Finite Group Databasebeta

← #27469884

Subgroups — #27469884

Order 13568.

730 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20280485C211yesmaximal, characteristic
2#20277339C211yesmaximal, characteristic
3#20276137C211yesmaximal, characteristic, complemented
4#20276136C211yesmaximal, characteristic, complemented
5#20276127C211yesmaximal, characteristic, complemented
6#20276126C211yesmaximal, characteristic, complemented
7#20275533C211yesmaximal, characteristic, complemented
8#20275515C211yesmaximal, characteristic, complemented
9#20275028C211yesmaximal, characteristic
10#20275025C211yesmaximal, characteristic
11#20275022C211yesmaximal, characteristic
12#20274847C211yesmaximal, characteristic, complemented
13#20274844C211yesmaximal, characteristic, complemented
14#20274483C211yesmaximal, characteristic
15C53×#1872C211yesmaximal, characteristic
16C2×#13133215C2211yescharacteristic
17C2×#13133196C2211yescharacteristic
18C2×#13133176C2211yescharacteristic
19C2×#13133176C2211yescharacteristic
20C2×#13133142C2211yescharacteristic
21C2×#13133142C2211yescharacteristic
22C2×#13133141C2211yescharacteristic
23C2×#13133141C2211yescharacteristic
24C2×#13133140C2211yescharacteristic
25C2×#13133140C2211yescharacteristic
26C2×C22⋊#88322C2211yescharacteristic
27C2×#13133127C2211yescharacteristic
28C2×C8×Dic53C2211yescharacteristic
29#16810556—21no—
30#16810556—21no—
31#16810551—21no—
32#16810551—21no—
33#16810530C2211yescharacteristic
34#16810528C2211yescharacteristic
35#16810519—21no—
36#16810519—21no—
37#16810496—21no—
38#16810496—21no—
39#16810492C2211yescharacteristic
40#16810492C2211yescharacteristic
41#16810488C2211yescharacteristic
42#16810488C2211yescharacteristic
43#16810488C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×C22⋊Q8×C53C2211yescharacteristic
53C22×C4⋊C4×C53C2211yescharacteristic
54C2×#103×C53C2211yescharacteristic
55C2×#103×C53C2211yescharacteristic
56C2×C22⋊C8×C53C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×Dic106C2311yescharacteristic
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62#13133215—22no—
63#13133215—22no—
64#13133196—22no—
65#13133196—22no—
66#13133176—22no—
67#13133176—22no—
68#13133176—41no—
69#13133176—41no—
70C2×C4⋊Dic53D411yescharacteristic
71C2×C4⋊Dic53D411yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×C22⋊Dic53—21no—
83C2×C22⋊Dic53—21no—
84C2×C22⋊Dic53D411yescharacteristic
85C2×C22⋊Dic53D411yescharacteristic
86C2×C4×Dic53—21no—
87C2×C4×Dic53C2311yescharacteristic
88C2×C4×Dic53C2311yescharacteristic
89#13133142—41no—
90#13133142—41no—
91#13133141—41no—
92#13133141—41no—
93#13133140—41no—
94#13133140—41no—
95C22⋊#88322—22no—
96#13133127—22no—
97C8×Dic53—22no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108C22×Q8×C53C2311yescharacteristic
109C23×C4×C53—21no—
110C23×C4×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112C22×C8×C53C2311yescharacteristic
113C22⋊Q8×C53—22no—
114C22⋊Q8×C53—22no—
115C2×C4⋊C4×C53—21no—
116C2×C4⋊C4×C53—21no—
117C2×C4⋊C4×C53—41no—
118C2×C4⋊C4×C53—41no—
119C2×C4⋊C4×C53C2311yescharacteristic
120C2×C4⋊C4×C53C2311yescharacteristic
121C2×C4⋊C4×C53C2311yescharacteristic
122C2×C22⋊C4×C53—21no—
123#103×C53—41no—
124#103×C53—41no—
125C22⋊C8×C53—22no—
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—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—42no—
142C22×C4×C53—81no—
143C22×C4×C53—81no—
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53#4112yes—
146C22×C4×C53#4112yes—
147C22×C4×C53C2411yescharacteristic
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53C2×D411yescharacteristic
153C22×Dic53C2×D411yescharacteristic
154C22×Dic53C2×D411yescharacteristic
155C22×Dic53C2×D411yescharacteristic
156C2×Dic106—21no—
157C2×Dic106—21no—
158C2×Dic106—21no—
159C2×Dic106—21no—
160C2×Dic106—21no—
161C2×Dic106—21no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—42no—
175C4⋊C4×C53—42no—
176C22⋊C4×C53—42no—
177C22⋊Dic53—22no—
178C22⋊Dic53—22no—
179C22⋊Dic53—42no—
180C22⋊Dic53—42no—
181C4⋊Dic53—22no—
