Finite Group Databasebeta

← #12967822

Subgroups — #12967822

Order 1280.

2138 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#540C211yesmaximal, characteristic, complemented
2#10617632C211yesmaximal, characteristic, complemented
3#10615170C214yesmaximal, complemented
4#10615166C214yesmaximal, complemented
5#10615165C218yesmaximal, complemented
6#10614896C212yesmaximal, complemented
7#10614748C212yesmaximal, complemented
8#10614355C212yesmaximal, complemented
9#10614347C212yesmaximal, complemented
10C4×#4078C214yesmaximal, complemented
11#10613940C211yesmaximal, characteristic, complemented
12C22×#709C2211yescharacteristic
13C5×#540—22no—
14C5×#540—22no—
15C5×#540C2214yes—
16C5×#540C2214yescomplemented
17C2×C5×#127C2211yescharacteristic, complemented
18C2×C5×#125C2211yescharacteristic, complemented
19C2×C5×#123—22no—
20C2×C5×#121—22no—
21C2×C5×#121C2212yes—
22C2×C4×C5×D4C2212yescomplemented
23C2×C5×#117C2211yescharacteristic
24C2×#4078C2214yes—
25C2×#4072C2211yescharacteristic
26#65857C2214yescomplemented
27#65856—22no—
28#65855C2212yescomplemented
29D5×#117—22no—
30#65850C2212yescomplemented
31#65838—22no—
32#65820—22no—
33C4×#709C2212yes—
34C4×#709C2212yes—
35#65504C2214yescomplemented
36#65496C2212yes—
37#65364C2218yescomplemented
38#65361C2218yescomplemented
39#65355C2218yescomplemented
40#65352C2218yescomplemented
41#65329C2214yescomplemented
42#65326C2218yescomplemented
43#65325C2214yescomplemented
44#65324C2218yescomplemented
45#65323C2214yescomplemented
46#65322C2214yescomplemented
47#65307C2218yescomplemented
48C4×#687C2218yescomplemented
49#65303C2218yescomplemented
50C4×#686C2218yescomplemented
51#65289C2214yescomplemented
52#65289C2214yescomplemented
53#65255C2214yes—
54#65253C2214yescomplemented
55#65213C2212yescomplemented
56C4×#681C2212yescomplemented
57#60568—51nomaximal, Sylow
58C22×C5×D4—22no—
59C22×C5×D4—22no—
60C22×C5×D4C2312yes—
61C23×C4×C5—22no—
62C23×C4×C5C2311yescharacteristic
63C22×C22⋊D5—21no—
64C2×#709—21no—
65C2×#709—21no—
66C2×#709—28no—
67C2×#709C2312yes—
68C2×#709C2312yes—
69C22×D20C2311yescharacteristic
70C22×C4×D5—21no—
71C22×Dic10C2311yescharacteristic
72C5×#127—22no—
73C5×#127C2314yes—
74C5×#125—22no—
75C5×#125C2314yes—
76C5×#123—28no—
77C5×#123—28no—
78C5×#121—22no—
79C5×#121—22no—
80C5×#121—28no—
81C5×#121—28no—
82C5×#121C2314yes—
83C5×#121C2314yes—
84C4×C5×D4—24no—
85C4×C5×D4C2318yes—
86C5×#117D412yes—
87C5×#117D412yes—
88C5×#117C2312yes—
89C5×#117C2312yes—
90C2×C5×C4⋊C4—22no—
91C2×C5×C4⋊C4D412yes—
92C2×C5×C4⋊C4C2312yes—
93C2×C5×C22⋊C4—22no—
94C2×C5×C22⋊C4—22no—
95C2×C5×C22⋊C4D412yes—
96C2×C42×C5C2311yescharacteristic
