Finite Group Databasebeta

← #27465171

Subgroups — #27465171

Order 13568.

794 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277322C211yesmaximal, characteristic
2#20277301C211yesmaximal, characteristic
3#20276256C211yesmaximal, characteristic, complemented
4#20276251C211yesmaximal, characteristic, complemented
5#20276136C211yesmaximal, characteristic, complemented
6#20276136C211yesmaximal, characteristic, complemented
7#20275359C211yesmaximal, characteristic, complemented
8#20275358C211yesmaximal, characteristic, complemented
9#20275352C211yesmaximal, characteristic, complemented
10#20275351C211yesmaximal, characteristic, complemented
11#20274916C211yesmaximal, characteristic
12#20274912C211yesmaximal, characteristic
13#20274784C211yesmaximal, characteristic
14#20274650C211yesmaximal, characteristic
15C53×#1787C211yesmaximal, characteristic
16C2×#13133176C2211yescharacteristic
17C2×#13133176C2211yescharacteristic
18C2×#13133151C2211yescharacteristic
19C2×#13133150C2211yescharacteristic
20C2×#13133149C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133141C2211yescharacteristic
23C2×#13133141C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×C22⋊#88322C2211yescharacteristic
27#16810530—21no—
28#16810530—21no—
29#16810530C2211yescharacteristic
30#16810530C2211yescharacteristic
31#16810530C2211yescharacteristic
32#16810530C2211yescharacteristic
33#16810517—21no—
34#16810511—21no—
35#16810508—21no—
36#16810506—21no—
37#16810492C2211yescharacteristic
38#16810488C2211yescharacteristic
39#16810486—21no—
40#16810486—21no—
41#16810483C2211yescharacteristic
42C4⋊C4×Dic53C2211yescharacteristic
43C4⋊C4×Dic53C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810377C2211yescharacteristic
49#16810377C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C23×Q8×C53C2211yescharacteristic
53C2×#103×C53C2211yescharacteristic
54C2×#103×C53C2211yescharacteristic
55C2×#103×C53C2211yescharacteristic
56C2×#103×C53C2211yescharacteristic
57C2×C22⋊C8×C53C2211yescharacteristic
58C53×#380C2211yescharacteristic
59C22×#88322C2311yescharacteristic
60C22×#88322C2311yescharacteristic
61#13133176—22no—
62#13133176—22no—
63#13133176—41no—
64#13133176—41no—
65C2×C4⋊Dic53D411yescharacteristic
66C2×C4⋊Dic53D411yescharacteristic
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×#11719919—21no—
72C2×#11719919—21no—
73C2×C22⋊Dic53—21no—
74C2×C22⋊Dic53—21no—
75C2×C22⋊Dic53D411yescharacteristic
76C2×C22⋊Dic53D411yescharacteristic
77C2×C4×Dic53—21no—
78C2×C4×Dic53—21no—
79C2×C4×Dic53—21no—
80C2×C4×Dic53C2311yescharacteristic
81C2×C4×Dic53C2311yescharacteristic
82#13133151—41no—
83#13133150—41no—
84#13133149—41no—
85#13133148—41no—
86#13133141—22no—
87#13133141—22no—
88#13133141—41no—
89#13133141—41no—
90C22⋊#88322—22no—
91#13133124—21no—
92#13133124—21no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104C22×Q8×C53—22no—
105C22×Q8×C53—22no—
106C22×Q8×C53—21no—
107C22×Q8×C53—22no—
108C22×Q8×C53—22no—
109C22×Q8×C53—41no—
110C22×Q8×C53—41no—
111C22×Q8×C53C2311yescharacteristic
112C22×Q8×C53C2311yescharacteristic
113C23×C4×C53—21no—
114C23×C4×C53C2311yescharacteristic
115C22×C8×C53C2311yescharacteristic
116C22×C8×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C22⋊C4×C53—21no—
120#103×C53—22no—
