Finite Group Databasebeta

← #27452553

Subgroups — #27452553

Order 13568.

720 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279058C211yesmaximal, characteristic
2#20277746C211yesmaximal, characteristic
3#20276035C211yesmaximal, characteristic, complemented
4#20275915C211yesmaximal, characteristic
5#20275864C211yesmaximal, characteristic, complemented
6#20275669C211yesmaximal, characteristic, complemented
7#20275407C211yesmaximal, characteristic, complemented
8#20275359C211yesmaximal, characteristic, complemented
9#20275325C211yesmaximal, characteristic
10#20275310C211yesmaximal, characteristic, complemented
11#20275058C211yesmaximal, characteristic, complemented
12#20275028C211yesmaximal, characteristic
13#20274811C211yesmaximal, characteristic
14#20274547C211yesmaximal, characteristic, complemented
15C53×#2001C211yesmaximal, characteristic
16C2×#13133208C2211yescharacteristic
17C2×Q8×Dic53C2211yescharacteristic
18C2×#13133176C2211yescharacteristic
19C2×#13133150C2211yescharacteristic
20C2×#13133146C2211yescharacteristic
21C2×#13133141C2211yescharacteristic
22C2×#13133141C2211yescharacteristic
23C2×#13133141C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×C22⋊#88322C2211yescharacteristic
26C2×#13133127C2211yescharacteristic
27#16810561—21no—
28#16810548—21no—
29#16810530C2211yescharacteristic
30#16810530C2211yescharacteristic
31#16810530C2211yescharacteristic
32#16810522C2211yescharacteristic
33#16810517—21no—
34#16810516—21no—
35#16810516—21no—
36#16810509—21no—
37#16810488—21no—
38#16810483C2211yescharacteristic
39C4⋊C4×Dic53—21no—
40C4⋊C4×Dic53C2211yescharacteristic
41C4⋊C4×Dic53C2211yescharacteristic
42C4×C22⋊Dic53C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C22×Q16×C53C2211yescharacteristic
53C2×C22⋊Q8×C53C2211yescharacteristic
54C2×C22⋊Q8×C53C2211yescharacteristic
55C2×#103×C53C2211yescharacteristic
56C2×#103×C53C2211yescharacteristic
57C2×C22⋊C8×C53C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×#88322C2311yescharacteristic
60C22×#88322C2311yescharacteristic
61#13133208—22no—
62#13133208—41no—
63Q8×Dic53—22no—
64Q8×Dic53—41no—
65#13133176—22no—
66#13133176—41no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×#11719919—21no—
75C2×C22⋊Dic53—21no—
76C2×C22⋊Dic53—21no—
77C2×C22⋊Dic53D411yescharacteristic
78C2×C22⋊Dic53D411yescharacteristic
79C2×C4×Dic53—21no—
80C2×C4×Dic53—21no—
81C2×C4×Dic53—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53C2311yescharacteristic
84C2×C4×Dic53C2311yescharacteristic
85#13133150—22no—
86#13133146—41no—
87#13133141—22no—
88#13133141—22no—
89#13133141—41no—
90#13133141—41no—
91C22⋊#88322—41no—
92#13133127—41no—
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#13133124D411yescharacteristic
104#13133124D411yescharacteristic
105C22×Q8×C53C2311yescharacteristic
106C22×Q8×C53C2311yescharacteristic
107C23×C4×C53C2311yescharacteristic
108C2×Q16×C53—22no—
109C2×Q16×C53—41no—
110C2×Q16×C53—41no—
111C22×C8×C53C2311yescharacteristic
112C22×C8×C53C2311yescharacteristic
113C22⋊Q8×C53—22no—
114C22⋊Q8×C53—22no—
115C22⋊Q8×C53—22no—
116C22⋊Q8×C53—41no—
117C2×C4⋊C4×C53—21no—
118C2×C4⋊C4×C53—21no—
119C2×C4⋊C4×C53C2311yescharacteristic
120C2×C4⋊C4×C53C2311yescharacteristic
121C2×C22⋊C4×C53—21no—
122C2×C22⋊C4×C53—21no—
123#103×C53—22no—
124#103×C53—22no—
125C22⋊C8×C53—41no—
126C24×C53C2×D411yescharacteristic
127C2×Q8×C53—21no—
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#4112yes—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2411yescharacteristic
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53C2×D411yescharacteristic
156C22×Dic53C2×D411yescharacteristic
157C22×Dic53C2×D411yescharacteristic
158C22×Dic53C2×D411yescharacteristic
159Q16×C53—44no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C4⋊C4×C53—24no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—42no—
170C4⋊C4×C53#4112yes—
171C4⋊C4×C53#4112yes—
172C22⋊C4×C53—24no—
173C22⋊C4×C53—42no—
174C22⋊Dic53—41no—
175C22⋊Dic53—41no—
176C22⋊Dic53—42no—
