Finite Group Databasebeta

← #27368088

Subgroups — #27368088

Order 13568.

630 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285040C211yesmaximal, characteristic, complemented
2#20284866C211yesmaximal, characteristic
3#41×#88322C211yesmaximal, characteristic, complemented
4#20274175C211yesmaximal, characteristic, complemented
5#20274168C211yesmaximal, characteristic, complemented
6#20274161C211yesmaximal, characteristic, complemented
7#20274154C211yesmaximal, characteristic, complemented
8#20274147C211yesmaximal, characteristic, complemented
9#20274140C211yesmaximal, characteristic, complemented
10#20274133C211yesmaximal, characteristic, complemented
11#20274126C211yesmaximal, characteristic, complemented
12#20273476C211yesmaximal, characteristic, complemented
13#20273425C211yesmaximal, characteristic, complemented
14#20273421C211yesmaximal, characteristic, complemented
15C53×#3116C211yesmaximal, characteristic, complemented
16#16811173C2211yescharacteristic
17#16810980—21no—
18#16810980—21no—
19#16810975C2211yescharacteristic
20#16810975C2211yescharacteristic
21#16810971C2211yescharacteristic
22#16810971C2211yescharacteristic
23#16810947—21no—
24#16810908C2211yescharacteristic
25#16810903C2211yescharacteristic
26#16810823C2211yescharacteristic
27#16810818C2211yescharacteristic
28#16810768—21no—
29Q8×#88322C2211yescharacteristic
30Q8×#88322C2211yescharacteristic
31#16810733—21no—
32#16810721—21no—
33D4×#88322—21no—
34D4×#88322—21no—
35D4×#88322C2211yescharacteristic
36D4×#88322C2211yescharacteristic
37#16810680C2211yescharacteristic
38#16810676—21no—
39#16810676C2211yescharacteristic
40#16810617—21no—
41#16810617—21no—
42C2×C4×#88322—21no—
43C2×C4×#88322C2211yescharacteristic
44#16810375C2211yescharacteristic
45#16810375C2211yescharacteristic
46#16810373C2211yescharacteristic
47#16810373C2211yescharacteristic
48#16810372C2211yescharacteristic
49#16810372C2211yescharacteristic
50#16810369C2211yescharacteristic
51#16810369C2211yescharacteristic
52#16810368C2211yescharacteristic
53#16810368C2211yescharacteristic
54#16810366C2211yescharacteristic
55#16810366C2211yescharacteristic
56C53×#533C2211yescharacteristic
57C4×#41×C53C2211yescharacteristic
58C53×#485—21no—
59C53×#477C2211yescharacteristic
60C53×#476C2211yescharacteristic
61C53×#462C2211yescharacteristic
62C53×#461C2211yescharacteristic
63C53×#452—21no—
64C53×#441—21no—
65C53×#440—21no—
66C53×#432C2211yescharacteristic
67C2×#11719945—21no—
68C2×#11719924—21no—
69C2×#11719922—21no—
70C22×#88322—21no—
71C22×#88322—21no—
72C22×#88322D411yescharacteristic
73C22×#88322D411yescharacteristic
74#13133217C2311yescharacteristic
75#13133214C2311yescharacteristic
76#13133205—21no—
77#13133191—21no—
78#13133191—21no—
79C4×D212—21no—
80C4×D212—21no—
81#13133150—21no—
82#13133148—21no—
83C4⋊#88322—21no—
84C4⋊#88322—21no—
85C4⋊#88322C2311yescharacteristic
86C4⋊#88322C2311yescharacteristic
87C4⋊#88322C2311yescharacteristic
88C4⋊#88322C2311yescharacteristic
89#13133141—21no—
90#13133140—21no—
91#13133140—21no—
92#13133140—21no—
93#13133140—21no—
94#13133138—21no—
95#13133137—21no—
96#13133137—21no—
97#13133137—21no—
98#13133137—21no—
99#13133137—21no—
100#13133137—21no—
101C22⋊#88322—21no—
102C22⋊#88322—21no—
103C22⋊#88322—21no—
104C22⋊#88322—21no—
105#13133128—21no—
106C4×#88322—21no—
107C4×#88322—21no—