121#103×C53—22no—
122#103×C53—41no—
123#103×C53—41no—
124C22⋊C8×C53—22no—
125#95×C53—21no—
126C24×C53C2×D411yescharacteristic
127C2×Q8×C53—24no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53—21no—
131C2×Q8×C53—21no—
132C2×Q8×C53—21no—
133C2×Q8×C53—21no—
134C2×Q8×C53—21no—
135C2×Q8×C53—21no—
136C2×Q8×C53—21no—
137C2×Q8×C53—21no—
138C2×Q8×C53—44no—
139C2×Q8×C53—44no—
140C2×Q8×C53—44no—
141C2×Q8×C53—44no—
142C2×Q8×C53—42no—
143C2×Q8×C53—42no—
144C2×Q8×C53#4112yes—
145C2×Q8×C53#4112yes—
146C22×C4×C53—21no—
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C22×C4×C53—21no—
150C22×C4×C53—24no—
151C22×C4×C53—21no—
152C22×C4×C53—21no—
153C22×C4×C53—21no—
154C22×C4×C53—21no—
155C22×C4×C53—42no—
156C22×C4×C53—42no—
157C22×C4×C53C2×D411yescharacteristic
158C22×C4×C53#4112yes—
159C22×C4×C53#4112yes—
160C22×C4×C53C2411yescharacteristic
161C22×Dic53—21no—
162C22×Dic53—21no—
163C22×Dic53—21no—
164C22×Dic53—21no—
165C22×Dic53C2×D411yescharacteristic
166C22×Dic53C2×D411yescharacteristic
167C22×Dic53C2×D411yescharacteristic
168C22×Dic53C2×D411yescharacteristic
169C2×C8×C53—21no—
170C2×C8×C53—21no—
171C2×C8×C53—21no—
172C2×C8×C53—21no—
173C2×C8×C53—21no—
174C2×C8×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—21no—
177C4⋊C4×C53—21no—
178C4⋊C4×C53—21no—
179C22⋊C4×C53—81no—
180C22⋊Dic53—22no—
181C22⋊Dic53—22no—
182C22⋊Dic53—42no—
183C22⋊Dic53—81no—
184C4⋊Dic53—22no—
185C4⋊Dic53—22no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194#11719919—81no—
195#11719919—81no—
196C4×Dic53—21no—
197C4×Dic53—21no—
198C4×Dic53—21no—
199C4×Dic53—21no—
200C4×Dic53—42no—
201C4×Dic53—81no—
202C4×Dic53—81no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53#12312yes—
216C23×C53C22×D411yescharacteristic
217Q8×C53—22no—
218Q8×C53—22no—
219Q8×C53—22no—
220Q8×C53—22no—
221Q8×C53—48no—
222Q8×C53—41no—
223Q8×C53—41no—
224Q8×C53—41no—
225Q8×C53—41no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—22no—
231C2×C4×C53—22no—
232C2×C4×C53—22no—
233C2×C4×C53—22no—
234C2×C4×C53—48no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53#12312yes—
246C2×C4×C53C2×#4111yescharacteristic
247C2×C4×C53C2×#4111yescharacteristic
248C2×C4×C53C2×#4111yescharacteristic
249C2×C4×C53C2×#4111yescharacteristic
250C2×C4×C53H(5,𝔽2)11yescharacteristic
251C2×C4×C53H(5,𝔽2)11yescharacteristic
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—21no—
256C2×Dic53—21no—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—21no—
261C2×Dic53—21no—
262C2×Dic53—21no—
263C2×Dic53—21no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C8×C53—41no—
277C8×C53—41no—
278#88322—41no—
279#88322—41no—
280#27596—531nomaximal, Sylow
281C22×C53—22no—
282C22×C53—22no—
283C22×C53—22no—
284C22×C53—41no—
285C22×C53—41no—
286C22×C53—41no—
287C22×C53—41no—
288C22×C53C2×#12311yescharacteristic
289C22×C53#52511yescharacteristic
290C22×C53#53511yescharacteristic
291C22×C53#57711yescharacteristic
292C22×C53#57711yescharacteristic
293C22×C53#57811yescharacteristic
294C22×C53#57811yescharacteristic
295C4×C53—22no—
296C4×C53—42no—
297C4×C53—42no—
298C4×C53—81no—
299C4×C53—81no—
300C4×C53#55111yescharacteristic
301C4×C53#55111yescharacteristic
302C4×C53#55111yescharacteristic
303C4×C53#55111yescharacteristic
304Dic53—41no—
305Dic53—41no—
306Dic53—41no—
307Dic53—41no—
