Finite Group Databasebeta

← #27483694

Subgroups — #27483694

Order 13568.

794 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20280549C211yesmaximal, characteristic
2#20277301C211yesmaximal, characteristic
3#20276126C211yesmaximal, characteristic, complemented
4#20276126C211yesmaximal, characteristic, complemented
5#20275864C211yesmaximal, characteristic, complemented
6#20275864C211yesmaximal, characteristic, complemented
7#20275407C211yesmaximal, characteristic, complemented
8#20275402C211yesmaximal, characteristic, complemented
9#20275393C211yesmaximal, characteristic, complemented
10#20275388C211yesmaximal, characteristic, complemented
11#20274811C211yesmaximal, characteristic
12#20274807C211yesmaximal, characteristic
13#20274650C211yesmaximal, characteristic
14#20274483C211yesmaximal, characteristic
15C53×#1787C211yesmaximal, characteristic
16C2×#13133208C2211yescharacteristic
17C2×#13133176C2211yescharacteristic
18C2×#13133176C2211yescharacteristic
19C2×#13133141C2211yescharacteristic
20C2×#13133141C2211yescharacteristic
21C2×#13133141C2211yescharacteristic
22C2×#13133141C2211yescharacteristic
23C2×C22⋊#88322C2211yescharacteristic
24C2×#13133127C2211yescharacteristic
25C2×C8×Dic53C2211yescharacteristic
26#16810557—21no—
27#16810555—21no—
28#16810546—21no—
29#16810546—21no—
30#16810530—21no—
31#16810530—21no—
32#16810530C2211yescharacteristic
33#16810530C2211yescharacteristic
34#16810530C2211yescharacteristic
35#16810530C2211yescharacteristic
36#16810522C2211yescharacteristic
37#16810519C2211yescharacteristic
38#16810492C2211yescharacteristic
39#16810486—21no—
40#16810486—21no—
41#16810483C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
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#13133208—41no—
62#13133208—41no—
63#13133176—22no—
64#13133176—22no—
65#13133176—41no—
66#13133176—41no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53D411yescharacteristic
70C2×C4⋊Dic53D411yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53D411yescharacteristic
82C2×C22⋊Dic53D411yescharacteristic
83C2×C4×Dic53—21no—
84C2×C4×Dic53C2311yescharacteristic
85C2×C4×Dic53C2311yescharacteristic
86#13133141—22no—
87#13133141—22no—
88#13133141—41no—
89#13133141—41no—
90C22⋊#88322—22no—
91#13133127—41no—
92C8×Dic53—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#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—41no—
181C22⋊Dic53—41no—
182C22⋊Dic53—42no—
183C22⋊Dic53—81no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—41no—
193C4⋊Dic53—41no—
194C4⋊Dic53—42no—
195C4⋊Dic53—81no—
196#11719919—42no—
197#11719919—81no—
198#11719919—81no—
199#11719919—81no—
200C4×Dic53—21no—
201C4×Dic53—21no—
202C4×Dic53—21no—
203C4×Dic53—21no—
204C4×Dic53—42no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53#12412yes—
218C23×C53C22×D411yescharacteristic
219Q8×C53—22no—
220Q8×C53—22no—
221Q8×C53—22no—
222Q8×C53—22no—
223Q8×C53—48no—
224Q8×C53—41no—
225Q8×C53—41no—
226Q8×C53—41no—
227Q8×C53—41no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—22no—
231C2×C4×C53—22no—
232C2×C4×C53—22no—
233C2×C4×C53—22no—
234C2×C4×C53—22no—
235C2×C4×C53—22no—
236C2×C4×C53—48no—
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—41no—
246C2×C4×C53—41no—
247C2×C4×C53#12412yes—
248C2×C4×C53C2×#4111yescharacteristic
249C2×C4×C53C2×#4111yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C53C2×#4111yescharacteristic
252C2×C4×C53H(5,𝔽2)11yescharacteristic
253C2×C4×C53H(5,𝔽2)11yescharacteristic
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—21no—
