Finite Group Databasebeta

← #12966955

Subgroups — #12966955

Order 1280.

2090 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10617790C211yesmaximal, characteristic, complemented
2#10617787C211yesmaximal, characteristic
3C5×#2913C211yesmaximal, characteristic, complemented
4#10615528C211yesmaximal, characteristic
5#10615514C211yesmaximal, characteristic, complemented
6#10614294C211yesmaximal, characteristic
7#10614293C211yesmaximal, characteristic, complemented
8#10610974C213yesmaximal, complemented
9#10610973C213yesmaximal, complemented
10#10610971C213yesmaximal, complemented
11#10610970C213yesmaximal, complemented
12#10610928C213yesmaximal, complemented
13#10610927C213yesmaximal, complemented
14#10610893C213yesmaximal, complemented
15C2×#65179C213yesmaximal, complemented
16C2×#4149C2211yescharacteristic
17C2×C5×#131C2211yescharacteristic
18C5×#520C2211yescharacteristic
19C5×#520C2211yescharacteristic
20#66161C2211yescharacteristic
21#41×Dic5C2211yescharacteristic
22C2×#4094C2211yescharacteristic
23#65908—21no—
24C4×#713—21no—
25#65903—21no—
26Q8×Dic10—21no—
27#65879—23no—
28C4×D4×D5—23no—
29#65873—23no—
30D4×Dic10—23no—
31#65856C2213yes—
32#65855C2213yes—
33#65853C2213yescomplemented
34#65851C2213yescomplemented
35#65838C2213yes—
36C2×#4039C2213yes—
37#65834C2213yescomplemented
38C2×#4034C2213yescomplemented
39C5×#428C2213yescomplemented
40C5×#428C2213yescomplemented
41C5×#427C2213yescomplemented
42C2×C5×#107C2213yescomplemented
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#65180C2216yescomplemented
60#65180C2213yescomplemented
61#65179C2216yescomplemented
62#65179C2216yescomplemented
63#65179C2213yescomplemented
64#65179C2216yescomplemented
65#65166C2213yescomplemented
66#65164C2216yescomplemented
67#65164C2211yescharacteristic, complemented
68#65160C2216yescomplemented
69#65160C2213yescomplemented
70#65159C2213yescomplemented
71#65159C2216yescomplemented
72#65146C2213yescomplemented
73#65143C2213yescomplemented
74#60494—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—
83C22×Dic10—23no—
84C5×#131—24no—
85C2×C5×M16C2313yes—
86C22×C5×C8C2313yes—
87C4×C5×Q8—21no—
88C4×C5×Q8—21no—
89C4×C5×D4—23no—
90C4×C5×D4—23no—
91C5×#117C2313yes—
92C5×#117C2313yes—
93C2×C5×C4⋊C4C2313yes—
94C2×C5×C4⋊C4C2313yes—
95#4094—24no—
96Q8×Dic5—21no—
97Q8×Dic5D411yescharacteristic
98Q8×Dic5D411yescharacteristic
99D4×Dic5—23no—
100D4×Dic5D413yes—
101D4×Dic5D413yes—
102C4×C22⋊D5—26no—
103C2×#685—23no—
104#4072C2313yes—
105#4072C2313yes—
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#4036C2313yes—
121#4036C2313yes—
122#4034—23no—
123#4034—23no—
124#4034—23no—
125#4034C2313yes—
126#4034C2316yes—
127#4034C2311yescharacteristic
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×#107C2316yes—
146C5×#107C2311yescharacteristic
147C5×#106C2313yes—
148C5×#106C2313yes—
149#3953C2313yes—
150#3953C2316yes—
151#3953C2313yes—
152#3948C2313yes—
153#3948C2311yescharacteristic
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#3503—51no—
170C4×H(5,𝔽2)—51no—
171#3083—53no—
172#3083—53no—
173#3081—53no—
174#3081—53no—
175#3079—53no—
176#3077—53no—
177#3076—53no—
178C2×#474—53no—
179#2913—51no—
180#2913D511yescharacteristic, 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×Q8112yescomplemented
200C2×D20—23no—
201C2×D20—26no—
202C2×D20C2×Q8112yescomplemented
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—21no—
211C2×Dic10—26no—
212C2×Dic10—86no—
213C2×Dic10C2411yescharacteristic
214C5×M16—26no—
215C2×C5×C8—26no—
216C2×C5×C8C2411yescharacteristic
217C2×C5×C8C2413yes—
218C5×C4⋊C4—23no—
219C5×C4⋊C4—23no—
220C5×C4⋊C4C2413yes—
221C5×C4⋊C4C2413yes—
222C5×C4⋊C4C2411yescharacteristic