308Dic53—81no—
309Dic53—81no—
310Dic53—81no—
311Dic53—81no—
312#2314—531no—
313#2310—531no—
314#2096—531no—
315#2095—531no—
316#2074—531no—
317#2074—531no—
318#1928—531no—
319#1928—531no—
320#1927—531no—
321#1927—531no—
322#1854—531no—
323#1854—531no—
324#1818—531no—
325#1787—531no—
326#1787D5311yescharacteristic
327C2×C53—42no—
328C2×C53#248211yescharacteristic
329C2×C53#309711yescharacteristic
330C2×C53#309711yescharacteristic
331C2×C53#312111yescharacteristic
332C2×C53#312111yescharacteristic
333C2×C53#322111yescharacteristic
334C2×C53#322311yescharacteristic
335C23×Q8C2×D5311yescharacteristic
336C2×#117—531no—
337C2×#117—531no—
338C2×#107—531no—
339C2×#107—531no—
340C2×#106—531no—
341C2×#106—531no—
342C2×#103—531no—
343C2×#103—531no—
344C2×#103—531no—
345C2×#103—531no—
346C2×#103C2×D5311yescharacteristic
347C2×#103C2×D5311yescharacteristic
348C2×#103C2×D5311yescharacteristic
349C2×#103C2×D5311yescharacteristic
350C2×C22⋊C8—531no—
351C2×C22⋊C8C2×D5311yescharacteristic
352#391—531no—
353#391—531no—
354#391—531no—
355#391—531no—
356#391—1061no—
357#391—1061no—
358#388—1061no—
359#388—1061no—
360#387—1061no—
361#387—1061no—
362#384—531no—
363#382—531no—
364#381—1061no—
365#381—1061no—
366#380—531no—
367#380C2×D5311yescharacteristic
368C4×C4⋊C4—531no—
369C4×C4⋊C4—531no—
370#340—531no—
371#340—531no—
372#340—531no—
373#340—531no—
374#336—531no—
375#336—531no—
376#336—531no—
377#336—531no—
378C53#2759611yescharacteristic, complemented, Sylow
379C22×Q8—22no—
380C22×Q8—22no—
381C22×Q8—21no—
382C22×Q8—22no—
383C22×Q8—22no—
384C22×Q8—41no—
385C22×Q8—41no—
386C22×Q8C22×D5311yescharacteristic
387C22×Q8C22×D5311yescharacteristic
388C23×C4—21no—
389C23×C4C22×D5311yescharacteristic
390C22×C8—531no—
391C22×C8—531no—
392C22×C8C22×D5311yescharacteristic
393C22×C8C22×D5311yescharacteristic
394#117—1062no—
395#117—1062no—
396#117—2121no—
397#117—2121no—
398C2×C4⋊C4—531no—
399C2×C4⋊C4—531no—
400C2×C4⋊C4—531no—
401C2×C4⋊C4—531no—
402C2×C4⋊C4—531no—
403C2×C4⋊C4—531no—
404C2×C4⋊C4—1061no—
405C2×C4⋊C4—1061no—
406C2×C4⋊C4C22×D5311yescharacteristic
407C2×C4⋊C4C22×D5311yescharacteristic
408C2×C22⋊C4—21no—
409C2×C22⋊C4—531no—
410C2×C22⋊C4—531no—
411C2×C22⋊C4—1061no—
412C2×C22⋊C4—1061no—
413C2×C42—531no—
414C2×C42—531no—
415C2×C42—1061no—
416C2×C42—1061no—
417C2×C42—1061no—
418#107—2121no—
419#107—2121no—
420#106—2121no—
421#106—2121no—
422#103—22no—
423#103—22no—
424#103—41no—
425#103—41no—
426#103—1062no—
427#103—1062no—
428#103—2121no—
429#103—2121no—
430C22⋊C8—22no—
431C22⋊C8—1062no—
432#95—21no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438#95—1061no—
439#95—1061no—
440#95—1061no—
441#95—1061no—
442#95—1061no—
443#95—1061no—
444#95—1061no—
445#95—1061no—
446C24D4×D5311yescharacteristic
447C2×Q8—24no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—44no—
459C2×Q8—44no—
460C2×Q8—44no—
461C2×Q8—44no—
462C2×Q8—42no—
463C2×Q8—42no—
464C2×Q8#1171994712yes—
465C2×Q8#1171994712yes—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—24no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—531no—
478C22×C4—531no—
479C22×C4—531no—
480C22×C4—531no—
481C22×C4—1061no—
482C22×C4—1061no—
483C22×C4—1061no—
484C22×C4—1061no—
485C22×C4#1171994512yes—
486C22×C4D4×D5311yescharacteristic
487C22×C4#1171994712yes—
