Finite Group Databasebeta

← #27457213

Subgroups — #27457213

Order 13568.

672 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279111C211yesmaximal, characteristic
2#20277742C211yesmaximal, characteristic
3#20276137C211yesmaximal, characteristic, complemented
4#20276097C211yesmaximal, characteristic, complemented
5#20275918C211yesmaximal, characteristic
6#20275912C211yesmaximal, characteristic
7#20275482C211yesmaximal, characteristic, complemented
8#20275452C211yesmaximal, characteristic, complemented
9#20275325C211yesmaximal, characteristic
10#20275305C211yesmaximal, characteristic, complemented
11#20275299C211yesmaximal, characteristic, complemented
12#20274970C211yesmaximal, characteristic, complemented
13#20274914C211yesmaximal, characteristic
14#20274544C211yesmaximal, characteristic, complemented
15C53×#1825C211yesmaximal, characteristic
16C2×#13133196C2211yescharacteristic
17C2×#13133182C2211yescharacteristic
18C2×#13133150C2211yescharacteristic
19C2×#13133150C2211yescharacteristic
20C2×#13133149C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133146C2211yescharacteristic
23C2×#13133142C2211yescharacteristic
24C2×#13133142C2211yescharacteristic
25C2×#13133140C2211yescharacteristic
26C2×#13133140C2211yescharacteristic
27C2×C22⋊#88322C2211yescharacteristic
28C2×#13133127C2211yescharacteristic
29C53⋊#401—21no—
30#16810551—21no—
31#16810528C2211yescharacteristic
32#16810522C2211yescharacteristic
33#16810519C2211yescharacteristic
34#16810519C2211yescharacteristic
35#16810517—21no—
36#16810516—21no—
37#16810511—21no—
38#16810508—21no—
39#16810507—21no—
40#16810488C2211yescharacteristic
41#16810483C2211yescharacteristic
42C4⋊C4×Dic53C2211yescharacteristic
43C4⋊C4×Dic53C2211yescharacteristic
44C4×#11719919—21no—
45C4×C22⋊Dic53C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×#117×C53C2211yescharacteristic
53C2×C22⋊C8×C53C2211yescharacteristic
54C53×#380C2211yescharacteristic
55C53×#340C2211yescharacteristic
56C53×#340C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×Dic106C2311yescharacteristic
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62#13133196—22no—
63#13133196—22no—
64#13133182—22no—
65#13133182—41no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×#11719919—21no—
71C2×#11719919—21no—
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×C22⋊Dic53—21no—
75C2×C22⋊Dic53—21no—
76C2×C22⋊Dic53D411yescharacteristic
77C2×C22⋊Dic53D411yescharacteristic
78C2×C4×Dic53—21no—
79C2×C4×Dic53—21no—
80C2×C4×Dic53—21no—
81C2×C4×Dic53C2311yescharacteristic
82C2×C4×Dic53C2311yescharacteristic
83#13133150—22no—
84#13133150—22no—
85#13133149—22no—
86#13133148—41no—
87#13133146—22no—
88#13133142—22no—
89#13133142—41no—
90#13133140—22no—
91#13133140—22no—
92C22⋊#88322—22no—
93#13133127—41no—
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—
104#13133124D411yescharacteristic
105#13133124D411yescharacteristic
106C23×C4×C53C2311yescharacteristic
107C22×C8×C53C2311yescharacteristic
108C22×C8×C53C2311yescharacteristic
109#117×C53—22no—
110#117×C53—41no—
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C22⋊C4×C53—21no—
116C2×C22⋊C4×C53—21no—
117C2×C42×C53—21no—
118C22⋊C8×C53—41no—
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#4112yes—
161C22⋊C4×C53—42no—
162C22⋊C4×C53—42no—
163C42×C53—42no—
164C22⋊Dic53—24no—
165C22⋊Dic53—41no—
166C22⋊Dic53—41no—
167C22⋊Dic53—81no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—42no—
174C4⋊Dic53#4112yes—
175#11719919—24no—
176#11719919—42no—
177#11719919—42no—
178#11719919—81no—
179C4×Dic53—21no—
180C4×Dic53—21no—
181C4×Dic53—21no—
182C4×Dic53—21no—
183C4×Dic53—42no—
184C4×Dic53—42no—
185C4×Dic53—81no—
186C2×#88322—21no—
187C2×#88322—21no—
188C2×#88322—21no—
189C2×#88322—21no—
190C2×#88322—21no—
191C2×#88322#4112yes—
192C23×C53—21no—
193C23×C53—21no—
194C23×C53—21no—
195C23×C53—21no—
