Finite Group Databasebeta

← #12966977

Subgroups — #12966977

Order 1280.

2054 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10617791C211yesmaximal, characteristic, complemented
2#10617787C211yesmaximal, characteristic
3C5×#2912C211yesmaximal, characteristic, complemented
4#10615528C211yesmaximal, characteristic
5#10615514C211yesmaximal, characteristic, complemented
6#10614294C211yesmaximal, characteristic
7#10614293C211yesmaximal, characteristic, complemented
8#10610974C213yesmaximal, complemented
9#10610973C213yesmaximal, complemented
10#10610972C213yesmaximal, complemented
11#10610970C213yesmaximal, complemented
12#10610928C213yesmaximal, complemented
13#10610927C213yesmaximal, complemented
14#10610861C213yesmaximal, complemented
15C2×#65160C213yesmaximal, complemented
16C2×#4149C2211yescharacteristic
17C2×C5×#131C2211yescharacteristic
18C5×#520C2211yescharacteristic
19C5×#520C2211yescharacteristic
20#66161C2211yescharacteristic
21#41×Dic5C2211yescharacteristic
22C2×#4094C2211yescharacteristic
23#65908—21no—
24C4×#713—21no—
25#65904—21no—
26#65903—21no—
27#65879—23no—
28C4×D4×D5—23no—
29#65877—23no—
30#65873—23no—
31#65856C2213yes—
32#65855C2213yes—
33#65853C2213yescomplemented
34#65851C2213yescomplemented
35#65838C2213yes—
36C2×#4039C2213yes—
37#65834C2213yescomplemented
38C2×#4036C2213yescomplemented
39C5×#428C2213yescomplemented
40C5×#428C2213yescomplemented
41C5×#427C2213yescomplemented
42C2×C5×#106C2213yescomplemented
43#65422C2213yes—
44#65420C2213yes—
45#65408C2213yescomplemented
46#65407C2213yescomplemented
47#65402C2213yes—
48C2×#3953C2213yes—
49#65391C2213yescomplemented
50C2×#3946C2213yescomplemented
51#65291C2213yes—
52#65290C2213yes—
53#65287C2213yescomplemented
54#65284C2213yescomplemented
55#65255C2213yes—
56C2×#3941C2213yes—
57#65253C2213yescomplemented
58C2×#3939C2213yescomplemented
59#65180C2213yescomplemented
60#65180C2216yescomplemented
61#65179C2216yescomplemented
62#65179C2213yescomplemented
63#65166C2213yescomplemented
64#65164C2213yescomplemented
65#65160C2216yescomplemented
66#65160C2216yescomplemented
67#65160C2213yescomplemented
68#65160C2216yescomplemented
69#65159C2216yescomplemented
70#65159C2213yescomplemented
71#65146C2216yescomplemented
72#65146C2211yescharacteristic, complemented
73#65143C2213yescomplemented
74#60496—51nomaximal, Sylow
75C2×C5×#41C2311yescharacteristic
76#4149—24no—
77#4149Q818yescomplemented
78C2×#713—21no—
79C2×D4×D5—23no—
80C2×#709C2313yes—
81C22×D20C2313yes—
82C22×C4×D5—23no—
83C5×#131—24no—
84C2×C5×M16C2313yes—
85C22×C5×C8C2313yes—
86C4×C5×Q8—21no—
87C4×C5×Q8—21no—
88C4×C5×D4—23no—
89C4×C5×D4—23no—
90C5×#117C2313yes—
91C5×#117C2313yes—
92C2×C5×C4⋊C4C2313yes—
93C2×C5×C4⋊C4C2313yes—
94#4094—24no—
95Q8×Dic5—21no—
96Q8×Dic5D411yescharacteristic
97Q8×Dic5D411yescharacteristic
98D4×Dic5—23no—
99D4×Dic5D413yes—
100D4×Dic5D413yes—
101C4×C22⋊D5—26no—
102C2×#685—23no—
103#4072C2313yes—
104#4072C2313yes—
105C2×C4⋊Dic5—23no—
106C2×C4⋊Dic5C2313yes—
107#4070—26no—
108C2×#683—23no—
109C2×C4×Dic5C2313yes—
110C2×#681C2313yes—
111C22×#168C2313yes—
112#4039—23no—
113#4039—23no—
114#4039C2313yes—
