Finite Group Databasebeta

← #13044909

Subgroups — #13044909

Order 1280.

1878 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10618368C211yesmaximal, characteristic
2#10617362C211yesmaximal, characteristic, complemented
3#10617361C211yesmaximal, characteristic, complemented
4#10617330C211yesmaximal, characteristic, complemented
5#10617329C211yesmaximal, characteristic, complemented
6#10617295C214yesmaximal, complemented
7#10617294C214yesmaximal, complemented
8#10617236C214yesmaximal, complemented
9#10617235C214yesmaximal, complemented
10#10617226C212yesmaximal, complemented
11#10617225C212yesmaximal, complemented
12#10617140C212yesmaximal, complemented
13#10617136C212yesmaximal, complemented
14#10617110C211yesmaximal, characteristic, complemented
15#10617105C211yesmaximal, characteristic, complemented
16C2×#4131C2212yes—
17C2×#4131C2212yes—
18C2×D5⋊M16C2211yescharacteristic
19C22×D5⋊C8C2211yescharacteristic
20C5×#540C411yescharacteristic
21#66053C411yescharacteristic
22#66050C2211yescharacteristic
23#66031C411yescharacteristic
24#66022C2211yescharacteristic
25#65994C414yes—
26#65983C2214yes—
27#65947C414yes—
28#65940C2212yes—
29#65937C412yes—
30#65934C2214yes—
31#65895C412yes—
32C4×#711C2212yes—
33D5×#117C2211yescharacteristic
34#65852C411yescharacteristic
35#65776C2212yes—
36#65776C2212yes—
37#65774C2214yes—
38#65774C2214yescomplemented
39#65773C2214yes—
40#65773C2214yes—
41#65773C2214yescomplemented
42#65773C2214yescomplemented
43D4×#170C2214yes—
44D4×#170C2214yescomplemented
45#65756C2212yes—
46#65754C2214yescomplemented
47C4×#704C2214yescomplemented
48C2×#4002C2214yescomplemented
49#65748C2214yescomplemented
50#65709C2214yescomplemented
51#65708C2212yes—
52#65708C2212yes—
53#65707C2214yescomplemented
54#65707C2214yescomplemented
55#65706C2214yes—
56#65703C2214yescomplemented
57#65701C2212yes—
58#65701C2212yes—
59#65701C2212yes—
60#65701C2212yes—
61#65697C2214yes—
62#65697C2214yescomplemented
63#65697C2214yescomplemented
64#65695C2214yes—
65#65694C2214yes—
66#65693C2214yes—
67#65692C2214yes—
68#65691C2212yes—
69#65691C2212yes—
70#65691C2214yescomplemented
71#65691C2214yescomplemented
72#65682C2211yescharacteristic
73#65681C2211yescharacteristic
74C4×#700C2212yes—
75#65676C2211yescharacteristic
76#65676C2211yescharacteristic
77C4×D5⋊C8C2211yescharacteristic
78C4×D5⋊C8C2211yescharacteristic
79#60209—51nomaximal, Sylow
80C2×#711C2312yes—
81C22×C4×D5C2311yescharacteristic
82C2×#704C2311yescharacteristic
83C2×#704C2311yescharacteristic
