Finite Group Databasebeta

← #12966971

Subgroups — #12966971

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
1C5×#3504C211yesmaximal, characteristic, complemented
2C4×#4149C211yesmaximal, characteristic
3#10615207C213yesmaximal, complemented
4#10615206C213yesmaximal, complemented
5#10615205C213yesmaximal, complemented
6#10615205C213yesmaximal, complemented
7#10615081C213yesmaximal, complemented
8#10615080C213yesmaximal, complemented
9#10614748C213yesmaximal, complemented
10C2×#65355C213yesmaximal, complemented
11#10614294C211yesmaximal, characteristic
12#10614294C211yesmaximal, characteristic
13#10614293C211yesmaximal, characteristic, complemented
14#10614292C211yesmaximal, characteristic, complemented
15#10614243C211yesmaximal, characteristic, complemented
16C2×#4149C2211yescharacteristic
17C5×#557—21no—
18C5×#557—21no—
19C5×#554—23no—
20C5×#554—23no—
21C5×#544C2213yescomplemented
22C5×#544C2213yescomplemented
23C5×#533C2213yescomplemented
24C2×C5×#125C2213yescomplemented
25C5×#520C2211yescharacteristic
26C5×#520C2211yescharacteristic
27C4×C5×#41C2211yescharacteristic
28#66161C2211yescharacteristic
29C2×#4094C2211yescharacteristic
30C2×#4094C2211yescharacteristic
31#65908—21no—
32C4×#713—21no—
33#65879—23no—
34C4×D4×D5—23no—
35#65857C2213yes—
36#65855C2213yes—
37C4×#709C2213yes—
38C2×C4×D20C2213yes—
39#65383C2213yescomplemented
40#65380C2213yescomplemented
41#65376C2213yescomplemented
42#65376C2216yescomplemented
43#65373C2213yescomplemented
44#65373C2216yescomplemented
45#65355C2213yescomplemented
46#65355C2213yescomplemented
47#65355C2216yescomplemented
48#65355C2216yescomplemented
49#65355C2216yescomplemented
50#65355C2216yescomplemented
51#65352C2211yescharacteristic, complemented
52#65352C2213yescomplemented
53#65352C2216yescomplemented
54#65291C2213yes—
55#65291C2213yes—
56#65290C2213yes—
57#65290C2213yes—
58#65288C2213yescomplemented
59#65288C2213yescomplemented
60#65287C2213yescomplemented
61#65287C2213yescomplemented
62#65284C2213yescomplemented
63#65284C2213yescomplemented
64#65255C2213yes—
65#65255C2213yes—
66C2×#3941C2213yes—
67C2×#3941C2213yes—
68#65253C2213yescomplemented
69#65253C2213yescomplemented
70C2×#3940C2213yescomplemented
71C2×#3939C2213yescomplemented
72#65213C2213yescomplemented
73C2×C4⋊#168C2213yescomplemented
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×C4⋊Q8—23no—
84C5×C4⋊Q8—23no—
85C5×C4⋊Q8C2313yes—
86C5×C4⋊Q8C2313yes—
87C5×#125—23no—
88C5×#125—23no—
89C5×#125—23no—
90C5×#125—23no—
91C5×#125C2313yes—
92C5×#125C2316yes—
93C5×#125C2311yescharacteristic
94C5×C22⋊Q8—23no—
95C5×C22⋊Q8—23no—
96C5×C22⋊Q8—26no—
97C5×C22⋊Q8—26no—
98C4×C5×Q8—21no—
99C4×C5×Q8—21no—
100C4×C5×Q8—23no—
101C4×C5×Q8—23no—
102C4×C5×Q8D411yescharacteristic
103C4×C5×Q8D411yescharacteristic
104C4×C5×D4—23no—
105C4×C5×D4—23no—
106C4×C5×D4D413yes—
107C4×C5×D4D413yes—
108C5×#117C2313yes—
109C5×#117C2313yes—
110C5×#117C2313yes—
111C2×C5×C4⋊C4—23no—
112C2×C5×C4⋊C4—23no—
113C2×C5×C4⋊C4C2313yes—
114C2×C5×C4⋊C4C2313yes—
