Finite Group Databasebeta

← #27368335

Subgroups — #27368335

Order 13568.

678 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284802C211yesmaximal, characteristic
2#20284508C211yesmaximal, characteristic, complemented
3M16×D212C211yesmaximal, characteristic, complemented
4#20274287C211yesmaximal, characteristic, complemented
5#20274245C211yesmaximal, characteristic, complemented
6#20274235C211yesmaximal, characteristic, complemented
7#20274177C211yesmaximal, characteristic, complemented
8#20274163C211yesmaximal, characteristic, complemented
9#20274121C211yesmaximal, characteristic, complemented
10#20274111C211yesmaximal, characteristic, complemented
11#20274053C211yesmaximal, characteristic, complemented
12#20273482C211yesmaximal, characteristic, complemented
13#20273450C211yesmaximal, characteristic, complemented
14#20273445C211yesmaximal, characteristic, complemented
15C53×#3182C211yesmaximal, characteristic, complemented
16C2×M16×D53—21no—
17C2×C4×D212C2211yescharacteristic
18#16810939—21no—
19#16810920—21no—
20#16810920—21no—
21#16810910C2211yescharacteristic
22#16810910C2211yescharacteristic
23#16810910C2211yescharacteristic
24#16810910C2211yescharacteristic
25#16810900C2211yescharacteristic
26#16810900C2211yescharacteristic
27#16810815C2211yescharacteristic
28#16810815C2211yescharacteristic
29#16810756—21no—
30#16810733—21no—
31#16810733—21no—
32#16810702C2211yescharacteristic
33#16810702C2211yescharacteristic
34#16810696—21no—
35#16810696—21no—
36C8×D212C2211yescharacteristic
37C8×D212C2211yescharacteristic
38#16810680C2211yescharacteristic
39#16810666—21no—
40#16810647C2211yescharacteristic
41#16810647C2211yescharacteristic
42#16810627—21no—
43#16810586—21no—
44#16810376C2211yescharacteristic, complemented
45#16810376C2211yescharacteristic, complemented
46#16810374C2211yescharacteristic, complemented
47#16810374C2211yescharacteristic, complemented
48#16810372C2211yescharacteristic, complemented
49#16810372C2211yescharacteristic, complemented
50#16810372C2211yescharacteristic, complemented
51#16810372C2211yescharacteristic, complemented
52#16810369C2211yescharacteristic, complemented
53#16810369C2211yescharacteristic, complemented
54#16810367C2211yescharacteristic, complemented
55#16810367C2211yescharacteristic, complemented
56C53×#533C2211yescharacteristic
57C53×#532C2211yescharacteristic
58C53×#500C2211yescharacteristic, complemented
59C53×#492C2211yescharacteristic, complemented
60C53×#487C2211yescharacteristic, complemented
61C53×#486C2211yescharacteristic, complemented
62C53×#468—21no—
63C53×#468—21no—
64C53×#440—21no—
65C53×#440—21no—
66C4×M16×C53C2211yescharacteristic
67C22×D212D411yescharacteristic
68C22×C4×D53—21no—
69C2×#11719922—21no—
70C2×#11719922—21no—
71C2×#11719917—21no—
72C2×#11719917—21no—
73C2×#11719912—21no—
74C2×#11719912—21no—
75M16×D53—44no—
76C2×C8×D53—21no—
77C2×C8×D53—21no—
78C4×D212—21no—
79C4×D212—21no—
80C4×D212D411yescharacteristic
81C4×D212D411yescharacteristic
82C4×D212C2311yescharacteristic
83C4×D212C2311yescharacteristic
84C2×C4⋊Dic53D411yescharacteristic
85C2×#11719921—21no—
86#13133150—21no—
87#13133150—21no—
88#13133148—21no—
89#13133148—21no—
90#13133148—21no—
91#13133148—21no—
92C4⋊#88322—21no—
93C4⋊#88322—21no—
94C4⋊#88322C2311yescharacteristic
95C4⋊#88322C2311yescharacteristic
96C4⋊#88322C2311yescharacteristic