488C22×C4C23×D5311yescharacteristic
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—1061no—
496C2×C8—1061no—
497C2×C8—1061no—
498C2×C8—1061no—
499C2×C8—1061no—
500C2×C8—1061no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—1062no—
506C4⋊C4—1062no—
507C4⋊C4—1061no—
508C4⋊C4—1061no—
509C4⋊C4—1061no—
510C4⋊C4—1061no—
511C4⋊C4—1061no—
512C4⋊C4—1061no—
513C4⋊C4—1061no—
514C4⋊C4—1061no—
515C4⋊C4—4241no—
516C4⋊C4—4241no—
517C22⋊C4—81no—
518C22⋊C4—1062no—
519C22⋊C4—1062no—
520C22⋊C4—4241no—
521C22⋊C4—2122no—
522C42—1061no—
523C42—1061no—
524C42—1061no—
525C42—1061no—
526C42—4241no—
527C42—4241no—
528C42—2122no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23#1313319912yes—
536C23C2×D4×D5311yescharacteristic
537Q8—22no—
538Q8—22no—
539Q8—22no—
540Q8—22no—
541Q8—48no—
542Q8—41no—
543Q8—41no—
544Q8—41no—
545Q8—41no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—48no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—1061no—
566C2×C4—1061no—
567C2×C4—1061no—
568C2×C4—1061no—
569C2×C4—1061no—
570C2×C4—1061no—
571C2×C4—1061no—
572C2×C4—1061no—
573C2×C4—1061no—
574C2×C4—1061no—
575C2×C4—1061no—
576C2×C4—1061no—
577C2×C4—2121no—
578C2×C4—2121no—
579C2×C4—2121no—
580C2×C4—2121no—
581C2×C4—2121no—
582C2×C4—2121no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4—2121no—
587C2×C4—2121no—
588C2×C4—2121no—
589C2×C4#1313319912yes—
590C2×C4C2×#1171994711yescharacteristic
591C2×C4C2×#1171994711yescharacteristic
592C2×C4C2×#1171994711yescharacteristic
593C2×C4C2×#1171994511yescharacteristic
594C2×C4#1313326211yescharacteristic
595C2×C4#1313326211yescharacteristic
596C8—41no—
597C8—41no—
598C8—2121no—
599C8—2121no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22C2×#1313319911yescharacteristic
608C22#1681112911yescharacteristic
609C22#1681118811yescharacteristic
610C22#1681142911yescharacteristic
611C22#1681143211yescharacteristic
612C22#1681143611yescharacteristic
613C22#1681143911yescharacteristic
614C4—22no—
615C4—42no—
616C4—42no—
617C4—81no—
618C4—81no—
619C4—4241no—
620C4—4241no—
621C4—4241no—
622C4—4241no—
623C4—2121no—
624C4—2121no—
625C4—2121no—
626C4—2121no—
627C4#1681129911yescharacteristic
628C4#1681129911yescharacteristic
629C4#1681130411yescharacteristic
630C4#1681130411yescharacteristic
631C2—42no—
632C2#2027898611yescharacteristic
633C2#2028468011yescharacteristic
634C2#2028468211yescharacteristic
635C2#2028493711yescharacteristic
636C2#2028494511yescharacteristic
637C2#2028577211yescharacteristic
638C2#2028579411yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1787C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1787D53
hypocenterC53#27596
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1787

Complements

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

NH ≅ G/Ntype
3. #20276256C2semidirect
4. #20276251C2semidirect
5. #20276136C2semidirect
6. #20276136C2semidirect
7. #20275359C2semidirect
8. #20275358C2semidirect
9. #20275352C2semidirect
10. #20275351C2semidirect
378. C53#27596semidirect

Maximal subgroups

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

subgroupindex
#202773222
#202773012
#202762562
#202762512
#202761362
#202753592
#202753582
#202753522
#202753512
#202749162
#202749122
#202747842
#202746502
C53×#17872
#2759653