Finite Group Databasebeta

← #13044898

Subgroups — #13044898

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#10618396C211yesmaximal, characteristic
2#10617372C211yesmaximal, characteristic, complemented
3#10617371C211yesmaximal, characteristic, complemented
4#10617350C214yesmaximal, complemented
5#10617349C214yesmaximal, complemented
6#10617333C214yesmaximal, complemented
7#10617332C214yesmaximal, complemented
8#10617330C211yesmaximal, characteristic, complemented
9#10617329C211yesmaximal, characteristic, complemented
10#10617258C212yesmaximal, complemented
11#10617257C212yesmaximal, complemented
12#10617188C212yesmaximal, complemented
13C4×#4133C212yesmaximal, complemented
14#10617110C211yesmaximal, characteristic, complemented
15#10617105C211yesmaximal, characteristic, complemented
16C2×#4133C2212yes—
17C2×#4133C2212yes—
18C2×D5⋊M16C2211yescharacteristic
19C22×D5⋊C8C2211yescharacteristic
20C5×#541C411yescharacteristic
21#66065C411yescharacteristic
22#66059C2211yescharacteristic
23#66040C2214yes—
24#66038C414yes—
25#66032C411yescharacteristic
26#66026C414yes—
27#66025C2214yes—
28#66022C2211yescharacteristic
29#65977C412yes—
30#65961C2212yes—
31#65916C412yes—
32C4×#713C2212yes—
33#65869C411yescharacteristic
34D5×#117C2211yescharacteristic
35#65787C2212yes—
36#65787C2212yes—
37#65756C2212yes—
38#65709C2214yescomplemented
39#65708C2212yes—
40#65708C2212yes—
41#65707C2214yescomplemented
42#65707C2214yescomplemented
43#65706C2214yes—
44#65706C2214yescomplemented
45#65706C2214yescomplemented
46#65705C2214yes—
47#65705C2214yescomplemented
48#65704C2214yes—
49#65704C2214yes—
50#65704C2214yescomplemented
51#65704C2214yescomplemented
52#65703C2214yescomplemented
53#65702C2214yes—
54#65702C2214yescomplemented
55#65701C2212yes—
56#65701C2212yes—
57#65701C2212yes—
58#65701C2212yes—
59#65701C2214yescomplemented
60#65701C2214yescomplemented
61#65697C2214yes—
62#65695C2214yes—
63#65694C2214yes—
64#65693C2214yes—
65#65692C2214yes—
66#65691C2212yes—
67#65691C2212yes—
68#65684C2214yescomplemented
69#65683C2214yescomplemented
70#65682C2211yescharacteristic
71#65681C2211yescharacteristic
72C4×#700C2212yes—
73C4×#700C2214yescomplemented
74#65676C2211yescharacteristic
75#65676C2211yescharacteristic
76C4×D5⋊C8C2211yescharacteristic
77C4×D5⋊C8C2211yescharacteristic
78C4×D5⋊C8C2214yescomplemented
79#60206—51nomaximal, Sylow
80C2×#713C2312yes—
81C22×C4×D5C2311yescharacteristic
82C2×#704C2312yes—
83C22×#170C2311yescharacteristic
84C22×#170C2311yescharacteristic
85#4133—28no—
86#4133—28no—
87D5⋊M16—24no—
88C2×#700C2312yes—
89C2×#700C2312yes—
90C2×#700C2314yes—
91C2×#700C2314yes—
92C2×#700C2312yes—
93C2×D5⋊C8—24no—
94C2×D5⋊C8C2311yescharacteristic
95C2×D5⋊C8C2314yes—
96C2×D5⋊C8C2314yes—
97C2×D5⋊C8C2311yescharacteristic
98C2×D5⋊C8C2311yescharacteristic
99C2×D5⋊C8C2311yescharacteristic
100C2×D5⋊C8C2311yescharacteristic
101C2×D5⋊C8C2311yescharacteristic
102C5×C4⋊Q8C2×C411yescharacteristic
103C5×#126C2×C414yes—
104C5×#125C2×C411yescharacteristic
105C5×#125C2×C414yes—
106C5×C22⋊Q8C2×C412yes—
107C4×C5×Q8C2×C412yes—