115C2×C42×C5C2313yes—
116#4094—24no—
117#4094—24no—
118Q8×Dic5—21no—
119D4×Dic5—23no—
120C4×C22⋊D5—23no—
121C4×C22⋊D5—23no—
122C2×#685—23no—
123#4072C2313yes—
124C2×C4⋊Dic5C2313yes—
125C2×#681C2313yes—
126C2×#681C2313yes—
127C22×#168C2313yes—
128C22×#168C2313yes—
129#4039—23no—
130#4039—23no—
131#4038—23no—
132D5×C4⋊C4—23no—
133#4027—26no—
134D5×C22⋊C4—26no—
135C4×D20—23no—
136C4×D20—23no—
137C4×D20C2313yes—
138C4×D20C2316yes—
139C4×D20C2313yes—
140#4018—23no—
141C42×D5—23no—
142C4×Dic10C2311yescharacteristic
143C4×Dic10C2313yes—
144#3942C2311yescharacteristic
145#3942C2311yescharacteristic
146#3942C2313yes—
147#3942C2313yes—
148#3941C2313yes—
149#3941C2313yes—
150#3941C2316yes—
151#3941C2316yes—
152#3941C2313yes—
153#3941C2313yes—
154#3940C2313yes—
155#3940C2316yes—
156#3940C2311yescharacteristic
157#3940C2313yes—
158#3940C2313yes—
159#3939C2313yes—
160#3939C2316yes—
161#3939C2311yescharacteristic
162#3939C2313yes—
163#3939C2313yes—
164C4⋊#168C2316yes—
165C4⋊#168C2313yes—
166C4⋊#168C2313yes—
167C4⋊#168C2311yescharacteristic
168C4⋊#168C2313yes—
169#3504D511yescharacteristic, complemented
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#2912—51no—
181#2906—51no—
182#2897—51no—
183#2897—51no—
184C23×D5—23no—
185C5×#41C2×Q814yes—
186C5×#41#4114yes—
187C2×C5×Q8—23no—
188C2×C5×Q8—23no—
189C2×C5×Q8C2×D411yescharacteristic
190C2×C5×D4C2×D413yes—
191C22×C4×C5—23no—
192C22×C4×C5—23no—
193C22×C4×C5C2×D413yes—
194C22×C4×C5C2×D413yes—
195C22×C4×C5C2413yes—
196C2×C22⋊D5—23no—
197C22×Dic5—23no—
198#713—28no—
199D4×D5—224no—
200#709—26no—
201#709C2×Q8112yes—
202C2×D20—23no—
203C2×D20—26no—
204C2×D20C2×Q8112yes—
205C2×D20C2413yes—
206C2×C4×D5—23no—
207C2×C4×D5—23no—
208C2×C4×D5—26no—
209C2×C4×D5—23no—
210C2×C4×D5—86no—
211C2×Dic10C2411yescharacteristic
212C5×C4⋊C4—23no—
213C5×C4⋊C4—23no—
214C5×C4⋊C4—26no—
215C5×C4⋊C4—26no—
216C5×C4⋊C4—23no—
217C5×C4⋊C4—23no—
218C5×C4⋊C4—21no—
219C5×C4⋊C4—21no—
220C5×C4⋊C4—23no—
221C5×C4⋊C4—23no—
222C5×C4⋊C4C2×D413yes—
223C5×C4⋊C4C2×D413yes—
224C5×C4⋊C4C2413yes—
225C5×C4⋊C4C2413yes—
226C5×C4⋊C4C2411yescharacteristic
227C5×C4⋊C4C2411yescharacteristic
228C5×C22⋊C4—23no—
229C5×C22⋊C4—23no—
230C5×C22⋊C4C2×D416yes—
231C42×C5—23no—
232C42×C5—23no—
233C42×C5C2×D413yes—
234C42×C5C2×D413yes—
235C42×C5C2411yescharacteristic
236C42×C5C2413yes—
237C22⋊Dic5—23no—
238#685—23no—
239#685—23no—
240#685—26no—
241C4⋊Dic5—23no—
242C4⋊Dic5C2413yes—
243C4⋊Dic5C2411yescharacteristic
244#683—21no—
245#683—23no—
246C4×Dic5—21no—
247C4×Dic5—23no—
248C4×Dic5—23no—
249#681—26no—
250#681—26no—
251C2×#168—26no—
252C2×#168—26no—
253C2×#168C2411yescharacteristic
254C2×#168C2413yes—
255C2×#168C2413yes—
256C2×#168C2411yescharacteristic
257C2×H(5,𝔽2)—51no—
258C2×#131—51no—
259C2×#131—51no—
260#557—21no—
261#557—21no—
262#554—23no—
263#554—23no—
264#544C2×D513yescomplemented
265#544C2×D513yescomplemented
266#533C2×D513yescomplemented
267C2×#125C2×D513yescomplemented
268#520—51no—
269#520—53no—
270#520—101no—
271#520C2×D511yescharacteristic
