Finite Group Databasebeta

← #13044988

Subgroups — #13044988

Order 1280.

1950 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10618612C211yesmaximal, characteristic
2#10617547C211yesmaximal, characteristic, complemented
3#41×#170C211yesmaximal, characteristic, complemented
4Q8×#704C211yesmaximal, characteristic, complemented
5C2×Q8×#170C211yesmaximal, characteristic, complemented
6#10617315C213yesmaximal, complemented
7#10617312C213yesmaximal, complemented
8#10617268C213yesmaximal, complemented
9#10617265C213yesmaximal, complemented
10#10617232C213yesmaximal, complemented
11#10617227C213yesmaximal, complemented
12#10617198C211yesmaximal, characteristic, complemented
13#10617195C211yesmaximal, characteristic, complemented
14#10617143C213yesmaximal, complemented
15C4×#4131C213yesmaximal, complemented
16C2×#4131C2213yes—
17C2×#4131C2213yes—
18C5×#548C411yescharacteristic
19#66168C411yescharacteristic
20#41×Dic5C2211yescharacteristic
21#66151C411yescharacteristic
22C2×Q8×Dic5C2211yescharacteristic
23#66015C413yes—
24#66014C2213yes—
25#65975C413yes—
26#65958C2213yes—
27#65952C413yes—
28#65930C2213yes—
29C4×#713C2211yescharacteristic
30#65903C411yescharacteristic
31#65899C413yes—
32C4×#711C2213yes—
33#65788C2213yes—
34#65788C2213yes—
35Q8×#170C2211yescharacteristic
36Q8×#170C2211yescharacteristic
37Q8×#170C2211yescharacteristic
38Q8×#170C2211yescharacteristic
39Q8×#170C2213yes—
40Q8×#170C2213yes—
41#65775C2216yes—
42#65774C2213yes—
43#65774C2213yes—
44D4×#170C2213yes—
45D4×#170C2213yes—
46#65759C2213yes—
47#65758C2213yes—
48C4×#704C2213yes—
49C4×#704C2213yes—
50#65748C2213yes—
51C2×C4⋊#170C2213yes—
52C2×C4×#170C2213yes—
53C2×C4×#170C2213yes—
54#65706C2216yes—
55#65705C2213yes—
56#65705C2213yes—
57#65702C2213yes—
58#65702C2213yes—
59#65701C2213yes—
60#65701C2213yes—
61#65697C2216yes—
62#65695C2216yes—
63#65694C2216yes—
64#65693C2216yes—
65#65692C2216yes—
66#65691C2211yescharacteristic
67#65691C2213yes—
68#65691C2213yes—
69#65691C2213yes—
70#65683C2213yes—
71#65680C2213yes—
72C4×#700C2213yes—
73C4×D5⋊C8C2213yes—
74#60218—51nomaximal, Sylow
75C2×C5×#41C2×C411yescharacteristic
76C22×C5×Q8C2×C411yescharacteristic
77C2×#713C2×C411yescharacteristic
78C2×#711C2313yes—
79C2×#704C2313yes—
80C2×#704C2313yes—
81C2×#704C2313yes—
82C2×#704C2311yescharacteristic
83C22×#170C2313yes—
84C22×#170C2313yes—
85C22×#170C2313yes—
86C22×#170C2311yescharacteristic
87#4131—212no—
88#4131—212no—
89C2×#700C2313yes—
90C2×#700C2313yes—
91C2×D5⋊C8C2313yes—
92C2×D5⋊C8C2313yes—
93C5×#124C2×C413yes—
94C5×C22⋊Q8C2×C413yes—
95C5×#121C2×C413yes—
96C4×C5×Q8C2×C411yescharacteristic
97C4×C5×D4C2×C413yes—
98#4093C2×C413yes—
99#4092C2×C413yes—
100Q8×Dic5C2×C411yescharacteristic