196C23×C53—21no—
197C23×C53—21no—
198C23×C53—21no—
199C23×C53C22×D411yescharacteristic
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—22no—
203C2×C4×C53—21no—
204C2×C4×C53—41no—
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×C53C2×#4111yescharacteristic
220C2×C4×C53C2×#4111yescharacteristic
221C2×C4×C53C2×#4111yescharacteristic
222C2×C4×C53C2×#4111yescharacteristic
223C2×C4×C53H(5,𝔽2)11yescharacteristic
224C2×C4×C532−1+411yescharacteristic
225C2×Dic53—22no—
226C2×Dic53—22no—
227C2×Dic53—21no—
228C2×Dic53—21no—
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—21no—
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—
249Dic106—22no—
250Dic106—22no—
251Dic106—41no—
252Dic106—41no—
253C8×C53—41no—
254C8×C53—41no—
255#88322—22no—
256#88322—41no—
257#27061—531nomaximal, Sylow
258C22×C53—41no—
259C22×C53—41no—
260C22×C53—41no—
261C22×C53—41no—
262C22×C53—41no—
263C22×C53—41no—
264C22×C53—41no—
265C22×C53#52511yescharacteristic
266C22×C53#54611yescharacteristic
267C22×C53#54711yescharacteristic
268C22×C53#57511yescharacteristic
269C22×C53#57511yescharacteristic
270C22×C53#57711yescharacteristic
271C22×C53#57811yescharacteristic
272C4×C53—41no—
273C4×C53—42no—
274C4×C53—81no—
275C4×C53—81no—
276C4×C53—81no—
277C4×C53#52911yescharacteristic
278C4×C53#53911yescharacteristic
279C4×C53#55311yescharacteristic
280C4×C53#55611yescharacteristic
281Dic53—41no—
282Dic53—41no—
283Dic53—42no—
284Dic53—42no—
285Dic53—41no—
286Dic53—41no—
287Dic53—81no—
288Dic53—81no—
289#2494—531no—
290#2362—531no—
291#2074—531no—
292#2066—531no—
293#2036—531no—
294#2035—531no—
295#1949—531no—
296#1943—531no—
297#1924—531no—
298#1921—531no—
299#1920—531no—
300#1863—531no—
301#1854—531no—
302#1825D5311yescharacteristic
303#1763—531no—
304C2×C53—81no—
305C2×C53#246611yescharacteristic
306C2×C53#311111yescharacteristic
307C2×C53#312111yescharacteristic
308C2×C53#330711yescharacteristic
309C2×C53#332711yescharacteristic
310C2×C53#334911yescharacteristic
311C2×C53#335011yescharacteristic
312C2×C22⋊Q8—531no—
313C2×C4×Q8—531no—
314C2×#117C2×D5311yescharacteristic
315C2×#107—531no—
316C2×#107—531no—
317C2×#106—531no—
318C2×#106—531no—
319C2×C4⋊C8—531no—
320C2×#103—531no—
321C2×#103—531no—
322C2×#103—531no—
323C2×#103—531no—
324C2×C22⋊C8—531no—
325C2×C22⋊C8C2×D5311yescharacteristic
326C2×#97—531no—
327#401—1061no—
328#397—1061no—
329#391—531no—
330#389—531no—
331#389—531no—
332#389—531no—
333#388—1061no—
334#388—1061no—
335#388—1061no—
336#387—1061no—
337#387—1061no—
338#382—531no—
339#380—531no—
340#380C2×D5311yescharacteristic
341C4×C4⋊C4—531no—
342C4×C4⋊C4—531no—
343C4×C4⋊C4—1061no—
344C4×C22⋊C4—531no—
345#340—531no—
346#340—531no—
347#340—531no—
348#340—531no—
349#340C2×D5311yescharacteristic
350#340C2×D5311yescharacteristic
351#340C2×D5311yescharacteristic
352#340C2×D5311yescharacteristic
353#336—531no—
354#336—531no—
355C53#2706111yescharacteristic, complemented, Sylow
356C22×Q8—531no—
357C23×C4C22×D5311yescharacteristic
358C22×C8—531no—
359C22×C8—531no—
360C22×C8C22×D5311yescharacteristic
361C22×C8C22×D5311yescharacteristic
362C22⋊Q8—1062no—
363C22⋊Q8—1062no—
364C4×Q8—1062no—
365C4×Q8—2121no—
366#117—22no—
367#117—41no—
368C2×C4⋊C4—531no—
369C2×C4⋊C4—531no—
370C2×C4⋊C4—531no—
371C2×C4⋊C4—1061no—
372C2×C4⋊C4—1061no—
373C2×C4⋊C4—1061no—
374C2×C4⋊C4—1061no—
375C2×C4⋊C4—1061no—
376C2×C4⋊C4C22×D5311yescharacteristic
377C2×C4⋊C4C22×D5311yescharacteristic
378C2×C4⋊C4C22×D5311yescharacteristic
379C2×C4⋊C4C22×D5311yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—531no—
383C2×C22⋊C4—531no—
384C2×C22⋊C4—1061no—
385C2×C22⋊C4—1061no—
386C2×C42—21no—
387C2×C42—531no—
388C2×C42—531no—
389C2×C42—1061no—
390C2×C42—1061no—
391C2×C42—1061no—
392#107—1062no—
393#107—1062no—
394#106—1062no—
395#106—2121no—
396C4⋊C8—1062no—
397#103—1062no—
398#103—1062no—
399#103—1062no—
400#103—2121no—