223C5×C4⋊C4C2411yescharacteristic
224C5×C22⋊C4—23no—
225C5×C22⋊C4—23no—
226C42×C5—23no—
227C42×C5—23no—
228C22⋊Dic5—23no—
229C22⋊Dic5C2×D416yes—
230#685—23no—
231#685—23no—
232#685—26no—
233C4⋊Dic5—23no—
234C4⋊Dic5—23no—
235C4⋊Dic5—23no—
236C4⋊Dic5C2×D413yes—
237C4⋊Dic5C2×D413yes—
238C4⋊Dic5C2413yes—
239C4⋊Dic5C2411yescharacteristic
240#683—21no—
241#683—23no—
242#683—23no—
243#683—23no—
244#683—26no—
245#683—21no—
246#683—23no—
247C4×Dic5—23no—
248C4×Dic5—21no—
249C4×Dic5—23no—
250C4×Dic5C2×D413yes—
251C4×Dic5C2×D413yes—
252C4×Dic5C2411yescharacteristic
253C4×Dic5C2413yes—
254#681—26no—
255C2×#168—26no—
256C2×#168C2411yescharacteristic
257C2×#168C2413yes—
258C2×H(5,𝔽2)—51no—
259C2×#131—51no—
260C2×#131C2×D511yescharacteristic
261Q82—101no—
262#557—101no—
263#554—103no—
264D4×Q8—103no—
265#544—53no—
266#544—53no—
267#533—53no—
268C2×C4⋊Q8—53no—
269#520—51no—
270#520—53no—
271#520—101no—
272#520C2×D511yescharacteristic
273#520C2×D511yescharacteristic
274#518—53no—
275#518—103no—
276C4×#41—51no—
277C4×#41—53no—
278C4×#41—101no—
279C2×C4×D4—53no—
280C2×C4×D4—103no—
281#479—53no—
282#479—53no—
283#478—53no—
284#478—56no—
285#478—53no—
286#478—56no—
287#477—51no—
288#477—56no—
289#476—53no—
290#476—56no—
291#475—53no—
292#474—53no—
293#474—56no—
294#474—56no—
295#474—56no—
296#428—53no—
297#428—53no—
298#428C2×D513yescomplemented
299#428C2×D513yescomplemented
300#427—53no—
301#427C2×D513yescomplemented
302C2×#107—53no—
303C2×#107C2×D513yescomplemented
304#424—53no—
305#424—53no—
306#423—53no—
307C2×C4⋊C8—53no—
308#418—53no—
309#418—53no—
310#417—53no—
311#417—53no—
312#416—53no—
313#416—53no—
314C2×#102—53no—
315C2×#102—53no—
316C23×C5C22×D413yes—
317C22×D5—23no—
318C22×D5—212no—
319C22×D5—83no—
320C22×D5—83no—
321C5×Q8C22⋊Q814yes—
322C5×D4C22⋊Q8112yes—
323C2×C4×C5—21no—
324C2×C4×C5—23no—
325C2×C4×C5—21no—
326C2×C4×C5—23no—
327C2×C4×C5—83no—
328C2×C4×C5—83no—
329C2×C4×C5C22×D413yes—
330C2×C4×C5C22×Q816yes—
331C2×C4×C5C2×#4116yes—
332C2×C4×C5C2511yescharacteristic
333C22⋊D5—212no—
334C2×Dic5—21no—
335C2×Dic5—26no—
336C2×Dic5—21no—
337C2×Dic5—23no—
338C2×Dic5—43no—
339C2×Dic5—83no—
340C2×Dic5C22×D413yes—
341C2×Dic5C22×D411yescharacteristic
342C2×Dic5C22×D413yes—
343C2×Dic5C22×D411yescharacteristic
344D20—212no—
345D20—23no—
346D20C22×Q816yes—
347C4×D5—212no—
348C4×D5—46no—
349C4×D5—83no—
350C4×D5—86no—
351Dic10—21no—
352Dic10—42no—
353Dic10—46no—
354Dic10—86no—
355Dic10C22×Q812yes—
356C5×C8—21no—
357C5×C8—23no—
358#168—21no—
359#168—23no—
360H(5,𝔽2)—58no—
361H(5,𝔽2)—104no—
362C2×#41—53no—
363C2×#41—101no—
364C2×#41C22×D511yescharacteristic
365C22×Q8—103no—
366C22×D4—53no—
367C22×D4—103no—
368C23×C4—103no—
369#131—24no—
370#131—104no—
371C2×M16—53no—
372C2×M16C22×D513yes—
373C22×C8—53no—
374C22×C8C22×D513yes—
375C4⋊Q8—53no—
376C4⋊Q8—56no—
377C4⋊Q8—51no—
378C4⋊Q8—103no—
379C4⋊Q8—103no—
380C4⋊Q8—103no—
381C4⋊Q8—103no—
382#125—53no—
383#125—53no—
384#125—103no—
385#125—103no—
386C22⋊Q8—106no—
387C22⋊Q8—103no—
388C22⋊Q8—103no—
389C22⋊Q8—106no—
390C4×Q8—21no—
391C4×Q8—21no—
392C4×Q8—51no—
393C4×Q8—51no—
394C4×Q8—51no—
395C4×Q8—53no—
396C4×Q8—103no—
397C4×Q8—103no—
398C4×Q8—101no—
399C4×D4—23no—
400C4×D4—23no—
401C4×D4—53no—
402C4×D4—53no—
403C4×D4—53no—
404C4×D4—56no—
405C4×D4—53no—
406C4×D4—103no—
407C4×D4—103no—
408C4×D4—103no—
409C4×D4—103no—
410C4×D4—103no—
411C4×D4—103no—
412C4×D4—106no—
413C4×D4—103no—
414#117—53no—
415#117—53no—
416#117—103no—
417#117—103no—
418#117C22×D513yes—
419#117C22×D513yes—
420C2×C4⋊C4—53no—
421C2×C4⋊C4—103no—
422C2×C4⋊C4—103no—
423C2×C4⋊C4C22×D513yes—
424C2×C4⋊C4C22×D513yes—
425C2×C22⋊C4—106no—
426C2×C22⋊C4—103no—
427C2×C42—53no—
428C2×C42—103no—
429#107—53no—
430#107—56no—