84C2×#704C2314yes—
85C2×#704C2314yes—
86C22×#170C2312yes—
87C22×#170C2314yes—
88C22×#170C2314yes—
89#4131—28no—
90#4131—28no—
91D5⋊M16—24no—
92C2×#700C2312yes—
93C2×#700C2312yes—
94C2×#700C2311yescharacteristic
95C2×#700C2311yescharacteristic
96C2×D5⋊C8—24no—
97C2×D5⋊C8C2311yescharacteristic
98C2×D5⋊C8C2311yescharacteristic
99C2×D5⋊C8C2312yes—
100C2×D5⋊C8C2311yescharacteristic
101C2×D5⋊C8C2311yescharacteristic
102C5×#127C2×C411yescharacteristic
103C5×#125C2×C411yescharacteristic
104C5×#123C2×C414yes—
105C5×#121C2×C414yes—
106C5×#121C2×C412yes—
107C4×C5×D4C2×C412yes—
108C5×#117C2×C411yescharacteristic
109#4085C2×C414yes—
110#4084C2×C414yes—
111#4084C2×C412yes—
112#4083C2×C414yes—
113#4082C2×C412yes—
114#4082C2×C414yes—
115#4081C2×C414yes—
116#4081C2×C414yes—
117D4×Dic5C2×C412yes—
118D4×Dic5C2314yes—
119D4×Dic5C2314yes—
120#4075C2×C414yes—
121C4×C22⋊D5C2×C414yes—
122#4072C2×C411yescharacteristic
123#4070C2×C412yes—
124#4070C2×C414yes—
125C2×C4×Dic5C2314yes—
126#4044C2×C414yes—
127#4043C2×C412yes—
128#4042C2×C414yes—
129#4038C2312yes—
130#4038C2312yes—
131D5×C4⋊C4C2×C412yes—
132#4036C2×C414yes—
133#4035C2×C414yes—
134#4034C2×C412yes—
135#4033C2×C412yes—
136#4033C2314yes—
137#4031C2×C414yes—
138#4029C2×C414yes—
139#4027C2×C414yes—
140#4027C2314yes—
141D5×C22⋊C4C2×C412yes—
142#4025C2×C414yes—
143#4025C2×C414yes—
144#4025C2×C414yes—
145#4024C2×C414yes—
146#4023C2312yes—
147#4023C2314yes—
148#4018C2×C411yescharacteristic
149#4018C2×C411yescharacteristic
150C42×D5C2311yescharacteristic
151C42×D5C2311yescharacteristic
152#4016C2×C411yescharacteristic
153#4015C2×C411yescharacteristic
154C4×Dic10C2×C412yes—
155C22⋊#170C2314yes—
156C22⋊#170C2314yes—
157C22⋊#170C2314yes—
158C22⋊#170C2314yes—
159C22⋊#170C2314yes—
160#4004C2314yes—
161#4004C2314yes—
162#4004C2314yes—
163#4003C2314yes—
164#4002C2312yes—
165#4002C2312yes—
166#4002C2312yes—
167#4002C2312yes—
168#4002C2314yes—
169#4002C2314yes—
170#4002C2314yes—
171C4⋊#170C2314yes—
172C4⋊#170C2314yes—
173C4⋊#170C2314yes—
174C4×#170C2312yes—
175C4×#170C2312yes—
176C4×#170C2312yes—
177C4×#170C2312yes—
178C4×#170C2314yes—
179C4×#170C2314yes—
180C4×#170C2314yes—
181#3455—51no—
182#2969—54no—
183#2969—54no—
184#2968—54no—
185#2968—54no—
186#2961—51no—
187#2960—51no—
188#2932—51no—
189#2931—51no—
190#2930—52no—
191#2929—52no—
192#2922—52no—
193#2921—51no—
194#2875—52no—
195C2×#405—51no—
196C23×D5C22×C411yescharacteristic
197C2×C5×D4C22×C412yes—
198C2×C5×D4C22×C414yes—
199C2×C5×D4C22×C414yes—
200C22×C4×C5C22×C411yescharacteristic
