Finite Group Databasebeta

← #27488243

Subgroups — #27488243

Order 13568.

724 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20280680C211yesmaximal, characteristic, complemented
2#20276192C211yesmaximal, characteristic, complemented
3#20276124C211yesmaximal, characteristic, complemented
4#20275951C211yesmaximal, characteristic
5#20275809C211yesmaximal, characteristic
6#20275585C211yesmaximal, characteristic, complemented
7#20275568C211yesmaximal, characteristic
8#20275415C211yesmaximal, characteristic
9#20275394C211yesmaximal, characteristic, complemented
10#20275056C211yesmaximal, characteristic, complemented
11#20275049C211yesmaximal, characteristic
12#20274983C211yesmaximal, characteristic, complemented
13#20274976C211yesmaximal, characteristic
14#20274467C211yesmaximal, characteristic, complemented
15C53×#2326C211yesmaximal, characteristic
16C22×#11719925C2211yescharacteristic
17C22×#11719923C2211yescharacteristic
18C2×#13133148C2211yescharacteristic
19C2×#13133148C2211yescharacteristic
20C2×#13133141C2211yescharacteristic
21C2×#13133141C2211yescharacteristic
22C2×#13133141C2211yescharacteristic
23C2×#13133140C2211yescharacteristic
24C2×#13133140C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×#13133138C2211yescharacteristic
28C2×#13133128C2211yescharacteristic
29C2×C4×#88322C2211yescharacteristic
30C53⋊#401—21no—
31#16810564—21no—
32#16810564—21no—
33#16810564—21no—
34#16810529C2211yescharacteristic
35#16810529C2211yescharacteristic
36#16810529C2211yescharacteristic
37#16810521C2211yescharacteristic
38#16810521C2211yescharacteristic
39#16810521C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810378C2211yescharacteristic
45#16810378C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×#124×C53C2211yescharacteristic
49C2×C4×Q8×C53C2211yescharacteristic
50C2×C4×D4×C53C2211yescharacteristic
51C53×#391C2211yescharacteristic
52C53×#390C2211yescharacteristic
53C53×#388—21no—
54C53×#388—21no—
55C53×#385—21no—
56C53×#385—21no—
57C53×#384C2211yescharacteristic
58C53×#376C2211yescharacteristic
59C22×Dic106C2311yescharacteristic
60C2×#11719925—22no—
61C2×#11719925—22no—
62C2×#11719925—41no—
63C2×#11719923—22no—
64C2×#11719923—22no—
65C2×#11719923—41no—
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719919—21no—
76#13133148—41no—
77#13133148—41no—
78#13133141—22no—
79#13133141—22no—
80#13133141—41no—
81#13133140—41no—
82#13133140—41no—
83#13133138—22no—
84#13133138—22no—
85#13133138—41no—
86#13133128—22no—
87C4×#88322—22no—
88#13133124—21no—
89#13133124—21no—
90#13133124—21no—
91#13133124—21no—
92#13133124—21no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97C22×Q8×C53C2311yescharacteristic
98C22×D4×C53C2311yescharacteristic
99C23×C4×C53—21no—
100#124×C53—22no—
101#124×C53—22no—
102C4×Q8×C53—41no—
103C4×Q8×C53—41no—
104C4×D4×C53—41no—
105C4×D4×C53—41no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53—21no—
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C22⋊C4×C53—21no—
112C2×C22⋊C4×C53—21no—
113C2×C22⋊C4×C53—21no—
114C2×C22⋊C4×C53—21no—
115C2×C22⋊C4×C53—21no—
116C2×C42×C53—21no—
117C2×C42×C53C2311yescharacteristic
118C2×C42×C53C2311yescharacteristic
119C2×C42×C53C2311yescharacteristic
120#95×C53—21no—
121#95×C53—21no—
122#95×C53D411yescharacteristic
123#95×C53D411yescharacteristic
124#95×C53D411yescharacteristic
125#95×C53D411yescharacteristic
126C24×C53—21no—
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53—21no—
131C2×Q8×C53—21no—
132C2×Q8×C53#4112yes—
133C2×D4×C53—21no—
134C2×D4×C53—21no—
135C2×D4×C53—21no—
136C2×D4×C53—21no—
137C2×D4×C53—21no—
138C2×D4×C53#4112yes—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—42no—
143C22×C4×C53—81no—
144C22×C4×C53—81no—
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2411yescharacteristic
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C2×Dic106—21no—
157C2×Dic106—21no—
158C2×Dic106—21no—
159C2×Dic106—21no—
160C2×Dic106—21no—
161C2×Dic106—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—42no—
167C4⋊C4×C53—81no—
168C4⋊C4×C53—81no—