115#4039C2316yes—
116#4039C2313yes—
117#4038—23no—
118D5×C4⋊C4—23no—
119#4036—23no—
120#4036—23no—
121#4036—23no—
122#4036C2313yes—
123#4036C2316yes—
124#4036C2311yescharacteristic
125#4034—23no—
126#4034C2313yes—
127#4034C2313yes—
128#4033C2313yes—
129#4033C2311yescharacteristic
130#4027—23no—
131#4027—23no—
132D5×C22⋊C4—26no—
133#4024—23no—
134#4024—23no—
135#4024—26no—
136C4×D20—23no—
137C4×D20—23no—
138#4018—23no—
139C42×D5—23no—
140#4016—23no—
141#4015—23no—
142C4×Dic10—23no—
143C4×Dic10—23no—
144C5×#107C2313yes—
145C5×#107C2313yes—
146C5×#106C2313yes—
147C5×#106C2316yes—
148C5×#106C2311yescharacteristic
149#3953C2313yes—
150#3953C2316yes—
151#3953C2313yes—
152#3948C2311yescharacteristic
153#3948C2313yes—
154#3946C2311yescharacteristic
155#3946C2313yes—
156#3946C2313yes—
157#3946C2313yes—
158#3946C2316yes—
159#3942C2311yescharacteristic
160#3942C2313yes—
161#3941C2313yes—
162#3941C2316yes—
163#3941C2313yes—
164#3940C2313yes—
165#3940C2313yes—
166#3939C2316yes—
167#3939C2311yescharacteristic
168#3939C2313yes—
169#3504—51no—
170C4×H(5,𝔽2)—51no—
171#3084—53no—
172#3083—53no—
173#3081—53no—
174#3081—53no—
175#3080—53no—
176#3077—53no—
177#3076—53no—
178C2×#478—53no—
179#2913—51no—
180#2912D511yescharacteristic, complemented
181#2906—51no—
182#2897—51no—
183#2897—51no—
184C23×D5—23no—
185C5×#41C2×Q814yes—
186C5×#41#4114yes—
187C2×C5×Q8C2×D411yescharacteristic
188C2×C5×D4C2×D413yes—
189C22×C4×C5—23no—
190C22×C4×C5—23no—
191C22×C4×C5C2413yes—
192C2×C22⋊D5—23no—
193C22×Dic5—23no—
194C22×Dic5C2×D413yes—
195C22×Dic5C2×D413yes—
196#713—28no—
197D4×D5—224no—
198#709—26no—
199#709C2×Q8112yes—
200C2×D20—23no—
201C2×D20—26no—
202C2×D20C2×Q8112yes—
203C2×D20C2413yes—
204C2×C4×D5—23no—
205C2×C4×D5—23no—
206C2×C4×D5—26no—
207C2×C4×D5—23no—
208C2×C4×D5—86no—
209C2×Dic10—23no—
210C2×Dic10—23no—
211C2×Dic10C2411yescharacteristic
212C5×M16—26no—
213C2×C5×C8—26no—
214C2×C5×C8C2411yescharacteristic
215C2×C5×C8C2413yes—
216C5×C4⋊C4—23no—
217C5×C4⋊C4—23no—
218C5×C4⋊C4C2413yes—
219C5×C4⋊C4C2413yes—
220C5×C4⋊C4C2411yescharacteristic
221C5×C4⋊C4C2411yescharacteristic
222C5×C22⋊C4—23no—
223C5×C22⋊C4—23no—
224C42×C5—23no—
225C42×C5—23no—
226C22⋊Dic5—23no—
227C22⋊Dic5C2×D416yes—
228#685—23no—
229#685—23no—
230#685—26no—
231C4⋊Dic5—23no—
232C4⋊Dic5—21no—
233C4⋊Dic5—23no—
234C4⋊Dic5—26no—
235C4⋊Dic5C2×D413yes—
236C4⋊Dic5C2×D413yes—
237C4⋊Dic5C2413yes—
238C4⋊Dic5C2411yescharacteristic
239#683—21no—
240#683—23no—
241#683—23no—
242#683—23no—
243#683—26no—
244#683—21no—
245#683—23no—
246C4×Dic5—23no—
247C4×Dic5—21no—
248C4×Dic5—23no—
249C4×Dic5C2×D413yes—
250C4×Dic5C2×D413yes—
251C4×Dic5C2413yes—
252C4×Dic5C2411yescharacteristic
253#681—26no—
254C2×#168—26no—
255C2×#168C2411yescharacteristic
256C2×#168C2413yes—
257C2×H(5,𝔽2)—51no—
258C2×#131—51no—
259C2×#131C2×D511yescharacteristic
260#557—101no—
261#557—101no—
262#554—103no—
263#554—103no—
264#544—53no—
265#544—53no—
266#533—53no—
267C2×#125—53no—
268#520—51no—
269#520—53no—
270#520—101no—
271#520C2×D511yescharacteristic
272#520C2×D511yescharacteristic
273#518—53no—
274#518—103no—
275C4×#41—51no—
276C4×#41—53no—
277C4×#41—101no—