101Q8×Dic5C2×C413yes—
102Q8×Dic5C2×C411yescharacteristic
103Q8×Dic5C2311yescharacteristic
104Q8×Dic5C2313yes—
105#4086C2×C413yes—
106#4085C2×C416yes—
107#4084C2×C413yes—
108D4×Dic5C2×C413yes—
109D4×Dic5C2×C413yes—
110D4×Dic5C2313yes—
111#4076C2×C413yes—
112#4076C2×C413yes—
113C4×C22⋊D5C2×C413yes—
114C4×C22⋊D5C2×C413yes—
115#4072C2×C413yes—
116C2×C4⋊Dic5C2×C413yes—
117C2×C4×Dic5C2313yes—
118C2×C4×Dic5C2313yes—
119#4044C2×C416yes—
120#4039C2313yes—
121#4038C2×C413yes—
122#4038C2313yes—
123#4036C2×C413yes—
124#4034C2×C413yes—
125#4033C2×C413yes—
126#4032C2×C416yes—
127#4031C2×C416yes—
128#4029C2×C416yes—
129#4027C2×C413yes—
130#4027C2×C413yes—
131#4027C2316yes—
132#4025C2×C416yes—
133#4023C2311yescharacteristic
134#4023C2313yes—
135C4×D20C2×C413yes—
136C42×D5C2313yes—
137#4016C2×C413yes—
138C4×Dic10C2×C413yes—
139C22⋊#170C2316yes—
140C22⋊#170C2316yes—
141#4004C2316yes—
142#4003C2316yes—
143#4003C2316yes—
144#4002C2311yescharacteristic
145#4002C2313yes—
146#4002C2313yes—
147#4002C2313yes—
148#4002C2311yescharacteristic
149#4002C2313yes—
150C4⋊#170C2313yes—
151C4⋊#170C2313yes—
152C4⋊#170C2313yes—
153C4⋊#170C2313yes—
154C4⋊#170C2316yes—
155C4×#170C2311yescharacteristic
156C4×#170C2313yes—
157C4×#170C2313yes—
158C4×#170C2313yes—
159C4×#170C2313yes—
160C4×#170C2313yes—
161C4×#170C2313yes—
162C4×#170C2313yes—
163C4×#170C2311yescharacteristic
164C4×#170C2313yes—
165#3480—51no—
166#2971—51no—
167#2971—51no—
168#2969—53no—
169#2969—53no—
170#2966—53no—
171#2966—53no—
172C8×#41—51no—
173C8×#41—51no—
174Q8×M16—51no—
175C2×C8×Q8—51no—
176#2935—53no—
177#2931—53no—
178#2925—53no—
179C4×#131—53no—
180C5×#41—24no—
181C2×C5×Q8—24no—
182C2×C5×Q8C22×C413yes—
183C2×C5×Q8C22×C411yescharacteristic
184C2×C5×D4C22×C413yes—
185C2×C5×D4C22×C413yes—
186C22×C4×C5C22×C413yes—
187C22×C4×C5C22×C413yes—
188C2×C22⋊D5C22×C413yes—
189C2×C22⋊D5C22×C413yes—
190C2×C22⋊D5C22×C416yes—
191C22×Dic5C22×C413yes—
192C22×Dic5C22×C413yes—
193C22×Dic5C2413yes—
194C22×Dic5C2411yescharacteristic
195#713—24no—
196#711—26no—
197#711—26no—
198C2×D20C22×C413yes—
199C2×C4×D5C22×C413yes—
200C2×C4×D5C22×C413yes—
201C2×C4×D5C2413yes—
202C2×Dic10C22×C413yes—
203#704—26no—
204#704—26no—
205#704—26no—
206#704#4114yescomplemented
207C2×#170—26no—
208C2×#170—26no—
209C2×#170—26no—
210C2×#170#4114yescomplemented
211C2×#170C2413yes—
212C2×#170C2413yes—
213C2×#170C2413yes—
214C2×#170C2413yes—
215C2×#170C2411yescharacteristic
216C2×#170C2411yescharacteristic
217C2×#170C2411yescharacteristic
218C2×#170C2411yescharacteristic
219#700—26no—
220#700—26no—
221D5⋊C8—26no—
222D5⋊C8—26no—
223C5×C4⋊C4C22×C413yes—
224C5×C4⋊C4C22×C413yes—
225C5×C22⋊C4C22×C411yescharacteristic
226C5×C22⋊C4C22×C413yes—
227C5×C22⋊C4C22×C416yes—