169C22⋊C4×C53—24no—
170C22⋊C4×C53—24no—
171C22⋊C4×C53—42no—
172C22⋊C4×C53—81no—
173C22⋊C4×C53—81no—
174C42×C53—21no—
175C42×C53—21no—
176C42×C53—21no—
177C42×C53—21no—
178C42×C53—21no—
179C42×C53—21no—
180C42×C53—42no—
181#11719925—44no—
182#11719923—44no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189#11719919—81no—
190#11719919—81no—
191#11719919—81no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322#4112yes—
203C2×#88322#4112yes—
204C23×C53—22no—
205C23×C53—41no—
206C23×C53—41no—
207C23×C53—41no—
208C23×C53—41no—
209C23×C53—41no—
210C23×C53—41no—
211C23×C53C22×D411yescharacteristic
212Q8×C53—22no—
213Q8×C53—22no—
214Q8×C53—41no—
215Q8×C53—41no—
216D4×C53—22no—
217D4×C53—22no—
218D4×C53—41no—
219D4×C53—41no—
220C2×C4×C53—22no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
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—84no—
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—41no—
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—
266Dic106—41no—
267Dic106—41no—
268Dic106—41no—
269Dic106—41no—
270#88322—22no—
271#88322—22no—
272#88322—41no—
273#88322—41no—
274#29506—531nomaximal, Sylow
275C22×C53—42no—
276C22×C53—42no—
277C22×C53—42no—
278C22×C53—81no—
279C22×C53—81no—
280C22×C53—81no—
281C22×C53—81no—
282C22×C53#52511yescharacteristic
283C22×C53#53211yescharacteristic
284C22×C53#53511yescharacteristic
285C22×C53#57711yescharacteristic
286C22×C53#57711yescharacteristic
287C22×C53#57711yescharacteristic
288C22×C53#57811yescharacteristic
289C4×C53—42no—
290C4×C53—41no—
291C4×C53—41no—
292C4×C53—41no—
293C4×C53—41no—
294C4×C53—41no—
295C4×C53—41no—
296C4×C53—81no—
297C4×C53—81no—
298C4×C53#55111yescharacteristic
299C4×C53#55111yescharacteristic
300C4×C53#55311yescharacteristic
301C4×C53#55311yescharacteristic
302Dic53—81no—
303Dic53—81no—
304Dic53—81no—
305Dic53—81no—
306#2645—531no—
307#2326D5311yescharacteristic
308#2084—531no—
309#2072—531no—
310#2042—531no—
311#2015—531no—
312#1971—531no—
313#1968—531no—
314#1936—531no—
315#1933—531no—
316#1878—531no—
317#1877—531no—
318#1865—531no—
319#1864—531no—
320#1746—531no—
321C2×C53—82no—
322C2×C53#260311yescharacteristic
323C2×C53#312011yescharacteristic
324C2×C53#312511yescharacteristic
325C2×C53#317311yescharacteristic
326C2×C53#317811yescharacteristic
327C2×C53#322211yescharacteristic
328C2×C53#322511yescharacteristic
329C22×Q16—531no—
330C22×SD16—531no—
331C2×#124C2×D5311yescharacteristic
332C2×C4×Q8C2×D5311yescharacteristic
333C2×C4×D4C2×D5311yescharacteristic
334C2×#106—531no—
335C2×#106—531no—
336C2×#103—531no—
337C2×#103—531no—
338C2×#103—531no—
339C2×#103—531no—
340C2×#103—531no—
341C2×#102—531no—
342C2×#102—531no—
343C2×#102—531no—
344C2×#97—531no—
345C2×C4×C8—531no—
346#401—1061no—
347#400—1061no—
348#400—1061no—
349#400—1061no—
350#391—531no—
351#391—531no—
352#391—531no—
353#391C2×D5311yescharacteristic
354#390C2×D5311yescharacteristic
355#389—531no—
356#389—531no—
357#389—531no—
358#388—21no—
359#388—21no—
360#385—21no—
361#385—21no—
362#384C2×D5311yescharacteristic
363#376C2×D5311yescharacteristic
364#340—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368#336—531no—
369#336—531no—
370#336—531no—
371#336—531no—
372C53#2950611yescharacteristic, complemented, Sylow
373C22×Q8—531no—
374C22×Q8C22×D5311yescharacteristic
375C22×D4C22×D5311yescharacteristic
376C23×C4—21no—
377C2×Q16—1062no—
378C2×Q16—1062no—
379C2×Q16—2121no—
380C2×SD16—1062no—
381C2×SD16—1062no—
382C2×SD16—2121no—
383C22×C8—531no—
384C22×C8—531no—
385C22×C8—531no—
386C22×C8—531no—
387#124—22no—
388#124—22no—
389C4×Q8—41no—
390C4×Q8—41no—
391C4×D4—41no—
392C4×D4—41no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—531no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—531no—
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—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C42—21no—
410C2×C42C22×D5311yescharacteristic
411C2×C42C22×D5311yescharacteristic
412C2×C42C22×D5311yescharacteristic
413#106—2121no—
414#106—2121no—
415#103—1062no—
416#103—1062no—
417#103—2121no—
418#103—2121no—