272#520C2×D511yescharacteristic
273#518—53no—
274#518—103no—
275C4×#41—53no—
276C4×#41—101no—
277C4×#41C2×D511yescharacteristic
278C2×C4×D4—53no—
279C2×C4×D4—103no—
280#479—51no—
281#479—53no—
282#479—56no—
283#478—53no—
284#478—56no—
285#478—56no—
286#478—56no—
287#478—53no—
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#428—53no—
298#428—53no—
299#427—53no—
300#427—53no—
301C2×#107—53no—
302C2×#106—53no—
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×Q8—83no—
321C5×Q8—83no—
322C5×Q8C22⋊Q814yes—
323C5×D4C22⋊Q8112yes—
324C2×C4×C5—21no—
325C2×C4×C5—21no—
326C2×C4×C5—23no—
327C2×C4×C5—23no—
328C2×C4×C5—43no—
329C2×C4×C5—43no—
330C2×C4×C5—83no—
331C2×C4×C5—83no—
332C2×C4×C5C22×D413yes—
333C2×C4×C5C22×D411yescharacteristic
334C2×C4×C5C22×D411yescharacteristic
335C2×C4×C5C22×D413yes—
336C2×C4×C5C22×D413yes—
337C2×C4×C5C22×Q816yes—
338C2×C4×C5C2×#4116yes—
339C2×C4×C5C2511yescharacteristic
340C22⋊D5—212no—
341C2×Dic5—21no—
342C2×Dic5—21no—
343C2×Dic5—23no—
344C2×Dic5—83no—
345D20—212no—
346D20—23no—
347D20C22×Q816yes—
348C4×D5—212no—
349C4×D5—46no—
350C4×D5—83no—
351C4×D5—86no—
352Dic10—21no—
353Dic10C22×Q812yes—
354#168—21no—
355#168—21no—
356#168—23no—
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—104no—
367#131—104no—
368C2×M16—53no—
369C2×M16—53no—
370C22×C8—53no—
371C22×C8—53no—
372C4⋊Q8—23no—
373C4⋊Q8—23no—
374C4⋊Q8C22×D513yes—
375C4⋊Q8C22×D513yes—
376#125—23no—
377#125—23no—
378#125—23no—
379#125—23no—
380#125C22×D513yes—
381#125C22×D516yes—
382#125C22×D511yescharacteristic
383C22⋊Q8—23no—
384C22⋊Q8—23no—
385C22⋊Q8—26no—
386C22⋊Q8—26no—
387C4×Q8—21no—
388C4×Q8—21no—
389C4×Q8—23no—
390C4×Q8—23no—
391C4×Q8—51no—
392C4×Q8—53no—
393C4×Q8—101no—
394C4×Q8C22⋊D511yescharacteristic
395C4×Q8C22⋊D511yescharacteristic
396C4×D4—23no—
397C4×D4—23no—
398C4×D4—53no—
399C4×D4—56no—
400C4×D4—53no—
401C4×D4—103no—
402C4×D4—103no—
403C4×D4—103no—
404C4×D4—103no—
405C4×D4—103no—
406C4×D4—106no—
407C4×D4—103no—
408C4×D4—103no—
409C4×D4C22⋊D513yes—
410C4×D4C22⋊D513yes—
411#117—53no—
412#117—103no—
413#117—103no—
414#117C22×D513yes—
415#117C22×D513yes—
416#117C22×D513yes—
417C2×C4⋊C4—23no—
418C2×C4⋊C4—23no—
419C2×C4⋊C4—53no—
420C2×C4⋊C4—103no—
421C2×C4⋊C4C22×D513yes—
422C2×C4⋊C4C22×D513yes—
423C2×C22⋊C4—103no—
424C2×C22⋊C4—106no—
425C2×C42—103no—
426C2×C42C22×D513yes—
427#107—53no—
428#107—56no—
429#107—51no—
430#107—53no—
431#107—53no—
432#106—53no—
433#106—56no—
434#106—51no—
435#106—53no—
436#106—53no—
437C4⋊C8—51no—
438C4⋊C8—53no—
439C4⋊C8—53no—
440C4⋊C8—53no—
441C4⋊C8—56no—
442#103—51no—
443#103—51no—
444#103—53no—
445#103—53no—
446#102—53no—
447#102—53no—
448#102—56no—
449#102—56no—
450#102—53no—
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—23no—
461C4×C5—21no—
462C4×C5—21no—
463C4×C5—23no—
464C4×C5—43no—
465C4×C5—41no—
466C4×C5—43no—
467C4×C5—41no—
468C4×C5C2×C22⋊Q816yes—
469C4×C5C23×Q811yescharacteristic
470C4×C5C22×#4111yescharacteristic
471Dic5—22no—
472Dic5—41no—
473Dic5—43no—
474C24—103no—