97C4⋊#88322C2311yescharacteristic
98C4⋊#88322C2311yescharacteristic
99C4⋊#88322C2311yescharacteristic
100#13133138—21no—
101#13133138—21no—
102#13133137—21no—
103#13133137—21no—
104#13133137—21no—
105#13133137—21no—
106#13133137—21no—
107#13133137—21no—
108C22⋊#88322—21no—
109#13133129—21no—
110#13133129—21no—
111#13133129—21no—
112#13133129—21no—
113#13133128—21no—
114C2×#41×C53—21no—
115C2×SD16×C53—21no—
116C2×SD16×C53—21no—
117C2×M16×C53D411yescharacteristic
118C2×M16×C53D411yescharacteristic
119C4⋊Q8×C53C2311yescharacteristic
120C4⋊Q8×C53C2311yescharacteristic
121C4⋊Q8×C53C2311yescharacteristic
122#127×C53C2311yescharacteristic
123#125×C53—21no—
124#124×C53—21no—
125C22⋊Q8×C53—21no—
126C22⋊Q8×C53—21no—
127#121×C53—21no—
128#121×C53—21no—
129C4×Q8×C53—21no—
130C4×Q8×C53—21no—
131C4×D4×C53—21no—
132C4×D4×C53—21no—
133#117×C53—21no—
134C2×C42×C53C2311yescharacteristic
135#107×C53—21no—
136#107×C53—21no—
137#103×C53—21no—
138#103×C53—21no—
139#102×C53—21no—
140#102×C53—21no—
141#97×C53D411yescharacteristic
142#97×C53D411yescharacteristic
143C4×C8×C53C2311yescharacteristic
144C4×C8×C53C2311yescharacteristic
145C23×D53—21no—
146#41×C53—82no—
147C2×Q8×C53—21no—
148C2×Q8×C53—21no—
149C2×Q8×C53—21no—
150C2×D4×C53—21no—
151C2×D4×C53—21no—
152C2×D4×C53—21no—
153C2×D4×C53—41no—
154C22×C4×C53—41no—
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53#4111yescharacteristic
158C22×Dic53—21no—
159C2×D212—22no—
160C2×D212—21no—
161C2×D212—41no—
162C2×D212C2×D411yescharacteristic
163C2×D212C2×D411yescharacteristic
164C2×C4×D53—21no—
165C2×C4×D53—21no—
166C2×C4×D53—21no—
167C2×C4×D53—21no—
168C2×C4×D53—42no—
169C2×C4×D53—42no—
170SD16×C53—42no—
171SD16×C53—42no—
172M16×C53—22no—
173M16×C53—41no—
174C2×C8×C53C2×D411yescharacteristic
175C2×C8×C53C2×D411yescharacteristic
176C2×C8×C53C2×D411yescharacteristic
177C2×C8×C53C2×D411yescharacteristic
178C4⋊C4×C53—21no—
179C4⋊C4×C53—21no—
180C4⋊C4×C53—21no—
181C4⋊C4×C53—21no—
182C4⋊C4×C53—21no—
183C4⋊C4×C53—21no—
184C4⋊C4×C53—41no—
185C4⋊C4×C53—41no—
186C22⋊C4×C53—41no—
187C22⋊C4×C53—41no—
188C22⋊C4×C53—41no—
189C42×C53—41no—
190C42×C53C2×D411yescharacteristic
191C42×C53C2×D411yescharacteristic
192C42×C53#4111yescharacteristic
193C42×C53C2411yescharacteristic
194#11719922—42no—
195#11719922—42no—
196#11719921—21no—
197#11719921—21no—
198#11719921—21no—
199#11719921—21no—
200C4⋊Dic53—21no—
201C4⋊Dic53C2×D411yescharacteristic
202C4⋊Dic53C2×D411yescharacteristic
203#11719917—42no—
204#11719917—42no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322—21no—
211#11719912—42no—
212#11719912—42no—
213C8×D53—42no—
214C8×D53—42no—
215C23×C53—41no—
216C23×C53#12111yescharacteristic
217C22×D53—21no—
218C22×D53—21no—
219C22×D53—42no—
220C22×D53—42no—
221C22×D53—81no—
222Q8×C53—41no—
223Q8×C53—41no—
224Q8×C53—81no—
225Q8×C53—81no—
226D4×C53—41no—
227D4×C53—41no—
228D4×C53—41no—
229D4×C53—41no—
230D4×C53—81no—
231D4×C53—82no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—82no—
241C2×C4×C53#12111yescharacteristic
242C2×C4×C53#12111yescharacteristic
243C2×C4×C53#12111yescharacteristic
244C2×C4×C53C22×D411yescharacteristic
245C2×C4×C53C22×D411yescharacteristic
246C2×C4×C53C2×#4111yescharacteristic
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—42no—
250D212—21no—
251D212—21no—
252D212—21no—
253D212—42no—
254D212#13712yes—
255C4×D53—42no—