201C22×C4×C5C22×C414yes—
202C2×C22⋊D5C22×C414yes—
203C2×C22⋊D5C22×C414yes—
204C22×Dic5C22×C414yes—
205C22×Dic5C2411yescharacteristic
206C22×Dic5C2414yes—
207#711—24no—
208#711—24no—
209C2×C4×D5—22no—
210C2×C4×D5—22no—
211C2×C4×D5C22×C411yescharacteristic
212C2×C4×D5C22×C411yescharacteristic
213C2×C4×D5C22×C412yes—
214C2×C4×D5C22×C412yes—
215C2×C4×D5C2411yescharacteristic
216C2×C4×D5C2411yescharacteristic
217C2×Dic10C22×C412yes—
218C2×Dic10C22×C414yes—
219#704—22no—
220#704—22no—
221#704—28no—
222#704—28no—
223C2×#170—24no—
224C2×#170—28no—
225C2×#170—28no—
226C2×#170C2412yes—
227C2×#170C2411yescharacteristic
228C2×#170C2411yescharacteristic
229C2×#170C2414yes—
230C2×#170C2414yes—
231C2×#170C2412yes—
232C2×#170C2411yescharacteristic
233C2×#170C2411yescharacteristic
234#700—24no—
235#700—24no—
236#700—22no—
237#700—22no—
238D5⋊C8—22no—
239D5⋊C8—22no—
240D5⋊C8—24no—
241D5⋊C8#4114yes—
242D5⋊C8#4114yes—
243C5×C4⋊C4C22×C412yes—
244C5×C4⋊C4C22×C412yes—
245C5×C4⋊C4C22×C414yes—
246C5×C22⋊C4C22×C412yes—
247C5×C22⋊C4C22×C414yes—
248C5×C22⋊C4C22×C414yes—
249C42×C5C22×C411yescharacteristic
250C42×C5C22×C411yescharacteristic
251C22⋊Dic5C22×C414yes—
252C22⋊Dic5C22×C412yes—
253C22⋊Dic5C22×C414yes—
254C22⋊Dic5C22×C414yes—
255C22⋊Dic5C22×C414yes—
256C22⋊Dic5C22×C414yes—
257#685C22×C412yes—
258#685C22×C412yes—
259#685C22×C414yes—
260#685C22×C414yes—
261C4⋊Dic5C22×C412yes—
262C4⋊Dic5C22×C414yes—
263C4⋊Dic5C22×C412yes—
264C4⋊Dic5C22×C414yes—
265#683C22×C414yes—
266#683C22×C414yes—
267#683C22×C412yes—
268#683C22×C412yes—
269#683C22×C414yes—
270#683C22×C414yes—
271#683C22×C414yes—
272C4×Dic5C22×C414yes—
273C4×Dic5C22×C411yescharacteristic
274C4×Dic5C22×C411yescharacteristic
275C4×Dic5C22×C414yes—
276C4×Dic5C2412yes—
277C4×Dic5C2412yes—
278C4×Dic5C2414yes—
279C2×#131—52no—
280C2×#131—52no—
281C22×M16—51no—
282C23×C8—51no—
283#542—52no—
284#542—52no—
285#541—51no—
286#541—51no—
287#540—51no—
288#540AGL(1,𝔽5)11yescharacteristic
289#539—54no—
290#538—54no—
291#533—51no—
292#532—51no—
293#529—54no—
294#529—54no—
295#525—52no—
296C4×#41—52no—
297C2×#117—51no—
298#446—54no—
299#446—54no—
300#446—54no—
301C8×Q8—54no—
302#436—54no—
303#436—54no—
304#436—54no—
305#435—54no—
306#435—54no—
307#435—54no—
308#435—54no—
309#435—54no—
310#435—54no—
311C8×D4—54no—
312C8×D4—54no—
313C8×D4—54no—
314#433—54no—
315#433—54no—
316#433—54no—
317#433—54no—
318#431—54no—
319#431—54no—
320#424—52no—
321#424—52no—
322#423—51no—
323#423—51no—
324#408—52no—
325#408—52no—
326#407—52no—