278C2×C4×D4—53no—
279C2×C4×D4—103no—
280#479—53no—
281#479—51no—
282#479—56no—
283#478—53no—
284#478—56no—
285#478—53no—
286#478—56no—
287#478—56no—
288#478—56no—
289#477—53no—
290#476—53no—
291#476—56no—
292#475—53no—
293#474—53no—
294#474—56no—
295#428—53no—
296#428—53no—
297#428C2×D513yescomplemented
298#428C2×D513yescomplemented
299#427—53no—
300#427C2×D513yescomplemented
301C2×#107—53no—
302C2×#106C2×D513yescomplemented
303#424—53no—
304#424—53no—
305#423—53no—
306C2×C4⋊C8—53no—
307#418—53no—
308#418—53no—
309#417—53no—
310#417—53no—
311#416—53no—
312#416—53no—
313C2×#102—53no—
314C2×#102—53no—
315C23×C5C22×D413yes—
316C22×D5—23no—
317C22×D5—212no—
318C22×D5—83no—
319C22×D5—83no—
320C5×Q8C22⋊Q814yes—
321C5×D4C22⋊Q8112yes—
322C2×C4×C5—21no—
323C2×C4×C5—23no—
324C2×C4×C5—23no—
325C2×C4×C5—21no—
326C2×C4×C5—83no—
327C2×C4×C5—83no—
328C2×C4×C5C22×D413yes—
329C2×C4×C5C22×Q816yes—
330C2×C4×C5C2×#4116yes—
331C2×C4×C5C2511yescharacteristic
332C22⋊D5—212no—
333C2×Dic5—21no—
334C2×Dic5—21no—
335C2×Dic5—23no—
336C2×Dic5—43no—
337C2×Dic5—43no—
338C2×Dic5—83no—
339C2×Dic5C22×D411yescharacteristic
340C2×Dic5C22×D413yes—
341C2×Dic5C22×D411yescharacteristic
342C2×Dic5C22×D413yes—
343D20—212no—
344D20—23no—
345D20C22×Q816yes—
346C4×D5—212no—
347C4×D5—46no—
348C4×D5—83no—
349C4×D5—86no—
350Dic10—21no—
351Dic10—83no—
352Dic10—83no—
353Dic10C22×Q812yes—
354C5×C8—21no—
355C5×C8—23no—
356#168—21no—
357#168—23no—
358H(5,𝔽2)—58no—
359H(5,𝔽2)—104no—
360C2×#41—53no—
361C2×#41—101no—
362C2×#41C22×D511yescharacteristic
363C22×D4—53no—
364C22×D4—103no—
365C23×C4—103no—
366#131—24no—
367#131—104no—
368C2×M16—53no—
369C2×M16C22×D513yes—
370C22×C8—53no—
371C22×C8C22×D513yes—
372C4⋊Q8—53no—
373C4⋊Q8—53no—
374C4⋊Q8—103no—
375C4⋊Q8—103no—
376#125—53no—
377#125—56no—
378#125—51no—
379#125—103no—
380#125—103no—
381#125—103no—
382#125—103no—
383C22⋊Q8—106no—
384C22⋊Q8—103no—
385C22⋊Q8—103no—
386C22⋊Q8—106no—
387C4×Q8—21no—
388C4×Q8—21no—
389C4×Q8—51no—
390C4×Q8—51no—
391C4×Q8—51no—
392C4×Q8—53no—
393C4×Q8—103no—
394C4×Q8—103no—
395C4×Q8—101no—
396C4×D4—23no—
397C4×D4—23no—
398C4×D4—53no—
399C4×D4—53no—
400C4×D4—53no—
401C4×D4—56no—
402C4×D4—53no—
403C4×D4—103no—
404C4×D4—103no—
405C4×D4—103no—
406C4×D4—103no—
407C4×D4—103no—
408C4×D4—103no—
409C4×D4—106no—
410C4×D4—103no—
411#117—53no—
412#117—53no—
413#117—103no—
414#117—103no—
415#117C22×D513yes—
416#117C22×D513yes—
417C2×C4⋊C4—53no—
418C2×C4⋊C4—103no—
419C2×C4⋊C4—103no—
420C2×C4⋊C4—103no—
421C2×C4⋊C4C22×D513yes—
422C2×C4⋊C4C22×D513yes—
423C2×C22⋊C4—106no—
424C2×C22⋊C4—103no—
425C2×C42—53no—
426C2×C42—103no—
427#107—53no—
428#107—56no—
429#107—51no—
430#107C22×D513yes—
431#107C22×D513yes—
432#106—53no—
433#106—53no—
434#106C22×D513yes—
435#106C22×D516yes—
436#106C22×D511yescharacteristic
437C4⋊C8—51no—
438C4⋊C8—53no—
439C4⋊C8—53no—
440C4⋊C8—53no—
441C4⋊C8—56no—
442#103—51no—
443#103—53no—
444#103—51no—
445#103—53no—
446#102—53no—
447#102—56no—
448#102—53no—
449#102—53no—
450#102—56no—
451#102—53no—
452C22×C5—23no—
453C22×C5—23no—
454C22×C5C2×C22⋊Q816yes—
455C22×C5C23×D411yescharacteristic
456C2×D5—26no—
457C2×D5—43no—
458C2×D5—43no—