256C4×D53—42no—
257C4×D53—41no—
258C4×D53—41no—
259C4×D53—41no—
260C4×D53—41no—
261C8×C53—21no—
262C8×C53—21no—
263C8×C53—21no—
264C8×C53AGL(1,ℤ/8)12yes—
265#88322—41no—
266#88322—41no—
267#88322—41no—
268#88322—41no—
269#88322—41no—
270#88322—41no—
271#19305—531nomaximal, Sylow
272C22×C53—21no—
273C22×C53—21no—
274C22×C53—21no—
275C22×C53—81no—
276C22×C53—81no—
277C22×C53—81no—
278C22×C53C2×#12111yescharacteristic
279C2×D53—22no—
280C2×D53—41no—
281C2×D53—41no—
282C2×D53—41no—
283C2×D53—41no—
284C2×D53—41no—
285C2×D53—84no—
286C4×C53—21no—
287C4×C53—21no—
288C4×C53—21no—
289C4×C53—21no—
290C4×C53—81no—
291C4×C53—81no—
292C4×C53—81no—
293C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
294C4×C53C2×#13711yescharacteristic
295C4×C53#57511yescharacteristic
296C4×C53#57511yescharacteristic
297Dic53—41no—
298Dic53—41no—
299#3182D5311yescharacteristic, complemented
300#3109—531no—
301#3077—531no—
302D4×M16—531no—
303#1703—531no—
304#1697—531no—
305#1695—531no—
306#1685—531no—
307#1683—531no—
308#1677—531no—
309#1675—531no—
310#1665—531no—
311#1569—531no—
312#1560—531no—
313#1559—531no—
314C2×C53—41no—
315C2×C53—161no—
316C2×C53#303611yescharacteristic
317C2×C53#305611yescharacteristic
318C2×C53#306811yescharacteristic
319D53—42no—
320D53—81no—
321C22×M16—1061no—
322#533C2×D5311yescharacteristic
323#532C2×D5311yescharacteristic
324C2×C4×D4—531no—
325#500C2×D5311yescharacteristic, complemented
326#492C2×D5311yescharacteristic, complemented
327#487C2×D5311yescharacteristic, complemented
328#486C2×D5311yescharacteristic, complemented
329#481—1061no—
330#478—1061no—
331#478—1061no—
332#476—531no—
333#476—531no—
334#476—531no—
335#476—531no—
336#474—531no—
337#474—531no—
338#468—21no—
339#468—21no—
340#459—531no—
341#459—531no—
342#447—1061no—
343#442—1061no—
344#442—1061no—
345#440—21no—
346#440—21no—
347#436—531no—
348#436—531no—
349#435—1061no—
350#435—1061no—
351C8×D4—531no—
352C8×D4—531no—
353#432—531no—
354#428—1061no—
355#423—531no—
356#423—531no—
357#418—1061no—
358#407—1061no—
359C4×M16C2×D5311yescharacteristic
360#335—531no—
361#335—531no—
362#334—531no—
363#334—531no—
364#333—531no—
365#333—531no—
366#333—531no—
367#333—531no—
368#332—531no—
369#332—531no—
370#331—531no—
371#331—531no—
372C53#1930511yescharacteristic, complemented, Sylow
373C2×#41—21no—
374C22×D4—531no—
375C23×C4—1061no—
376C2×SD16—21no—
377C2×SD16—21no—
378C2×D8—1061no—
379C2×D8—1061no—
380C2×M16—1061no—
381C2×M16—1061no—
382C2×M16—1061no—
383C2×M16—1061no—
384C2×M16—2124no—
385C2×M16C22⋊D5311yescharacteristic
386C2×M16C22⋊D5311yescharacteristic
387C22×C8—1061no—
388C22×C8—1061no—
389C4⋊Q8C22×D5311yescharacteristic
390C4⋊Q8C22×D5311yescharacteristic
391C4⋊Q8C22×D5311yescharacteristic
392#127C22×D5311yescharacteristic
393#125—21no—
394#124—21no—
395C22⋊Q8—21no—
396C22⋊Q8—21no—
397#121—21no—
398#121—21no—
399C4×Q8—21no—
400C4×Q8—21no—
401C4×D4—21no—
402C4×D4—21no—
403C4×D4—531no—
404C4×D4—531no—
405C4×D4—531no—
406C4×D4—531no—
407C4×D4—1061no—
408C4×D4—1061no—
409#117—21no—
410C2×C4⋊C4—531no—
411C2×C22⋊C4—1061no—
412C2×C42C22×D5311yescharacteristic
413#107—21no—
414#107—21no—
415#107—1061no—
416#107—1061no—
417#106—1061no—
418#106—1061no—
419#106—1061no—
420#106—1061no—
421C4⋊C8—531no—
422C4⋊C8—531no—
423C4⋊C8—531no—
424C4⋊C8—531no—
425C4⋊C8—531no—
426C4⋊C8—531no—
427C4⋊C8—1061no—