97C2×C42×C5C2311yescharacteristic
98C2×C22⋊Dic5—21no—
99C2×#687C2318yes—
100#4078—216no—
101C2×#686C2318yes—
102C4×C22⋊D5—24no—
103C4×C22⋊D5—24no—
104#4072—21no—
105#4072—21no—
106#4072C2312yes—
107#4072C2312yes—
108C2×C4⋊Dic5C2312yes—
109C2×C4×Dic5—21no—
110C2×#681C2314yes—
111#4039—24no—
112#4038—24no—
113D5×C4⋊C4—24no—
114#4033—24no—
115#4027—24no—
116#4027—24no—
117D5×C22⋊C4—24no—
118#4023—24no—
119C4×D20C2314yes—
120C4×D20C2314yes—
121#4018—22no—
122#4018—22no—
123C42×D5—22no—
124C42×D5—22no—
125C4×Dic10C2314yes—
126C4×Dic10C2314yes—
127#3964C2318yes—
128#3964C2318yes—
129C22⋊#168C2314yes—
130C22⋊#168C2314yes—
131#3942C2318yes—
132#3941C2318yes—
133#3940C2318yes—
134#3939C2318yes—
135C4⋊#168C2318yes—
136#3935C2314yes—
137C4×#168C2314yes—
138C2×#540D511yescharacteristic, complemented
139#3428—51no—
140#3126—54no—
141#3122—54no—
142#3119—58no—
143#3094—52no—
144#3080—52no—
145#3038—52no—
146#3006—52no—
147C4×AGL(1,ℤ/8)—54no—
148#2921—51no—
149C24×C5—22no—
150C24×C5C2×D411yescharacteristic
151C23×D5—21no—
152C2×C5×D4—24no—
153C2×C5×D4—24no—
154C2×C5×D4—22no—
155C2×C5×D4—28no—
156C2×C5×D4—28no—
157C2×C5×D4—84no—
158C2×C5×D4—84no—
159C2×C5×D4C2414yes—
160C22×C4×C5—24no—
161C22×C4×C5—22no—
162C22×C4×C5—28no—
163C22×C4×C5—84no—
164C22×C4×C5C2×D411yescharacteristic
165C22×C4×C5C2×D411yescharacteristic
166C22×C4×C5C2×D411yescharacteristic
167C22×C4×C5C2×D412yes—
168C22×C4×C5C2×D412yes—
169C22×C4×C5C2411yescharacteristic
170C22×C4×C5C2411yescharacteristic
171C22×C4×C5C2411yescharacteristic
172C2×C22⋊D5—28no—
173C2×C22⋊D5—22no—
174C2×C22⋊D5—22no—
175C22×Dic5—21no—
176C22×Dic5—22no—
177#709—28no—
178#709—28no—
179#709—24no—
180#709—24no—
181#709—24no—
182#709#4118yes—
183C2×D20—21no—
184C2×D20—24no—
185C2×D20C2412yes—
186C2×C4×D5—28no—
187C2×C4×D5—22no—
188C2×C4×D5—22no—
189C2×C4×D5—24no—
190C2×C4×D5—24no—
191C2×Dic10—21no—
192C2×Dic10—24no—
193C2×Dic10C2412yes—
194C5×C4⋊C4—28no—
195C5×C4⋊C4—22no—
196C5×C4⋊C4C2×D414yes—
197C5×C4⋊C4C2×D414yes—
198C5×C4⋊C4C2414yes—
199C5×C22⋊C4—28no—
200C5×C22⋊C4—24no—
201C5×C22⋊C4—24no—
202C5×C22⋊C4C2×D414yes—
203C5×C22⋊C4C2×D414yes—
204C42×C5C2×D412yes—
205C42×C5C2×D412yes—
206C42×C5C2412yes—
207C42×C5C2412yes—
208C22⋊Dic5—22no—
209C22⋊Dic5—22no—
210#687—216no—
211#686—216no—
212#685—24no—
213#685—24no—
214C4⋊Dic5—22no—
215C4⋊Dic5C2414yes—
216#683—24no—
217#683—24no—
218C4×Dic5—24no—
219C4×Dic5—24no—
220C4×Dic5—22no—
221C4×Dic5—22no—
222#681—28no—
223C2×#168C2418yes—
224C22×#41—51no—
225C2×AGL(1,ℤ/8)—54no—
226#540—22no—
227#540—22no—
228#540C2×D514yes—
229#540C2×D514yescomplemented
230C2×#127C2×D511yescharacteristic, complemented
231C2×#125C2×D511yescharacteristic, complemented
