Finite Group Databasebeta

← #28252046

Subgroups — #28252046

Order 13568.

2116 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20287918C211yesmaximal, characteristic, complemented
2#20284700C213yesmaximal, complemented
3#20284689C213yesmaximal, complemented
4#20284678C213yesmaximal, complemented
5#20284678C213yesmaximal, complemented
6#20284659C213yesmaximal, complemented
7#20284644C213yesmaximal, complemented
8#20284624C213yesmaximal, complemented
9C2×#16810961C213yesmaximal, complemented
10#20283313C211yesmaximal, characteristic, complemented
11#20283308C211yesmaximal, characteristic, complemented
12#20283248C211yesmaximal, characteristic, complemented
13#20283248C211yesmaximal, characteristic, complemented
14#20283164C211yesmaximal, characteristic, complemented
15C53×#3489C211yesmaximal, characteristic
16C2×#13133224C2211yescharacteristic
17C2×#13133224C2211yescharacteristic
18#16811106C2213yes—
19#16811106C2213yes—
20#16811106C2211yescharacteristic
21#16811106C2211yescharacteristic
22#16811101—21no—
23#16811098—23no—
24#41×Dic53—23no—
25#41×Dic53—21no—
26C2×#13133176C2213yes—
27C2×#13133176C2213yes—
28#16810961C2213yes—
29#16810961C2216yes—
30#16810961C2213yes—
31#16810961C2216yescomplemented
32#16810961C2216yescomplemented
33#16810961C2216yescomplemented
34#16810956C2213yescomplemented
35#16810956C2216yescomplemented
36#16810951C2213yescomplemented
37#16810951C2216yescomplemented
38#16810944C2211yescharacteristic
39#16810944C2213yes—
40#16810944C2216yescomplemented
41#16810939C2213yescomplemented
42#16810934C2213yescomplemented
43#16810666C2213yescomplemented
44#16810666C2213yescomplemented
45#16810663C2213yescomplemented
46#16810663C2213yescomplemented
47#16810659C2213yescomplemented
48#16810659C2213yescomplemented
49C2×#13133150C2213yescomplemented
50C2×#13133148C2213yescomplemented
51#16810632C2213yescomplemented
52#16810632C2213yescomplemented
53#16810624C2213yescomplemented
54#16810624C2213yescomplemented
55#16810619C2213yescomplemented
56#16810619C2213yescomplemented
57C2×#13133141C2213yescomplemented
58C2×#13133141C2213yescomplemented
59#16810590C2213yes—
60#16810590C2213yes—
61#16810587C2213yes—
62C2×C22⋊#88322C2213yes—
63C2×2−1+4×C53C2211yescharacteristic
64C22×#41×C53C2211yescharacteristic
65D4×Q8×C53—21no—
66C53×#548—21no—
67C53×#547—23no—
68C53×#546—23no—
69C53×#537C2213yes—
70C53×#536C2213yes—
71C53×#525C2213yes—
72C2×C22⋊Q8×C53C2213yes—
73C53×#520C2211yescharacteristic
74#13133224—24no—
75#13133224—24no—
76C2×#11719917C2313yes—
77C2×#11719917C2313yes—
78C22×#88322C2313yes—
79C22×#88322C2313yes—
80Q8×Dic53—23no—
81Q8×Dic53—23no—