428C4⋊C8—1061no—
429#103—21no—
430#103—21no—
431#102—21no—
432#102—21no—
433#102—1061no—
434#102—1061no—
435#102—1061no—
436#102—1061no—
437#102—1061no—
438#102—1061no—
439#102—1061no—
440#102—1061no—
441C22⋊C8—1061no—
442C22⋊C8—1061no—
443C22⋊C8—1061no—
444C22⋊C8—1061no—
445C22⋊C8—1061no—
446#97—1061no—
447#97C22⋊D5311yescharacteristic
448#97C22⋊D5311yescharacteristic
449C4×C8C22×D5311yescharacteristic
450C4×C8C22×D5311yescharacteristic
451C24—1061no—
452#41—82no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—41no—
460C2×D4—531no—
461C2×D4—531no—
462C2×D4—1062no—
463C2×D4—1061no—
464C2×D4—2121no—
465C22×C4—41no—
466C22×C4—1061no—
467C22×C4—1061no—
468C22×C4—1061no—
469C22×C4—1061no—
470C22×C4—1061no—
471C22×C4—2122no—
472C22×C4—2122no—
473C22×C4D4×D5311yescharacteristic
474C22×C4#1171994711yescharacteristic
475C22×C4C2×C22⋊D5311yescharacteristic
476SD16—42no—
477SD16—42no—
478D8—2122no—
479D8—2122no—
480M16—22no—
481M16—41no—
482M16—2122no—
483M16—2122no—
484M16—2122no—
485M16—2122no—
486C2×C8—1061no—
487C2×C8—1061no—
488C2×C8—1061no—
489C2×C8—1061no—
490C2×C8—1061no—
491C2×C8—1061no—
492C2×C8—2122no—
493C2×C8—2122no—
494C2×C8C2×C22⋊D5311yescharacteristic
495C2×C8C2×C22⋊D5311yescharacteristic
496C2×C8C2×C22⋊D5311yescharacteristic
497C2×C8C2×C22⋊D5311yescharacteristic
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—531no—
507C4⋊C4—531no—
508C4⋊C4—1061no—
509C22⋊C4—41no—
510C22⋊C4—41no—
511C22⋊C4—41no—
512C22⋊C4—1061no—
513C22⋊C4—1061no—
514C22⋊C4—1061no—
515C22⋊C4—1061no—
516C42—41no—
517C42D4×D5311yescharacteristic
518C42#1171994711yescharacteristic
519C42C2×C22⋊D5311yescharacteristic
520C42C23×D5311yescharacteristic
521C23—41no—
522C23—1061no—
523C23—1061no—
524C23—4241no—
525C23—2122no—
526C23—2122no—
527C23#1313318811yescharacteristic
528Q8—41no—
529Q8—41no—
530Q8—81no—
531Q8—81no—
532D4—41no—
533D4—41no—
534D4—41no—
535D4—41no—
536D4—81no—
537D4—82no—
538D4—532no—
539D4—1061no—
540D4—1061no—
541D4—1061no—
542D4—2122no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—82no—
552C2×C4—1061no—
553C2×C4—1061no—
554C2×C4—2122no—
555C2×C4—2122no—
556C2×C4—2122no—
557C2×C4—2121no—
558C2×C4—2121no—
559C2×C4—2121no—
560C2×C4—2121no—
561C2×C4#1313318811yescharacteristic
562C2×C4#1313318811yescharacteristic
563C2×C4#1313318811yescharacteristic
564C2×C4C2×D4×D5311yescharacteristic
565C2×C4C22×C22⋊D5311yescharacteristic
566C2×C4C2×#1171994711yescharacteristic
567C8—21no—
568C8—21no—
569C8—21no—
570C8—2121no—
571C8—2121no—
572C8—2121no—
573C8—2121no—
574C8—2121no—
575C8—2121no—
576C8#1313324212yes—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22—1062no—
584C22—4244no—
585C22—2121no—
586C22—2121no—
587C22—2121no—
588C22—2121no—
589C22—2121no—
590C22C2×#1313318811yescharacteristic
591C4—21no—
592C4—21no—
593C4—21no—
594C4—21no—
595C4—81no—
596C4—81no—
597C4—81no—
598C4—2121no—
599C4—2121no—
600C4C2×#1313324211yescharacteristic
601C4#137×D5311yescharacteristic
602C4#1681142111yescharacteristic
603C4#1681142211yescharacteristic
604C2—41no—
605C2—161no—
606C2—4241no—
607C2—2122no—
608C2#2028417511yescharacteristic
609C2#2028433911yescharacteristic
610C2#2028443911yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3182C2
socleC22×C53C2×#121
prime radical#20284802C2
hypercenter#3182D53
hypocenterC53#19305
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3182

