Finite Group Databasebeta

← #28252040

Subgroups — #28252040

Order 13568.

2184 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#20288384C211yesmaximal, characteristic
2#20284704C213yesmaximal, complemented
3#20284693C213yesmaximal, complemented
4#20284673C213yesmaximal, complemented
5#20284673C213yesmaximal, complemented
6#20284655C213yesmaximal, complemented
7#20284646C213yesmaximal, complemented
8#20284628C213yesmaximal, complemented
9C2×#16810947C213yesmaximal, complemented
10#20283313C211yesmaximal, characteristic, complemented
11#20283308C211yesmaximal, characteristic, complemented
12#20283238C211yesmaximal, characteristic, complemented
13#20283238C211yesmaximal, characteristic, complemented
14#20283164C211yesmaximal, characteristic, complemented
15C53×#3428C211yesmaximal, characteristic, complemented
16C2×#13133263C2211yescharacteristic
17C2×#13133224C2211yescharacteristic
18C2×#13133224C2211yescharacteristic
19#16811292—21no—
20#16811284—21no—
21#16811261—23no—
22D4×C22⋊D53—23no—
23#16811201C2213yes—
24#16811187C2213yes—
25#16811134C2213yes—
26C2×#13133191C2213yes—
27#16811106C2211yescharacteristic
28#16810964C2216yescomplemented
29#16810964C2211yescharacteristic, complemented
30#16810964C2213yescomplemented
31#16810958C2213yescomplemented
32#16810953C2213yescomplemented
33#16810947C2216yescomplemented
34#16810947C2216yescomplemented
35#16810947C2216yescomplemented
36#16810947C2213yescomplemented
37#16810947C2216yescomplemented
38#16810947C2213yescomplemented
39#16810941C2213yescomplemented
40#16810941C2216yescomplemented
41#16810936C2213yescomplemented
42#16810936C2216yescomplemented
43#16810666C2213yescomplemented
44#16810666C2213yescomplemented
45#16810663C2213yescomplemented
46#16810663C2213yescomplemented
47#16810659C2213yescomplemented
48#16810659C2213yescomplemented
49C2×#13133150C2213yescomplemented
50C2×#13133148C2213yescomplemented
51#16810627C2213yescomplemented
52#16810627C2213yescomplemented
53#16810622C2213yescomplemented
54#16810622C2213yescomplemented
55#16810617C2213yescomplemented
56#16810617C2213yescomplemented
57C2×#13133137C2213yescomplemented
58C2×#13133137C2213yescomplemented
59#16810590C2213yescomplemented
60#16810590C2213yescomplemented
61#16810587C2213yescomplemented
62C2×C22⋊#88322C2213yescomplemented
63C22×#41×C53C2211yescharacteristic
64C53×#520C2211yescharacteristic
65C53×#520C2211yescharacteristic
66C53×#520C2213yescomplemented
67C53×#520C2213yescomplemented
68C53×#519—21no—
69C53×#518—23no—
70C4×#41×C53—23no—
71C4×#41×C53—21no—
72C2×#117×C53C2213yescomplemented
73C2×#117×C53C2213yescomplemented
74#13133263—24no—
75#13133263—24no—
76C2×#11719949—21no—
77C2×#11719945C2313yes—
78C22×D212C2313yes—
79C22×C22⋊D53—23no—
80C2×D4×D53—23no—
81#13133224—24no—