82Q8×Dic53—21no—
83Q8×Dic53—21no—
84D4×Dic53—26no—
85D4×Dic53—26no—
86D4×Dic53—23no—
87D4×Dic53—23no—
88D4×Dic53—23no—
89D4×Dic53—23no—
90#13133176—23no—
91#13133176—23no—
92#13133176—23no—
93#13133176—23no—
94#13133176C2313yes—
95#13133176C2311yescharacteristic
96#13133176C2313yes—
97#13133176C2313yes—
98#13133176C2316yes—
99C2×C4⋊Dic53C2313yes—
100C2×C4⋊Dic53C2313yes—
101C2×C4⋊Dic53C2313yes—
102C2×C4⋊Dic53C2313yes—
103C2×C22⋊Dic53—23no—
104C2×C22⋊Dic53—23no—
105C2×C4×Dic53—23no—
106C2×C4×Dic53—23no—
107#13133150C2313yes—
108#13133150C2316yes—
109#13133150C2311yescharacteristic
110#13133150C2313yes—
111#13133150C2313yes—
112#13133148C2313yes—
113#13133148C2316yes—
114#13133148C2311yescharacteristic
115#13133148C2313yes—
116#13133148C2313yes—
117#13133141C2313yes—
118#13133141C2313yes—
119#13133141C2316yes—
120#13133141C2316yes—
121#13133141C2313yes—
122#13133141C2313yes—
123#13133138C2311yescharacteristic
124#13133138C2311yescharacteristic
125#13133138C2313yes—
126#13133138C2313yes—
127C22⋊#88322C2311yescharacteristic
128C22⋊#88322C2313yes—
129C22⋊#88322C2313yes—
130C22⋊#88322C2313yes—
131C22⋊#88322C2316yes—
1322−1+4×C53—24no—
1332−1+4×C53—24no—
134C2×#41×C53—24no—
135C2×#41×C53—24no—
136C2×#41×C53—23no—
137C2×#41×C53D413yes—
138C2×#41×C53D411yescharacteristic
139C2×#41×C53C2313yes—
140C2×#41×C53C2313yes—
141C2×#41×C53C2311yescharacteristic
142C22×Q8×C53—21no—
143C22×Q8×C53D411yescharacteristic
144C22×Q8×C53C2313yes—
145C22×D4×C53D413yes—
146C23×C4×C53C2313yes—
147C4⋊Q8×C53—23no—
148#124×C53—23no—
149#123×C53—26no—
150#123×C53—23no—
151C22⋊Q8×C53—23no—
152C22⋊Q8×C53—23no—
153C22⋊Q8×C53—23no—
154C22⋊Q8×C53C2313yes—
155C22⋊Q8×C53C2316yes—
156C22⋊Q8×C53C2313yes—
157#121×C53—26no—
158#121×C53—23no—
159#121×C53C2311yescharacteristic
160#121×C53C2313yes—
161#120×C53—23no—
162C4×Q8×C53—21no—
163C4×D4×C53—23no—
164C4×D4×C53—23no—
165C4×D4×C53—23no—
166#117×C53C2313yes—
167C2×C4⋊C4×C53—23no—
168C2×C4⋊C4×C53C2313yes—
169C2×C22⋊C4×C53—23no—
170C24×C53C2×D413yes—
171#41×C53—24no—
172#41×C53—212no—
173#41×C53—26no—
174#41×C53—26no—
175#41×C53—26no—
176#41×C53—26no—
177#41×C53—412no—
178#41×C53#4114yes—
179#41×C53#4114yes—
180C2×Q8×C53—24no—
181C2×Q8×C53—23no—
182C2×Q8×C53—26no—
183C2×Q8×C53—26no—
184C2×Q8×C53—44no—
185C2×Q8×C53C2×D413yes—
186C2×Q8×C53C2×D411yescharacteristic
187C2×Q8×C53C2413yes—
188C2×D4×C53—212no—
189C2×D4×C53—21no—
190C2×D4×C53—23no—
191C2×D4×C53—23no—
192C2×D4×C53C2×D416yes—
193C2×D4×C53C2×D413yes—
194C2×D4×C53C2×D413yes—
195C2×D4×C53C2411yescharacteristic
196C22×C4×C53—26no—
197C22×C4×C53—26no—
198C22×C4×C53—23no—
199C22×C4×C53—23no—