327#405—52no—
328#405—52no—
329#405—52no—
330#405—52no—
331#405—52no—
332#405—52no—
333#405—54no—
334#405—54no—
335C4×M16—52no—
336C4×M16—54no—
337C2×#97—51no—
338C2×#97—51no—
339C2×#97—54no—
340C2×C4×C8—51no—
341C2×C4×C8—51no—
342C23×C5C23×C411yescharacteristic
343C23×C5C23×C414yes—
344C22×D5—22no—
345C22×D5—42no—
346C22×D5C23×C411yescharacteristic
347C22×D5C23×C411yescharacteristic
348C5×D4—24no—
349C5×D4—44no—
350C5×D4—44no—
351C2×C4×C5—22no—
352C2×C4×C5—44no—
353C2×C4×C5C23×C412yes—
354C2×C4×C5C23×C414yes—
355C2×C4×C5C23×C412yes—
356C2×C4×C5C23×C411yescharacteristic
357C2×C4×C5C23×C411yescharacteristic
358C22⋊D5—28no—
359C22⋊D5—44no—
360C2×Dic5—21no—
361C2×Dic5—24no—
362C2×Dic5—24no—
363C2×Dic5—21no—
364C2×Dic5—44no—
365C2×Dic5C23×C412yes—
366C2×Dic5C23×C414yes—
367C2×Dic5C23×C414yes—
368C2×Dic5C23×C412yes—
369C2×Dic5C23×C411yescharacteristic
370C2×Dic5C2511yescharacteristic
371C4×D5—22no—
372C4×D5—22no—
373C4×D5—21no—
374C4×D5—21no—
375C4×D5—42no—
376C4×D5—42no—
377C4×D5C2×#4112yes—
378C4×D5C2×#4112yes—
379Dic10—24no—
380Dic10—44no—
381#170—22no—
382#170—21no—
383#170—21no—
384#170—24no—
385#170—24no—
386#170C2×#4114yes—
387#170C2×#4112yes—
388#170C2×#4112yes—
389C2×#41—52no—
390C23×C4—51no—
391#131—108no—
392#131—108no—
393C2×M16—51no—
394C2×M16—51no—
395C2×M16—52no—
396C2×M16—52no—
397C2×M16—51no—
398C2×M16—51no—
399C2×M16—54no—
400C2×M16—54no—
401C2×M16—104no—
402C22×C8—51no—
403C22×C8—51no—
404C22×C8—51no—
405C22×C8—52no—
406C22×C8—51no—
407C22×C8—52no—
408C22×C8—54no—
409C22×C8—54no—
410C22×C8—104no—
411C4⋊Q8—52no—
412C4⋊Q8—51no—
413#127C2×AGL(1,𝔽5)11yescharacteristic
414#126—54no—
415#126—54no—
416#126—54no—
417#126—54no—
418#125—54no—
419#125—54no—
420#125—51no—
421#125C2×AGL(1,𝔽5)11yescharacteristic
422#124—54no—
423#124—52no—
424#124—54no—
425#123—54no—
426#123—54no—
427#123—54no—
428#123—54no—
429#123C2×AGL(1,𝔽5)14yes—
430C22⋊Q8—52no—
431C22⋊Q8—54no—
432C22⋊Q8—52no—
433C22⋊Q8—54no—
434C22⋊Q8—54no—
435#121—52no—
436#121—54no—
437#121—54no—
438#121C2×AGL(1,𝔽5)12yes—
439#121C2×AGL(1,𝔽5)14yes—
440#120—54no—
441C4×Q8—52no—
442C4×Q8—54no—
443C4×Q8—52no—
444C4×D4—52no—
445C4×D4—54no—
446C4×D4—54no—
447C4×D4—54no—
448C4×D4—54no—
449C4×D4—54no—
450C4×D4C2×AGL(1,𝔽5)12yes—
451#117—51no—
452#117—52no—
453#117—52no—
454#117—52no—
455#117—54no—
456#117—51no—
457#117—51no—
458#117C2×AGL(1,𝔽5)11yescharacteristic
459C2×C4⋊C4—52no—
460C2×C22⋊C4—52no—
461C2×C42—54no—
462C2×C42—51no—
463C2×C42—51no—
464C4⋊C8—54no—
465C4⋊C8—54no—
466C4⋊C8—54no—