108C4×#88322—21no—
109C4×#88322D411yescharacteristic
110C4×#88322D411yescharacteristic
111C4×#88322C2311yescharacteristic
112C4×#88322C2311yescharacteristic
113C2×#41×C53D411yescharacteristic
114C2×Q16×C53—21no—
115C2×SD16×C53—21no—
116C4⋊Q8×C53C2311yescharacteristic
117C4⋊Q8×C53C2311yescharacteristic
118#125×C53—21no—
119C22⋊Q8×C53—21no—
120C22⋊Q8×C53—21no—
121C4×Q8×C53—21no—
122C4×Q8×C53—21no—
123C4×Q8×C53D411yescharacteristic
124C4×Q8×C53C2311yescharacteristic
125C4×D4×C53—21no—
126C4×D4×C53—21no—
127C4×D4×C53D411yescharacteristic
128C4×D4×C53C2311yescharacteristic
129#117×C53—21no—
130#117×C53—21no—
131#117×C53D411yescharacteristic
132C2×C42×C53—21no—
133C2×C42×C53C2311yescharacteristic
134#107×C53—21no—
135#106×C53—21no—
136C4⋊C8×C53C2311yescharacteristic
137C4⋊C8×C53C2311yescharacteristic
138#103×C53—21no—
139#103×C53—21no—
140#103×C53—21no—
141#102×C53—21no—
142C22⋊C8×C53—21no—
143#97×C53—21no—
144#41×C53—22no—
145#41×C53—22no—
146C2×Q8×C53—21no—
147C2×Q8×C53—21no—
148C2×Q8×C53C2×D411yescharacteristic
149C2×D4×C53—21no—
150C2×D4×C53C2×D411yescharacteristic
151C22×C4×C53—21no—
152C22×C4×C53—21no—
153C22×C4×C53—21no—
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53#4111yescharacteristic
157C2×C22⋊D53—41no—
158#11719945—42no—
159#11719945—42no—
160C2×D212—21no—
161C2×D212—21no—
162C2×D212—21no—
163C2×C4×D53—41no—
164C2×Dic106—21no—
165Q16×C53—42no—
166SD16×C53—42no—
167C2×C8×C53—21no—
168C2×C8×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—21no—
175C4⋊C4×C53—41no—
176C4⋊C4×C53—41no—
177C4⋊C4×C53C2×D411yescharacteristic
178C4⋊C4×C53C2×D411yescharacteristic
179C22⋊C4×C53—21no—
180C22⋊C4×C53—21no—
181C22⋊C4×C53—21no—
182C22⋊C4×C53—41no—
183C42×C53—21no—
184C42×C53—21no—
185C42×C53—21no—
186C42×C53—41no—
187C42×C53C2×D411yescharacteristic
188C42×C53C2×D411yescharacteristic
189C42×C53#4111yescharacteristic
190C42×C53C2411yescharacteristic
191#11719924—42no—
192#11719922—42no—
193#11719921—41no—
194#11719921—41no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—41no—
202C2×#88322—41no—
203C2×#88322—42no—
204C2×#88322—42no—
205C2×#88322C2×D411yescharacteristic
206C2×#88322C2×D411yescharacteristic
207C2×#88322C2×D411yescharacteristic
208C2×#88322C2×D411yescharacteristic
209C23×C53—21no—
210C23×C53#12111yescharacteristic
211C22×D53—41no—
212Q8×C53—21no—
213Q8×C53—41no—
214Q8×C53—41no—
215Q8×C53—81no—
216Q8×C53#13512yes—
217D4×C53—24no—
218D4×C53—21no—
219D4×C53#13512yes—
220C2×C4×C53—24no—
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—41no—
229C2×C4×C53—42no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53#12111yescharacteristic
235C2×C4×C53#12111yescharacteristic
236C2×C4×C53#12111yescharacteristic
237C2×C4×C53C22×D411yescharacteristic
238C2×C4×C53C22×D411yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C22⋊D53—82no—
241C2×Dic53—41no—
242D212—41no—
243D212—41no—
244D212—41no—
245D212—41no—
246D212—81no—
247C4×D53—82no—
248Dic106—41no—
249Dic106—41no—
250C8×C53—41no—
251C8×C53—41no—
252#88322—21no—
253#88322—21no—
254#88322—21no—
255#88322—21no—
256#88322—41no—
257#88322—41no—
258#88322—41no—
259#88322—41no—
260#19278—531nomaximal, Sylow
261C22×C53—21no—
262C22×C53—21no—
263C22×C53—22no—
264C22×C53—21no—
265C22×C53—41no—
266C22×C53—41no—
267C22×C53C2×#12111yescharacteristic
268C2×D53—81no—
269C2×D53—81no—