182C4⋊Dic53—22no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189#11719919—42no—
190#11719919—42no—
191#11719919—42no—
192#11719919—81no—
193#11719919—81no—
194#11719919—81no—
195#11719919—81no—
196C4×Dic53—21no—
197C4×Dic53—21no—
198C4×Dic53—42no—
199C4×Dic53#4112yes—
200C4×Dic53#4112yes—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53#12412yes—
214C23×C53C22×D411yescharacteristic
215Q8×C53—41no—
216Q8×C53—41no—
217Q8×C53—41no—
218Q8×C53—41no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
225C2×C4×C53—41no—
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—84no—
238C2×C4×C53#12412yes—
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—21no—
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—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
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#12312yes—
268C2×Dic53#12312yes—
269Dic106—41no—
270Dic106—41no—
271Dic106—41no—
272Dic106—41no—
273C8×C53—41no—
274C8×C53—41no—
275#88322—41no—
276#88322—41no—
277#27985—531nomaximal, Sylow
278C22×C53—22no—
279C22×C53—22no—
280C22×C53—22no—
281C22×C53—41no—
282C22×C53—41no—
283C22×C53—41no—
284C22×C53—41no—
285C22×C53C2×#12311yescharacteristic
286C22×C53C2×#12411yescharacteristic
287C22×C53#53511yescharacteristic
288C22×C53#57711yescharacteristic
289C22×C53#57711yescharacteristic
290C22×C53#57811yescharacteristic
291C22×C53#57811yescharacteristic
292C4×C53—22no—
293C4×C53—81no—
294C4×C53—81no—
295C4×C53—81no—
296C4×C53—81no—
297C4×C53#55311yescharacteristic
298C4×C53#55311yescharacteristic
299C4×C53#55311yescharacteristic
300C4×C53#55311yescharacteristic
301Dic53—22no—
302Dic53—22no—
303Dic53—41no—
304Dic53—41no—
305Dic53—81no—
306Dic53—81no—
307Dic53—81no—
308Dic53—81no—
309#2621—531no—
310#2317—531no—
311#2074—531no—
312#2074—531no—
313#2072—531no—
314#2072—531no—
315#1960—531no—
316#1956—531no—
317#1873—531no—
318#1873—531no—
319#1872—531no—
320#1872D5311yescharacteristic
321#1836—531no—
322#1835—531no—
323#1750—531no—
324C2×C53—42no—
325C2×C53#250411yescharacteristic
326C2×C53#309711yescharacteristic
327C2×C53#309711yescharacteristic
328C2×C53#313711yescharacteristic
329C2×C53#313711yescharacteristic
330C2×C53#322211yescharacteristic
331C2×C53#322311yescharacteristic
332C2×C4⋊Q8—531no—
333C2×C22⋊Q8—531no—
334C2×C22⋊Q8C2×D5311yescharacteristic
335C2×#117—531no—
336C2×#117—531no—
337C22×C4⋊C4C2×D5311yescharacteristic
338C2×#103—531no—
339C2×#103—531no—
340C2×#103—531no—
341C2×#103—531no—
342C2×#103—531no—
343C2×#103—531no—
344C2×#103C2×D5311yescharacteristic
345C2×#103C2×D5311yescharacteristic
346C2×C22⋊C8—531no—
347C2×C22⋊C8C2×D5311yescharacteristic
348C2×#97—531no—
349C2×C4×C8—531no—
350#398—1061no—
351#398—1061no—
352#397—1061no—
353#397—1061no—
354#391—531no—
355#391—531no—
356#389—1061no—
357#389—1061no—
358#385—1061no—
359#385—1061no—
360#384—531no—
361#384—531no—
362#382—531no—
363#382—531no—
364#382—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368#340—531no—
369#340—531no—
370#340—531no—
371#340C2×D5311yescharacteristic
372#340C2×D5311yescharacteristic
373#336—531no—
374#336—531no—
375C53#2798511yescharacteristic, complemented, Sylow
376C22×Q8—531no—
377C22×Q8C22×D5311yescharacteristic
378C23×C4—21no—
379C23×C4C22×D5311yescharacteristic
380C22×C8—531no—
381C22×C8—531no—
382C22×C8C22×D5311yescharacteristic
383C22×C8C22×D5311yescharacteristic
384C4⋊Q8—1062no—
385C4⋊Q8—1062no—
386C22⋊Q8—22no—
387C22⋊Q8—22no—
388C22⋊Q8—1062no—
389C22⋊Q8—1062no—
390#117—1062no—
391#117—1062no—
392#117—2121no—
393#117—2121no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—41no—
397C2×C4⋊C4—41no—
398C2×C4⋊C4—531no—
399C2×C4⋊C4—531no—
400C2×C4⋊C4—531no—
401C2×C4⋊C4—531no—
402C2×C4⋊C4—531no—
403C2×C4⋊C4—1061no—
404C2×C4⋊C4—1061no—
405C2×C4⋊C4—1061no—
406C2×C4⋊C4—1061no—
407C2×C4⋊C4—1061no—
408C2×C4⋊C4—1061no—
409C2×C4⋊C4—1061no—
410C2×C4⋊C4C22×D5311yescharacteristic
411C2×C4⋊C4C22×D5311yescharacteristic
412C2×C4⋊C4C22×D5311yescharacteristic
413C2×C22⋊C4—21no—
414C2×C22⋊C4—531no—