467C4⋊C8—54no—
468C4⋊C8—54no—
469C4⋊C8—54no—
470C22⋊C8—54no—
471C22⋊C8—54no—
472C22⋊C8—54no—
473C22⋊C8—54no—
474C22⋊C8—54no—
475C22⋊C8—54no—
476#97—52no—
477#97—52no—
478#97—54no—
479#97—52no—
480#97—52no—
481#97—54no—
482#97—54no—
483C4×C8—52no—
484C4×C8—52no—
485C4×C8—52no—
486C4×C8—52no—
487C4×C8—54no—
488C4×C8—54no—
489C4×C8—54no—
490C22×C5—24no—
491C22×C5—21no—
492C22×C5—24no—
493C22×C5—24no—
494C22×C5—22no—
495C22×C5C24×C411yescharacteristic
496C2×D5—22no—
497C2×D5—21no—
498C2×D5—22no—
499C2×D5—44no—
500C2×D5C4×#4112yes—
501C4×C5—21no—
502C4×C5—24no—
503C4×C5—22no—
504C4×C5—22no—
505C4×C5C4×#4112yes—
506Dic5—24no—
507Dic5—21no—
508Dic5—22no—
509Dic5—24no—
510Dic5—22no—
511Dic5C22×#4111yescharacteristic
512Dic5C22×#4111yescharacteristic
513C24—51no—
514#41—104no—
515#41—104no—
516C2×Q8—52no—
517C2×Q8—54no—
518C2×D4—54no—
519C2×D4—54no—
520C2×D4C22×AGL(1,𝔽5)12yes—
521C2×D4C22×AGL(1,𝔽5)14yes—
522C2×D4C22×AGL(1,𝔽5)14yes—
523C22×C4—51no—
524C22×C4—54no—
525C22×C4—51no—
526C22×C4—54no—
527C22×C4—51no—
528C22×C4—51no—
529C22×C4—51no—
530C22×C4—52no—
531C22×C4—52no—
532C22×C4—102no—
533C22×C4—102no—
534C22×C4C22×AGL(1,𝔽5)11yescharacteristic
535C22×C4C22×AGL(1,𝔽5)14yes—
536M16—102no—
537M16—102no—
538M16—102no—
539M16—102no—
540M16—108no—
541M16—108no—
542M16—104no—
543M16—104no—
544C2×C8—51no—
545C2×C8—51no—
546C2×C8—52no—
547C2×C8—54no—
548C2×C8—54no—
549C2×C8—52no—
550C2×C8—51no—
551C2×C8—51no—
552C2×C8—54no—
553C2×C8—54no—
554C2×C8—104no—
555C2×C8—104no—
556C2×C8—108no—
557C2×C8—108no—
558C2×C8—102no—
559C2×C8—102no—
560C4⋊C4—52no—
561C4⋊C4—52no—
562C4⋊C4—52no—
563C4⋊C4—52no—
564C4⋊C4—54no—
565C4⋊C4—54no—
566C4⋊C4—54no—
567C4⋊C4—54no—
568C4⋊C4—54no—
569C4⋊C4—54no—
570C4⋊C4—54no—
571C4⋊C4C22×AGL(1,𝔽5)12yes—
572C4⋊C4C22×AGL(1,𝔽5)12yes—
573C4⋊C4C22×AGL(1,𝔽5)14yes—
574C22⋊C4—52no—
575C22⋊C4—54no—
576C22⋊C4—54no—
577C22⋊C4—54no—
578C22⋊C4—54no—
579C22⋊C4—54no—
580C22⋊C4—52no—
581C22⋊C4—52no—
582C22⋊C4—54no—
583C22⋊C4—54no—
584C22⋊C4C22×AGL(1,𝔽5)12yes—
585C22⋊C4C22×AGL(1,𝔽5)14yes—
586C22⋊C4C22×AGL(1,𝔽5)14yes—
587C42—52no—
588C42—52no—
589C42—54no—
590C42—51no—
591C42—51no—
592C42—54no—
593C42—54no—
594C42C22×AGL(1,𝔽5)11yescharacteristic
595C42C22×AGL(1,𝔽5)11yescharacteristic
596C2×C5—41no—
597C2×C5—44no—
598C2×C5C2×C4×#4111yescharacteristic
599C2×C5#357312yes—
600D5—22no—
601D5—41no—
602C23—51no—
603C23—51no—
604C23—102no—
605C23—202no—
606C23C23×AGL(1,𝔽5)11yescharacteristic
607C23C23×AGL(1,𝔽5)14yes—
608Q8—104no—
609Q8—204no—
610D4—24no—