82#13133224—24no—
83C2×#11719917C2313yes—
84C2×#11719917C2313yes—
85C22×#88322C2313yes—
86C22×#88322C2313yes—
87#13133213—23no—
88#13133206—23no—
89#13133200—23no—
90#13133196C2311yescharacteristic
91#13133196C2313yes—
92#13133193—23no—
93#13133191—23no—
94#13133191—23no—
95#13133191C2313yes—
96#13133191C2316yes—
97#13133191C2313yes—
98#13133189—26no—
99#13133188—23no—
100#13133186—23no—
101#13133185—26no—
102Q8×Dic53—21no—
103D4×Dic53—23no—
104C4×C22⋊D53—23no—
105C4×C22⋊D53—23no—
106#13133176C2313yes—
107C2×C4⋊Dic53C2313yes—
108C2×#11719921—23no—
109#13133150C2313yes—
110#13133150C2316yes—
111#13133150C2311yescharacteristic
112#13133150C2313yes—
113#13133150C2313yes—
114#13133148C2313yes—
115#13133148C2316yes—
116#13133148C2311yescharacteristic
117#13133148C2313yes—
118#13133148C2313yes—
119#13133140C2311yescharacteristic
120#13133140C2311yescharacteristic
121#13133140C2313yes—
122#13133140C2313yes—
123#13133137C2313yes—
124#13133137C2313yes—
125#13133137C2316yes—
126#13133137C2316yes—
127#13133137C2313yes—
128#13133137C2313yes—
129C22⋊#88322C2311yescharacteristic
130C22⋊#88322C2313yes—
131C22⋊#88322C2313yes—
132C22⋊#88322C2313yes—
133C22⋊#88322C2316yes—
134C2×#41×C53—24no—
135C2×#41×C53—24no—
136C2×#41×C53D411yescharacteristic
137C2×#41×C53D413yes—
138C2×#41×C53C2313yes—
139C2×#41×C53C2311yescharacteristic
140C22×Q8×C53D411yescharacteristic
141C22×D4×C53D413yes—
142C23×C4×C53C2313yes—
143C4×Q8×C53—21no—
144C4×Q8×C53—21no—
145C4×Q8×C53—23no—
146C4×Q8×C53—23no—
147C4×D4×C53—23no—
148C4×D4×C53—23no—
149C4×D4×C53—26no—
150C4×D4×C53—26no—
151C4×D4×C53—23no—
152C4×D4×C53—23no—
153#117×C53—23no—
154#117×C53—23no—
155#117×C53—23no—
156#117×C53—23no—
157#117×C53C2313yes—
158#117×C53C2311yescharacteristic
159#117×C53C2313yes—
160#117×C53C2313yes—
161#117×C53C2316yes—
162C2×C4⋊C4×C53C2313yes—
163C2×C4⋊C4×C53C2313yes—
164C2×C4⋊C4×C53C2313yes—
165C2×C4⋊C4×C53C2313yes—
166C2×C22⋊C4×C53—23no—
167C2×C22⋊C4×C53—23no—
168C2×C42×C53—23no—
169C2×C42×C53—23no—
170C24×C53C2×D413yes—
171C23×D53—23no—
172#41×C53—24no—
173#41×C53—212no—
174#41×C53—26no—
175#41×C53—26no—
176#41×C53#4114yes—
177#41×C53#4114yes—
178C2×Q8×C53—24no—
179C2×Q8×C53C2×D413yes—
180C2×Q8×C53C2×D411yescharacteristic
181C2×D4×C53—212no—
182C2×D4×C53C2×D416yes—
183C2×D4×C53C2×D413yes—
184C2×D4×C53C2×D413yes—
185C22×C4×C53—26no—
186C22×C4×C53—26no—
187C22×C4×C53—23no—
188C22×C4×C53—23no—
189C22×C4×C53—23no—
190C22×C4×C53—23no—
191C22×C4×C53C2×D413yes—
192C22×C4×C53C2×D413yes—
193C22×C4×C53C2×D413yes—
194C22×C4×C53C2413yes—
195C22×C4×C53C2411yescharacteristic
196C22×C4×C53C2413yes—
197C2×C22⋊D53—23no—
198C2×C22⋊D53—23no—
199C2×C22⋊D53—23no—
200C2×C22⋊D53—26no—
201C2×C22⋊D53—86no—
202C22×Dic53—23no—
203#11719949—44no—
204D4×D53—412no—
205#11719945—26no—