228C42×C5C22×C413yes—
229C22⋊Dic5C22×C413yes—
230C22⋊Dic5C22×C411yescharacteristic
231C22⋊Dic5C22×C416yes—
232#685C22×C416yes—
233#685C22×C413yes—
234#685C22×C413yes—
235#685C22×C416yes—
236C4⋊Dic5C22×C413yes—
237C4⋊Dic5C22×C413yes—
238C4⋊Dic5C22×C413yes—
239C4⋊Dic5C22×C413yes—
240C4⋊Dic5C22×C416yes—
241#683C22×C411yescharacteristic
242#683C22×C413yes—
243#683C22×C416yes—
244#683C22×C413yes—
245#683C22×C411yescharacteristic
246C4×Dic5C22×C413yes—
247C4×Dic5C22×C413yes—
248C4×Dic5C22×C413yes—
249C4×Dic5C22×C413yes—
250C4×Dic5C22×C413yes—
251C4×Dic5C2413yes—
252C4×Dic5C2411yescharacteristic
253C4×Dic5C2411yescharacteristic
254C4×Dic5C2413yes—
255C2×#131—53no—
256C2×#131—53no—
257#559—53no—
258#557—51no—
259#556—53no—
260#553—53no—
261#553—53no—
262#550—51no—
263#548—51no—
264#548AGL(1,𝔽5)11yescharacteristic
265#532—53no—
266#529—53no—
267#529—53no—
268C4×#41—51no—
269C4×#41—51no—
270C4×#41—53no—
271C2×C4×Q8—51no—
272#446—53no—
273#446—53no—
274C8×Q8—51no—
275C8×Q8—53no—
276C8×Q8—53no—
277C8×Q8—51no—
278C8×Q8—51no—
279C8×Q8—51no—
280#436—56no—
281#436—53no—
282#436—53no—
283#436—53no—
284#436—53no—
285#435—56no—
286#435—56no—
287C8×D4—53no—
288C8×D4—53no—
289C8×D4—56no—
290C8×D4—53no—
291C8×D4—53no—
292#433—56no—
293#433—56no—
294#432—53no—
295#432—53no—
296#431—53no—
297#431—53no—
298#423—53no—
299C2×C4⋊C8—53no—
300#408—56no—
301#405—51no—
302#405—53no—
303#405—53no—
304#405—53no—
305#405—53no—
306#405—53no—
307C4×M16—53no—
308C4×M16—53no—
309C4×M16—53no—
310C2×C4×C8—53no—
311C2×C4×C8—53no—
312C2×C4×C8—53no—
313C23×C5C23×C413yes—
314C23×C5C23×C411yescharacteristic
315C22×D5C23×C413yes—
316C5×Q8—26no—
317C5×Q8#13114yescomplemented
318C5×D4—26no—
319C5×D4—26no—
320C2×C4×C5—26no—
321C2×C4×C5—26no—
322C2×C4×C5C23×C411yescharacteristic
323C2×C4×C5C23×C413yes—
324C2×C4×C5C23×C413yes—
325C2×C4×C5C23×C411yescharacteristic
326C2×C4×C5C23×C413yes—
327C22⋊D5—26no—
328C22⋊D5—26no—
329C22⋊D5—46no—
330C2×Dic5—23no—
331C2×Dic5—23no—
332C2×Dic5—43no—
333C2×Dic5—43no—
334C2×Dic5C23×C411yescharacteristic
335C2×Dic5C23×C411yescharacteristic
336C2×Dic5C23×C413yes—
337C2×Dic5C23×C413yes—
338C2×Dic5C23×C413yes—
339C2×Dic5C23×C411yescharacteristic
340C2×Dic5C2×#4112yes—
341C2×Dic5C2×#4112yes—
342C2×Dic5C2511yescharacteristic
343D20—26no—
344C4×D5—26no—
345C4×D5—23no—
346C4×D5—23no—
347C4×D5—43no—
348Dic10—26no—
349#170—23no—
350#170—23no—
351#170—23no—
352#170—23no—
353#170C2×#4112yes—
354#170C2×#4112yes—
355#170C2×#4112yes—
356#170C2×#4112yes—
357C2×#41—51no—
358C2×#41—53no—
359C2×#41C2×AGL(1,𝔽5)11yescharacteristic
360C22×Q8C2×AGL(1,𝔽5)11yescharacteristic
361#131—1012no—
362#131—1012no—
363C2×M16—51no—
364C2×M16—53no—
365C2×M16—53no—
366C2×M16—53no—
367C2×M16—53no—
