Finite Group Databasebeta

← #12966980

Subgroups — #12966980

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
1C4×C5×H(5,𝔽2)C211yesmaximal, characteristic
2#10617730C211yesmaximal, characteristic, complemented
3#10615830C211yesmaximal, characteristic
4#10615830C211yesmaximal, characteristic
5#10614293C211yesmaximal, characteristic, complemented
6#10614292C211yesmaximal, characteristic, complemented
7#10614291C213yesmaximal, complemented
8#10614290C213yesmaximal, complemented
9#10614270C213yesmaximal, complemented
10#10614270C213yesmaximal, complemented
11#10614243C211yesmaximal, characteristic, complemented
12#10614240C213yesmaximal, complemented
13#10614238C213yesmaximal, complemented
14#10614027C213yesmaximal, complemented
15C2×#65298C213yesmaximal, complemented
16C2×C5×H(5,𝔽2)C2211yescharacteristic
17C5×#520—21no—
18C5×#520C2213yes—
19C5×#520C2211yescharacteristic
20C5×#518—23no—
21C5×#518C2213yes—
22C4×C5×#41—21no—
23C4×C5×#41C2213yes—
24C4×C5×#41C2211yescharacteristic
25C2×C4×C5×D4—23no—
26C2×C4×C5×D4C2213yes—
27#66161C2211yescharacteristic
28#66161C2211yescharacteristic
29C2×#4094C2211yescharacteristic
30C2×#4094C2211yescharacteristic
31#65903—21no—
32#65903—21no—
33#65873—23no—
34#65873—23no—
35#65853C2213yescomplemented
36#65853C2213yescomplemented
37#65804C2213yescomplemented
38C2×#4016C2213yescomplemented
39#65522C2213yes—
40#65522C2213yes—
41#65521C2213yes—
42#65521C2213yes—
43#65504C2213yes—
44#65504C2213yes—
45C2×#3964C2213yes—
46C2×#3964C2213yes—
47#65311C2216yescomplemented
48#65311C2211yescharacteristic, complemented
49#65311C2213yescomplemented
50#65310C2213yescomplemented
51#65309C2213yescomplemented
52#65298C2213yescomplemented
53#65298C2216yescomplemented
54#65298C2216yescomplemented
55#65298C2216yescomplemented
56#65298C2213yescomplemented
57#65298C2216yescomplemented
58#65297C2216yescomplemented
59#65297C2213yescomplemented
60#65296C2216yescomplemented
61#65296C2213yescomplemented
62#65288C2213yescomplemented
63#65288C2213yescomplemented
64#65287C2213yescomplemented
65#65287C2213yescomplemented
66#65284C2213yescomplemented
67#65284C2213yescomplemented
68#65253C2213yescomplemented
69#65253C2213yescomplemented
70C2×#3940C2213yescomplemented
71C2×#3939C2213yescomplemented
72#65213C2213yescomplemented
73C2×C4⋊#168C2213yescomplemented
74#60496—51nomaximal, Sylow
75C5×H(5,𝔽2)—24no—
76C5×H(5,𝔽2)Q818yescomplemented
77C2×C5×#41—21no—
78C2×C5×#41C2313yes—
79C2×C5×#41C2311yescharacteristic
80C22×C5×D4—23no—
81C22×C5×D4C2313yes—
82C23×C4×C5—23no—
83C4×C5×Q8—21no—
84C4×C5×Q8D411yescharacteristic
85C4×C5×Q8D411yescharacteristic
86C4×C5×Q8C2313yes—
87C4×C5×Q8C2311yescharacteristic
88C4×C5×D4—23no—
89C4×C5×D4—23no—
90C4×C5×D4—23no—
91C4×C5×D4—23no—
92C4×C5×D4—23no—
93C4×C5×D4—26no—
94C4×C5×D4—23no—
95C4×C5×D4—23no—
96C4×C5×D4D413yes—
97C4×C5×D4D413yes—
98C4×C5×D4C2316yes—
99C4×C5×D4C2313yes—
100C4×C5×D4C2313yes—
101C5×#117—23no—
102C5×#117—23no—
103C5×#117C2313yes—
104C5×#117C2313yes—
105C2×C5×C4⋊C4—23no—
106C2×C5×C4⋊C4C2313yes—
107C2×C5×C22⋊C4—23no—