206#11719945—26no—
207C2×D212—26no—
208C2×D212—26no—
209C2×D212—23no—
210C2×D212C2413yes—
211C2×C4×D53—23no—
212C2×Dic106C2411yescharacteristic
213C4⋊C4×C53—23no—
214C4⋊C4×C53—23no—
215C4⋊C4×C53—23no—
216C4⋊C4×C53—23no—
217C4⋊C4×C53C2413yes—
218C4⋊C4×C53C2411yescharacteristic
219C4⋊C4×C53C2413yes—
220C4⋊C4×C53C2411yescharacteristic
221C22⋊C4×C53—23no—
222C22⋊C4×C53—21no—
223C22⋊C4×C53—26no—
224C22⋊C4×C53—23no—
225C22⋊C4×C53—23no—
226C42×C53—23no—
227C42×C53—21no—
228C42×C53—26no—
229C42×C53—23no—
230C42×C53—23no—
231C22⋊Dic53—23no—
232C22⋊Dic53—21no—
233C22⋊Dic53—23no—
234#11719921—23no—
235#11719921—26no—
236#11719921—23no—
237C4⋊Dic53—23no—
238C4⋊Dic53C2413yes—
239C4⋊Dic53C2411yescharacteristic
240#11719919—21no—
241#11719919—23no—
242C4×Dic53—23no—
243#11719917—26no—
244#11719917—26no—
245C2×#88322—26no—
246C2×#88322—26no—
247C2×#88322C2411yescharacteristic
248C2×#88322C2413yes—
249C2×#88322C2413yes—
250C2×#88322C2411yescharacteristic
251C23×C53—26no—
252C23×C53—43no—
253C23×C53—43no—
254C23×C53C22×D413yes—
255C23×C53C22×D411yescharacteristic
256C23×C53C22×D413yes—
257C22×D53—23no—
258C22×D53—46no—
259C22×D53—83no—
260C22×D53—83no—
261Q8×C53—26no—
262Q8×C53#12314yes—
263D4×C53—212no—
264D4×C53—26no—
265D4×C53#123112yes—
266C2×C4×C53—26no—
267C2×C4×C53—26no—
268C2×C4×C53—26no—
269C2×C4×C53—21no—
270C2×C4×C53—21no—
271C2×C4×C53—23no—
272C2×C4×C53—23no—
273C2×C4×C53—21no—
274C2×C4×C53—23no—
275C2×C4×C53—21no—
276C2×C4×C53—23no—
277C2×C4×C53—83no—
278C2×C4×C53—83no—
279C2×C4×C53—83no—
280C2×C4×C53—83no—
281C2×C4×C53C22×D411yescharacteristic
282C2×C4×C53C22×D413yes—
283C2×C4×C53C22×D413yes—
284C2×C4×C53C2×#4116yes—
285C2×C4×C53C2×#4116yes—
286C2×C4×C53C2511yescharacteristic
287C22⋊D53—46no—
288C22⋊D53—46no—
289C22⋊D53—83no—
290C22⋊D53—86no—
291C2×Dic53—21no—
292C2×Dic53—23no—
293C2×Dic53—21no—
294C2×Dic53—83no—
295D212—23no—
296D212—23no—
297D212—46no—
298C4×D53—46no—
299Dic106—21no—
300Dic106—21no—
301#88322—21no—
302#88322—21no—
303#88322—23no—
304#88322—23no—
305#60523—531nomaximal, Sylow
306C22×C53—23no—
307C22×C53—23no—
308C22×C53—23no—
309C22×C53—21no—
310C22×C53—21no—
311C22×C53—23no—
312C22×C53—21no—
313C22×C53—23no—
314C22×C53—86no—
315C22×C53C2×#12316yes—
316C22×C53C23×D411yescharacteristic
317C2×D53—43no—
318C2×D53—43no—
319C2×D53—43no—
320C2×D53—166no—
321C4×C53—21no—
322C4×C53—23no—
323C4×C53—43no—
324C4×C53—41no—
325C4×C53—43no—
326C4×C53—41no—
327C4×C53C2×#12316yes—
328C4×C53C22×#4111yescharacteristic
329C4×C53C22×#4111yescharacteristic
330Dic53—41no—
331Dic53—43no—
332Dic53—41no—
333#3488—531no—
334#3428D5311yescharacteristic, complemented
335#3099—533no—
336#3098—533no—
337#3096—533no—
338#3096—533no—
339#3094—533no—
340#3093—533no—
341#3091—533no—
342C2×#482—533no—
343#2913—531no—