368C2×M16—53no—
369C22×C8—51no—
370C22×C8—53no—
371C22×C8—53no—
372C22×C8—53no—
373C22×C8—53no—
374C22×C8—53no—
375C4⋊Q8—53no—
376#127—53no—
377#126—56no—
378#126—56no—
379#125—53no—
380#125—53no—
381#124—53no—
382#124—56no—
383#124C2×AGL(1,𝔽5)13yes—
384#123—56no—
385#123—56no—
386C22⋊Q8—53no—
387C22⋊Q8C2×AGL(1,𝔽5)13yes—
388#121—53no—
389#121—56no—
390#121—53no—
391#121—53no—
392#121C2×AGL(1,𝔽5)13yes—
393C4×Q8—51no—
394C4×Q8—53no—
395C4×Q8—51no—
396C4×Q8—53no—
397C4×Q8—51no—
398C4×Q8—53no—
399C4×Q8—53no—
400C4×Q8C2×AGL(1,𝔽5)11yescharacteristic
401C4×D4—53no—
402C4×D4—53no—
403C4×D4—53no—
404C4×D4—53no—
405C4×D4—53no—
406C4×D4—53no—
407C4×D4—53no—
408C4×D4—53no—
409C4×D4—56no—
410C4×D4—53no—
411C4×D4C2×AGL(1,𝔽5)13yes—
412#117—53no—
413#117—53no—
414#117—53no—
415#117—51no—
416#117—53no—
417C2×C4⋊C4—53no—
418C2×C42—53no—
419C2×C42—53no—
420C2×C42—53no—
421C4⋊C8—53no—
422C4⋊C8—53no—
423C4⋊C8—56no—
424C4⋊C8—53no—
425C4⋊C8—53no—
426C4⋊C8—56no—
427C22⋊C8—56no—
428C22⋊C8—56no—
429C22⋊C8—56no—
430C22⋊C8—56no—
431#97—51no—
432#97—51no—
433#97—53no—
434#97—53no—
435#97—53no—
436#97—53no—
437C4×C8—51no—
438C4×C8—51no—
439C4×C8—53no—
440C4×C8—53no—
441C4×C8—53no—
442C4×C8—53no—
443C4×C8—53no—
444C4×C8—53no—
445C4×C8—53no—
446C4×C8—53no—
447C22×C5—21no—
448C22×C5—21no—
449C22×C5—23no—
450C22×C5—23no—
451C22×C5—23no—
452C22×C5C4×#4112yes—
453C22×C5C24×C411yescharacteristic
454C2×D5—23no—
455C2×D5—23no—
456C2×D5—23no—
457C4×C5—23no—
458C4×C5—21no—
459C4×C5—23no—
460C4×C5—21no—
461C4×C5C2×#13116yes—
462Dic5—23no—
463Dic5—21no—
464Dic5—23no—
465Dic5—21no—
466Dic5—23no—
467Dic5C4×#4112yes—
468Dic5C22×#4111yescharacteristic
469Dic5C22×#4111yescharacteristic
470#41—24no—
471#41—106no—
472#41—106no—
473#41—104no—
474C2×Q8—24no—
475C2×Q8—53no—
476C2×Q8C22×AGL(1,𝔽5)13yes—
477C2×Q8C22×AGL(1,𝔽5)11yescharacteristic
478C2×D4—53no—
479C2×D4—53no—
480C2×D4—56no—
481C2×D4—53no—
482C2×D4C22×AGL(1,𝔽5)13yes—
483C2×D4C22×AGL(1,𝔽5)13yes—
484C22×C4—51no—
485C22×C4—53no—
486C22×C4—53no—
487C22×C4—53no—
488C22×C4—53no—
489C22×C4—53no—
490C22×C4—53no—
491C22×C4C22×AGL(1,𝔽5)13yes—
492C22×C4C22×AGL(1,𝔽5)13yes—
493M16—54no—
494M16—106no—
495M16—106no—
496M16—106no—
497M16—106no—
498M16—106no—
499C2×C8—51no—
500C2×C8—51no—
501C2×C8—51no—
502C2×C8—51no—
503C2×C8—54no—
504C2×C8—53no—
505C2×C8—53no—
506C2×C8—53no—
507C2×C8—53no—
508C2×C8—106no—
509C2×C8—106no—
510C2×C8—106no—
511C2×C8—106no—
512C2×C8—106no—
513C4⋊C4—53no—
514C4⋊C4—53no—
515C4⋊C4—56no—
516C4⋊C4—51no—
517C4⋊C4—51no—
518C4⋊C4—53no—
519C4⋊C4—53no—
520C4⋊C4—53no—
521C4⋊C4—53no—
522C4⋊C4—56no—
523C4⋊C4C22×AGL(1,𝔽5)13yes—
524C4⋊C4C22×AGL(1,𝔽5)13yes—
525C22⋊C4—51no—
526C22⋊C4—53no—
527C22⋊C4—56no—
528C22⋊C4—53no—
529C22⋊C4—56no—
