Finite Group Databasebeta

← #27369198

Subgroups — #27369198

Order 13568.

636 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285730C211yesmaximal, characteristic
2#20285091C211yesmaximal, characteristic, complemented
3#20283720C211yesmaximal, characteristic, complemented
4#20274285C211yesmaximal, characteristic, complemented
5#20274278C211yesmaximal, characteristic, complemented
6#20274257C211yesmaximal, characteristic, complemented
7#20274250C211yesmaximal, characteristic, complemented
8#20274161C211yesmaximal, characteristic, complemented
9#20274154C211yesmaximal, characteristic, complemented
10#20274133C211yesmaximal, characteristic, complemented
11#20274126C211yesmaximal, characteristic, complemented
12#20273480C211yesmaximal, characteristic, complemented
13#20273425C211yesmaximal, characteristic, complemented
14#20273384C211yesmaximal, characteristic, complemented
15C53×#3252C211yesmaximal, characteristic, complemented
16C2×#13133214C2211yescharacteristic
17#16811013—21no—
18#16810989—21no—
19#16810989—21no—
20#16810989—21no—
21#16810971C2211yescharacteristic
22#16810971C2211yescharacteristic
23#16810971C2211yescharacteristic
24#16810971C2211yescharacteristic
25#16810823C2211yescharacteristic
26#16810823C2211yescharacteristic
27#16810818C2211yescharacteristic
28#16810818C2211yescharacteristic
29#16810773—21no—
30#16810756—21no—
31#16810754C2211yescharacteristic, complemented
32Q8×#88322C2211yescharacteristic, complemented
33#16810701—21no—
34#16810701C2211yescharacteristic, complemented
35#16810695—21no—
36D4×#88322C2211yescharacteristic, complemented
37#16810684—21no—
38#16810680C2211yescharacteristic
39#16810676C2211yescharacteristic
40#16810627—21no—
41#16810627—21no—
42#16810579—21no—
43C4×#11719917C2211yescharacteristic
44#16810375C2211yescharacteristic, complemented
45#16810375C2211yescharacteristic, complemented
46#16810373C2211yescharacteristic, complemented
47#16810373C2211yescharacteristic, complemented
48#16810372C2211yescharacteristic, complemented
49#16810372C2211yescharacteristic, complemented
50#16810369C2211yescharacteristic, complemented
51#16810369C2211yescharacteristic, complemented
52#16810366C2211yescharacteristic
53#16810366C2211yescharacteristic
54#16810366C2211yescharacteristic
55#16810366C2211yescharacteristic
56C53×#533C2211yescharacteristic
57C53×#529C2211yescharacteristic
58C53×#479—21no—
59C53×#477C2211yescharacteristic, complemented
60C53×#476C2211yescharacteristic, complemented
61C53×#470—21no—
62C53×#468—21no—
63C53×#464—21no—
64C53×#462C2211yescharacteristic, complemented
65C53×#461C2211yescharacteristic, complemented
66C53×#423C2211yescharacteristic
67C22×D212—21no—
68C22×D212—21no—
69C22×D212—21no—
70C2×#11719924—21no—
71C2×#11719924—21no—
72C2×#11719922—21no—
73C2×#11719922—21no—
74C2×#11719917—21no—
75C2×#11719917D411yescharacteristic
76C2×#11719917D411yescharacteristic
77C22×#88322—21no—
78#13133214—21no—
79#13133214—21no—
80#13133214—21no—
81#13133214C2311yescharacteristic
82#13133214C2311yescharacteristic
83C4⋊#88322—21no—
84C4⋊#88322—21no—
85C4⋊#88322C2311yescharacteristic
86C4⋊#88322C2311yescharacteristic
87C4⋊#88322C2311yescharacteristic
88C4⋊#88322C2311yescharacteristic
89#13133137—21no—
90#13133137—21no—
91#13133137—21no—
92#13133137—21no—
93#13133137—21no—
94#13133137—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—
106#13133128—21no—
107#13133128D411yescharacteristic
108#13133128D411yescharacteristic
109C4×#88322—21no—
110C4×#88322—21no—
111C4×#88322C2311yescharacteristic
112C4×#88322C2311yescharacteristic
113C2×Q16×C53—21no—
114C2×SD16×C53—21no—
115C2×M16×C53—21no—
116C4⋊Q8×C53C2311yescharacteristic
117C4⋊Q8×C53C2311yescharacteristic
118#126×C53—21no—
119#125×C53—21no—
120#125×C53D411yescharacteristic
121#124×C53D411yescharacteristic
122#123×C53—21no—
123C22⋊Q8×C53—21no—
124C22⋊Q8×C53—21no—
125C22⋊Q8×C53D411yescharacteristic
126#121×C53D411yescharacteristic
127C4×Q8×C53—21no—
128C4×Q8×C53—21no—
129C4×Q8×C53C2311yescharacteristic
130C4×D4×C53—21no—
131C4×D4×C53C2311yescharacteristic
132#117×C53—21no—
133#117×C53—21no—
134C2×C42×C53C2311yescharacteristic
135#107×C53—21no—
136#106×C53—21no—
137C4⋊C8×C53—21no—
138C4⋊C8×C53C2311yescharacteristic
139C4⋊C8×C53C2311yescharacteristic
140#103×C53—21no—
141#103×C53—21no—
142#103×C53—21no—
143#102×C53—21no—
144C23×D53—41no—
