Finite Group Databasebeta

← #27465638

Subgroups — #27465638

Order 13568.

674 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277393C211yesmaximal, characteristic
2#20277322C211yesmaximal, characteristic
3#20276146C211yesmaximal, characteristic
4#20275912C211yesmaximal, characteristic
5#20275911C211yesmaximal, characteristic
6#20275484C211yesmaximal, characteristic, complemented
7#20275477C211yesmaximal, characteristic, complemented
8#20275354C211yesmaximal, characteristic, complemented
9#20275348C211yesmaximal, characteristic, complemented
10#20275298C211yesmaximal, characteristic, complemented
11#20275292C211yesmaximal, characteristic, complemented
12#20274729C211yesmaximal, characteristic, complemented
13#20274719C211yesmaximal, characteristic, complemented
14#20274697C211yesmaximal, characteristic
15C53×#2077C211yesmaximal, characteristic
16C2×#13133196C2211yescharacteristic
17C2×#13133182C2211yescharacteristic
18C2×#13133176C2211yescharacteristic
19C2×#13133150C2211yescharacteristic
20C2×#13133149C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133146C2211yescharacteristic
23C2×#13133146C2211yescharacteristic
24C2×#13133142C2211yescharacteristic
25C2×#13133142C2211yescharacteristic
26C2×#13133140C2211yescharacteristic
27C2×#13133140C2211yescharacteristic
28C2×C22⋊#88322C2211yescharacteristic
29#16810528C2211yescharacteristic
30#16810520—21no—
31#16810519C2211yescharacteristic
32#16810517—21no—
33#16810517—21no—
34#16810511—21no—
35#16810511—21no—
36#16810510—21no—
37#16810508—21no—
38#16810506—21no—
39#16810492C2211yescharacteristic
40#16810488C2211yescharacteristic
41#16810483C2211yescharacteristic
42C4⋊C4×Dic53C2211yescharacteristic
43C4⋊C4×Dic53C2211yescharacteristic
44C4⋊C4×Dic53C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×#106×C53C2211yescharacteristic
53C2×C22⋊C8×C53C2211yescharacteristic
54C53×#380C2211yescharacteristic
55C53×#380C2211yescharacteristic
56C53×#340C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×Dic106C2311yescharacteristic
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62#13133196—22no—
63#13133196—22no—
64#13133182—22no—
65#13133182—22no—
66#13133176—22no—
67#13133176—22no—
68C2×C4⋊Dic53D411yescharacteristic
69C2×C4⋊Dic53D411yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×C22⋊Dic53—21no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53D411yescharacteristic
81C2×C22⋊Dic53D411yescharacteristic
82C2×C4×Dic53—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53C2311yescharacteristic
87C2×C4×Dic53C2311yescharacteristic
88#13133150—41no—
89#13133149—22no—
90#13133148—22no—
91#13133146—22no—
92#13133146—22no—
93#13133142—22no—
94#13133142—41no—
95#13133140—22no—
96#13133140—41no—
97C22⋊#88322—41no—
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—
107C23×C4×C53C2311yescharacteristic
108C22×C8×C53C2311yescharacteristic
109C22×C8×C53C2311yescharacteristic
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C22⋊C4×C53—21no—
115C2×C22⋊C4×C53—21no—
116#106×C53—41no—
