, characteristic, complemented
12#10601863C211yesmaximal, characteristic, complemented
13C4×#3959C211yesmaximal, characteristic, complemented
14#10601681C211yesmaximal, characteristic, complemented
15D4×#672C211yesmaximal, characteristic, complemented
16C2×C5×#135C2211yescharacteristic
17C22×C5×D8C2211yescharacteristic
18C5×#546—21no—
19#65875—21no—
20C5×#495C2211yescharacteristic
21C5×#467C2211yescharacteristic
22C5×#466C2211yescharacteristic
23C5×#463—21no—
24C5×#450—21no—
25C5×#449—21no—
26C4×C5×D8C2211yescharacteristic
27C5×C8×D4C2211yescharacteristic
28C2×#3961C2211yescharacteristic, complemented
29C2×#3959C2211yescharacteristic
30C2×#3959C2211yescharacteristic
31#65405C2211yescharacteristic
32#65405C2211yescharacteristic, complemented
33#65395C2211yescharacteristic, complemented
34C2×#3950C2211yescharacteristic
35C22×#672C2211yescharacteristic
36C22×#672C2211yescharacteristic, complemented
37#65118—21no—
38#65012—21no—
39#65010—21no—
40C4×Dic20C2211yescharacteristic, complemented
41#64972C2211yescharacteristic, complemented
42#64784C2211yescharacteristic, complemented
43#64784C2211yescharacteristic, complemented
44#64784C2211yescharacteristic, complemented
45#64782C2211yescharacteristic
46#64782C2211yescharacteristic
47#64782C2211yescharacteristic
48#64782C2211yescharacteristic
49#64782C2211yescharacteristic
50#64782C2211yescharacteristic
51C22⋊#672C2211yescharacteristic
52C22⋊#672C2211yescharacteristic, complemented
53#64712C2211yescharacteristic, complemented
54#64710C2211yescharacteristic, complemented
55#64710C2211yescharacteristic, complemented
56#64709C2211yescharacteristic, complemented
57C4⋊#672C2211yescharacteristic, complemented
58C4×#672C2211yescharacteristic, complemented
59#33863—51nomaximal, Sylow
60C2×C5×#41—21no—
61C22×C5×D4—21no—
62C5×#135—22no—
63C5×#135—41no—
64C2×C5×Q16C2311yescharacteristic
65C2×C5×SD16—21no—
66C2×C5×SD16—21no—
67C2×C5×D8—22no—
68C2×C5×D8—21no—
69C2×C5×D8—41no—
70C2×C5×D8C2311yescharacteristic
71C2×C5×D8C2311yescharacteristic
72C22×C5×C8C2311yescharacteristic
73C22×C5×C8C2311yescharacteristic
74C5×#124—21no—
75C5×#123—41no—
76C5×C22⋊Q8—21no—
77C5×#121—21no—
78C5×#121—41no—
79C5×#120—41no—
80C4×C5×D4—21no—
81C4×C5×D4D411yescharacteristic
82C2×C5×C22⋊C4—41no—
83#4081—41no—
84#4072—21no—
85C2×C4⋊Dic5—21no—
86#4070—21no—
87#4070—21no—
88C2×Dic20C2311yescharacteristic
89#4035—41no—
90#4025—41no—
91#4023—41no—
92#4016—21no—
93C4×Dic10—21no—
94C5×#107C2311yescharacteristic
95C5×C4⋊C8D411yescharacteristic
96C5×#103—21no—
97C5×#102—21no—
98C5×#102—21no—
99C5×C22⋊C8D411yescharacteristic
100C5×C22⋊C8D411yescharacteristic
101C4×C5×C8C2311yescharacteristic
102#3961—41no—
103#3959—22no—
104#3959—41no—
105#3950—21no—
106#3950C2311yescharacteristic
107#3950C2311yescharacteristic
108#3950C2311yescharacteristic
109#3949—21no—
110#3949—21no—
111#3948—21no—
112#3948—21no—
113#3948—21no—
114C2×#672—41no—
115C2×#672—41no—
116C2×#672C2311yescharacteristic
117C2×#672C2311yescharacteristic