530C22⋊C4—53no—
531C22⋊C4—56no—
532C22⋊C4C22×AGL(1,𝔽5)11yescharacteristic
533C22⋊C4C22×AGL(1,𝔽5)13yes—
534C22⋊C4C22×AGL(1,𝔽5)16yes—
535C42—51no—
536C42—51no—
537C42—53no—
538C42—53no—
539C42—53no—
540C42—53no—
541C42—53no—
542C42—53no—
543C42—53no—
544C42C22×AGL(1,𝔽5)13yes—
545C2×C5—22no—
546C2×C5—43no—
547C2×C5C2×C4×#4111yescharacteristic
548C2×C5C22×#13111yescharacteristic
549C2×C5#357311yescharacteristic
550D5—43no—
551C23—53no—
552C23C23×AGL(1,𝔽5)13yes—
553C23C23×AGL(1,𝔽5)11yescharacteristic
554Q8—26no—
555Q8—106no—
556Q8#413114yescomplemented
557D4—26no—
558D4—26no—
559D4—106no—
560D4—106no—
561D4—106no—
562D4—206no—
563C2×C4—26no—
564C2×C4—26no—
565C2×C4—52no—
566C2×C4—52no—
567C2×C4—51no—
568C2×C4—51no—
569C2×C4—51no—
570C2×C4—51no—
571C2×C4—53no—
572C2×C4—53no—
573C2×C4—53no—
574C2×C4—103no—
575C2×C4—103no—
576C2×C4—103no—
577C2×C4—103no—
578C2×C4—106no—
579C2×C4—203no—
580C2×C4—203no—
581C2×C4—203no—
582C2×C4C23×AGL(1,𝔽5)11yescharacteristic
583C2×C4C23×AGL(1,𝔽5)13yes—
584C2×C4C23×AGL(1,𝔽5)11yescharacteristic
585C2×C4C23×AGL(1,𝔽5)13yes—
586C2×C4C23×AGL(1,𝔽5)13yes—
587C8—52no—
588C8—52no—
589C8—52no—
590C8—52no—
591C8—103no—
592C8—103no—
593C8—103no—
594C8—103no—
595C5#6021811yescharacteristic, complemented, Sylow
596C22—21no—
597C22—21no—
598C22—23no—
599C22—23no—
600C22—23no—
601C22—103no—
602C22—103no—
603C22—103no—
604C22#41×AGL(1,𝔽5)12yes—
605C22C24×AGL(1,𝔽5)11yescharacteristic
606C4—21no—
607C4—23no—
608C4—23no—
609C4—21no—
610C4—51no—
611C4—51no—
612C4—52no—
613C4—101no—
614C4—103no—
615C4—103no—
616C4—101no—
617C4—103no—
618C4C2×#413116yes—
619C2—22no—
620C2—43no—
621C2—203no—
622C2C22×#413111yescharacteristic
623C2C2×#41×AGL(1,𝔽5)11yescharacteristic
624C2#1061941511yescharacteristic
subgroupG/H
center, FrattiniC22C24×AGL(1,𝔽5)
commutator, socleC22×C5C24×C4
FittingC5×#548C4
prime radical#10618612C2
cosocleC2×Dic5C25
hypercenter#548AGL(1,𝔽5)
hypocenterC5#60218
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
2. #10617547C2semidirect
3. #41×#170C2semidirect
4. Q8×#704C2semidirect
5. C2×Q8×#170C2semidirect
6. #10617315C2semidirect
7. #10617312C2semidirect
8. #10617268C2semidirect
9. #10617265C2semidirect
10. #10617232C2semidirect
11. #10617227C2semidirect
12. #10617198C2semidirect
13. #10617195C2semidirect
14. #10617143C2semidirect
15. C4×#4131C2semidirect
206. #704#41semidirect
210. C2×#170#41semidirect
317. C5×Q8#131semidirect
556. Q8#4131semidirect
595. C5#60218semidirect

Maximal subgroups

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

subgroupindex
#106186122
#106175472
#41×#1702
Q8×#7042
C2×Q8×#1702
#106173152
#106173122
#106172682
#106172652
#106172322
#106172272
#106171982
#106171952
#106171432
C4×#41312
#602185