265C2×Dic53—21no—
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—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C8×C53—41no—
279C8×C53—41no—
280#88322—41no—
281#88322—41no—
282#29117—531nomaximal, Sylow
283C22×C53—22no—
284C22×C53—22no—
285C22×C53—22no—
286C22×C53—41no—
287C22×C53—41no—
288C22×C53—41no—
289C22×C53—41no—
290C22×C53#52511yescharacteristic
291C22×C53C2×#12411yescharacteristic
292C22×C53#53411yescharacteristic
293C22×C53#57711yescharacteristic
294C22×C53#57711yescharacteristic
295C22×C53#57811yescharacteristic
296C22×C53#57811yescharacteristic
297C4×C53—22no—
298C4×C53—42no—
299C4×C53—42no—
300C4×C53—81no—
301C4×C53—81no—
302C4×C53#55111yescharacteristic
303C4×C53#55111yescharacteristic
304C4×C53#55111yescharacteristic
305C4×C53#55111yescharacteristic
306Dic53—41no—
307Dic53—41no—
308Dic53—41no—
309Dic53—41no—
310Dic53—81no—
311Dic53—81no—
312Dic53—81no—
313Dic53—81no—
314#2629—531no—
315#2310—531no—
316#2072—531no—
317#2072—531no—
318#2026—531no—
319#2026—531no—
320#1935—531no—
321#1935—531no—
322#1933—531no—
323#1933—531no—
324#1825—531no—
325#1825—531no—
326#1787—531no—
327#1787D5311yescharacteristic
328#1750—531no—
329C2×C53—42no—
330C2×C53#256111yescharacteristic
331C2×C53#312611yescharacteristic
332C2×C53#312711yescharacteristic
333C2×C53#313711yescharacteristic
334C2×C53#313711yescharacteristic
335C2×C53#320111yescharacteristic
336C2×C53#320111yescharacteristic
337C23×Q8C2×D5311yescharacteristic
338C2×#125—531no—
339C2×#117—531no—
340C2×#117—531no—
341C2×#103—531no—
342C2×#103—531no—
343C2×#103—531no—
344C2×#103—531no—
345C2×#103C2×D5311yescharacteristic
346C2×#103C2×D5311yescharacteristic
347C2×#103C2×D5311yescharacteristic
348C2×#103C2×D5311yescharacteristic
349C2×C22⋊C8—531no—
350C2×C22⋊C8C2×D5311yescharacteristic
351C2×#97—531no—
352C2×C4×C8—531no—
353#398—1061no—
354#398—1061no—
355#396—1061no—
356#396—1061no—
357#391—531no—
358#391—531no—
359#391—531no—
360#391—531no—
361#391—1061no—
362#391—1061no—
363#389—531no—
364#389—531no—
365#384—531no—
366#381—1061no—
367#381—1061no—
368#380—531no—
369#380C2×D5311yescharacteristic
370#340—531no—
371#340—531no—
372#340—531no—
373#340—531no—
374#340—531no—
375#340—531no—
376#340—531no—
377#340—531no—
378#336—531no—
379#336—531no—
380C53#2911711yescharacteristic, complemented, Sylow
381C22×Q8—22no—
382C22×Q8—22no—
383C22×Q8—21no—
384C22×Q8—22no—
385C22×Q8—22no—
386C22×Q8—41no—
387C22×Q8—41no—
388C22×Q8C22×D5311yescharacteristic
389C22×Q8C22×D5311yescharacteristic
390C23×C4—21no—
391C23×C4C22×D5311yescharacteristic
392C22×C8—531no—
393C22×C8—531no—
394C22×C8C22×D5311yescharacteristic
395C22×C8C22×D5311yescharacteristic
396#125—2121no—
397#125—2121no—
398#117—1062no—
399#117—1062no—
400#117—2121no—
401#117—2121no—
402C2×C4⋊C4—531no—
403C2×C4⋊C4—531no—
404C2×C4⋊C4—531no—
405C2×C4⋊C4—531no—
406C2×C4⋊C4—531no—
407C2×C4⋊C4—531no—
408C2×C4⋊C4—1061no—
409C2×C4⋊C4—1061no—
410C2×C4⋊C4—1061no—
411C2×C4⋊C4—1061no—
412C2×C4⋊C4—1061no—
413C2×C4⋊C4—1061no—
414C2×C4⋊C4C22×D5311yescharacteristic
415C2×C4⋊C4C22×D5311yescharacteristic
416C2×C22⋊C4—21no—
417C2×C22⋊C4—531no—
418C2×C22⋊C4—531no—
419C2×C22⋊C4—1061no—
420C2×C22⋊C4—1061no—
421C2×C42—531no—
422C2×C42—531no—
423C2×C42—1061no—
424#103—22no—
425#103—22no—
426#103—41no—
427#103—41no—
428#103—1062no—
429#103—1062no—
430#103—2121no—
431#103—2121no—
432C22⋊C8—22no—
433C22⋊C8—1062no—
434#97—2121no—
435C4×C8—2121no—
436#95—21no—
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—
446#95—1061no—
447#95—1061no—