108C5×#117C2×C411yescharacteristic
109#4093C2×C412yes—
110#4092C2×C412yes—
111Q8×Dic5C2×C412yes—
112#4076C2×C412yes—
113#4072C2×C411yescharacteristic
114#4044C2×C414yes—
115#4044C2×C414yes—
116#4043C2×C414yes—
117#4042C2×C414yes—
118#4042C2×C414yes—
119#4041C2×C414yes—
120#4041C2×C412yes—
121#4041C2×C414yes—
122#4040C2×C414yes—
123#4040C2×C414yes—
124#4040C2×C414yes—
125#4040C2×C414yes—
126#4039C2×C412yes—
127#4039C2×C414yes—
128#4039C2314yes—
129#4039C2314yes—
130#4038C2312yes—
131#4038C2312yes—
132#4038C2314yes—
133D5×C4⋊C4C2×C412yes—
134#4035C2×C414yes—
135#4034C2×C412yes—
136#4033C2314yes—
137#4031C2×C414yes—
138#4030C2×C414yes—
139#4029C2×C414yes—
140#4028C2×C414yes—
141#4027C2314yes—
142D5×C22⋊C4C2×C412yes—
143#4023C2312yes—
144#4022C2×C414yes—
145#4021C2×C414yes—
146#4020C2×C411yescharacteristic
147C4×D20C2×C412yes—
148C4×D20C2×C414yes—
149#4018C2×C411yescharacteristic
150#4018C2×C411yescharacteristic
151C42×D5C2311yescharacteristic
152C42×D5C2311yescharacteristic
153C42×D5C2314yes—
154#4016C2×C411yescharacteristic
155C22⋊#170C2314yes—
156#4004C2314yes—
157#4004C2314yes—
158#4004C2314yes—
159#4003C2314yes—
160#4003C2314yes—
161#4003C2314yes—
162#4003C2314yes—
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#2967—54no—
183#2967—54no—
184#2966—54no—
185#2966—54no—
186#2961—51no—
187#2960—51no—
188#2932—51no—
189#2931—51no—
190#2930—52no—
191#2929—52no—
192#2923—52no—
193#2921—51no—
194C4×#131—52no—
195C2×#405—51no—
196C23×D5C22×C411yescharacteristic
197C2×C5×Q8C22×C412yes—
198C22×C4×C5C22×C411yescharacteristic
199C2×C22⋊D5C22×C414yes—
200C22×Dic5C2411yescharacteristic
201#713—24no—
202#713—24no—
203C2×D20C22×C412yes—
204C2×D20C22×C414yes—
205C2×D20C22×C414yes—
206C2×D20C22×C414yes—
207C2×C4×D5—22no—
208C2×C4×D5—22no—
209C2×C4×D5C22×C411yescharacteristic
210C2×C4×D5C22×C411yescharacteristic
211C2×C4×D5C22×C412yes—
212C2×C4×D5C22×C414yes—
213C2×C4×D5C22×C412yes—
214C2×C4×D5C22×C414yes—
215C2×C4×D5C2411yescharacteristic
216C2×C4×D5C2414yes—
217C2×C4×D5C2411yescharacteristic
218C2×Dic10C22×C414yes—
219#704—24no—
220C2×#170—22no—
221C2×#170—22no—
222C2×#170C2411yescharacteristic
223C2×#170C2411yescharacteristic
224C2×#170C2414yes—
225C2×#170C2414yes—
226C2×#170C2412yes—
227C2×#170C2411yescharacteristic
228C2×#170C2412yes—
229C2×#170C2411yescharacteristic
230#700—24no—
231#700—24no—
232#700—24no—
233#700—28no—
234#700—28no—
235D5⋊C8—22no—
236D5⋊C8—22no—
237D5⋊C8—22no—
238D5⋊C8—22no—
239D5⋊C8—28no—
240D5⋊C8—28no—
241D5⋊C8#4114yes—
242D5⋊C8#4114yes—
243C5×C4⋊C4C22×C412yes—
244C5×C4⋊C4C22×C412yes—
245C5×C4⋊C4C22×C414yes—
246C5×C4⋊C4C22×C414yes—
247C5×C4⋊C4C22×C414yes—
248C5×C4⋊C4C22×C414yes—
249C5×C22⋊C4C22×C412yes—
250C5×C22⋊C4C22×C414yes—
251C42×C5C22×C411yescharacteristic
252C42×C5C22×C411yescharacteristic
253C42×C5C22×C414yes—
254C22⋊Dic5C22×C412yes—
255#685C22×C412yes—
256#685C22×C412yes—
257#685C22×C414yes—
258#685C22×C414yes—
259#685C22×C414yes—
260#685C22×C414yes—
261#685C22×C414yes—
262#685C22×C414yes—
263#685C22×C414yes—
264#685C22×C414yes—
265C4⋊Dic5C22×C412yes—
266C4⋊Dic5C22×C414yes—