415C2×C22⋊C4—531no—
416C2×C22⋊C4—1061no—
417C2×C22⋊C4—1061no—
418C2×C42—531no—
419C2×C42—531no—
420C2×C42—1061no—
421#103—41no—
422#103—41no—
423#103—2121no—
424#103—2121no—
425#103—2121no—
426#103—2121no—
427#103—2121no—
428#103—2121no—
429C22⋊C8—22no—
430C22⋊C8—1062no—
431#97—1062no—
432C4×C8—1062no—
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—
443C24D4×D5311yescharacteristic
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—1061no—
451C2×Q8—1061no—
452C2×Q8—1061no—
453C2×Q8—1061no—
454C2×Q8—1061no—
455C2×Q8—1061no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—42no—
465C22×C4—81no—
466C22×C4—81no—
467C22×C4—531no—
468C22×C4—531no—
469C22×C4—531no—
470C22×C4—531no—
471C22×C4—1061no—
472C22×C4—1061no—
473C22×C4—1061no—
474C22×C4—1061no—
475C22×C4#1171994512yes—
476C22×C4D4×D5311yescharacteristic
477C22×C4#1171994712yes—
478C22×C4C23×D5311yescharacteristic
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—1061no—
486C2×C8—1061no—
487C2×C8—1061no—
488C2×C8—1061no—
489C2×C8—1061no—
490C2×C8—1061no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—1062no—
500C4⋊C4—1062no—
501C4⋊C4—1061no—
502C4⋊C4—1061no—
503C4⋊C4—1061no—
504C4⋊C4—1061no—
505C4⋊C4—1061no—
506C4⋊C4—1061no—
507C4⋊C4—4241no—
508C4⋊C4—4241no—
509C4⋊C4—4241no—
510C4⋊C4—4241no—
511C4⋊C4—2122no—
512C4⋊C4—2122no—
513C4⋊C4—2122no—
514C22⋊C4—42no—
515C22⋊C4—1062no—
516C22⋊C4—1062no—
517C22⋊C4—2122no—
518C22⋊C4—2122no—
519C42—532no—
520C42—532no—
521C42—1061no—
522C42—1061no—
523C42—2122no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23#1313320312yes—
531C23C2×D4×D5311yescharacteristic
532Q8—41no—
533Q8—41no—
534Q8—41no—
535Q8—41no—
536Q8—2121no—
537Q8—2121no—
538Q8—2121no—
539Q8—2121no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—84no—
559C2×C4—532no—
560C2×C4—532no—
561C2×C4—1061no—
562C2×C4—1061no—
563C2×C4—1061no—
564C2×C4—1061no—
565C2×C4—1061no—
566C2×C4—1061no—
567C2×C4—1061no—
568C2×C4—1061no—
569C2×C4—1061no—
570C2×C4—1061no—
571C2×C4—2121no—
572C2×C4—2121no—
573C2×C4—2121no—
574C2×C4—2121no—
575C2×C4—2121no—
576C2×C4—2121no—
577C2×C4—2121no—
578C2×C4—2121no—
579C2×C4—2121no—
580C2×C4—2121no—
581C2×C4—2121no—
582C2×C4—2121no—
583C2×C4#1313320312yes—
584C2×C4#41×D5311yescharacteristic
585C2×C4#41×D5311yescharacteristic
586C2×C4C2×#1171994711yescharacteristic
587C2×C4C2×#1171994511yescharacteristic
588C2×C4#1313326211yescharacteristic
589C2×C4#1313326211yescharacteristic
590C8—41no—
591C8—41no—
592C8—2121no—
593C8—2121no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22#123×D5311yescharacteristic
602C22C2×#1313320311yescharacteristic
603C22#1681118811yescharacteristic
604C22#1681142911yescharacteristic
605C22#1681143311yescharacteristic
606C22#1681143611yescharacteristic
607C22#1681143911yescharacteristic
608C4—22no—
609C4—81no—
610C4—81no—
611C4—81no—
612C4—81no—
613C4—1062no—
614C4—1062no—
615C4—4241no—
616C4—4241no—
617C4—4241no—
618C4—4241no—
619C4—2121no—
620C4—2121no—
621C4#1681131611yescharacteristic
622C4#1681131611yescharacteristic
623C4#1681132211yescharacteristic
624C4#1681132211yescharacteristic
625C2—42no—
626C2#2027920511yescharacteristic
627C2#2028467711yescharacteristic
628C2#2028468311yescharacteristic
629C2#2028507811yescharacteristic
630C2#2028508311yescharacteristic
631C2#2028578311yescharacteristic
632C2#2028579411yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1872C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1872D53
hypocenterC53#27985
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
3. #20276137C2semidirect
4. #20276136C2semidirect
5. #20276127C2semidirect
6. #20276126C2semidirect
7. #20275533C2semidirect
8. #20275515C2semidirect
12. #20274847C2semidirect
13. #20274844C2semidirect
375. C53#27985semidirect

Maximal subgroups

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

subgroupindex
#202804852
#202773392
#202761372
#202761362
#202761272
#202761262
#202755332
#202755152
#202750282
#202750252
#202750222
#202748472
#202748442
#202744832
C53×#18722
#2798553