177C22⋊Dic53—42no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—42no—
186C4⋊Dic53—42no—
187C4⋊Dic53—42no—
188C4⋊Dic53#4112yes—
189#11719919—42no—
190C4×Dic53—21no—
191C4×Dic53—21no—
192C4×Dic53—21no—
193C4×Dic53—21no—
194C4×Dic53—42no—
195C4×Dic53—42no—
196C4×Dic53—42no—
197C4×Dic53—42no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53C22×D411yescharacteristic
212Q8×C53—22no—
213Q8×C53—22no—
214Q8×C53—41no—
215Q8×C53—41no—
216Q8×C53—41no—
217Q8×C53—41no—
218Q8×C53—41no—
219Q8×C53—41no—
220C2×C4×C53—21no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—22no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×C4×C53C2×#4111yescharacteristic
242C2×C4×C53C2×#4111yescharacteristic
243C2×C4×C53H(5,𝔽2)11yescharacteristic
244C2×C4×C53H(5,𝔽2)11yescharacteristic
245C2×Dic53—22no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—21no—
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—21no—
256C2×Dic53—21no—
257C2×Dic53—21no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C8×C53—41no—
270C8×C53—41no—
271#88322—41no—
272#88322—41no—
273#26749—531nomaximal, Sylow
274C22×C53—41no—
275C22×C53—41no—
276C22×C53—41no—
277C22×C53—41no—
278C22×C53—41no—
279C22×C53—41no—
280C22×C53—41no—
281C22×C53#52511yescharacteristic
282C22×C53#54611yescharacteristic
283C22×C53#54611yescharacteristic
284C22×C53#57511yescharacteristic
285C22×C53#57511yescharacteristic
286C22×C53#57811yescharacteristic
287C22×C53#57811yescharacteristic
288C4×C53—42no—
289C4×C53—41no—
290C4×C53—42no—
291C4×C53—42no—
292C4×C53—81no—
293C4×C53#52911yescharacteristic
294C4×C53#53811yescharacteristic
295C4×C53#55511yescharacteristic
296C4×C53#55911yescharacteristic
297Dic53—41no—
298Dic53—41no—
299Dic53—41no—
300Dic53—42no—
301Dic53—41no—
302Dic53—81no—
303Dic53—81no—
304Dic53—81no—
305#2489—531no—
306#2362—531no—
307#2056—531no—
308#2036—531no—
309#2026—531no—
310#2001D5311yescharacteristic
311#1989—531no—
312#1935—531no—
313#1928—531no—
314#1924—531no—
315#1921—531no—
316#1878—531no—
317#1873—531no—
318#1825—531no—
319#1763—531no—
320C2×C53—81no—
321C2×C53#245911yescharacteristic
322C2×C53#311111yescharacteristic
323C2×C53#312711yescharacteristic
324C2×C53#331311yescharacteristic
325C2×C53#331511yescharacteristic
326C2×C53#331511yescharacteristic
327C2×C53#332711yescharacteristic
328C22×Q16C2×D5311yescharacteristic
329C2×#125—531no—
330C2×C22⋊Q8C2×D5311yescharacteristic
331C2×C22⋊Q8C2×D5311yescharacteristic
332C2×C4×Q8—531no—
333C2×#117—531no—
334C2×#107—531no—
335C2×C4⋊C8—531no—
336C2×#103—531no—
337C2×#103—531no—
338C2×#103—531no—
339C2×#103—531no—
340C2×#103C2×D5311yescharacteristic
341C2×#103C2×D5311yescharacteristic
342C2×C22⋊C8—531no—
343C2×C22⋊C8C2×D5311yescharacteristic
344C2×#97—531no—
345#400—1061no—
346#397—1061no—
347#391—531no—
348#391—531no—
349#391—531no—
350#389—531no—
351#388—1061no—
352#388—1061no—
353#388—1061no—
354#387—1061no—
355#382—1061no—
356#380—531no—
357C4×C4⋊C4—531no—
358C4×C4⋊C4—531no—
359C4×C4⋊C4—1061no—
360C4×C22⋊C4—531no—
361#340—531no—
362#340—531no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368#340C2×D5311yescharacteristic
369#336—531no—
370#336—531no—
371C53#2674911yescharacteristic, complemented, Sylow
372C22×Q8C22×D5311yescharacteristic
373C22×Q8C22×D5311yescharacteristic
374C23×C4C22×D5311yescharacteristic
375C2×Q16—22no—
376C2×Q16—41no—
377C2×Q16—41no—
378C22×C8—531no—
379C22×C8—531no—
380C22×C8C22×D5311yescharacteristic
381C22×C8C22×D5311yescharacteristic
382#125—1062no—
383#125—2121no—
384C22⋊Q8—22no—
385C22⋊Q8—22no—
386C22⋊Q8—22no—
387C22⋊Q8—41no—
388C4×Q8—1062no—
389C4×Q8—2121no—
390#117—1062no—
391#117—2121no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—531no—
395C2×C4⋊C4—531no—
396C2×C4⋊C4—531no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—1061no—
399C2×C4⋊C4—1061no—
400C2×C4⋊C4—1061no—
401C2×C4⋊C4—1061no—
402C2×C4⋊C4C22×D5311yescharacteristic
403C2×C4⋊C4C22×D5311yescharacteristic
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—531no—
407C2×C22⋊C4—531no—
408C2×C22⋊C4—1061no—