200C22×C4×C53—23no—
201C22×C4×C53C2×D413yes—
202C22×C4×C53C2×D413yes—
203C22×C4×C53C2×D413yes—
204C22×C4×C53C2411yescharacteristic
205C22×C4×C53C2413yes—
206C22×C4×C53C2413yes—
207C22×Dic53—23no—
208C22×Dic53—23no—
209C22×Dic53—23no—
210C22×Dic53—23no—
211C4⋊C4×C53—23no—
212C4⋊C4×C53—26no—
213C4⋊C4×C53—23no—
214C4⋊C4×C53—23no—
215C4⋊C4×C53C2413yes—
216C4⋊C4×C53C2411yescharacteristic
217C22⋊C4×C53—23no—
218C22⋊C4×C53—26no—
219C22⋊C4×C53—23no—
220C22⋊C4×C53—23no—
221C22⋊C4×C53—21no—
222C22⋊C4×C53—23no—
223C42×C53—23no—
224C22⋊Dic53—23no—
225C22⋊Dic53—21no—
226C22⋊Dic53—26no—
227C22⋊Dic53—23no—
228C22⋊Dic53—23no—
229C4⋊Dic53—23no—
230C4⋊Dic53—23no—
231C4⋊Dic53—23no—
232C4⋊Dic53—23no—
233C4⋊Dic53C2411yescharacteristic
234C4⋊Dic53C2413yes—
235C4⋊Dic53C2411yescharacteristic
236C4⋊Dic53C2413yes—
237C4×Dic53—23no—
238C4×Dic53—21no—
239C4×Dic53—26no—
240C4×Dic53—23no—
241C4×Dic53—23no—
242#11719917—26no—
243#11719917—26no—
244C2×#88322—26no—
245C2×#88322—26no—
246C2×#88322C2411yescharacteristic
247C2×#88322C2413yes—
248C2×#88322C2413yes—
249C2×#88322C2411yescharacteristic
250C23×C53—26no—
251C23×C53—21no—
252C23×C53—43no—
253C23×C53—43no—
254C23×C53C22×D413yes—
255C23×C53C22×D411yescharacteristic
256C23×C53C22×D413yes—
257Q8×C53—26no—
258Q8×C53—23no—
259Q8×C53—23no—
260Q8×C53—46no—
261Q8×C53#12314yes—
262D4×C53—212no—
263D4×C53—26no—
264D4×C53—21no—
265D4×C53—21no—
266D4×C53—42no—
267D4×C53—46no—
268D4×C53—83no—
269D4×C53#123112yes—
270C2×C4×C53—26no—
271C2×C4×C53—26no—
272C2×C4×C53—26no—
273C2×C4×C53—21no—
274C2×C4×C53—21no—
275C2×C4×C53—23no—
276C2×C4×C53—23no—
277C2×C4×C53—23no—
278C2×C4×C53—23no—
279C2×C4×C53—21no—
280C2×C4×C53—46no—
281C2×C4×C53—46no—
282C2×C4×C53—83no—
283C2×C4×C53C22×D411yescharacteristic
284C2×C4×C53C22×D413yes—
285C2×C4×C53C22×D413yes—
286C2×C4×C53C2×#4116yes—
287C2×C4×C53C2×#4116yes—
288C2×C4×C53C2511yescharacteristic
289C2×Dic53—21no—
290C2×Dic53—23no—
291C2×Dic53—21no—
292C2×Dic53—23no—
293C2×Dic53—83no—
294C2×Dic53—83no—
295C2×Dic53—83no—
296C2×Dic53—83no—
297#88322—21no—
298#88322—21no—
299#88322—23no—
300#88322—23no—
301#60524—531nomaximal, Sylow
302C22×C53—23no—
303C22×C53—23no—
304C22×C53—23no—
305C22×C53—21no—
306C22×C53—23no—
307C22×C53—21no—
308C22×C53—21no—
309C22×C53—23no—
310C22×C53—41no—
311C22×C53—41no—
312C22×C53—41no—
313C22×C53—86no—
314C22×C53C2×#12316yes—
315C22×C53C23×D411yescharacteristic
316C4×C53—21no—
317C4×C53—23no—
318C4×C53—43no—
319C4×C53—41no—
320C4×C53—43no—
321C4×C53C2×#12316yes—
322C4×C53C22×#4111yescharacteristic
323C4×C53C22×#4111yescharacteristic
324Dic53—43no—
325Dic53—41no—
326Dic53—43no—