431#107—51no—
432#107C22×D513yes—
433#107C22×D516yes—
434#107C22×D511yescharacteristic
435#106—53no—
436#106—53no—
437#106C22×D513yes—
438#106C22×D513yes—
439C4⋊C8—51no—
440C4⋊C8—53no—
441C4⋊C8—53no—
442C4⋊C8—53no—
443C4⋊C8—56no—
444#103—51no—
445#103—53no—
446#103—51no—
447#103—53no—
448#102—53no—
449#102—56no—
450#102—53no—
451#102—53no—
452#102—56no—
453#102—53no—
454C22×C5—23no—
455C22×C5—23no—
456C22×C5C2×C22⋊Q816yes—
457C22×C5C23×D411yescharacteristic
458C2×D5—26no—
459C2×D5—43no—
460C2×D5—43no—
461C2×D5—812no—
462C4×C5—41no—
463C4×C5—41no—
464C4×C5—43no—
465C4×C5—43no—
466C4×C5C2×C22⋊Q816yes—
467C4×C5C23×Q811yescharacteristic
468C4×C5C22×#4111yescharacteristic
469Dic5—21no—
470Dic5—23no—
471Dic5—22no—
472Dic5—23no—
473Dic5—41no—
474Dic5—43no—
475Dic5#57612yes—
476C24—103no—
477#41—512no—
478#41—108no—
479#41—106no—
480#41#70914yes—
481#41Q8×D514yes—
482C2×Q8—51no—
483C2×Q8—101no—
484C2×Q8—103no—
485C2×Q8—106no—
486C2×Q8—406no—
487C2×Q8D4×D511yescharacteristic
488C2×D4—512no—
489C2×D4—53no—
490C2×D4—103no—
491C2×D4—1024no—
492C2×D4—106no—
493C2×D4—103no—
494C2×D4D4×D513yes—
495C22×C4—23no—
496C22×C4—23no—
497C22×C4—53no—
498C22×C4—53no—
499C22×C4—103no—
500C22×C4—103no—
501C22×C4—106no—
502C22×C4—103no—
503C22×C4—103no—
504C22×C4—406no—
505C22×C4C23×D513yes—
506M16—26no—
507M16—106no—
508C2×C8—26no—
509C2×C8—53no—
510C2×C8—51no—
511C2×C8—106no—
512C2×C8C23×D513yes—
513C2×C8C23×D511yescharacteristic
514C4⋊C4—23no—
515C4⋊C4—23no—
516C4⋊C4—53no—
517C4⋊C4—53no—
518C4⋊C4—51no—
519C4⋊C4—53no—
520C4⋊C4—103no—
521C4⋊C4—103no—
522C4⋊C4—106no—
523C4⋊C4—101no—
524C4⋊C4—101no—
525C4⋊C4—103no—
526C4⋊C4—103no—
527C4⋊C4—103no—
528C4⋊C4—103no—
529C4⋊C4—103no—
530C4⋊C4C23×D511yescharacteristic
531C4⋊C4C23×D513yes—
532C4⋊C4C23×D511yescharacteristic
533C4⋊C4C23×D513yes—
534C22⋊C4—23no—
535C22⋊C4—23no—
536C22⋊C4—56no—
537C22⋊C4—103no—
538C22⋊C4—103no—
539C22⋊C4—106no—
540C22⋊C4—103no—
541C42—23no—
542C42—23no—
543C42—53no—
544C42—53no—
545C42—51no—
546C42—53no—
547C42—101no—
548C42—103no—
549C42—103no—
550C2×C5—26no—
551C2×C5C22×C22⋊Q811yescharacteristic
552C2×C5C2×#57611yescharacteristic
553C2×C5#358611yescharacteristic
554D5—46no—
555C23—103no—
556C23—1012no—
557C23—403no—
558C23—403no—
559C23C2×D4×D513yes—
560Q8—52no—
561Q8—101no—
562Q8—206no—
563Q8—202no—
564Q8—406no—
565Q8#404214yes—
566D4—56no—
567D4—1012no—
568D4—1012no—
569D4—103no—
570D4#4042112yes—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—23no—
574C2×C4—23no—
575C2×C4—51no—
576C2×C4—51no—
577C2×C4—53no—
578C2×C4—53no—
579C2×C4—83no—
580C2×C4—83no—
581C2×C4—106no—
582C2×C4—101no—
583C2×C4—1012no—
584C2×C4—101no—
585C2×C4—103no—
586C2×C4—206no—
587C2×C4—203no—
588C2×C4—406no—
589C2×C4—403no—
590C2×C4—403no—
591C2×C4C2×#70916yes—
592C2×C4C2×D4×D513yes—
593C2×C4C2×Q8×D516yes—
594C2×C4C24×D511yescharacteristic
595C8—21no—
596C8—23no—
597C8—101no—
598C8—103no—
599C5#6049411yescharacteristic, complemented, Sylow
600C22—23no—
601C22—23no—
602C22—106no—
603C22—203no—
604C22—203no—
605C22—4012no—
606C22C2×#404216yes—
607C22C22×D4×D511yescharacteristic
608C4—43no—
609C4—43no—
610C4—41no—
611C4—41no—
612C4—52no—
613C4—102no—
614C4—103no—
615C4—101no—
616C4—103no—
617C4—203no—
618C4—201no—
619C4C2×#404216yes—
620C4C22×#70911yescharacteristic
621C4C22×Q8×D511yescharacteristic
622C2—26no—
623C2—206no—
624C2C22×#404211yescharacteristic
625C2D5×#57611yescharacteristic
626C2#1061911211yescharacteristic
subgroupG/H
centerC22C22×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC5×#2913C2
socleC22×C5C23×D4
prime radicalC2×#4149C22
hypercenter#2913D5
hypocenterC5#60494
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×#41 ◁ #2913