270C2×D53—81no—
271C4×C53—21no—
272C4×C53—21no—
273C4×C53—22no—
274C4×C53—21no—
275C4×C53—21no—
276C4×C53—41no—
277C4×C53—41no—
278C4×C53—81no—
279C4×C53—81no—
280C4×C53C2×#13511yescharacteristic
281C4×C53C2×#13511yescharacteristic
282C4×C53#57511yescharacteristic
283C4×C53#57511yescharacteristic
284Dic53—81no—
285#3131—531no—
286#3116D5311yescharacteristic, complemented
287#3115—531no—
288C8×#41—531no—
289#1684—531no—
290#1683—531no—
291#1682—531no—
292#1681—531no—
293#1680—531no—
294#1679—531no—
295#1678—531no—
296#1677—531no—
297#1568—531no—
298#1554—531no—
299#1553—531no—
300C2×C53—41no—
301C2×C53—42no—
302C2×C53#304011yescharacteristic
303C2×C53#305011yescharacteristic
304C2×C53#306711yescharacteristic
305D53—161no—
306#533C2×D5311yescharacteristic
307#532—531no—
308C4×#41C2×D5311yescharacteristic
309#488—1061no—
310#488—1061no—
311#487—531no—
312#487—531no—
313#486—531no—
314#486—531no—
315#485—21no—
316#482—1061no—
317#477C2×D5311yescharacteristic
318#476C2×D5311yescharacteristic
319#475—531no—
320#474—531no—
321#462C2×D5311yescharacteristic
322#461C2×D5311yescharacteristic
323#460—531no—
324#459—531no—
325#452—21no—
326#449—1061no—
327C8×Q8—531no—
328C8×Q8—531no—
329#442—1061no—
330#441—21no—
331#440—21no—
332#440—1061no—
333C8×D4—531no—
334C8×D4—531no—
335C8×D4—1061no—
336C8×D4—1061no—
337#432—531no—
338#432C2×D5311yescharacteristic
339#431—531no—
340#431—1061no—
341#416—1061no—
342#416—1061no—
343C2×C4×C8—531no—
344C2×C4×C8—1061no—
345#335—531no—
346#335—531no—
347#334—531no—
348#334—531no—
349#333—531no—
350#333—531no—
351#332—531no—
352#332—531no—
353#332—531no—
354#332—531no—
355#331—531no—
356#331—531no—
357C53#1927811yescharacteristic, complemented, Sylow
358C2×#41—1061no—
359C2×#41C22⋊D5311yescharacteristic
360C2×Q16—21no—
361C2×SD16—21no—
362C2×SD16—1061no—
363C2×D8—1061no—
364C22×C8—531no—
365C22×C8—531no—
366C22×C8—1061no—
367C22×C8—1061no—
368C4⋊Q8—531no—
369C4⋊Q8C22×D5311yescharacteristic
370C4⋊Q8C22×D5311yescharacteristic
371#127—531no—
372#125—21no—
373#124—1061no—
374C22⋊Q8—21no—
375C22⋊Q8—21no—
376#121—1061no—
377#121—1061no—
378C4×Q8—21no—
379C4×Q8—21no—
380C4×Q8C22⋊D5311yescharacteristic
381C4×Q8C22×D5311yescharacteristic
382C4×D4—21no—
383C4×D4—21no—
384C4×D4—1061no—
385C4×D4—1061no—
386C4×D4C22⋊D5311yescharacteristic
387C4×D4C22×D5311yescharacteristic
388#117—21no—
389#117—21no—
390#117C22⋊D5311yescharacteristic
391C2×C42—21no—
392C2×C42C22×D5311yescharacteristic
393#107—21no—
394#107—1061no—
395#106—21no—
396#106—1061no—
397C4⋊C8—531no—
398C4⋊C8—531no—
399C4⋊C8—531no—
400C4⋊C8—531no—
401C4⋊C8—1061no—
402C4⋊C8—1061no—
403C4⋊C8C22×D5311yescharacteristic
404C4⋊C8C22×D5311yescharacteristic
405#103—21no—
406#103—21no—
407#103—21no—
408#103—1061no—
409#103—1061no—
410#103—1061no—
411#103—1061no—
412#103—1061no—
413#102—21no—
414#102—1061no—
415#102—1061no—
416#102—1061no—
417#102—1061no—
418#102—1061no—
419#102—1061no—
420#102—1061no—
421C22⋊C8—21no—
422C22⋊C8—1061no—
423C22⋊C8—1061no—
424C22⋊C8—1061no—
425C22⋊C8—1061no—
426#97—21no—
427#97—1061no—
428C4×C8—531no—
429C4×C8—531no—
430C4×C8—531no—
431C4×C8—531no—
432C4×C8—1061no—
433C4×C8—1061no—
434C4×C8—1061no—
435#41—22no—
436#41—22no—
437#41—2122no—
438#41—2122no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—1061no—
442C2×Q8C2×C22⋊D5311yescharacteristic