232C2×#123—22no—
233C2×#121—22no—
234C2×#121C2×D512yes—
235#520—54no—
236#520—102no—
237#519—52no—
238#518—52no—
239#518—102no—
240C4×#41—52no—
241C4×#41—52no—
242C4×#41—102no—
243C2×C4×D4C2×D512yescomplemented
244C2×#117—51no—
245C2×#117—102no—
246C2×#117C2×D511yescharacteristic
247#482—58no—
248#481—54no—
249#480—54no—
250#479—58no—
251#478—58no—
252#460—58no—
253#459—58no—
254#449—58no—
255#448—54no—
256#447—54no—
257#442—58no—
258#440—58no—
259C4×SD16—58no—
260C4×D8—58no—
261#433—54no—
262#433—54no—
263#428—54no—
264#423—52no—
265#418—54no—
266#417—54no—
267#407—52no—
268C4×M16—52no—
269C23×C5—22no—
270C23×C5—24no—
271C23×C5—42no—
272C23×C5—44no—
273C23×C5—44no—
274C23×C5—44no—
275C23×C5C22×D411yescharacteristic
276C23×C5C22×D411yescharacteristic
277C23×C5C22×D411yescharacteristic
278C22×D5—22no—
279C22×D5—24no—
280C22×D5—82no—
281C5×D4—28no—
282C5×D4—44no—
283C5×D4—88no—
284C5×D4—84no—
285C5×D4—84no—
286C2×C4×C5—22no—
287C2×C4×C5—22no—
288C2×C4×C5—22no—
289C2×C4×C5—22no—
290C2×C4×C5—22no—
291C2×C4×C5—24no—
292C2×C4×C5—42no—
293C2×C4×C5—42no—
294C2×C4×C5—44no—
295C2×C4×C5—88no—
296C2×C4×C5C22×D411yescharacteristic
297C2×C4×C5C22×D414yes—
298C2×C4×C5C22×D414yes—
299C2×C4×C5C22×D411yescharacteristic
300C2×C4×C5C22×D411yescharacteristic
301C2×C4×C5C2×#4114yes—
302C2×C4×C5C2511yescharacteristic
303C22⋊D5—28no—
304C22⋊D5—28no—
305C2×Dic5—24no—
306C2×Dic5—22no—
307C2×Dic5—24no—
308C2×Dic5—82no—
309D20—24no—
310D20—22no—
311D20C2×#4114yes—
312C4×D5—28no—
313C4×D5—28no—
314C4×D5—84no—
315C4×D5—84no—
316Dic10—24no—
317Dic10—22no—
318Dic10C2×#4114yes—
319#168—28no—
320C2×#41—52no—
321C2×#41—52no—
322C2×#41—108no—
323C2×#41—101no—
324C2×#41—101no—
325C22×Q8—51no—
326C22×D4—22no—
327C22×D4—22no—
328C22×D4—51no—
329C22×D4—101no—
330C22×D4C22×D512yes—
331C23×C4—22no—
332C23×C4—101no—
333C23×C4C22×D511yescharacteristic
334AGL(1,ℤ/8)—1016no—
335C2×SD16—58no—
336C2×D8—58no—
337C2×M16—54no—
338#127—22no—
339#127C22×D514yes—
340#125—22no—
341#125C22×D514yes—
342#123—28no—
343#123—28no—
344#121—22no—
345#121—22no—
346#121—28no—
347#121—28no—
348#121C22×D514yes—
349#121C22×D514yes—
350C4×Q8—54no—
351C4×Q8—54no—
352C4×Q8—104no—
353C4×D4—24no—
354C4×D4—54no—
355C4×D4—54no—
356C4×D4—104no—
357C4×D4—104no—
358C4×D4—104no—
359C4×D4—104no—
360C4×D4—104no—
361C4×D4C22×D518yes—
362#117—52no—
363#117—52no—
364#117—101no—
365#117—101no—
366#117—104no—
367#117—104no—
368#117—102no—
369#117—102no—
370#117C22⋊D512yes—
371#117C22⋊D512yes—
372#117C22×D512yes—
373#117C22×D512yes—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—52no—
376C2×C4⋊C4—104no—
377C2×C4⋊C4C22⋊D512yes—