145C2×Q8×C53—21no—
146C2×Q8×C53—21no—
147C2×Q8×C53C2×D411yescharacteristic
148C2×D4×C53—21no—
149C2×D4×C53C2×D411yescharacteristic
150C22×C4×C53—21no—
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53#4111yescharacteristic
154C2×D212—21no—
155C2×D212—21no—
156C2×D212—21no—
157C2×D212—21no—
158C2×D212—21no—
159C2×D212—21no—
160C2×D212—42no—
161C2×D212—42no—
162C2×D212—41no—
163C2×D212—41no—
164C2×D212—41no—
165C2×D212—81no—
166C2×D212—81no—
167C2×D212—82no—
168Q16×C53—81no—
169SD16×C53—81no—
170M16×C53—81no—
171C2×C8×C53—21no—
172C2×C8×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—21no—
177C4⋊C4×C53—21no—
178C4⋊C4×C53—21no—
179C4⋊C4×C53—21no—
180C4⋊C4×C53—21no—
181C4⋊C4×C53—41no—
182C4⋊C4×C53—41no—
183C4⋊C4×C53C2×D411yescharacteristic
184C4⋊C4×C53C2×D411yescharacteristic
185C22⋊C4×C53—21no—
186C22⋊C4×C53—21no—
187C22⋊C4×C53—21no—
188C22⋊C4×C53—21no—
189C22⋊C4×C53—21no—
190C22⋊C4×C53—41no—
191C42×C53—21no—
192C42×C53—41no—
193C42×C53C2×D411yescharacteristic
194C42×C53C2×D411yescharacteristic
195C42×C53#4111yescharacteristic
196C42×C53C2411yescharacteristic
197#11719924—81no—
198#11719924—81no—
199#11719922—81no—
200#11719922—81no—
201#11719917—41no—
202#11719917—41no—
203#11719917—42no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—42no—
209C2×#88322C2×D411yescharacteristic
210C2×#88322C2×D411yescharacteristic
211C2×#88322C2×D411yescharacteristic
212C2×#88322C2×D411yescharacteristic
213C23×C53—21no—
214C23×C53#12111yescharacteristic
215C22×D53—41no—
216C22×D53—41no—
217C22×D53—82no—
218C22×D53—82no—
219C22×D53—82no—
220Q8×C53—21no—
221Q8×C53—21no—
222Q8×C53—41no—
223Q8×C53—41no—
224Q8×C53—81no—
225D4×C53—21no—
226D4×C53—21no—
227D4×C53—81no—
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—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53#12111yescharacteristic
241C2×C4×C53#12111yescharacteristic
242C2×C4×C53#12111yescharacteristic
243C2×C4×C53C22×D411yescharacteristic
244C2×C4×C53C22×D411yescharacteristic
245C2×C4×C53C2×#4111yescharacteristic
246D212—41no—
247D212—41no—
248D212—41no—
249D212—41no—
250D212—41no—
251D212—41no—
252D212—41no—
253D212—41no—
254D212—81no—
255D212—81no—
256D212—82no—
257D212—82no—
258D212—82no—
259C8×C53—41no—
260C8×C53—41no—
261#88322—21no—
262#88322—21no—
263#88322—21no—
264#88322—21no—
265#88322—41no—
266#88322—41no—
267#88322—41no—
268#88322—41no—
269#19380—531nomaximal, Sylow
270C22×C53—21no—
271C22×C53—21no—
272C22×C53—21no—
273C22×C53—41no—
274C22×C53—41no—
275C22×C53—41no—
276C22×C53C2×#12111yescharacteristic
277C2×D53—81no—
278C2×D53—81no—
279C2×D53—81no—
280C2×D53—81no—
281C2×D53—81no—
282C2×D53—81no—
283C2×D53—164no—
284C4×C53—21no—
285C4×C53—21no—
286C4×C53—21no—
287C4×C53—21no—
288C4×C53—41no—
289C4×C53—41no—
290C4×C53—41no—
291C4×C53—81no—
292C4×C53—81no—
293C4×C53#57511yescharacteristic
294C4×C53#57511yescharacteristic
295C4×C53#57611yescharacteristic
296C4×C53#57711yescharacteristic
297#3252D5311yescharacteristic, complemented
298#3217—531no—
299#3138—531no—
300#2969—531no—
301#1702—531no—
302#1701—531no—
303#1698—531no—
304#1697—531no—
305#1682—531no—
306#1681—531no—
307#1678—531no—
308#1677—531no—
309#1569—531no—
310#1554—531no—
311#1545—531no—
312C2×C53—41no—
313C2×C53—81no—
314C2×C53#304011yescharacteristic
315C2×C53#306211yescharacteristic
316C2×C53#306811yescharacteristic
317D53—161no—
318D53—161no—
319#533C2×D5311yescharacteristic
320C2×#127—531no—
321#529C2×D5311yescharacteristic
322#496—1061no—
323#490—1061no—
324#490—1061no—
325#490—1061no—
326#486—531no—
327#486—531no—
328#486—531no—
329#486—531no—
330#479—21no—
331#477C2×D5311yescharacteristic, complemented
332#476C2×D5311yescharacteristic, complemented
333#470—21no—
334#468—21no—
335#464—21no—
336#462C2×D5311yescharacteristic, complemented
337#461C2×D5311yescharacteristic, complemented
338#460—531no—
339#460—531no—
340#459—531no—
341#459—531no—
342#450—1061no—
343#447—1061no—
344#446—531no—
345C8×Q8—531no—
346#436—531no—
347#436—1061no—
348#435—1061no—
349C8×D4—531no—
350#433—1061no—
351#432—531no—
352#431—531no—
353#423C2×D5311yescharacteristic
354#418—1061no—
355#418—1061no—
356#405—1061no—
357C4×M16—531no—
358#335—531no—
359#335—531no—
360#334—531no—
361#334—531no—
362#333—531no—
363#333—531no—
364#332—531no—
365#332—531no—
366#331—531no—
367#331—531no—
368#331—531no—
369#331—531no—