108C2×C5×C22⋊C4—26no—
109C2×C42×C5—23no—
110C2×C42×C5C2313yes—
111#4094—24no—
112#4094—24no—
113Q8×Dic5—21no—
114Q8×Dic5—21no—
115D4×Dic5—23no—
116D4×Dic5—23no—
117#4072C2313yes—
118#4072C2313yes—
119C2×C4⋊Dic5C2313yes—
120C2×C4⋊Dic5C2313yes—
121#4070—23no—
122#4070—23no—
123C2×#683—23no—
124C2×#683—23no—
125C2×#681C2313yes—
126C2×#681C2313yes—
127C22×#168C2313yes—
128C22×#168C2313yes—
129#4036—23no—
130#4036—23no—
131#4034—23no—
132#4034—23no—
133#4024—26no—
134#4024—26no—
135#4016—23no—
136#4016—23no—
137#4016C2313yes—
138#4016C2316yes—
139#4016C2311yescharacteristic
140#4015C2313yes—
141#4015C2313yes—
142C4×Dic10—23no—
143C4×Dic10—23no—
144#3967C2311yescharacteristic
145#3967C2313yes—
146#3967C2311yescharacteristic
147#3967C2313yes—
148#3964C2313yes—
149#3964C2313yes—
150#3964C2313yes—
151#3964C2316yes—
152#3964C2313yes—
153#3964C2316yes—
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#3504—51no—
170C4×H(5,𝔽2)D511yescharacteristic
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—
184C24×C5—23no—
185C5×#41—28no—
186C5×#41—26no—
187C5×#41C2×Q8112yes—
188C5×#41C2×Q814yes—
189C5×#41#4114yes—
190C2×C5×Q8C2×D411yescharacteristic
191C2×C5×Q8C2411yescharacteristic
192C2×C5×D4—23no—
193C2×C5×D4—224no—
194C2×C5×D4—23no—
195C2×C5×D4—26no—
196C2×C5×D4C2×D413yes—
197C2×C5×D4C2×Q8112yes—
198C2×C5×D4C2413yes—
199C22×C4×C5—23no—
200C22×C4×C5—23no—
201C22×C4×C5—23no—
202C22×C4×C5—23no—
203C22×C4×C5—26no—
204C22×C4×C5—86no—
205C22×C4×C5C2×D413yes—
206C22×C4×C5C2×D413yes—
207C22×C4×C5C2413yes—
208C22×Dic5—23no—
209C22×Dic5—23no—
210C2×Dic10—23no—
211C2×Dic10—23no—
212C5×C4⋊C4—23no—
213C5×C4⋊C4—23no—
214C5×C4⋊C4—21no—
215C5×C4⋊C4C2×D413yes—
216C5×C4⋊C4C2×D413yes—
217C5×C4⋊C4C2413yes—
218C5×C4⋊C4C2411yescharacteristic
219C5×C22⋊C4—23no—
220C5×C22⋊C4—23no—
221C5×C22⋊C4—23no—
222C5×C22⋊C4—26no—
223C5×C22⋊C4C2×D416yes—
224C42×C5—21no—
225C42×C5—23no—
226C42×C5—23no—
227C42×C5C2×D413yes—
228C42×C5C2×D413yes—
229C42×C5C2411yescharacteristic
230C42×C5C2413yes—
231C22⋊Dic5—23no—
232C22⋊Dic5—23no—
233C4⋊Dic5—23no—
234C4⋊Dic5—23no—
235C4⋊Dic5—23no—
236C4⋊Dic5—23no—
237C4⋊Dic5C2411yescharacteristic
238C4⋊Dic5C2413yes—
239C4⋊Dic5C2411yescharacteristic
240C4⋊Dic5C2413yes—
241#683—21no—
242#683—21no—
243#683—23no—
244#683—23no—
245#683—26no—
246#683—26no—
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)C2×D511yescharacteristic
258C2×#131—51no—
259C2×#131—51no—
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—21no—
269#520—51no—
270#520—51no—
271#520C2×D513yes—
272#520C2×D511yescharacteristic
273#518—23no—
274#518C2×D513yes—
275C4×#41—21no—
276C4×#41C2×D513yes—
277C4×#41C2×D511yescharacteristic
278C2×C4×D4—23no—
279C2×C4×D4C2×D513yes—
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×C5—23no—
316C23×C5—212no—
317C23×C5—83no—
318C23×C5—83no—
319C23×C5C22×D413yes—
320C5×Q8—21no—
321C5×Q8C22⋊Q814yes—
322C5×Q8C22×Q812yes—
323C5×D4—212no—
324C5×D4—212no—
325C5×D4—23no—
326C5×D4C22⋊Q8112yes—
327C5×D4C22×Q816yes—
328C2×C4×C5—21no—
329C2×C4×C5—212no—
330C2×C4×C5—21no—
331C2×C4×C5—23no—