327Dic53—41no—
328#3489D5311yescharacteristic
329#3428—531no—
330#3099—533no—
331#3098—533no—
332#3097—533no—
333#3097—533no—
334#3095—533no—
335#3093—533no—
336#3091—533no—
337C2×#485—533no—
338#2913—531no—
339#2912—531no—
340#2898—531no—
341#2898—531no—
342#2880—531no—
343C2×C53—26no—
344C2×C53—43no—
345C2×C53—41no—
346C2×C53—81no—
347C2×C53C22×#12311yescharacteristic
348C2×C53#358411yescharacteristic
349C2×C53#358611yescharacteristic
350C2×2−1+4C2×D5311yescharacteristic
351C22×#41C2×D5311yescharacteristic
352C2×#131—531no—
353C2×#131—531no—
354D4×Q8—21no—
355#548—21no—
356#547—23no—
357#546—23no—
358#537C2×D5313yes—
359#536C2×D5313yes—
360#525C2×D5313yes—
361C2×C22⋊Q8C2×D5313yes—
362#520—533no—
363#520—533no—
364#520—531no—
365#520—531no—
366#520C2×D5311yescharacteristic
367#519—1061no—
368#518—1063no—
369C4×#41—1063no—
370C4×#41—1061no—
371C2×#117—533no—
372C2×#117—533no—
373#485—533no—
374#485—536no—
375#485—536no—
376#485—536no—
377#485—533no—
378#485—536no—
379#484—533no—
380#484—536no—
381#483—533no—
382#483—536no—
383#482—531no—
384#482—533no—
385#482—536no—
386#481—533no—
387#480—533no—
388#428—533no—
389#428—533no—
390#428—533no—
391#428—533no—
392#427—533no—
393#427—533no—
394C2×#107—533no—
395C2×#106—533no—
396#419—533no—
397#419—533no—
398#417—533no—
399#417—533no—
400#416—533no—
401#416—533no—
402C2×#103—533no—
403C2×#103—533no—
404#408—533no—
405#408—533no—
406#407—533no—
407C2×C22⋊C8—533no—
408C53#6052411yescharacteristic, complemented, Sylow
4092−1+4—24no—
4102−1+4—24no—
411C2×#41—24no—
412C2×#41—24no—
413C2×#41—23no—
414C2×#41C22⋊D5313yes—
415C2×#41C22⋊D5311yescharacteristic
416C2×#41C22×D5313yes—
417C2×#41C22×D5313yes—
418C2×#41C22×D5311yescharacteristic
419C22×Q8—21no—
420C22×Q8C22⋊D5311yescharacteristic
421C22×Q8C22×D5313yes—
422C22×D4C22⋊D5313yes—
423C23×C4C22×D5313yes—
424#131—1064no—
425#131—1064no—
426C2×M16—533no—
427C2×M16—533no—
428C22×C8—533no—
429C22×C8—533no—
430C4⋊Q8—23no—
431#124—23no—
432#123—26no—
433#123—23no—
434C22⋊Q8—23no—
435C22⋊Q8—23no—
436C22⋊Q8—23no—
437C22⋊Q8C22×D5313yes—
438C22⋊Q8C22×D5316yes—
439C22⋊Q8C22×D5313yes—
440#121—26no—
441#121—23no—
442#121C22×D5311yescharacteristic
443#121C22×D5313yes—
444#120—23no—
445C4×Q8—21no—
446C4×Q8—1063no—
447C4×Q8—1063no—
448C4×Q8—1061no—
449C4×Q8—1061no—
450C4×D4—23no—
451C4×D4—23no—
452C4×D4—23no—
453C4×D4—1066no—
454C4×D4—1066no—
455C4×D4—1063no—
456C4×D4—1063no—
457C4×D4—1063no—
458C4×D4—1063no—
459#117—533no—
460#117—531no—
461#117—533no—
462#117—533no—
463#117—536no—
464#117—1063no—
465#117—1063no—
466#117—1063no—
467#117—1063no—
468#117C22×D5313yes—
469C2×C4⋊C4—23no—