118C2×#672C2311yescharacteristic
119C2×#672C2311yescharacteristic
120#3336—51no—
121#3280D511yescharacteristic
122#2242—51no—
123#2235—51no—
124#2234—51no—
125#2228—51no—
126#2217—51no—
127#2216—51no—
128#2211—51no—
129#2189—51no—
130#2187—51no—
131C4×SD32—51no—
132D4×C16—51no—
133#2138—51no—
134C2×#357—51no—
135C24×C5—41no—
136C5×#41—42no—
137C5×#41—81no—
138C2×C5×Q8—21no—
139C2×C5×D4—21no—
140C2×C5×D4—21no—
141C2×C5×D4—42no—
142C2×C5×D4—41no—
143C2×C5×D4—41no—
144C2×C5×D4—41no—
145C2×C5×D4—81no—
146C2×C5×D4D811yescharacteristic
147C22×C4×C5—41no—
148C22×C4×C5—41no—
149C22×C4×C5C2×D411yescharacteristic
150C22×C4×C5C2×D411yescharacteristic
151C22×Dic5—41no—
152C2×Dic10—21no—
153C5×Q16—21no—
154C5×Q16—21no—
155C5×SD16—42no—
156C5×SD16—81no—
157C5×D8—21no—
158C5×D8—21no—
159C5×D8—21no—
160C5×D8—42no—
161C5×D8#4112yes—
162C2×C5×C8—21no—
163C2×C5×C8—21no—
164C2×C5×C8—21no—
165C2×C5×C8—21no—
166C2×C5×C8C2×D411yescharacteristic
167C2×C5×C8C2×D411yescharacteristic
168C2×C5×C8C2×D411yescharacteristic
169C2×C5×C8C2411yescharacteristic
170C5×C4⋊C4—21no—
171C5×C4⋊C4—41no—
172C5×C4⋊C4D811yescharacteristic
173C5×C22⋊C4—41no—
174C5×C22⋊C4—41no—
175C5×C22⋊C4—41no—
176C5×C22⋊C4—41no—
177C5×C22⋊C4—41no—
178C5×C22⋊C4D811yescharacteristic
179C5×C22⋊C4D811yescharacteristic
180C42×C5C2×D411yescharacteristic
181C22⋊Dic5—41no—
182C22⋊Dic5—41no—
183C22⋊Dic5—41no—
184C4⋊Dic5—21no—
185C4⋊Dic5—21no—
186C4⋊Dic5—21no—
187C4⋊Dic5—41no—
188#683—41no—
189#683—41no—
190#683—41no—
191#683—41no—
192#683—41no—
193C4×Dic5—41no—
194C4×Dic5—41no—
195Dic20—21no—
196Dic20—21no—
197#672—21no—
198#672—21no—
199#672—21no—
200#672—21no—
201C2×#135C2×D511yescharacteristic
202C22×D8C2×D511yescharacteristic
203#552—101no—
204#546—21no—
205C2×Q32—51no—
206C2×SD32—51no—
207C2×SD32—51no—
208C22×C16—51no—
209C22×C16—51no—
210#499—51no—
211#495C2×D511yescharacteristic
212#485—101no—
213#483—101no—
214#479—101no—
215#467—51no—
216#467—51no—
217#467C2×D511yescharacteristic
218#466C2×D511yescharacteristic
219#463—21no—
220#450—21no—
221#449—21no—
222C4×Q16—51no—
223C4×D8C2×D511yescharacteristic
224C8×D4C2×D511yescharacteristic
225#427—51no—
226C2×#107—51no—
227#367—51no—
228#367—51no—
229#366—51no—
230C4⋊C16—51no—
231#358—51no—
232#358—51no—
233#358—51no—
234#358—51no—
235#357—51no—
236#357—51no—
237#357—51no—
238#357—51no—
239#357—51no—
240#357—51no—
241C22⋊C16—51no—
242C22⋊C16—51no—
243C4×C16—51no—
244C23×C5—41no—
245C23×C5—41no—
246C23×C5—82no—
247C23×C5—82no—
248C23×C5—161no—
249C23×C5C2×D811yescharacteristic
250C23×C5C2×D811yescharacteristic
251C5×Q8—41no—
252C5×Q8—41no—
253C5×D4—22no—
254C5×D4—22no—
255C5×D4—41no—
256C5×D4—41no—
257C5×D4—41no—
258C5×D4—82no—
259C5×D4—82no—
260C5×D4—161no—
261C2×C4×C5—21no—
262C2×C4×C5—21no—
263C2×C4×C5—21no—
264C2×C4×C5—21no—
265C2×C4×C5—41no—
266C2×C4×C5—41no—
267C2×C4×C5—82no—
268C2×C4×C5—161no—
269C2×C4×C5C2×D811yescharacteristic
270C2×C4×C5C2×D811yescharacteristic
271C2×C4×C5C2×D811yescharacteristic