419#103—2121no—
420#102—1062no—
421#102—1062no—
422#102—2121no—
423#97—1062no—
424C4×C8—1062no—
425#95—21no—
426#95—21no—
427#95—1061no—
428#95—1061no—
429#95—1061no—
430#95—1061no—
431#95—1061no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436#95C22⋊D5311yescharacteristic
437#95C22⋊D5311yescharacteristic
438#95C22⋊D5311yescharacteristic
439#95C22⋊D5311yescharacteristic
440C24—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—1061no—
447C2×Q8—1061no—
448C2×Q8—1061no—
449C2×Q8—1061no—
450C2×Q8—1061no—
451C2×Q8—1061no—
452C2×Q8#1171994512yes—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4#1171994512yes—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—42no—
463C22×C4—81no—
464C22×C4—81no—
465C22×C4—1061no—
466C22×C4—1061no—
467C22×C4—1061no—
468C22×C4—1061no—
469C22×C4C2×C22⋊D5311yescharacteristic
470C22×C4C2×C22⋊D5311yescharacteristic
471C22×C4C2×C22⋊D5311yescharacteristic
472C22×C4C2×C22⋊D5311yescharacteristic
473C22×C4C2×C22⋊D5311yescharacteristic
474C22×C4C2×C22⋊D5311yescharacteristic
475C22×C4C23×D5311yescharacteristic
476Q16—2124no—
477SD16—2124no—
478C2×C8—532no—
479C2×C8—532no—
480C2×C8—1061no—
481C2×C8—1061no—
482C2×C8—1061no—
483C2×C8—1061no—
484C2×C8—1061no—
485C2×C8—1061no—
486C2×C8—1061no—
487C2×C8—1061no—
488C2×C8—1061no—
489C2×C8—1061no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—42no—
495C4⋊C4—81no—
496C4⋊C4—81no—
497C4⋊C4—1061no—
498C4⋊C4—1061no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4—1061no—
502C4⋊C4—1061no—
503C4⋊C4—4241no—
504C4⋊C4—4241no—
505C4⋊C4—4241no—
506C22⋊C4—24no—
507C22⋊C4—24no—
508C22⋊C4—42no—
509C22⋊C4—81no—
510C22⋊C4—81no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—42no—
518C23—22no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23C22×C22⋊D5311yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—41no—
529Q8—41no—
530Q8—2121no—
531Q8—2121no—
532Q8—2121no—
533Q8—2121no—
534D4—22no—
535D4—22no—
536D4—41no—
537D4—41no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
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—84no—
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—2121no—
577C2×C4—2121no—
578C2×C4#41×D5311yescharacteristic
579C2×C4#41×D5311yescharacteristic
580C2×C4C2×#1171994511yescharacteristic
581C2×C4C2×#1171994511yescharacteristic
582C2×C4#1313326211yescharacteristic
583C2×C4#1313326211yescharacteristic
584C8—1062no—
585C8—1062no—
586C8—2121no—
587C8—2121no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22#1681113411yescharacteristic
596C22#1681116911yescharacteristic
597C22#1681119011yescharacteristic
598C22#1681143111yescharacteristic
599C22#1681143411yescharacteristic
600C22#1681143411yescharacteristic
601C22#1681143811yescharacteristic
602C4—41no—
603C4—42no—
604C4—41no—
605C4—41no—
606C4—41no—
607C4—41no—
608C4—41no—
609C4—81no—
610C4—81no—
611C4—4241no—
612C4—4241no—
613C4—4241no—
614C4—4241no—
615C4#1681130211yescharacteristic
616C4#1681130211yescharacteristic
617C4#1681132211yescharacteristic
618C4#1681132211yescharacteristic
619C2—82no—
620C2#2028033611yescharacteristic
621C2#2028492911yescharacteristic
622C2#2028499211yescharacteristic
623C2#2028532611yescharacteristic
624C2#2028537711yescharacteristic
625C2#2028578711yescharacteristic
626C2#2028582411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2326C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#2326D53
hypocenterC53#29506
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ #376 ◁ #2326

Complements

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

NH ≅ G/Ntype
1. #20280680C2semidirect
2. #20276192C2semidirect
3. #20276124C2semidirect
6. #20275585C2semidirect
9. #20275394C2semidirect
10. #20275056C2semidirect
12. #20274983C2semidirect
14. #20274467C2semidirect
372. C53#29506semidirect

Maximal subgroups

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

subgroupindex
#202806802
#202761922
#202761242
#202759512
#202758092
#202755852
#202755682
#202754152
#202753942
#202750562
#202750492
#202749832
#202749762
#202744672
C53×#23262
#2950653