459C2×D5—812no—
460C4×C5—43no—
461C4×C5—41no—
462C4×C5—41no—
463C4×C5—43no—
464C4×C5C2×C22⋊Q816yes—
465C4×C5C23×Q811yescharacteristic
466C4×C5C22×#4111yescharacteristic
467Dic5—23no—
468Dic5—21no—
469Dic5—23no—
470Dic5—21no—
471Dic5—22no—
472Dic5—41no—
473Dic5—43no—
474C24—103no—
475#41—512no—
476#41—108no—
477#41—106no—
478#41#70914yes—
479#41Q8×D514yes—
480C2×Q8—51no—
481C2×Q8—103no—
482C2×Q8—103no—
483C2×Q8D4×D511yescharacteristic
484C2×D4—512no—
485C2×D4—53no—
486C2×D4—103no—
487C2×D4—1024no—
488C2×D4—106no—
489C2×D4—103no—
490C2×D4D4×D513yes—
491C22×C4—23no—
492C22×C4—23no—
493C22×C4—53no—
494C22×C4—53no—
495C22×C4—103no—
496C22×C4—103no—
497C22×C4—106no—
498C22×C4—103no—
499C22×C4—103no—
500C22×C4—406no—
501C22×C4C23×D513yes—
502M16—26no—
503M16—106no—
504C2×C8—26no—
505C2×C8—53no—
506C2×C8—51no—
507C2×C8—106no—
508C2×C8C23×D511yescharacteristic
509C2×C8C23×D513yes—
510C4⋊C4—23no—
511C4⋊C4—23no—
512C4⋊C4—53no—
513C4⋊C4—53no—
514C4⋊C4—51no—
515C4⋊C4—53no—
516C4⋊C4—103no—
517C4⋊C4—103no—
518C4⋊C4—103no—
519C4⋊C4—106no—
520C4⋊C4—106no—
521C4⋊C4—101no—
522C4⋊C4—101no—
523C4⋊C4—101no—
524C4⋊C4—103no—
525C4⋊C4—103no—
526C4⋊C4—103no—
527C4⋊C4C23×D513yes—
528C4⋊C4C23×D511yescharacteristic
529C4⋊C4C23×D511yescharacteristic
530C4⋊C4C23×D513yes—
531C22⋊C4—23no—
532C22⋊C4—23no—
533C22⋊C4—56no—
534C22⋊C4—103no—
535C22⋊C4—103no—
536C22⋊C4—106no—
537C22⋊C4—103no—
538C42—23no—
539C42—23no—
540C42—53no—
541C42—53no—
542C42—51no—
543C42—53no—
544C42—101no—
545C42—103no—
546C42—103no—
547C2×C5—26no—
548C2×C5C22×C22⋊Q811yescharacteristic
549C2×C5#358411yescharacteristic
550C2×C5#358511yescharacteristic
551D5—46no—
552C23—103no—
553C23—1012no—
554C23—403no—
555C23—403no—
556C23C2×D4×D513yes—
557Q8—52no—
558Q8—101no—
559Q8—403no—
560Q8—403no—
561Q8#404214yes—
562D4—56no—
563D4—1012no—
564D4—1012no—
565D4—103no—
566D4#4042112yes—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—23no—
570C2×C4—23no—
571C2×C4—51no—
572C2×C4—51no—
573C2×C4—53no—
574C2×C4—53no—
575C2×C4—83no—
576C2×C4—83no—
577C2×C4—101no—
578C2×C4—1012no—
579C2×C4—101no—
580C2×C4—103no—
581C2×C4—206no—
582C2×C4—203no—
583C2×C4—203no—
584C2×C4—406no—
585C2×C4—403no—
586C2×C4—403no—
587C2×C4C2×#70916yes—
588C2×C4C2×D4×D513yes—
589C2×C4C2×Q8×D516yes—
590C2×C4C24×D511yescharacteristic
591C8—21no—
592C8—23no—
593C8—101no—
594C8—103no—
595C5#6049611yescharacteristic, complemented, Sylow
596C22—23no—
597C22—23no—
598C22—106no—
599C22—203no—
600C22—203no—
601C22—4012no—
602C22C2×#404216yes—
603C22C22×D4×D511yescharacteristic
604C4—43no—
605C4—41no—
606C4—43no—
607C4—41no—
608C4—102no—
609C4—101no—
610C4—103no—
611C4—101no—
612C4—103no—
613C4—201no—
614C4—203no—
615C4C2×#404216yes—
616C4C22×#70911yescharacteristic
617C4C22×Q8×D511yescharacteristic
618C2—26no—
619C2—206no—
620C2C22×#404211yescharacteristic
621C2#1061910711yescharacteristic
622C2#1061910811yescharacteristic
subgroupG/H
centerC22C22×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC5×#2912C2
socleC22×C5C23×D4
prime radicalC2×#4149C22
hypercenter#2912D5
hypocenterC5#60496
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×#41 ◁ #2912