378C2×C4⋊C4C22×D512yes—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—101no—
382C2×C22⋊C4—104no—
383C2×C22⋊C4C22⋊D512yes—
384C2×C42—101no—
385C2×C42—102no—
386C2×C42—102no—
387C2×C42C22×D511yescharacteristic
388C2×C42C22×D511yescharacteristic
389#107—58no—
390#106—58no—
391C4⋊C8—58no—
392#103—58no—
393#102—58no—
394#102—58no—
395#102—58no—
396C22⋊C8—54no—
397C22⋊C8—54no—
398#97—54no—
399C4×C8—54no—
400C22×C5—22no—
401C22×C5—22no—
402C22×C5—22no—
403C22×C5—21no—
404C22×C5—21no—
405C22×C5—44no—
406C22×C5—44no—
407C22×C5—44no—
408C22×C5—82no—
409C22×C5—88no—
410C22×C5#52512yes—
411C22×C5C23×D411yescharacteristic
412C2×D5—24no—
413C2×D5—44no—
414C2×D5—84no—
415C4×C5—24no—
416C4×C5—24no—
417C4×C5—21no—
418C4×C5—21no—
419C4×C5—44no—
420C4×C5#52512yes—
421C4×C5C22×#4112yes—
422Dic5—24no—
423Dic5—44no—
424C24—22no—
425C24—101no—
426C24C2×C22⋊D511yescharacteristic
427#41—58no—
428#41—104no—
429#41—104no—
430#41—108no—
431#41—108no—
432#41—104no—
433C2×Q8—52no—
434C2×Q8—104no—
435C2×Q8—101no—
436C2×D4—24no—
437C2×D4—24no—
438C2×D4—28no—
439C2×D4—22no—
440C2×D4—28no—
441C2×D4—52no—
442C2×D4—84no—
443C2×D4—84no—
444C2×D4—104no—
445C2×D4—108no—
446C2×D4—102no—
447C2×D4—102no—
448C2×D4—101no—
449C2×D4C23×D514yes—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—28no—
453C22×C4—84no—
454C22×C4—108no—
455C22×C4—101no—
456C22×C4—102no—
457C22×C4—102no—
458C22×C4—102no—
459C22×C4—104no—
460C22×C4—104no—
461C22×C4C2×C22⋊D511yescharacteristic
462C22×C4C2×C22⋊D511yescharacteristic
463C22×C4C2×C22⋊D511yescharacteristic
464C22×C4C2×C22⋊D512yes—
465C22×C4C2×C22⋊D512yes—
466C22×C4C23×D511yescharacteristic
467C22×C4C23×D511yescharacteristic
468C22×C4C23×D511yescharacteristic
469SD16—1016no—
470D8—1016no—
471M16—108no—
472C2×C8—58no—
473C4⋊C4—28no—
474C4⋊C4—22no—
475C4⋊C4—54no—
476C4⋊C4—102no—
477C4⋊C4—104no—
478C4⋊C4—104no—
479C4⋊C4C2×C22⋊D514yes—
480C4⋊C4C2×C22⋊D514yes—
481C4⋊C4C23×D514yes—
482C22⋊C4—28no—
483C22⋊C4—24no—
484C22⋊C4—24no—
485C22⋊C4—102no—
486C22⋊C4—102no—
487C22⋊C4—104no—
488C22⋊C4—104no—
489C22⋊C4C2×C22⋊D514yes—
490C22⋊C4C2×C22⋊D514yes—
491C42—102no—
492C42—102no—
493C42—104no—
494C42—104no—
495C42C2×C22⋊D512yes—
496C42C2×C22⋊D512yes—
497C42C23×D512yes—
498C42C23×D512yes—
499C2×C5—22no—
500C2×C5—41no—
501C2×C5—41no—
502C2×C5—84no—
503C2×C5C2×#52511yescharacteristic
504C2×C5#358512yes—
505D5—44no—
506C23—22no—
507C23—24no—
508C23—42no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—104no—
513C23—102no—
514C23—402no—
515C23C22×C22⋊D511yescharacteristic
516C23C22×C22⋊D511yescharacteristic
517C23C22×C22⋊D511yescharacteristic
518Q8—54no—
519Q8—104no—
520Q8—102no—
521D4—28no—