Complements

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

NH ≅ G/Ntype
1. #10617790C2semidirect
3. C5×#2913C2semidirect
5. #10615514C2semidirect
7. #10614293C2semidirect
8. #10610974C2semidirect
9. #10610973C2semidirect
10. #10610971C2semidirect
11. #10610970C2semidirect
12. #10610928C2semidirect
13. #10610927C2semidirect
14. #10610893C2semidirect
15. C2×#65179C2semidirect
33. #65853C22semidirect
34. #65851C22semidirect
37. #65834C22semidirect
38. C2×#4034C22semidirect
39. C5×#428C22semidirect
40. C5×#428C22semidirect
41. C5×#427C22semidirect
42. C2×C5×#107C22semidirect
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. #65179C22semidirect
64. #65179C22semidirect
65. #65166C22semidirect
66. #65164C22semidirect
67. #65164C22semidirect
68. #65160C22semidirect
69. #65160C22semidirect
70. #65159C22semidirect
71. #65159C22semidirect
72. #65146C22semidirect
73. #65143C22semidirect
77. #4149Q8semidirect
180. #2913D5semidirect
199. #709C2×Q8semidirect
202. C2×D20C2×Q8semidirect
298. #428C2×D5semidirect
299. #428C2×D5semidirect
301. #427C2×D5semidirect
303. C2×#107C2×D5semidirect
599. C5#60494semidirect

Maximal subgroups

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

subgroupindex
#106177902
#106177872
C5×#29132
#106155282
#106155142
#106142942
#106142932
#106109742
#106109732
#106109712
#106109702
#106109282
#106109272
#106108932
C2×#651792
#604945