117C22⋊C8×C53—41no—
118#95×C53—21no—
119#95×C53—21no—
120C24×C53C2×D411yescharacteristic
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2411yescharacteristic
133C22×Dic53—21no—
134C22×Dic53—21no—
135C22×Dic53—21no—
136C22×Dic53—21no—
137C22×Dic53C2×D411yescharacteristic
138C22×Dic53C2×D411yescharacteristic
139C22×Dic53C2×D411yescharacteristic
140C22×Dic53C2×D411yescharacteristic
141C2×Dic106—21no—
142C2×Dic106—21no—
143C2×Dic106—21no—
144C2×Dic106—21no—
145C2×Dic106—21no—
146C2×Dic106#4112yes—
147C2×C8×C53—21no—
148C2×C8×C53—21no—
149C2×C8×C53—21no—
150C2×C8×C53—21no—
151C2×C8×C53—21no—
152C2×C8×C53—21no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C22⋊C4×C53—81no—
162C22⋊C4×C53—81no—
163C22⋊Dic53—24no—
164C22⋊Dic53—22no—
165C22⋊Dic53—22no—
166C22⋊Dic53—81no—
167C4⋊Dic53—21no—
168C4⋊Dic53—22no—
169C4⋊Dic53—22no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53#4112yes—
175#11719919—24no—
176#11719919—42no—
177#11719919—81no—
178#11719919—81no—
179#11719919—81no—
180C4×Dic53—21no—
181C4×Dic53—21no—
182C4×Dic53—42no—
183C4×Dic53—81no—
184C4×Dic53—81no—
185C4×Dic53—81no—
186C4×Dic53#4112yes—
187C4×Dic53#4112yes—
188C2×#88322—21no—
189C2×#88322—21no—
190C2×#88322—21no—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C23×C53—21no—
195C23×C53—21no—
196C23×C53—21no—
197C23×C53—21no—
198C23×C53—21no—
199C23×C53—21no—
200C23×C53—21no—
201C23×C53C22×D411yescharacteristic
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—21no—
205C2×C4×C53—41no—
206C2×C4×C53—41no—
207C2×C4×C53—41no—
208C2×C4×C53—41no—
209C2×C4×C53—41no—
210C2×C4×C53—41no—
211C2×C4×C53—41no—
212C2×C4×C53—41no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53C2×#4111yescharacteristic
222C2×C4×C53C2×#4111yescharacteristic
223C2×C4×C53C2×#4111yescharacteristic
224C2×C4×C53C2×#4111yescharacteristic
225C2×C4×C53H(5,𝔽2)11yescharacteristic
226C2×C4×C532−1+411yescharacteristic
227C2×Dic53—21no—
228C2×Dic53—21no—
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—22no—
234C2×Dic53—22no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53#12312yes—
250C2×Dic53#12312yes—
251Dic106—22no—
252Dic106—22no—
253Dic106—41no—
254Dic106—41no—
255C8×C53—41no—
256C8×C53—41no—
257#88322—41no—
258#88322—41no—
259#27638—531nomaximal, Sylow
260C22×C53—41no—
261C22×C53—41no—
262C22×C53—41no—
263C22×C53—41no—
264C22×C53—41no—
265C22×C53—41no—
266C22×C53—41no—
267C22×C53C2×#12311yescharacteristic
268C22×C53#52511yescharacteristic
269C22×C53#53711yescharacteristic
270C22×C53#57711yescharacteristic
271C22×C53#57711yescharacteristic
272C22×C53#57711yescharacteristic
273C22×C53#57811yescharacteristic
274C4×C53—41no—
275C4×C53—81no—
276C4×C53—81no—
277C4×C53—81no—
278C4×C53—81no—
279C4×C53#55211yescharacteristic
280C4×C53#55311yescharacteristic
281C4×C53#55511yescharacteristic
282C4×C53#55611yescharacteristic
283Dic53—22no—
284Dic53—22no—
285Dic53—41no—
286Dic53—41no—
287Dic53—42no—
288Dic53—42no—
289Dic53—81no—
290Dic53—81no—
291#2323—531no—
292#2314—531no—
293#2077D5311yescharacteristic