332C2×C4×C5—43no—
333C2×C4×C5—43no—
334C2×C4×C5—46no—
335C2×C4×C5—83no—
336C2×C4×C5—86no—
337C2×C4×C5—83no—
338C2×C4×C5C22×D411yescharacteristic
339C2×C4×C5C22×D411yescharacteristic
340C2×C4×C5C22×D413yes—
341C2×C4×C5C22×D413yes—
342C2×C4×C5C22×D413yes—
343C2×C4×C5C22×Q816yes—
344C2×C4×C5C2×#4116yes—
345C2×C4×C5C2511yescharacteristic
346C2×Dic5—21no—
347C2×Dic5—21no—
348C2×Dic5—23no—
349C2×Dic5—23no—
350C2×Dic5—83no—
351C2×Dic5—83no—
352Dic10—83no—
353Dic10—83no—
354#168—21no—
355#168—21no—
356#168—23no—
357#168—23no—
358H(5,𝔽2)—24no—
359H(5,𝔽2)Dic1018yescomplemented
360C2×#41—21no—
361C2×#41C22×D513yes—
362C2×#41C22×D511yescharacteristic
363C22×D4—23no—
364C22×D4C22×D513yes—
365C23×C4—23no—
366#131—104no—
367#131—104no—
368C2×M16—53no—
369C2×M16—53no—
370C22×C8—53no—
371C22×C8—53no—
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—103no—
384C22⋊Q8—103no—
385C22⋊Q8—106no—
386C22⋊Q8—106no—
387C4×Q8—21no—
388C4×Q8—101no—
389C4×Q8—101no—
390C4×Q8—103no—
391C4×Q8—103no—
392C4×Q8C22⋊D511yescharacteristic
393C4×Q8C22⋊D511yescharacteristic
394C4×Q8C22×D511yescharacteristic
395C4×Q8C22×D513yes—
396C4×D4—23no—
397C4×D4—23no—
398C4×D4—23no—
399C4×D4—23no—
400C4×D4—23no—
401C4×D4—26no—
402C4×D4—23no—
403C4×D4—23no—
404C4×D4—103no—
405C4×D4—103no—
406C4×D4C22⋊D513yes—
407C4×D4C22⋊D513yes—
408C4×D4C22×D513yes—
409C4×D4C22×D516yes—
410C4×D4C22×D513yes—
411#117—23no—
412#117—23no—
413#117—53no—
414#117—53no—
415#117C22×D513yes—
416#117C22×D513yes—
417C2×C4⋊C4—23no—
418C2×C4⋊C4—53no—
419C2×C4⋊C4—53no—
420C2×C4⋊C4—103no—
421C2×C4⋊C4—103no—
422C2×C4⋊C4C22×D513yes—
423C2×C22⋊C4—23no—
424C2×C22⋊C4—26no—
425C2×C42—23no—
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×C5—26no—
455C22×C5—43no—
456C22×C5—43no—
457C22×C5—812no—
458C22×C5C2×C22⋊Q816yes—
459C22×C5C23×D411yescharacteristic
460C4×C5—23no—
461C4×C5—21no—
462C4×C5—23no—
463C4×C5—21no—
464C4×C5—22no—
465C4×C5—41no—
466C4×C5—43no—
467C4×C5C2×C22⋊Q816yes—
468C4×C5C23×Q811yescharacteristic
469C4×C5C22×#4111yescharacteristic
470Dic5—41no—
471Dic5—43no—
472Dic5—41no—
473Dic5—43no—
474C24—23no—
475#41—28no—
476#41—26no—
477#41C2×Dic10112yes—
478#41C2×Dic1014yes—
479#41#70914yes—
480C2×Q8—103no—
481C2×Q8—103no—
482C2×Q8C2×C22⋊D511yescharacteristic
483C2×Q8C23×D511yescharacteristic
484C2×D4—23no—
485C2×D4—224no—
486C2×D4—26no—
487C2×D4—23no—
488C2×D4C2×Dic10112yes—
489C2×D4C2×C22⋊D513yes—
490C2×D4C23×D513yes—
491C22×C4—23no—
492C22×C4—23no—
493C22×C4—23no—
494C22×C4—23no—
495C22×C4—26no—
496C22×C4—86no—
497C22×C4—103no—
498C22×C4—103no—
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—21no—
513C4⋊C4—53no—
514C4⋊C4—51no—
515C4⋊C4—53no—
516C4⋊C4—51no—
517C4⋊C4—103no—
518C4⋊C4—103no—
519C4⋊C4—103no—
520C4⋊C4—103no—
521C4⋊C4—106no—
522C4⋊C4—106no—
523C4⋊C4—101no—
524C4⋊C4—101no—
525C4⋊C4—103no—
526C4⋊C4—103no—
527C4⋊C4C2×C22⋊D513yes—
528C4⋊C4C2×C22⋊D513yes—
529C4⋊C4C23×D513yes—
530C4⋊C4C23×D511yescharacteristic
531C22⋊C4—23no—
532C22⋊C4—23no—
533C22⋊C4—26no—
534C22⋊C4—23no—
535C22⋊C4—103no—
536C22⋊C4—103no—
537C22⋊C4C2×C22⋊D516yes—