344#2912—531no—
345#2897—531no—
346#2897—531no—
347#2880—531no—
348C2×C53—26no—
349C2×C53—43no—
350C2×C53—41no—
351C2×C53C22×#12311yescharacteristic
352C2×C53#358411yescharacteristic
353C2×C53#358511yescharacteristic
354D53—83no—
355C2×H(5,𝔽2)—531no—
356C22×#41C2×D5311yescharacteristic
357C2×#131—531no—
358C2×#131—531no—
359#550—1061no—
360#548—1061no—
361#546—1063no—
362D42—1063no—
363#537—533no—
364#535—533no—
365#525—533no—
366C2×#121—533no—
367#520—531no—
368#520C2×D5311yescharacteristic
369#520C2×D5311yescharacteristic
370#520C2×D5313yescomplemented
371#520C2×D5313yescomplemented
372#519—21no—
373#518—23no—
374C4×#41—23no—
375C4×#41—21no—
376C2×#117C2×D5313yescomplemented
377C2×#117C2×D5313yescomplemented
378#485—531no—
379#485—533no—
380#485—536no—
381#484—533no—
382#483—533no—
383#482—533no—
384#482—536no—
385#482—536no—
386#482—536no—
387#482—533no—
388#482—536no—
389#481—533no—
390#481—536no—
391#480—533no—
392#480—536no—
393#428—533no—
394#428—533no—
395#428—533no—
396#428—533no—
397#427—533no—
398#427—533no—
399C2×#107—533no—
400C2×#106—533no—
401#418—533no—
402#418—533no—
403#417—533no—
404#417—533no—
405#416—533no—
406#416—533no—
407C2×#102—533no—
408C2×#102—533no—
409#408—533no—
410#408—533no—
411#407—533no—
412C2×C22⋊C8—533no—
413C53#6052311yescharacteristic, complemented, Sylow
414H(5,𝔽2)—1064no—
415H(5,𝔽2)—1064no—
416C2×#41—24no—
417C2×#41—24no—
418C2×#41—533no—
419C2×#41—1061no—
420C2×#41C22⋊D5313yes—
421C2×#41C22⋊D5311yescharacteristic
422C2×#41C22×D5313yes—
423C2×#41C22×D5311yescharacteristic
424C22×Q8C22⋊D5311yescharacteristic
425C22×D4—533no—
426C22×D4—1063no—
427C22×D4—1063no—
428C22×D4C22⋊D5313yes—
429C23×C4C22×D5313yes—
430#131—1064no—
431#131—1064no—
432C2×M16—533no—
433C2×M16—533no—
434C22×C8—533no—
435C22×C8—533no—
436#127—1063no—
437#124—1063no—
438#123—1063no—
439C22⋊Q8—531no—
440C22⋊Q8—533no—
441C22⋊Q8—1063no—
442#121—533no—
443#121—536no—
444#121—533no—
445#121—1066no—
446#121—1063no—
447#121—1063no—
448#121—1063no—
449#120—1063no—
450#120—1066no—
451C4×Q8—21no—
452C4×Q8—21no—
453C4×Q8—23no—
454C4×Q8—23no—
455C4×Q8—1061no—
456C4×D4—23no—
457C4×D4—23no—
458C4×D4—26no—
459C4×D4—26no—
460C4×D4—23no—
461C4×D4—23no—
462C4×D4—1063no—
463C4×D4—1063no—
464C4×D4—1063no—
465#117—23no—
466#117—23no—
467#117—23no—
468#117—23no—
469#117—533no—
470#117C22×D5313yes—
471#117C22×D5311yescharacteristic
472#117C22×D5313yes—
473#117C22×D5313yes—
474#117C22×D5316yes—
475C2×C4⋊C4—533no—
476C2×C4⋊C4C22×D5313yes—
477C2×C4⋊C4C22×D5313yes—
478C2×C4⋊C4C22×D5313yes—
479C2×C4⋊C4C22×D5313yes—
480C2×C22⋊C4—23no—
481C2×C22⋊C4—23no—
482C2×C22⋊C4—1063no—
483C2×C42—23no—
484C2×C42—23no—
485#107—533no—
486#107—536no—
487#107—531no—
488#107—533no—
489#107—533no—
490#106—533no—
491#106—536no—