267C4⋊Dic5C22×C412yes—
268#683C22×C414yes—
269#683C22×C412yes—
270#683C22×C412yes—
271#683C22×C414yes—
272#683C22×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#541AGL(1,𝔽5)11yescharacteristic
287#540—51no—
288#540—51no—
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#432—54no—
319#432—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×C4×C8—51no—
340C2×C4×C8—51no—
341C2×C4×C8—54no—
342C23×C5C23×C411yescharacteristic
343C22×D5—22no—
344C22×D5—42no—
345C22×D5C23×C411yescharacteristic
346C22×D5C23×C414yes—
347C22×D5C23×C411yescharacteristic
348C5×Q8—24no—
349C2×C4×C5—22no—
350C2×C4×C5C23×C414yes—
351C2×C4×C5C23×C412yes—
352C2×C4×C5C23×C412yes—
353C2×C4×C5C23×C414yes—
354C2×C4×C5C23×C411yescharacteristic
355C2×C4×C5C23×C411yescharacteristic
356C22⋊D5—44no—
357C2×Dic5—21no—
358C2×Dic5—21no—
359C2×Dic5C23×C412yes—
360C2×Dic5C23×C414yes—
361C2×Dic5C23×C412yes—
362C2×Dic5C23×C411yescharacteristic
363C2×Dic5C2511yescharacteristic
364D20—28no—
365D20—24no—
366D20—44no—
367D20—44no—
368C4×D5—22no—
369C4×D5—22no—
370C4×D5—21no—
371C4×D5—24no—
372C4×D5—24no—
373C4×D5—21no—
374C4×D5—42no—
375C4×D5—44no—
376C4×D5—44no—
377C4×D5—42no—
378C4×D5C2×#4112yes—
379C4×D5C2×#4112yes—
380Dic10—44no—
381#170—21no—
382#170—21no—
383#170—24no—
384#170—24no—
385#170—22no—
386#170C2×#4112yes—
387#170C2×#4114yes—
388#170C2×#4112yes—
389C2×#41—52no—
390C23×C4—51no—
391#131—108no—
392#131—108no—
393C2×M16—52no—
394C2×M16—52no—
395C2×M16—52no—
396C2×M16—54no—
397C2×M16—54no—
398C2×M16—52no—
399C2×M16—104no—
400C22×C8—51no—
401C22×C8—51no—
402C22×C8—51no—
403C22×C8—51no—
404C22×C8—51no—
405C22×C8—51no—
406C22×C8—54no—
407C22×C8—54no—
408C22×C8—51no—
409C22×C8—51no—
410C22×C8—104no—
411C4⋊Q8—52no—
412C4⋊Q8C2×AGL(1,𝔽5)11yescharacteristic
413#127—51no—
414#126—54no—
415#126—54no—
416#126—54no—
417#126C2×AGL(1,𝔽5)14yes—
418#125—54no—
419#125—51no—
420#125C2×AGL(1,𝔽5)14yes—
421#125C2×AGL(1,𝔽5)11yescharacteristic
422#124—52no—
423#124—54no—
424#124—54no—
425#123—54no—
426#123—54no—
427#123—54no—
428#123—54no—
429#123—54no—
430C22⋊Q8—52no—
431C22⋊Q8—54no—
432C22⋊Q8—54no—
433C22⋊Q8—54no—
434C22⋊Q8C2×AGL(1,𝔽5)12yes—
435#121—52no—
436#121—52no—
437#121—54no—
438#121—54no—
439#121—54no—
440#120—54no—
441C4×Q8—52no—
442C4×Q8—54no—
443C4×Q8C2×AGL(1,𝔽5)12yes—
444C4×D4—54no—
445C4×D4—52no—
446C4×D4—54no—
447C4×D4—54no—
448C4×D4—54no—
449C4×D4—54no—
450C4×D4—52no—
451#117—51no—
452#117—52no—
453#117—52no—
454#117—54no—
455#117—52no—
456#117—51no—
457#117—51no—
458#117C2×AGL(1,𝔽5)11yescharacteristic
459C2×C4⋊C4—52no—
460C2×C22⋊C4—52no—
461C2×C42—51no—
462C2×C42—51no—
463C2×C42—54no—
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—52no—
479#97—52no—
480#97—54no—
481#97—54no—
482#97—54no—
483C4×C8—52no—
484C4×C8—52no—
485C4×C8—54no—
486C4×C8—52no—
487C4×C8—54no—
488C4×C8—52no—
489C4×C8—54no—
490C22×C5—21no—
491C22×C5—22no—
492C22×C5C24×C411yescharacteristic
493C2×D5—22no—
494C2×D5—21no—
495C2×D5—24no—
496C2×D5—24no—
497C2×D5—24no—
498C2×D5—22no—
499C2×D5—44no—
500C2×D5C4×#4112yes—
501C4×C5—21no—
502C4×C5—24no—
503C4×C5—24no—
504C4×C5—22no—