538C42—21no—
539C42—23no—
540C42—23no—
541C42—103no—
542C42—103no—
543C42C2×C22⋊D513yes—
544C42C2×C22⋊D513yes—
545C42C23×D511yescharacteristic
546C42C23×D513yes—
547C2×C5—26no—
548C2×C5—46no—
549C2×C5C22×C22⋊Q811yescharacteristic
550C2×C5#358411yescharacteristic
551C2×C5#358511yescharacteristic
552C23—23no—
553C23—212no—
554C23—83no—
555C23—83no—
556C23C22×C22⋊D513yes—
557Q8—21no—
558Q8—403no—
559Q8—403no—
560Q8#407014yes—
561Q8C22×Dic1012yes—
562D4—212no—
563D4—212no—
564D4—23no—
565D4#4070112yes—
566D4C22×Dic1016yes—
567C2×C4—21no—
568C2×C4—212no—
569C2×C4—21no—
570C2×C4—23no—
571C2×C4—43no—
572C2×C4—43no—
573C2×C4—46no—
574C2×C4—86no—
575C2×C4—83no—
576C2×C4—83no—
577C2×C4—101no—
578C2×C4—101no—
579C2×C4—103no—
580C2×C4—103no—
581C2×C4—403no—
582C2×C4—403no—
583C2×C4C22×Dic1016yes—
584C2×C4C2×#70916yes—
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—26no—
599C22—43no—
600C22—43no—
601C22—812no—
602C22C2×#407016yes—
603C22C23×C22⋊D511yescharacteristic
604C4—21no—
605C4—23no—
606C4—21no—
607C4—23no—
608C4—22no—
609C4—43no—
610C4—41no—
611C4—203no—
612C4—201no—
613C4—203no—
614C4—201no—
615C4C2×#407016yes—
616C4C23×Dic1011yescharacteristic
617C4C22×#70911yescharacteristic
618C2—26no—
619C2—46no—
620C2C22×#407011yescharacteristic
621C2#1061917811yescharacteristic
622C2#1061919811yescharacteristic
subgroupG/H
centerC22C23×C22⋊D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC4×C5×H(5,𝔽2)C2
socleC22×C5C23×D4
prime radicalC2×C5×H(5,𝔽2)C22
hypercenterC4×H(5,𝔽2)D5
hypocenterC5#60496
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C4×#41 ◁ C4×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
2. #10617730C2semidirect
5. #10614293C2semidirect
6. #10614292C2semidirect
7. #10614291C2semidirect
8. #10614290C2semidirect
9. #10614270C2semidirect
10. #10614270C2semidirect
11. #10614243C2semidirect
12. #10614240C2semidirect
13. #10614238C2semidirect
14. #10614027C2semidirect
15. C2×#65298C2semidirect
35. #65853C22semidirect
36. #65853C22semidirect
37. #65804C22semidirect
38. C2×#4016C22semidirect
47. #65311C22semidirect
48. #65311C22semidirect
49. #65311C22semidirect
50. #65310C22semidirect
51. #65309C22semidirect
52. #65298C22semidirect
53. #65298C22semidirect
54. #65298C22semidirect
55. #65298C22semidirect
56. #65298C22semidirect
57. #65298C22semidirect
58. #65297C22semidirect
59. #65297C22semidirect
60. #65296C22semidirect
61. #65296C22semidirect
62. #65288C22semidirect
63. #65288C22semidirect
64. #65287C22semidirect
65. #65287C22semidirect
66. #65284C22semidirect
67. #65284C22semidirect
68. #65253C22semidirect
69. #65253C22semidirect
70. C2×#3940C22semidirect
71. C2×#3939C22semidirect
72. #65213C22semidirect
73. C2×C4⋊#168C22semidirect
76. C5×H(5,𝔽2)Q8semidirect
359. H(5,𝔽2)Dic10semidirect
595. C5#60496semidirect

Maximal subgroups

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

subgroupindex
C4×C5×H(5,𝔽2)2
#106177302
#106158302
#106142932
#106142922
#106142912
#106142902
#106142702
#106142432
#106142402
#106142382
#106140272
C2×#652982
#604965