294#2076—531no—
295#2035—531no—
296#2035—531no—
297#1949—531no—
298#1948—531no—
299#1928—531no—
300#1927—531no—
301#1920—531no—
302#1919—531no—
303#1805—531no—
304#1803—531no—
305#1798—531no—
306C2×C53—81no—
307C2×C53#248511yescharacteristic
308C2×C53#309711yescharacteristic
309C2×C53#309911yescharacteristic
310C2×C53#312011yescharacteristic
311C2×C53#312611yescharacteristic
312C2×C53#325811yescharacteristic
313C2×C53#326311yescharacteristic
314C2×C22⋊Q8—531no—
315C2×C4×Q8—531no—
316C2×#117—531no—
317C2×#107—531no—
318C2×#106—531no—
319C2×#106—531no—
320C2×#106C2×D5311yescharacteristic
321C2×C4⋊C8—531no—
322C2×C4⋊C8—531no—
323C2×#103—531no—
324C2×#103—531no—
325C2×#103—531no—
326C2×#103—531no—
327C2×C22⋊C8—531no—
328C2×C22⋊C8C2×D5311yescharacteristic
329#391—531no—
330#389—531no—
331#389—1061no—
332#388—1061no—
333#388—1061no—
334#388—1061no—
335#388—1061no—
336#387—1061no—
337#387—1061no—
338#387—1061no—
339#384—531no—
340#382—531no—
341#380—531no—
342#380C2×D5311yescharacteristic
343#380C2×D5311yescharacteristic
344C4×C4⋊C4—531no—
345C4×C4⋊C4—531no—
346C4×C4⋊C4—531no—
347#340—531no—
348#340—531no—
349#340—531no—
350#340—531no—
351#340—531no—
352#340C2×D5311yescharacteristic
353#340C2×D5311yescharacteristic
354#340C2×D5311yescharacteristic
355#336—531no—
356#336—531no—
357C53#2763811yescharacteristic, complemented, Sylow
358C22×Q8—531no—
359C23×C4C22×D5311yescharacteristic
360C22×C8—531no—
361C22×C8—531no—
362C22×C8C22×D5311yescharacteristic
363C22×C8C22×D5311yescharacteristic
364C22⋊Q8—1062no—
365C22⋊Q8—1062no—
366C4×Q8—1062no—
367C4×Q8—1062no—
368#117—1062no—
369#117—1062no—
370C2×C4⋊C4—531no—
371C2×C4⋊C4—531no—
372C2×C4⋊C4—531no—
373C2×C4⋊C4—531no—
374C2×C4⋊C4—531no—
375C2×C4⋊C4—1061no—
376C2×C4⋊C4—1061no—
377C2×C4⋊C4—1061no—
378C2×C4⋊C4—1061no—
379C2×C4⋊C4—1061no—
380C2×C4⋊C4C22×D5311yescharacteristic
381C2×C4⋊C4C22×D5311yescharacteristic
382C2×C4⋊C4C22×D5311yescharacteristic
383C2×C4⋊C4C22×D5311yescharacteristic
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—531no—
387C2×C22⋊C4—531no—
388C2×C22⋊C4—1061no—
389C2×C22⋊C4—1061no—
390C2×C42—531no—
391C2×C42—531no—
392C2×C42—1061no—
393C2×C42—1061no—
394C2×C42—1061no—
395C2×C42—1061no—
396#107—2121no—
397#106—41no—
398#106—1062no—
399#106—1062no—
400C4⋊C8—1062no—
401C4⋊C8—1062no—
402#103—1062no—
403#103—1062no—
404#103—2121no—
405#103—2121no—
406C22⋊C8—41no—
407C22⋊C8—2121no—
408#95—21no—
409#95—21no—
410#95—1061no—
411#95—1061no—
412#95—1061no—
413#95—1061no—
414#95—1061no—
415#95—1061no—
416#95—1061no—
417#95—1061no—
418#95—1061no—
419C24D4×D5311yescharacteristic
420C2×Q8—532no—
421C2×Q8—1061no—
422C2×Q8—1061no—
423C2×Q8—1061no—
424C2×Q8—1061no—
425C2×Q8—1061no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—531no—
437C22×C4—531no—
438C22×C4—531no—
439C22×C4—531no—
440C22×C4—1061no—
441C22×C4—1061no—
442C22×C4—1061no—
443C22×C4—1061no—
444C22×C4D4×D5311yescharacteristic
445C22×C4C23×D5311yescharacteristic
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—1061no—