505C4×C5—22no—
506C4×C5C4×#4112yes—
507Dic5—21no—
508Dic5—22no—
509Dic5—24no—
510Dic5—22no—
511Dic5C22×#4111yescharacteristic
512Dic5C22×#4111yescharacteristic
513C24—51no—
514#41—104no—
515#41—104no—
516C2×Q8—54no—
517C2×Q8C22×AGL(1,𝔽5)12yes—
518C2×D4—54no—
519C2×D4—54no—
520C2×D4—52no—
521C2×D4—54no—
522C2×D4—54no—
523C22×C4—51no—
524C22×C4—51no—
525C22×C4—54no—
526C22×C4—51no—
527C22×C4—51no—
528C22×C4—51no—
529C22×C4—52no—
530C22×C4—52no—
531C22×C4—54no—
532C22×C4—54no—
533C22×C4—102no—
534C22×C4—102no—
535C22×C4C22×AGL(1,𝔽5)11yescharacteristic
536M16—104no—
537M16—104no—
538M16—108no—
539M16—108no—
540M16—104no—
541M16—104no—
542C2×C8—51no—
543C2×C8—52no—
544C2×C8—51no—
545C2×C8—54no—
546C2×C8—54no—
547C2×C8—54no—
548C2×C8—54no—
549C2×C8—52no—
550C2×C8—51no—
551C2×C8—51no—
552C2×C8—102no—
553C2×C8—102no—
554C2×C8—102no—
555C2×C8—102no—
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⋊C4C22×AGL(1,𝔽5)12yes—
569C4⋊C4C22×AGL(1,𝔽5)12yes—
570C4⋊C4C22×AGL(1,𝔽5)14yes—
571C4⋊C4C22×AGL(1,𝔽5)14yes—
572C4⋊C4C22×AGL(1,𝔽5)14yes—
573C4⋊C4C22×AGL(1,𝔽5)14yes—
574C22⋊C4—52no—
575C22⋊C4—52no—
576C22⋊C4—52no—
577C22⋊C4—54no—
578C22⋊C4—54no—
579C22⋊C4—54no—
580C22⋊C4—54no—
581C22⋊C4—54no—
582C22⋊C4—54no—
583C22⋊C4—54no—
584C22⋊C4—54no—
585C22⋊C4C22×AGL(1,𝔽5)12yes—
586C22⋊C4C22×AGL(1,𝔽5)14yes—
587C42—52no—
588C42—52no—
589C42—54no—
590C42—51no—
591C42—51no—
592C42—54no—
593C42C22×AGL(1,𝔽5)11yescharacteristic
594C42C22×AGL(1,𝔽5)11yescharacteristic
595C42C22×AGL(1,𝔽5)14yes—
596C2×C5—41no—
597C2×C5C2×C4×#4111yescharacteristic
598C2×C5#357312yes—
599D5—22no—
600D5—44no—
601D5—41no—
602C23—51no—
603C23—51no—
604C23—54no—
605C23—102no—
606C23—202no—
607C23C23×AGL(1,𝔽5)11yescharacteristic
608Q8—24no—
609Q8—204no—
610D4—108no—
611D4—104no—
612D4—204no—
613D4—204no—
614D4—204no—
615C2×C4—22no—
616C2×C4—51no—
617C2×C4—52no—
618C2×C4—52no—
619C2×C4—52no—
620C2×C4—52no—
621C2×C4—54no—
622C2×C4—51no—
623C2×C4—101no—
624C2×C4—104no—
625C2×C4—104no—
626C2×C4—101no—
627C2×C4—102no—
628C2×C4—101no—
629C2×C4—101no—
630C2×C4—102no—
631C2×C4—202no—
632C2×C4—204no—
633C2×C4—202no—
634C2×C4—204no—
635C2×C4C23×AGL(1,𝔽5)12yes—
636C2×C4C23×AGL(1,𝔽5)14yes—
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—52no—
642C8—52no—
643C8—54no—
644C8—101no—
645C8—101no—
646C8—104no—
647C8—104no—
648C8—102no—
649C5#6020611yescharacteristic, complemented, Sylow
650C22—21no—
651C22—22no—
652C22—52no—
653C22—102no—
654C22—101no—
655C22—104no—
656C22—104no—
657C22—104no—
658C22—102no—
659C22—204no—
660C22C24×AGL(1,𝔽5)11yescharacteristic
661C4—21no—
662C4—24no—
663C4—22no—
664C4—22no—
665C4—24no—
666C4—51no—
667C4—51no—
668C4—101no—
669C4—102no—
670C4—104no—
671C4—102no—
672C4#41×AGL(1,𝔽5)12yes—
673C2—41no—
674C2—102no—
675C2—201no—
676C2—204no—
677C2C2×#41×AGL(1,𝔽5)11yescharacteristic
678C2#1061941512yes—
subgroupG/H
center, FrattiniC22C24×AGL(1,𝔽5)
commutator, socleC22×C5C24×C4
FittingC5×#541C4
prime radical#10618396C2
cosocleC2×Dic5C25
hypercenter#541AGL(1,𝔽5)
hypocenterC5#60206
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ #541