475#41—512no—
476#41—108no—
477#41—106no—
478#41Q8×D514yes—
479#41#71314yes—
480C2×Q8—23no—
481C2×Q8—23no—
482C2×Q8—51no—
483C2×Q8C2×C22⋊D511yescharacteristic
484C2×D4—512no—
485C2×D4—53no—
486C2×D4—103no—
487C2×D4—1024no—
488C2×D4—106no—
489C2×D4—103no—
490C2×D4C2×C22⋊D513yes—
491C22×C4—23no—
492C22×C4—23no—
493C22×C4—103no—
494C22×C4—103no—
495C22×C4—103no—
496C22×C4—103no—
497C22×C4—106no—
498C22×C4—406no—
499C22×C4C2×C22⋊D513yes—
500C22×C4C2×C22⋊D513yes—
501C22×C4C23×D513yes—
502M16—106no—
503M16—106no—
504C2×C8—51no—
505C2×C8—53no—
506C2×C8—53no—
507C2×C8—51no—
508C2×C8—106no—
509C2×C8—106no—
510C4⋊C4—23no—
511C4⋊C4—23no—
512C4⋊C4—26no—
513C4⋊C4—26no—
514C4⋊C4—23no—
515C4⋊C4—23no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—23no—
519C4⋊C4—23no—
520C4⋊C4—53no—
521C4⋊C4—51no—
522C4⋊C4—103no—
523C4⋊C4—103no—
524C4⋊C4—101no—
525C4⋊C4C2×C22⋊D513yes—
526C4⋊C4C2×C22⋊D513yes—
527C4⋊C4C23×D513yes—
528C4⋊C4C23×D511yescharacteristic
529C4⋊C4C23×D513yes—
530C4⋊C4C23×D511yescharacteristic
531C22⋊C4—23no—
532C22⋊C4—23no—
533C22⋊C4—103no—
534C22⋊C4—103no—
535C22⋊C4—106no—
536C22⋊C4—103no—
537C22⋊C4C2×C22⋊D516yes—
538C42—23no—
539C42—23no—
540C42—101no—
541C42—103no—
542C42—103no—
543C42C2×C22⋊D513yes—
544C42C2×C22⋊D513yes—
545C42C23×D511yescharacteristic
546C42C23×D513yes—
547C2×C5—26no—
548C2×C5C22×C22⋊Q811yescharacteristic
549C2×C5#358411yescharacteristic
550C2×C5#358511yescharacteristic
551D5—46no—
552C23—103no—
553C23—1012no—
554C23—403no—
555C23—403no—
556C23C22×C22⋊D513yes—
557Q8—52no—
558Q8—83no—
559Q8—83no—
560Q8—101no—
561Q8#409214yes—
562D4—56no—
563D4—1012no—
564D4—1012no—
565D4—103no—
566D4#4092112yes—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—23no—
570C2×C4—23no—
571C2×C4—43no—
572C2×C4—43no—
573C2×C4—83no—
574C2×C4—83no—
575C2×C4—101no—
576C2×C4—1012no—
577C2×C4—101no—
578C2×C4—103no—
579C2×C4—206no—
580C2×C4—406no—
581C2×C4—403no—
582C2×C4—403no—
583C2×C4C2×Q8×D516yes—
584C2×C4C2×#71316yes—
585C2×C4C22×C22⋊D511yescharacteristic
586C2×C4C22×C22⋊D513yes—
587C2×C4C22×C22⋊D511yescharacteristic
588C2×C4C22×C22⋊D513yes—
589C2×C4C22×C22⋊D513yes—
590C2×C4C24×D511yescharacteristic
591C8—101no—
592C8—101no—
593C8—103no—
594C8—103no—
595C5#6049611yescharacteristic, complemented, Sylow
596C22—23no—
597C22—23no—
598C22—106no—
599C22—203no—
600C22—203no—
601C22—4012no—
602C22C2×#409216yes—
603C22C23×C22⋊D511yescharacteristic
604C4—21no—
605C4—23no—
606C4—21no—
607C4—23no—
608C4—43no—
609C4—41no—
610C4—43no—
611C4—41no—
612C4—102no—
613C4—201no—
614C4—203no—
615C4C2×#409216yes—
616C4C22×Q8×D511yescharacteristic
617C4C22×#71311yescharacteristic
618C2—26no—
619C2—206no—
620C2C22×#409211yescharacteristic
621C2#1061919811yescharacteristic
622C2#1061919911yescharacteristic
subgroupG/H
centerC22C23×C22⋊D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC5×#3504C2
socleC22×C5C23×D4
prime radicalC2×#4149C22
hypercenter#3504D5
hypocenterC5#60496
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C4×#41 ◁ #3504