448C24D4×D5311yescharacteristic
449C2×Q8—24no—
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—21no—
459C2×Q8—21no—
460C2×Q8—44no—
461C2×Q8—44no—
462C2×Q8—44no—
463C2×Q8—44no—
464C2×Q8—42no—
465C2×Q8—42no—
466C2×Q8#1171994712yes—
467C2×Q8#1171994712yes—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—24no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—531no—
480C22×C4—531no—
481C22×C4—531no—
482C22×C4—531no—
483C22×C4—1061no—
484C22×C4—1061no—
485C22×C4—1061no—
486C22×C4—1061no—
487C22×C4#1171994512yes—
488C22×C4D4×D5311yescharacteristic
489C22×C4#1171994712yes—
490C22×C4C23×D5311yescharacteristic
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—1061no—
498C2×C8—1061no—
499C2×C8—1061no—
500C2×C8—1061no—
501C2×C8—1061no—
502C2×C8—1061no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
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—
517C4⋊C4—4241no—
518C4⋊C4—4241no—
519C4⋊C4—2121no—
520C4⋊C4—2121no—
521C4⋊C4—2122no—
522C4⋊C4—2122no—
523C22⋊C4—81no—
524C22⋊C4—4241no—
525C22⋊C4—2121no—
526C22⋊C4—2121no—
527C22⋊C4—2122no—
528C42—1061no—
529C42—1061no—
530C42—1061no—
531C42—1061no—
532C42—2122no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23#1313320312yes—
540C23C2×D4×D5311yescharacteristic
541Q8—22no—
542Q8—22no—
543Q8—22no—
544Q8—22no—
545Q8—48no—
546Q8—41no—
547Q8—41no—
548Q8—41no—
549Q8—41no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—48no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
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—1061no—
578C2×C4—1061no—
579C2×C4—1061no—
580C2×C4—1061no—
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—2121no—
590C2×C4—2121no—
591C2×C4—2121no—
592C2×C4—2121no—
593C2×C4#1313320312yes—
594C2×C4C2×#1171994711yescharacteristic
595C2×C4C2×#1171994711yescharacteristic
596C2×C4C2×#1171994711yescharacteristic
597C2×C4C2×#1171994511yescharacteristic
598C2×C4#1313326211yescharacteristic
599C2×C4#1313326211yescharacteristic
600C8—41no—
601C8—41no—
602C8—2121no—
603C8—2121no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22#1681112911yescharacteristic
612C22C2×#1313320311yescharacteristic
613C22#1681118311yescharacteristic
614C22#1681142911yescharacteristic
615C22#1681143211yescharacteristic
616C22#1681143611yescharacteristic
617C22#1681143911yescharacteristic
618C4—22no—
619C4—42no—
620C4—42no—
621C4—81no—
622C4—81no—
623C4—4241no—
624C4—4241no—
625C4—4241no—
626C4—4241no—
627C4—2121no—
628C4—2121no—
629C4—2121no—
630C4—2121no—
631C4#1681129911yescharacteristic
632C4#1681129911yescharacteristic
633C4#1681130411yescharacteristic
634C4#1681130411yescharacteristic
635C2—42no—
636C2#2027987011yescharacteristic
637C2#2028500011yescharacteristic
638C2#2028500911yescharacteristic
639C2#2028507611yescharacteristic
640C2#2028507811yescharacteristic
641C2#2028559311yescharacteristic
642C2#2028560111yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1787C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1787D53
hypocenterC53#29117
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. #20276126C2semidirect
4. #20276126C2semidirect
5. #20275864C2semidirect
6. #20275864C2semidirect
7. #20275407C2semidirect
8. #20275402C2semidirect
9. #20275393C2semidirect
10. #20275388C2semidirect
380. C53#29117semidirect

Maximal subgroups

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

subgroupindex
#202805492
#202773012
#202761262
#202758642
#202754072
#202754022
#202753932
#202753882
#202748112
#202748072
#202746502
#202744832
C53×#17872
#2911753