492#106—531no—
493#106—533no—
494#106—533no—
495#103—531no—
496#103—531no—
497#103—533no—
498#103—533no—
499#102—533no—
500#102—533no—
501#102—536no—
502#102—536no—
503#102—533no—
504#102—533no—
505C22⋊C8—531no—
506C22⋊C8—533no—
507C22⋊C8—533no—
508C22⋊C8—533no—
509C22⋊C8—536no—
510C24—1063no—
511C24C2×C22⋊D5313yes—
512#41—24no—
513#41—212no—
514#41—26no—
515#41—26no—
516#41—1066no—
517#41—1066no—
518#41—2124no—
519#41#1171994514yes—
520#41#1171994514yes—
521C2×Q8—24no—
522C2×Q8—531no—
523C2×Q8C2×C22⋊D5313yes—
524C2×Q8C2×C22⋊D5311yescharacteristic
525C2×D4—212no—
526C2×D4—533no—
527C2×D4—1063no—
528C2×D4—1066no—
529C2×D4—1063no—
530C2×D4—1063no—
531C2×D4—1063no—
532C2×D4—1066no—
533C2×D4—1066no—
534C2×D4—4246no—
535C2×D4—21212no—
536C2×D4C2×C22⋊D5316yes—
537C2×D4C2×C22⋊D5313yes—
538C2×D4C2×C22⋊D5313yes—
539C22×C4—26no—
540C22×C4—26no—
541C22×C4—23no—
542C22×C4—23no—
543C22×C4—23no—
544C22×C4—23no—
545C22×C4—1063no—
546C22×C4—1063no—
547C22×C4C2×C22⋊D5313yes—
548C22×C4C2×C22⋊D5313yes—
549C22×C4C2×C22⋊D5313yes—
550C22×C4C23×D5313yes—
551C22×C4C23×D5311yescharacteristic
552C22×C4C23×D5313yes—
553M16—1066no—
554M16—1066no—
555C2×C8—531no—
556C2×C8—533no—
557C2×C8—533no—
558C2×C8—531no—
559C2×C8—1066no—
560C2×C8—1066no—
561C4⋊C4—23no—
562C4⋊C4—23no—
563C4⋊C4—23no—
564C4⋊C4—23no—
565C4⋊C4—533no—
566C4⋊C4—531no—
567C4⋊C4—1063no—
568C4⋊C4—1061no—
569C4⋊C4—1063no—
570C4⋊C4C23×D5313yes—
571C4⋊C4C23×D5311yescharacteristic
572C4⋊C4C23×D5313yes—
573C4⋊C4C23×D5311yescharacteristic
574C22⋊C4—23no—
575C22⋊C4—21no—
576C22⋊C4—26no—
577C22⋊C4—23no—
578C22⋊C4—23no—
579C22⋊C4—1063no—
580C22⋊C4—1061no—
581C22⋊C4—1063no—
582C22⋊C4—1063no—
583C22⋊C4—1066no—
584C22⋊C4—1063no—
585C42—23no—
586C42—21no—
587C42—26no—
588C42—23no—
589C42—23no—
590C42—1063no—
591C23—26no—
592C23—43no—
593C23—43no—
594C23—1063no—
595C23—4243no—
596C23—4243no—
597C23—2126no—
598C23C22×C22⋊D5311yescharacteristic
599C23C22×C22⋊D5313yes—
600C23C22×C22⋊D5313yes—
601Q8—26no—
602Q8—1061no—
603Q8—1061no—
604Q8#1313320014yes—
605D4—212no—
606D4—26no—
607D4—1063no—
608D4—1063no—
609D4—4243no—
610D4—4246no—
611D4—2126no—
612D4—2126no—
613D4—2126no—
614D4#13133200112yes—
615C2×C4—26no—
616C2×C4—26no—
617C2×C4—26no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—23no—
621C2×C4—23no—
622C2×C4—23no—
623C2×C4—21no—
624C2×C4—23no—
625C2×C4—21no—
626C2×C4—83no—
627C2×C4—83no—
628C2×C4—83no—
629C2×C4—83no—
630C2×C4—1061no—
631C2×C4—1063no—
632C2×C4—1061no—
633C2×C4—4243no—
634C2×C4—2126no—
635C2×C4C22×C22⋊D5311yescharacteristic
636C2×C4C22×C22⋊D5313yes—
637C2×C4C22×C22⋊D5313yes—
638C2×C4C2×#1171994516yes—
639C2×C4C2×#1171994516yes—
640C2×C4C24×D5311yescharacteristic
641C8—1061no—
642C8—1061no—
643C8—1063no—
644C8—1063no—
645C22—23no—
646C22—23no—
647C22—21no—