272C2×C4×C5C2×D811yescharacteristic
273C2×C4×C5C22×D411yescharacteristic
274C2×Dic5—41no—
275C2×Dic5—41no—
276C2×Dic5—41no—
277C2×Dic5—41no—
278C2×Dic5—82no—
279Dic10—41no—
280Dic10—41no—
281C5×C8—21no—
282C5×C8—21no—
283C5×C8—21no—
284C5×C8C2×#4111yescharacteristic
285C5×C8H(5,𝔽2)11yescharacteristic
286C2×#41—21no—
287C22×D4—21no—
288#135—22no—
289#135—41no—
290C2×Q16—51no—
291C2×Q16C22×D511yescharacteristic
292C2×SD16—21no—
293C2×SD16—21no—
294C2×D8—22no—
295C2×D8—21no—
296C2×D8—41no—
297C2×D8C22×D511yescharacteristic
298C2×D8C22×D511yescharacteristic
299C22×C8C22×D511yescharacteristic
300C22×C8C22×D511yescharacteristic
301#126—201no—
302#125—101no—
303#125—201no—
304#124—21no—
305#123—41no—
306#123—201no—
307C22⋊Q8—21no—
308C22⋊Q8—101no—
309C22⋊Q8—101no—
310#121—21no—
311#121—41no—
312#120—41no—
313C4×Q8—101no—
314C4×D4—21no—
315C4×D4C22⋊D511yescharacteristic
316#117—101no—
317#117—201no—
318C2×C4⋊C4—101no—
319C2×C22⋊C4—41no—
320Q32—201no—
321SD32—102no—
322SD32—201no—
323C2×C16—51no—
324C2×C16—51no—
325C2×C16—51no—
326C2×C16—51no—
327C2×C16—201no—
328C2×C16—201no—
329#107—51no—
330#107—51no—
331#107—51no—
332#107—101no—
333#107C22×D511yescharacteristic
334#106—101no—
335#106—101no—
336C4⋊C8C22⋊D511yescharacteristic
337#103—21no—
338#103—101no—
339#103—101no—
340#103—101no—
341#102—21no—
342#102—21no—
343C22⋊C8C22⋊D511yescharacteristic
344C22⋊C8C22⋊D511yescharacteristic
345C4×C8C22×D511yescharacteristic
346C22×C5—21no—
347C22×C5—21no—
348C22×C5—21no—
349C22×C5—21no—
350C22×C5—21no—
351C22×C5—42no—
352C22×C5—81no—
353C22×C5—81no—
354C22×C5—81no—
355C22×C5—81no—
356C22×C5—81no—
357C22×C5—164no—
358C22×C5#50812yes—
359C22×C5C22×D811yescharacteristic
360C4×C5—21no—
361C4×C5—21no—
362C4×C5—21no—
363C4×C5—21no—
364C4×C5—81no—
365C4×C5—81no—
366C4×C5#54611yescharacteristic
367C4×C5#57711yescharacteristic
368Dic5—81no—
369Dic5—81no—
370Dic5—81no—
371Dic5—81no—
372C24—41no—
373#41—42no—
374#41—81no—
375C2×Q8—21no—
376C2×Q8—101no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—42no—
380C2×D4—41no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4—81no—
384C2×D4#68611yescharacteristic
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—201no—
388C22×C4C2×C22⋊D511yescharacteristic
389C22×C4C2×C22⋊D511yescharacteristic
390Q16—21no—
391Q16—21no—
392Q16—101no—
393Q16—101no—
394SD16—42no—
395SD16—81no—
396D8—21no—
397D8—21no—
398D8—21no—
399D8—42no—
400D8#71112yes—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8C2×C22⋊D511yescharacteristic
406C2×C8C2×C22⋊D511yescharacteristic
407C2×C8C2×C22⋊D511yescharacteristic
408C2×C8C23×D511yescharacteristic
409C4⋊C4—21no—
410C4⋊C4—41no—
411C4⋊C4—101no—
412C4⋊C4—101no—
413C4⋊C4—101no—
414C4⋊C4—201no—
415C4⋊C4—201no—
416C4⋊C4—201no—
417C4⋊C4—201no—
418C4⋊C4—201no—
419C4⋊C4—201no—
420C4⋊C4#68611yescharacteristic
421C22⋊C4—41no—
422C22⋊C4—41no—
423C22⋊C4—41no—
424C22⋊C4—41no—
425C22⋊C4—41no—
426C22⋊C4—201no—
427C22⋊C4—201no—
428C22⋊C4—201no—
429C22⋊C4#68611yescharacteristic