Complements

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

NH ≅ G/Ntype
1. C5×#3504C2semidirect
3. #10615207C2semidirect
4. #10615206C2semidirect
5. #10615205C2semidirect
6. #10615205C2semidirect
7. #10615081C2semidirect
8. #10615080C2semidirect
9. #10614748C2semidirect
10. C2×#65355C2semidirect
13. #10614293C2semidirect
14. #10614292C2semidirect
15. #10614243C2semidirect
21. C5×#544C22semidirect
22. C5×#544C22semidirect
23. C5×#533C22semidirect
24. C2×C5×#125C22semidirect
39. #65383C22semidirect
40. #65380C22semidirect
41. #65376C22semidirect
42. #65376C22semidirect
43. #65373C22semidirect
44. #65373C22semidirect
45. #65355C22semidirect
46. #65355C22semidirect
47. #65355C22semidirect
48. #65355C22semidirect
49. #65355C22semidirect
50. #65355C22semidirect
51. #65352C22semidirect
52. #65352C22semidirect
53. #65352C22semidirect
58. #65288C22semidirect
59. #65288C22semidirect
60. #65287C22semidirect
61. #65287C22semidirect
62. #65284C22semidirect
63. #65284C22semidirect
68. #65253C22semidirect
69. #65253C22semidirect
70. C2×#3940C22semidirect
71. C2×#3939C22semidirect
72. #65213C22semidirect
73. C2×C4⋊#168C22semidirect
77. #4149Q8semidirect
169. #3504D5semidirect
264. #544C2×D5semidirect
265. #544C2×D5semidirect
266. #533C2×D5semidirect
267. C2×#125C2×D5semidirect
595. C5#60496semidirect

Maximal subgroups

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

subgroupindex
C5×#35042
C4×#41492
#106152072
#106152062
#106152052
#106150812
#106150802
#106147482
C2×#653552
#106142942
#106142932
#106142922
#106142432
#604965