470C2×C4⋊C4—533no—
471C2×C4⋊C4—533no—
472C2×C4⋊C4—533no—
473C2×C4⋊C4—533no—
474C2×C4⋊C4C22×D5313yes—
475C2×C22⋊C4—23no—
476C2×C22⋊C4—1063no—
477C2×C22⋊C4—1063no—
478C2×C42—1063no—
479C2×C42—1063no—
480#107—533no—
481#107—536no—
482#107—531no—
483#107—533no—
484#107—533no—
485#106—533no—
486#106—536no—
487#106—531no—
488#106—533no—
489#106—533no—
490#103—533no—
491#103—533no—
492#103—536no—
493#103—536no—
494#103—533no—
495#103—533no—
496#102—531no—
497#102—531no—
498#102—533no—
499#102—533no—
500C22⋊C8—531no—
501C22⋊C8—533no—
502C22⋊C8—533no—
503C22⋊C8—533no—
504C22⋊C8—536no—
505C24C2×C22⋊D5313yes—
506#41—24no—
507#41—212no—
508#41—26no—
509#41—26no—
510#41—26no—
511#41—26no—
512#41—412no—
513#41#1171994714yes—
514#41#1171994714yes—
515C2×Q8—24no—
516C2×Q8—23no—
517C2×Q8—26no—
518C2×Q8—26no—
519C2×Q8—44no—
520C2×Q8C2×C22⋊D5313yes—
521C2×Q8C2×C22⋊D5311yescharacteristic
522C2×Q8C23×D5313yes—
523C2×D4—212no—
524C2×D4—21no—
525C2×D4—23no—
526C2×D4—23no—
527C2×D4C2×C22⋊D5316yes—
528C2×D4C2×C22⋊D5313yes—
529C2×D4C2×C22⋊D5313yes—
530C2×D4C23×D5311yescharacteristic
531C22×C4—26no—
532C22×C4—26no—
533C22×C4—23no—
534C22×C4—23no—
535C22×C4—23no—
536C22×C4—1063no—
537C22×C4—1063no—
538C22×C4—1063no—
539C22×C4—1063no—
540C22×C4C2×C22⋊D5313yes—
541C22×C4C2×C22⋊D5313yes—
542C22×C4C2×C22⋊D5313yes—
543C22×C4C23×D5313yes—
544C22×C4C23×D5311yescharacteristic
545C22×C4C23×D5313yes—
546M16—1066no—
547M16—1066no—
548C2×C8—531no—
549C2×C8—533no—
550C2×C8—533no—
551C2×C8—531no—
552C2×C8—1066no—
553C2×C8—1066no—
554C4⋊C4—23no—
555C4⋊C4—26no—
556C4⋊C4—23no—
557C4⋊C4—23no—
558C4⋊C4—531no—
559C4⋊C4—533no—
560C4⋊C4—531no—
561C4⋊C4—533no—
562C4⋊C4—1063no—
563C4⋊C4—1063no—
564C4⋊C4—1063no—
565C4⋊C4—1063no—
566C4⋊C4C23×D5313yes—
567C4⋊C4C23×D5311yescharacteristic
568C22⋊C4—23no—
569C22⋊C4—26no—
570C22⋊C4—23no—
571C22⋊C4—23no—
572C22⋊C4—21no—
573C22⋊C4—23no—
574C22⋊C4—1063no—
575C22⋊C4—1061no—
576C22⋊C4—1066no—
577C22⋊C4—1063no—
578C22⋊C4—1063no—
579C42—23no—
580C42—1063no—
581C42—1061no—
582C42—1066no—
583C42—1063no—
584C42—1063no—
585C23—26no—
586C23—21no—
587C23—43no—
588C23—43no—
589C23C22×C22⋊D5311yescharacteristic
590C23C22×C22⋊D5313yes—
591C23C22×C22⋊D5313yes—
592Q8—26no—
593Q8—23no—
594Q8—23no—
595Q8—46no—
596Q8#1313319814yes—
597D4—212no—
598D4—26no—
599D4—21no—
600D4—21no—
601D4—42no—
602D4—46no—
603D4—83no—
604D4#13133198112yes—
605C2×C4—26no—
606C2×C4—26no—
607C2×C4—26no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—23no—
611C2×C4—23no—
612C2×C4—21no—
613C2×C4—23no—
614C2×C4—23no—
615C2×C4—46no—
616C2×C4—46no—
617C2×C4—83no—
618C2×C4—1063no—
619C2×C4—1061no—