430C22⋊C4#68611yescharacteristic
431C42—201no—
432C42—201no—
433C42C2×C22⋊D511yescharacteristic
434C16—101no—
435C16—101no—
436C16—101no—
437C16—101no—
438C2×C5—22no—
439C2×C5—41no—
440C2×C5—82no—
441C2×C5—161no—
442C2×C5#329411yescharacteristic
443C2×C5C2×#50811yescharacteristic
444C2×C5#341611yescharacteristic
445C23—41no—
446C23—41no—
447C23—82no—
448C23—82no—
449C23—161no—
450C23C2×#68611yescharacteristic
451C23C2×#68611yescharacteristic
452Q8—41no—
453Q8—41no—
454Q8—201no—
455Q8—201no—
456D4—22no—
457D4—22no—
458D4—41no—
459D4—41no—
460D4—41no—
461D4—82no—
462D4—82no—
463D4—161no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—82no—
471C2×C4—161no—
472C2×C4—201no—
473C2×C4—201no—
474C2×C4—201no—
475C2×C4—201no—
476C2×C4—402no—
477C2×C4C2×#68611yescharacteristic
478C2×C4C2×#68611yescharacteristic
479C2×C4C2×#68611yescharacteristic
480C2×C4C2×#68611yescharacteristic
481C2×C4C22×C22⋊D511yescharacteristic
482C8—21no—
483C8—21no—
484C8—21no—
485C8C2×#71111yescharacteristic
486C8#414411yescharacteristic
487C5#3386311yescharacteristic, complemented, Sylow
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—42no—
494C22—81no—
495C22—81no—
496C22—81no—
497C22—81no—
498C22—81no—
499C22—164no—
500C22#6548312yes—
501C22C22×#68611yescharacteristic
502C4—21no—
503C4—21no—
504C4—21no—
505C4—21no—
506C4—81no—
507C4—81no—
508C4—401no—
509C4—401no—
510C4—401no—
511C4—401no—
512C4#6614011yescharacteristic
513C4#6617011yescharacteristic
514C2—22no—
515C2—41no—
516C2—82no—
517C2—161no—
518C2#1061521311yescharacteristic
519C2C2×#6548311yescharacteristic
520C2#1061573911yescharacteristic
subgroupG/H
centerC22C22×#686
commutator, cosocleC2×C5×C8C24
FrattiniC2×C8C23×D5
Fitting, prime radicalC5×#3280C2
socleC22×C5C22×D8
hypercenter#3280D5
hypocenterC5#33863
LCSG ▷ C2×C5×C8 ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3280

Complements

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

NH ≅ G/Ntype
2. #10611144C2semidirect
3. #10602876C2semidirect
4. #10602870C2semidirect
5. #10602829C2semidirect
6. C2×#64782C2semidirect
7. #10602117C2semidirect
8. #10602111C2semidirect
9. #10601874C2semidirect
10. #10601871C2semidirect
11. #10601865C2semidirect
12. #10601863C2semidirect
13. C4×#3959C2semidirect
14. #10601681C2semidirect
15. D4×#672C2semidirect
28. C2×#3961C22semidirect
32. #65405C22semidirect
33. #65395C22semidirect
36. C22×#672C22semidirect
40. C4×Dic20C22semidirect
41. #64972C22semidirect
42. #64784C22semidirect
43. #64784C22semidirect
44. #64784C22semidirect
52. C22⋊#672C22semidirect
53. #64712C22semidirect
54. #64710C22semidirect
55. #64710C22semidirect
56. #64709C22semidirect
57. C4⋊#672C22semidirect
58. C4×#672C22semidirect
487. C5#33863semidirect

Maximal subgroups

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

subgroupindex
C5×#32802
#106111442
#106028762
#106028702
#106028292
C2×#647822
#106021172
#106021112
#106018742
#106018712
#106018652
#106018632
C4×#39592
#106016812
D4×#6722
#338635