370C53#1938011yescharacteristic, complemented, Sylow
371C22×D4—1061no—
372C22×D4—1061no—
373C22×D4—1061no—
374C2×Q16—21no—
375C2×SD16—21no—
376C2×SD16—1061no—
377C2×SD16—1061no—
378C2×D8—1061no—
379C2×D8—1061no—
380C2×M16—21no—
381C2×M16—531no—
382C2×M16—531no—
383C2×M16—1061no—
384C22×C8—1061no—
385C4⋊Q8C22×D5311yescharacteristic
386C4⋊Q8C22×D5311yescharacteristic
387#127—531no—
388#127—531no—
389#127—1061no—
390#127—1061no—
391#127—1061no—
392#126—21no—
393#125—21no—
394#125C22⋊D5311yescharacteristic
395#124C22⋊D5311yescharacteristic
396#123—21no—
397C22⋊Q8—21no—
398C22⋊Q8—21no—
399C22⋊Q8C22⋊D5311yescharacteristic
400#121C22⋊D5311yescharacteristic
401C4×Q8—21no—
402C4×Q8—21no—
403C4×Q8C22×D5311yescharacteristic
404C4×D4—21no—
405C4×D4C22×D5311yescharacteristic
406#117—21no—
407#117—21no—
408C2×C42C22×D5311yescharacteristic
409#107—21no—
410#106—21no—
411C4⋊C8—21no—
412C4⋊C8—531no—
413C4⋊C8—531no—
414C4⋊C8—531no—
415C4⋊C8—531no—
416C4⋊C8—1061no—
417C4⋊C8—1061no—
418C4⋊C8C22×D5311yescharacteristic
419C4⋊C8C22×D5311yescharacteristic
420#103—21no—
421#103—21no—
422#103—21no—
423#102—21no—
424#102—1061no—
425#102—1061no—
426#102—1061no—
427#102—1061no—
428#102—1061no—
429#102—1061no—
430#102—1061no—
431#102—1061no—
432#102—1061no—
433#102—1061no—
434#102—1061no—
435#102—1061no—
436C22⋊C8—1061no—
437C22⋊C8—1061no—
438C22⋊C8—1061no—
439C22⋊C8—1061no—
440#97—531no—
441#97—531no—
442#97—1061no—
443#97—1061no—
444C4×C8—531no—
445C4×C8—531no—
446C4×C8—1061no—
447C4×C8—1061no—
448C24—2121no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8C2×C22⋊D5311yescharacteristic
452C2×D4—21no—
453C2×D4—1061no—
454C2×D4—1061no—
455C2×D4—1061no—
456C2×D4—1061no—
457C2×D4—1061no—
458C2×D4—1061no—
459C2×D4—4241no—
460C2×D4—4241no—
461C2×D4—4242no—
462C2×D4—2122no—
463C2×D4—2122no—
464C2×D4—2121no—
465C2×D4—2121no—
466C2×D4—2121no—
467C2×D4C2×C22⋊D5311yescharacteristic
468C22×C4—21no—
469C22×C4D4×D5311yescharacteristic
470C22×C4#1171994711yescharacteristic
471C22×C4C2×C22⋊D5311yescharacteristic
472Q16—81no—
473SD16—81no—
474SD16—4241no—
475SD16—4241no—
476D8—4241no—
477D8—4241no—
478M16—81no—
479M16—2121no—
480M16—2121no—
481M16—2122no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—531no—
485C2×C8—531no—
486C2×C8—531no—
487C2×C8—531no—
488C2×C8—1061no—
489C2×C8—1061no—
490C2×C8—1061no—
491C2×C8—1061no—
492C2×C8—2122no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—41no—
502C4⋊C4—41no—
503C4⋊C4C2×C22⋊D5311yescharacteristic
504C4⋊C4C2×C22⋊D5311yescharacteristic
505C22⋊C4—21no—
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—41no—
511C42—21no—
512C42—41no—
513C42D4×D5311yescharacteristic
514C42#1171994711yescharacteristic
515C42C2×C22⋊D5311yescharacteristic
516C42C23×D5311yescharacteristic
517C23—21no—
518C23—4242no—
519C23—4242no—
520C23—4242no—
521C23—2121no—
522C23—2121no—
523C23#1313318911yescharacteristic
524Q8—21no—
525Q8—21no—
526Q8—41no—
527Q8—41no—
528Q8—81no—
529D4—21no—
530D4—21no—
531D4—81no—
532D4—4241no—
533D4—4241no—
534D4—4242no—
535D4—4242no—
536D4—4242no—
537D4—2121no—
538D4—2121no—
539D4—2121no—
540D4—2121no—
541D4—2121no—
542D4—2121no—
543D4—2121no—
544D4—2121no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4#1313318911yescharacteristic
558C2×C4#1313318911yescharacteristic
559C2×C4#1313318911yescharacteristic
560C2×C4C2×D4×D5311yescharacteristic
561C2×C4C22×C22⋊D5311yescharacteristic
562C2×C4C2×#1171994711yescharacteristic
563C8—41no—
564C8—41no—
565C8—1061no—
566C8—1061no—
567C8—1061no—
568C8—1061no—
569C8—2121no—
570C8—2121no—
571C8—2121no—
572C8—2121no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—8484no—
580C22—4241no—
581C22—4241no—
582C22—4241no—
583C22—4241no—
584C22—4241no—
585C22—4241no—
586C22C2×#1313318911yescharacteristic
587C4—21no—
588C4—21no—
589C4—21no—
590C4—21no—
591C4—41no—
592C4—41no—
593C4—41no—
594C4—81no—
595C4—81no—
596C4#1681142211yescharacteristic
597C4#1681142211yescharacteristic
598C4#1681142611yescharacteristic
599C4#1681143111yescharacteristic
600C2—41no—
601C2—81no—
602C2—8481no—
603C2—8481no—
604C2#2028421111yescharacteristic
605C2#2028439711yescharacteristic
606C2#2028444511yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3252C2
socleC22×C53C2×#121
prime radical#20285730C2
hypercenter#3252D53
hypocenterC53#19380
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3252