Complements

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

NH ≅ G/Ntype
2. #10617372C2semidirect
3. #10617371C2semidirect
4. #10617350C2semidirect
5. #10617349C2semidirect
6. #10617333C2semidirect
7. #10617332C2semidirect
8. #10617330C2semidirect
9. #10617329C2semidirect
10. #10617258C2semidirect
11. #10617257C2semidirect
12. #10617188C2semidirect
13. C4×#4133C2semidirect
14. #10617110C2semidirect
15. #10617105C2semidirect
38. #65709C22semidirect
41. #65707C22semidirect
42. #65707C22semidirect
44. #65706C22semidirect
45. #65706C22semidirect
47. #65705C22semidirect
50. #65704C22semidirect
51. #65704C22semidirect
52. #65703C22semidirect
54. #65702C22semidirect
59. #65701C22semidirect
60. #65701C22semidirect
68. #65684C22semidirect
69. #65683C22semidirect
73. C4×#700C22semidirect
78. C4×D5⋊C8C22semidirect
649. C5#60206semidirect

Maximal subgroups

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

subgroupindex
#106183962
#106173722
#106173712
#106173502
#106173492
#106173332
#106173322
#106173302
#106173292
#106172582
#106172572
#106171882
C4×#41332
#106171102
#106171052
#602065