522D4—44no—
523D4—54no—
524D4—88no—
525D4—84no—
526D4—84no—
527D4—104no—
528D4—108no—
529D4—108no—
530D4—102no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—88no—
541C2×C4—108no—
542C2×C4—104no—
543C2×C4—102no—
544C2×C4—108no—
545C2×C4—104no—
546C2×C4—404no—
547C2×C4—404no—
548C2×C4—402no—
549C2×C4D5×#4114yes—
550C2×C4C22×C22⋊D511yescharacteristic
551C2×C4C22×C22⋊D514yes—
552C2×C4C22×C22⋊D514yes—
553C2×C4C22×C22⋊D511yescharacteristic
554C2×C4C22×C22⋊D511yescharacteristic
555C2×C4C24×D511yescharacteristic
556C8—108no—
557C5#6056811yescharacteristic, complemented, Sylow
558C22—22no—
559C22—21no—
560C22—21no—
561C22—22no—
562C22—22no—
563C22—44no—
564C22—44no—
565C22—44no—
566C22—82no—
567C22—88no—
568C22—104no—
569C22—204no—
570C22—404no—
571C22#6616212yes—
572C22C23×C22⋊D511yescharacteristic
573C4—21no—
574C4—21no—
575C4—24no—
576C4—24no—
577C4—44no—
578C4—104no—
579C4—204no—
580C4#6616212yes—
581C4C2×D5×#4112yes—
582C2—22no—
583C2—41no—
584C2—41no—
585C2—84no—
586C2—204no—
587C2C2×#6616211yescharacteristic
588C2#1061917812yes—
subgroupG/H
centerC22C23×C22⋊D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC2×C5×#540C2
socleC22×C5C23×D4
hypercenterC2×#540D5
hypocenterC5#60568
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×#117 ◁ C2×#540

Complements

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

NH ≅ G/Ntype
1. C2×C5×#540C2semidirect
2. #10617632C2semidirect
3. #10615170C2semidirect
4. #10615166C2semidirect
5. #10615165C2semidirect
6. #10614896C2semidirect
7. #10614748C2semidirect
8. #10614355C2semidirect
9. #10614347C2semidirect
10. C4×#4078C2semidirect
11. #10613940C2semidirect
16. C5×#540C22semidirect
17. C2×C5×#127C22semidirect
18. C2×C5×#125C22semidirect
22. C2×C4×C5×D4C22semidirect
26. #65857C22semidirect
28. #65855C22semidirect
30. #65850C22semidirect
35. #65504C22semidirect
37. #65364C22semidirect
38. #65361C22semidirect
39. #65355C22semidirect
40. #65352C22semidirect
41. #65329C22semidirect
42. #65326C22semidirect
43. #65325C22semidirect
44. #65324C22semidirect
45. #65323C22semidirect
46. #65322C22semidirect
47. #65307C22semidirect
48. C4×#687C22semidirect
49. #65303C22semidirect
50. C4×#686C22semidirect
51. #65289C22semidirect
52. #65289C22semidirect
54. #65253C22semidirect
55. #65213C22semidirect
56. C4×#681C22semidirect
138. C2×#540D5semidirect
229. #540C2×D5semidirect
230. C2×#127C2×D5semidirect
231. C2×#125C2×D5semidirect
243. C2×C4×D4C2×D5semidirect
557. C5#60568semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#5402
#106176322
#106151702
#106151662
#106151652
#106148962
#106147482
#106143552
#106143472
C4×#40782
#106139402
#605685