409C2×C22⋊C4—1061no—
410C2×C42—531no—
411C2×C42—531no—
412C2×C42—1061no—
413C2×C42—1061no—
414C2×C42—1061no—
415C2×C42—1061no—
416#107—1062no—
417C4⋊C8—2121no—
418#103—22no—
419#103—22no—
420#103—1062no—
421#103—1062no—
422#103—2121no—
423#103—2121no—
424C22⋊C8—41no—
425C22⋊C8—2121no—
426#97—2121no—
427#95—531no—
428#95—531no—
429#95—1061no—
430#95—1061no—
431#95—1061no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438#95—1061no—
439C24D4×D5311yescharacteristic
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8#1171994712yes—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—531no—
463C22×C4—531no—
464C22×C4—531no—
465C22×C4—531no—
466C22×C4—1061no—
467C22×C4—1061no—
468C22×C4—1061no—
469C22×C4—1061no—
470C22×C4D4×D5311yescharacteristic
471C22×C4C23×D5311yescharacteristic
472Q16—44no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—1061no—
480C2×C8—1061no—
481C2×C8—1061no—
482C2×C8—1061no—
483C2×C8—1061no—
484C2×C8—1061no—
485C4⋊C4—24no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—42no—
489C4⋊C4—532no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—1061no—
497C4⋊C4—2122no—
498C4⋊C4—2122no—
499C4⋊C4—2122no—
500C4⋊C4—2122no—
501C4⋊C4#1171994712yes—
502C4⋊C4#1171994712yes—
503C22⋊C4—24no—
504C22⋊C4—42no—
505C22⋊C4—2121no—
506C22⋊C4—2121no—
507C22⋊C4—2122no—
508C22⋊C4—2122no—
509C42—1061no—
510C42—1061no—
511C42—1061no—
512C42—1061no—
513C42—2122no—
514C42—2122no—
515C42—2122no—
516C42—2122no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23C2×D4×D5311yescharacteristic
525Q8—22no—
526Q8—22no—
527Q8—41no—
528Q8—41no—
529Q8—41no—
530Q8—41no—
531Q8—41no—
532Q8—41no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—1061no—
553C2×C4—1061no—
554C2×C4—1061no—
555C2×C4—1061no—
556C2×C4—1061no—
557C2×C4—1061no—
558C2×C4—1061no—
559C2×C4—1061no—
560C2×C4—1061no—
561C2×C4—1061no—
562C2×C4—1061no—
563C2×C4—1061no—
564C2×C4—1062no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4—2121no—
568C2×C4—2121no—
569C2×C4—2121no—
570C2×C4—2121no—
571C2×C4—2121no—
572C2×C4—2121no—
573C2×C4—2121no—
574C2×C4—2121no—
575C2×C4—2121no—
576C2×C4#41×D5311yescharacteristic
577C2×C4C2×#1171994711yescharacteristic
578C2×C4C2×#1171994711yescharacteristic
579C2×C4C2×#1171994711yescharacteristic
580C2×C4#1313326211yescharacteristic
581C2×C4#1313326211yescharacteristic
582C8—41no—
583C8—41no—
584C8—2121no—
585C8—2121no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22#1681112911yescharacteristic
594C22#1681126011yescharacteristic
595C22#1681126211yescharacteristic
596C22#1681142011yescharacteristic
597C22#1681142211yescharacteristic
598C22#1681143611yescharacteristic
599C22#1681143911yescharacteristic
600C4—42no—
601C4—41no—
602C4—42no—
603C4—42no—
604C4—81no—
605C4—4241no—
606C4—4241no—
607C4—4241no—
608C4—2121no—
609C4—2121no—
610C4—2121no—
611C4—2121no—
612C4—2122no—
613C4#1681115411yescharacteristic
614C4#1681120811yescharacteristic
615C4#1681133711yescharacteristic
616C4#1681135911yescharacteristic
617C2—81no—
618C2#2027864711yescharacteristic
619C2#2028482511yescharacteristic
620C2#2028500911yescharacteristic
621C2#2028677411yescharacteristic
622C2#2028681511yescharacteristic
623C2#2028681711yescharacteristic
624C2#2028697811yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2001C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2001D53
hypocenterC53#26749
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2001

Complements

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

NH ≅ G/Ntype
3. #20276035C2semidirect
5. #20275864C2semidirect
6. #20275669C2semidirect
7. #20275407C2semidirect
8. #20275359C2semidirect
10. #20275310C2semidirect
11. #20275058C2semidirect
14. #20274547C2semidirect
371. C53#26749semidirect

Maximal subgroups

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

subgroupindex
#202790582
#202777462
#202760352
#202759152
#202758642
#202756692
#202754072
#202753592
#202753252
#202753102
#202750582
#202750282
#202748112
#202745472
C53×#20012
#2674953