401C22⋊C8—41no—
402C22⋊C8—1062no—
403#97—2121no—
404#95—21no—
405#95—531no—
406#95—531no—
407#95—1061no—
408#95—1061no—
409#95—1061no—
410#95—1061no—
411#95—1061no—
412#95—1061no—
413#95—1061no—
414#95—1061no—
415#95—1061no—
416#95—1061no—
417C24D4×D5311yescharacteristic
418C2×Q8—532no—
419C2×Q8—1061no—
420C2×Q8—1061no—
421C2×Q8—1061no—
422C2×Q8—1061no—
423C2×Q8—1061no—
424C22×C4—21no—
425C22×C4—21no—
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—531no—
435C22×C4—531no—
436C22×C4—531no—
437C22×C4—531no—
438C22×C4—1061no—
439C22×C4—1061no—
440C22×C4—1061no—
441C22×C4—1061no—
442C22×C4D4×D5311yescharacteristic
443C22×C4C23×D5311yescharacteristic
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—532no—
451C2×C8—1061no—
452C2×C8—1061no—
453C2×C8—1061no—
454C2×C8—1061no—
455C2×C8—1061no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—532no—
464C4⋊C4—1064no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
467C4⋊C4—1061no—
468C4⋊C4—1061no—
469C4⋊C4—1061no—
470C4⋊C4—4241no—
471C4⋊C4—2122no—
472C4⋊C4—2122no—
473C4⋊C4—2122no—
474C4⋊C4#1171994512yes—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—1064no—
478C22⋊C4—4241no—
479C22⋊C4—2121no—
480C22⋊C4—2121no—
481C42—42no—
482C42—1061no—
483C42—1061no—
484C42—1061no—
485C42—1061no—
486C42—4241no—
487C42—2122no—
488C42—2122no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23C2×D4×D5311yescharacteristic
497Q8—1062no—
498Q8—1062no—
499Q8—2121no—
500Q8—2121no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
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—1061no—
521C2×C4—1061no—
522C2×C4—1062no—
523C2×C4—1062no—
524C2×C4—1061no—
525C2×C4—1061no—
526C2×C4—1061no—
527C2×C4—1061no—
528C2×C4—1061no—
529C2×C4—1061no—
530C2×C4—1061no—
531C2×C4—1061no—
532C2×C4—1061no—
533C2×C4—1061no—
534C2×C4—2121no—
535C2×C4—2121no—
536C2×C4—2121no—
537C2×C4—2121no—
538C2×C4—2121no—
539C2×C4—2121no—
540C2×C4—2121no—
541C2×C4—2121no—
542C2×C4—2121no—
543C2×C4—2121no—
544C2×C4#41×D5311yescharacteristic
545C2×C4#41×D5311yescharacteristic
546C2×C4#41×D5311yescharacteristic
547C2×C4C2×#1171994511yescharacteristic
548C2×C4#1313326211yescharacteristic
549C2×C4#1313326411yescharacteristic
550C8—41no—
551C8—41no—
552C8—1062no—
553C8—2121no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22#1681113111yescharacteristic
562C22#1681126311yescharacteristic
563C22#1681127111yescharacteristic
564C22#1681141911yescharacteristic
565C22#1681142011yescharacteristic
566C22#1681143011yescharacteristic
567C22#1681143911yescharacteristic
568C4—41no—
569C4—42no—
570C4—81no—
571C4—81no—
572C4—81no—
573C4—4241no—
574C4—4241no—
575C4—2121no—
576C4—2121no—
577C4—2121no—
578C4—2122no—
579C4—2122no—
580C4—2121no—
581C4#1681115111yescharacteristic
582C4#1681121611yescharacteristic
583C4#1681131811yescharacteristic
584C4#1681134111yescharacteristic
585C2—81no—
586C2#2027875011yescharacteristic
587C2#2028481911yescharacteristic
588C2#2028494711yescharacteristic
589C2#2028668711yescharacteristic
590C2#2028698411yescharacteristic
591C2#2028724911yescharacteristic
592C2#2028725311yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1825C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1825D53
hypocenterC53#27061
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1825

Complements

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

NH ≅ G/Ntype
3. #20276137C2semidirect
4. #20276097C2semidirect
7. #20275482C2semidirect
8. #20275452C2semidirect
10. #20275305C2semidirect
11. #20275299C2semidirect
12. #20274970C2semidirect
14. #20274544C2semidirect
355. C53#27061semidirect

Maximal subgroups

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

subgroupindex
#202791112
#202777422
#202761372
#202760972
#202759182
#202759122
#202754822
#202754522
#202753252
#202753052
#202752992
#202749702
#202749142
#202745442
C53×#18252
#2706153