Complements

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

NH ≅ G/Ntype
1. #10617791C2semidirect
3. C5×#2912C2semidirect
5. #10615514C2semidirect
7. #10614293C2semidirect
8. #10610974C2semidirect
9. #10610973C2semidirect
10. #10610972C2semidirect
11. #10610970C2semidirect
12. #10610928C2semidirect
13. #10610927C2semidirect
14. #10610861C2semidirect
15. C2×#65160C2semidirect
33. #65853C22semidirect
34. #65851C22semidirect
37. #65834C22semidirect
38. C2×#4036C22semidirect
39. C5×#428C22semidirect
40. C5×#428C22semidirect
41. C5×#427C22semidirect
42. C2×C5×#106C22semidirect
45. #65408C22semidirect
46. #65407C22semidirect
49. #65391C22semidirect
50. C2×#3946C22semidirect
53. #65287C22semidirect
54. #65284C22semidirect
57. #65253C22semidirect
58. C2×#3939C22semidirect
59. #65180C22semidirect
60. #65180C22semidirect
61. #65179C22semidirect
62. #65179C22semidirect
63. #65166C22semidirect
64. #65164C22semidirect
65. #65160C22semidirect
66. #65160C22semidirect
67. #65160C22semidirect
68. #65160C22semidirect
69. #65159C22semidirect
70. #65159C22semidirect
71. #65146C22semidirect
72. #65146C22semidirect
73. #65143C22semidirect
77. #4149Q8semidirect
180. #2912D5semidirect
297. #428C2×D5semidirect
298. #428C2×D5semidirect
300. #427C2×D5semidirect
302. C2×#106C2×D5semidirect
595. C5#60496semidirect

Maximal subgroups

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

subgroupindex
#106177912
#106177872
C5×#29122
#106155282
#106155142
#106142942
#106142932
#106109742
#106109732
#106109722
#106109702
#106109282
#106109272
#106108612
C2×#651602
#604965