Complements

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

NH ≅ G/Ntype
2. #20284508C2semidirect
3. M16×D212C2semidirect
4. #20274287C2semidirect
5. #20274245C2semidirect
6. #20274235C2semidirect
7. #20274177C2semidirect
8. #20274163C2semidirect
9. #20274121C2semidirect
10. #20274111C2semidirect
11. #20274053C2semidirect
12. #20273482C2semidirect
13. #20273450C2semidirect
14. #20273445C2semidirect
15. C53×#3182C2semidirect
44. #16810376C22semidirect
45. #16810376C22semidirect
46. #16810374C22semidirect
47. #16810374C22semidirect
48. #16810372C22semidirect
49. #16810372C22semidirect
50. #16810372C22semidirect
51. #16810372C22semidirect
52. #16810369C22semidirect
53. #16810369C22semidirect
54. #16810367C22semidirect
55. #16810367C22semidirect
58. C53×#500C22semidirect
59. C53×#492C22semidirect
60. C53×#487C22semidirect
61. C53×#486C22semidirect
299. #3182D53semidirect
325. #500C2×D53semidirect
326. #492C2×D53semidirect
327. #487C2×D53semidirect
328. #486C2×D53semidirect
372. C53#19305semidirect

Maximal subgroups

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

subgroupindex
#202848022
#202845082
M16×D2122
#202742872
#202742452
#202742352
#202741772
#202741632
#202741212
#202741112
#202740532
#202734822
#202734502
#202734452
C53×#31822
#1930553