648C22—23no—
649C22—23no—
650C22—21no—
651C22—23no—
652C22—21no—
653C22—86no—
654C22—8486no—
655C22—2123no—
656C22—2123no—
657C22—2123no—
658C22C2×#1313320016yes—
659C22C23×C22⋊D5311yescharacteristic
660C4—21no—
661C4—23no—
662C4—43no—
663C4—41no—
664C4—43no—
665C4—41no—
666C4—2121no—
667C4—2121no—
668C4—2123no—
669C4C2×#1313320016yes—
670C4C22×#1171994511yescharacteristic
671C4C22×#1171994511yescharacteristic
672C2—26no—
673C2—41no—
674C2—43no—
675C2—4243no—
676C2C22×#1313320011yescharacteristic
677C2#2028948611yescharacteristic
678C2#2028949411yescharacteristic
subgroupG/H
centerC22C23×C22⋊D53
commutator, cosocleC2×C4×C53C25
FrattiniC2×C4C24×D53
FittingC53×#3428C2
socleC22×C53C23×D4
prime radical#20288384C2
hypercenter#3428D53
hypocenterC53#60523
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×#41 ◁ #3428

Complements

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

NH ≅ G/Ntype
2. #20284704C2semidirect
3. #20284693C2semidirect
4. #20284673C2semidirect
5. #20284673C2semidirect
6. #20284655C2semidirect
7. #20284646C2semidirect
8. #20284628C2semidirect
9. C2×#16810947C2semidirect
10. #20283313C2semidirect
11. #20283308C2semidirect
12. #20283238C2semidirect
13. #20283238C2semidirect
14. #20283164C2semidirect
15. C53×#3428C2semidirect
28. #16810964C22semidirect
29. #16810964C22semidirect
30. #16810964C22semidirect
31. #16810958C22semidirect
32. #16810953C22semidirect
33. #16810947C22semidirect
34. #16810947C22semidirect
35. #16810947C22semidirect
36. #16810947C22semidirect
37. #16810947C22semidirect
38. #16810947C22semidirect
39. #16810941C22semidirect
40. #16810941C22semidirect
41. #16810936C22semidirect
42. #16810936C22semidirect
43. #16810666C22semidirect
44. #16810666C22semidirect
45. #16810663C22semidirect
46. #16810663C22semidirect
47. #16810659C22semidirect
48. #16810659C22semidirect
49. C2×#13133150C22semidirect
50. C2×#13133148C22semidirect
51. #16810627C22semidirect
52. #16810627C22semidirect
53. #16810622C22semidirect
54. #16810622C22semidirect
55. #16810617C22semidirect
56. #16810617C22semidirect
57. C2×#13133137C22semidirect
58. C2×#13133137C22semidirect
59. #16810590C22semidirect
60. #16810590C22semidirect
61. #16810587C22semidirect
62. C2×C22⋊#88322C22semidirect
66. C53×#520C22semidirect
67. C53×#520C22semidirect
72. C2×#117×C53C22semidirect
73. C2×#117×C53C22semidirect
334. #3428D53semidirect
370. #520C2×D53semidirect
371. #520C2×D53semidirect
376. C2×#117C2×D53semidirect
377. C2×#117C2×D53semidirect
413. C53#60523semidirect

Maximal subgroups

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

subgroupindex
#202883842
#202847042
#202846932
#202846732
#202846552
#202846462
#202846282
C2×#168109472
#202833132
#202833082
#202832382
#202831642
C53×#34282
#6052353