443C2×D4—21no—
444C2×D4—1061no—
445C2×D4—1061no—
446C2×D4—1061no—
447C2×D4—2121no—
448C2×D4C2×C22⋊D5311yescharacteristic
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—2121no—
453C22×C4D4×D5311yescharacteristic
454C22×C4#1171994711yescharacteristic
455C22×C4C2×C22⋊D5311yescharacteristic
456Q16—42no—
457SD16—42no—
458SD16—2122no—
459D8—2122no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—531no—
463C2×C8—531no—
464C2×C8—531no—
465C2×C8—531no—
466C2×C8—1061no—
467C2×C8—1061no—
468C2×C8—1061no—
469C2×C8—1061no—
470C2×C8—2121no—
471C2×C8—2121no—
472C2×C8—2122no—
473C2×C8—2122no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—1061no—
483C4⋊C4—1061no—
484C4⋊C4C2×C22⋊D5311yescharacteristic
485C4⋊C4C2×C22⋊D5311yescharacteristic
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—21no—
489C22⋊C4—41no—
490C22⋊C4—2121no—
491C22⋊C4—2121no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—41no—
496C42D4×D5311yescharacteristic
497C42#1171994711yescharacteristic
498C42C2×C22⋊D5311yescharacteristic
499C42C23×D5311yescharacteristic
500C23—21no—
501C23—2121no—
502C23#1313318911yescharacteristic
503Q8—21no—
504Q8—41no—
505Q8—41no—
506Q8—81no—
507Q8—2121no—
508Q8—2121no—
509Q8#1313323712yes—
510D4—24no—
511D4—21no—
512D4—4241no—
513D4—4242no—
514D4—2121no—
515D4—2121no—
516D4—2121no—
517D4—2121no—
518D4#1313323712yes—
519C2×C4—24no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—4242no—
534C2×C4—2121no—
535C2×C4#1313318911yescharacteristic
536C2×C4#1313318911yescharacteristic
537C2×C4#1313318911yescharacteristic
538C2×C4C2×D4×D5311yescharacteristic
539C2×C4C22×C22⋊D5311yescharacteristic
540C2×C4C2×#1171994711yescharacteristic
541C8—41no—
542C8—41no—
543C8—1061no—
544C8—1061no—
545C8—1061no—
546C8—1061no—
547C8—2121no—
548C8—2121no—
549C8—2121no—
550C8—2121no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—22no—
555C22—41no—
556C22—41no—
557C22—4241no—
558C22—4241no—
559C22—4241no—
560C22C2×#1313318911yescharacteristic
561C4—21no—
562C4—21no—
563C4—21no—
564C4—21no—
565C4—22no—
566C4—41no—
567C4—41no—
568C4—81no—
569C4—81no—
570C4—4241no—
571C4C2×#1313323711yescharacteristic
572C4C2×#1313323711yescharacteristic
573C4#1681142211yescharacteristic
574C4#1681142211yescharacteristic
575C2—41no—
576C2—42no—
577C2—8481no—
578C2#2028421111yescharacteristic
579C2#2028429311yescharacteristic
580C2#2028443211yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3116C2
socleC22×C53C2×#121
prime radical#20284866C2
hypercenter#3116D53
hypocenterC53#19278
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3116

Complements

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

NH ≅ G/Ntype
1. #20285040C2semidirect
3. #41×#88322C2semidirect
4. #20274175C2semidirect
5. #20274168C2semidirect
6. #20274161C2semidirect
7. #20274154C2semidirect
8. #20274147C2semidirect
9. #20274140C2semidirect
10. #20274133C2semidirect
11. #20274126C2semidirect
12. #20273476C2semidirect
13. #20273425C2semidirect
14. #20273421C2semidirect
15. C53×#3116C2semidirect
286. #3116D53semidirect
357. C53#19278semidirect

Maximal subgroups

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

subgroupindex
#202850402
#202848662
#41×#883222
#202741752
#202741682
#202741612
#202741542
#202741472
#202741402
#202741332
#202741262
#202734762
#202734252
#202734212
C53×#31162
#1927853