Complements

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

NH ≅ G/Ntype
2. #20285091C2semidirect
3. #20283720C2semidirect
4. #20274285C2semidirect
5. #20274278C2semidirect
6. #20274257C2semidirect
7. #20274250C2semidirect
8. #20274161C2semidirect
9. #20274154C2semidirect
10. #20274133C2semidirect
11. #20274126C2semidirect
12. #20273480C2semidirect
13. #20273425C2semidirect
14. #20273384C2semidirect
15. C53×#3252C2semidirect
31. #16810754C22semidirect
32. Q8×#88322C22semidirect
34. #16810701C22semidirect
36. D4×#88322C22semidirect
44. #16810375C22semidirect
45. #16810375C22semidirect
46. #16810373C22semidirect
47. #16810373C22semidirect
48. #16810372C22semidirect
49. #16810372C22semidirect
50. #16810369C22semidirect
51. #16810369C22semidirect
59. C53×#477C22semidirect
60. C53×#476C22semidirect
64. C53×#462C22semidirect
65. C53×#461C22semidirect
297. #3252D53semidirect
331. #477C2×D53semidirect
332. #476C2×D53semidirect
336. #462C2×D53semidirect
337. #461C2×D53semidirect
370. C53#19380semidirect

Maximal subgroups

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

subgroupindex
#202857302
#202850912
#202837202
#202742852
#202742782
#202742572
#202742502
#202741612
#202741542
#202741332
#202741262
#202734802
#202734252
#202733842
C53×#32522
#1938053