620C2×C4—1063no—
621C2×C4—1061no—
622C2×C4—4243no—
623C2×C4—4243no—
624C2×C4—4243no—
625C2×C4—4243no—
626C2×C4C22×C22⋊D5311yescharacteristic
627C2×C4C22×C22⋊D5313yes—
628C2×C4C22×C22⋊D5313yes—
629C2×C4C2×#1171994716yes—
630C2×C4C2×#1171994716yes—
631C2×C4C24×D5311yescharacteristic
632C8—1061no—
633C8—1061no—
634C8—1063no—
635C8—1063no—
636C22—23no—
637C22—23no—
638C22—21no—
639C22—23no—
640C22—23no—
641C22—21no—
642C22—23no—
643C22—21no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22—86no—
648C22C2×#1313319816yes—
649C22C23×C22⋊D5311yescharacteristic
650C4—21no—
651C4—23no—
652C4—43no—
653C4—43no—
654C4—41no—
655C4—2123no—
656C4—2121no—
657C4—2123no—
658C4—2121no—
659C4C2×#1313319816yes—
660C4C22×#1171994711yescharacteristic
661C4C22×#1171994711yescharacteristic
662C2—26no—
663C2—41no—
664C2—43no—
665C2—81no—
666C2C22×#1313319811yescharacteristic
667C2#2028948611yescharacteristic
668C2#2028950211yescharacteristic
subgroupG/H
centerC22C23×C22⋊D53
commutator, cosocleC2×C4×C53C25
FrattiniC2×C4C24×D53
Fitting, prime radicalC53×#3489C2
socleC22×C53C23×D4
hypercenter#3489D53
hypocenterC53#60524
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×#41 ◁ #3489

Complements

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

NH ≅ G/Ntype
1. #20287918C2semidirect
2. #20284700C2semidirect
3. #20284689C2semidirect
4. #20284678C2semidirect
5. #20284678C2semidirect
6. #20284659C2semidirect
7. #20284644C2semidirect
8. #20284624C2semidirect
9. C2×#16810961C2semidirect
10. #20283313C2semidirect
11. #20283308C2semidirect
12. #20283248C2semidirect
13. #20283248C2semidirect
14. #20283164C2semidirect
31. #16810961C22semidirect
32. #16810961C22semidirect
33. #16810961C22semidirect
34. #16810956C22semidirect
35. #16810956C22semidirect
36. #16810951C22semidirect
37. #16810951C22semidirect
40. #16810944C22semidirect
41. #16810939C22semidirect
42. #16810934C22semidirect
43. #16810666C22semidirect
44. #16810666C22semidirect
45. #16810663C22semidirect
46. #16810663C22semidirect
47. #16810659C22semidirect
48. #16810659C22semidirect
49. C2×#13133150C22semidirect
50. C2×#13133148C22semidirect
51. #16810632C22semidirect
52. #16810632C22semidirect
53. #16810624C22semidirect
54. #16810624C22semidirect
55. #16810619C22semidirect
56. #16810619C22semidirect
57. C2×#13133141C22semidirect
58. C2×#13133141C22semidirect
408. C53#60524semidirect

Maximal subgroups

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

subgroupindex
#202879182
#202847002
#202846892
#202846782
#202846592
#202846442
#202846242
C2×#168109612
#202833132
#202833082
#202832482
#202831642
C53×#34892
#6052453