611D4—44no—
612D4—44no—
613D4—108no—
614D4—204no—
615C2×C4—22no—
616C2×C4—44no—
617C2×C4—51no—
618C2×C4—52no—
619C2×C4—52no—
620C2×C4—52no—
621C2×C4—52no—
622C2×C4—54no—
623C2×C4—54no—
624C2×C4—51no—
625C2×C4—101no—
626C2×C4—101no—
627C2×C4—102no—
628C2×C4—101no—
629C2×C4—104no—
630C2×C4—104no—
631C2×C4—101no—
632C2×C4—102no—
633C2×C4—202no—
634C2×C4—202no—
635C2×C4—204no—
636C2×C4C23×AGL(1,𝔽5)12yes—
637C2×C4C23×AGL(1,𝔽5)14yes—
638C2×C4C23×AGL(1,𝔽5)12yes—
639C2×C4C23×AGL(1,𝔽5)11yescharacteristic
640C2×C4C23×AGL(1,𝔽5)11yescharacteristic
641C8—54no—
642C8—52no—
643C8—52no—
644C8—102no—
645C8—104no—
646C8—104no—
647C8—101no—
648C8—101no—
649C5#6020911yescharacteristic, complemented, Sylow
650C22—21no—
651C22—24no—
652C22—24no—
653C22—24no—
654C22—22no—
655C22—52no—
656C22—102no—
657C22—101no—
658C22—102no—
659C22—204no—
660C22C24×AGL(1,𝔽5)11yescharacteristic
661C4—21no—
662C4—22no—
663C4—24no—
664C4—22no—
665C4—51no—
666C4—51no—
667C4—101no—
668C4—104no—
669C4—102no—
670C4—104no—
671C4—102no—
672C4#41×AGL(1,𝔽5)12yes—
673C2—41no—
674C2—44no—
675C2—102no—
676C2—201no—
677C2C2×#41×AGL(1,𝔽5)11yescharacteristic
678C2#1061941312yes—
subgroupG/H
center, FrattiniC22C24×AGL(1,𝔽5)
commutator, socleC22×C5C24×C4
FittingC5×#540C4
prime radical#10618368C2
cosocleC2×Dic5C25
hypercenter#540AGL(1,𝔽5)
hypocenterC5#60209
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ #540

Complements

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

NH ≅ G/Ntype
2. #10617362C2semidirect
3. #10617361C2semidirect
4. #10617330C2semidirect
5. #10617329C2semidirect
6. #10617295C2semidirect
7. #10617294C2semidirect
8. #10617236C2semidirect
9. #10617235C2semidirect
10. #10617226C2semidirect
11. #10617225C2semidirect
12. #10617140C2semidirect
13. #10617136C2semidirect
14. #10617110C2semidirect
15. #10617105C2semidirect
38. #65774C22semidirect
41. #65773C22semidirect
42. #65773C22semidirect
44. D4×#170C22semidirect
46. #65754C22semidirect
47. C4×#704C22semidirect
48. C2×#4002C22semidirect
49. #65748C22semidirect
50. #65709C22semidirect
53. #65707C22semidirect
54. #65707C22semidirect
56. #65703C22semidirect
62. #65697C22semidirect
63. #65697C22semidirect
70. #65691C22semidirect
71. #65691C22semidirect
649. C5#60209semidirect

Maximal subgroups

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

subgroupindex
#106183682
#106173622
#106173612
#106173302
#106173292
#106172952
#106172942
#106172362
#106172352
#106172262
#106172252
#106171402
#106171362
#106171102
#106171052
#602095