453C2×C8—1061no—
454C2×C8—1061no—
455C2×C8—1061no—
456C2×C8—1061no—
457C2×C8—1061no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—532no—
467C4⋊C4—1062no—
468C4⋊C4—1062no—
469C4⋊C4—1064no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4—1061no—
475C4⋊C4—4241no—
476C4⋊C4—4241no—
477C4⋊C4—4241no—
478C4⋊C4—2122no—
479C22⋊C4—81no—
480C22⋊C4—81no—
481C22⋊C4—1062no—
482C22⋊C4—1062no—
483C22⋊C4—1064no—
484C22⋊C4—4241no—
485C42—532no—
486C42—532no—
487C42—1061no—
488C42—1061no—
489C42—4241no—
490C42—4241no—
491C42—4241no—
492C42—2122no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23C2×D4×D5311yescharacteristic
501Q8—1062no—
502Q8—1062no—
503Q8—2121no—
504Q8—2121no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—532no—
525C2×C4—532no—
526C2×C4—1061no—
527C2×C4—1061no—
528C2×C4—1061no—
529C2×C4—1061no—
530C2×C4—1061no—
531C2×C4—1061no—
532C2×C4—1062no—
533C2×C4—1062no—
534C2×C4—1061no—
535C2×C4—1061no—
536C2×C4—1061no—
537C2×C4—1061no—
538C2×C4—2121no—
539C2×C4—2121no—
540C2×C4—2121no—
541C2×C4—2121no—
542C2×C4—2121no—
543C2×C4—2121no—
544C2×C4—2121no—
545C2×C4—2121no—
546C2×C4—2121no—
547C2×C4—2121no—
548C2×C4#41×D5311yescharacteristic
549C2×C4#41×D5311yescharacteristic
550C2×C4#41×D5311yescharacteristic
551C2×C4#41×D5311yescharacteristic
552C2×C4#1313326211yescharacteristic
553C2×C4#1313326411yescharacteristic
554C8—41no—
555C8—41no—
556C8—2121no—
557C8—2121no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22#123×D5311yescharacteristic
566C22#1681113111yescharacteristic
567C22#1681120011yescharacteristic
568C22#1681142911yescharacteristic
569C22#1681143011yescharacteristic
570C22#1681143311yescharacteristic
571C22#1681143911yescharacteristic
572C4—41no—
573C4—81no—
574C4—81no—
575C4—81no—
576C4—81no—
577C4—1062no—
578C4—1062no—
579C4—4241no—
580C4—4241no—
581C4—2122no—
582C4—2121no—
583C4—2121no—
584C4—2122no—
585C4#1681130911yescharacteristic
586C4#1681131811yescharacteristic
587C4#1681133511yescharacteristic
588C4#1681134011yescharacteristic
589C2—81no—
590C2#2027900711yescharacteristic
591C2#2028468111yescharacteristic
592C2#2028470511yescharacteristic
593C2#2028493211yescharacteristic
594C2#2028500211yescharacteristic
595C2#2028614211yescharacteristic
596C2#2028617911yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2077C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2077D53
hypocenterC53#27638
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2077

Complements

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

NH ≅ G/Ntype
6. #20275484C2semidirect
7. #20275477C2semidirect
8. #20275354C2semidirect
9. #20275348C2semidirect
10. #20275298C2semidirect
11. #20275292C2semidirect
12. #20274729C2semidirect
13. #20274719C2semidirect
357. C53#27638semidirect

Maximal subgroups

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

subgroupindex
#202773932
#202773222
#202761462
#202759122
#202759112
#202754842
#202754772
#202753542
#202753482
#202752982
#202752922
#202747292
#202747192
#202746972
C53×#20772
#2763853