Finite Group Databasebeta

← #28317442

Subgroups — #28317442

Order 13568.

616 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291624C211yesmaximal, characteristic, complemented
2#20291576C211yesmaximal, characteristic
3#20290359C211yesmaximal, characteristic, complemented
4#20290350C211yesmaximal, characteristic, complemented
5#20290342C211yesmaximal, characteristic, complemented
6#20290331C211yesmaximal, characteristic, complemented
7#20290323C211yesmaximal, characteristic, complemented
8#20290321C211yesmaximal, characteristic, complemented
9#20290307C211yesmaximal, characteristic, complemented
10#20290299C211yesmaximal, characteristic, complemented
11#20290286C211yesmaximal, characteristic, complemented
12#20290077C211yesmaximal, characteristic, complemented
13#20290016C211yesmaximal, characteristic
14#20289936C211yesmaximal, characteristic, complemented
15#20284720C211yesmaximal, characteristic
16#16811675—21no—
17#16811674C2211yescharacteristic, complemented
18#16811673C2211yescharacteristic, complemented
19#16811672—21no—
20#16811672C2211yescharacteristic, complemented
21Q8×#88324C2211yescharacteristic, complemented
22#16811657C2211yescharacteristic
23#16811657C2211yescharacteristic
24#16811653—21no—
25#16811653—21no—
26#16811651C2211yescharacteristic
27#16811651C2211yescharacteristic
28#16811647—21no—
29#16811647—21no—
30#16811644C2211yescharacteristic
31#16811640C2211yescharacteristic
32#16811639C2211yescharacteristic
33#16811639C2211yescharacteristic
34#16811594—21no—
35D53⋊(C22⋊C8)—21no—
36#16811487C2211yescharacteristic, complemented
37#16811486C2211yescharacteristic, complemented
38#16811485C2211yescharacteristic, complemented
39#16811484C2211yescharacteristic, complemented
40#16811473C2211yescharacteristic, complemented
41#16811473C2211yescharacteristic, complemented
42Q8⋊#88324C2211yescharacteristic, complemented
43Q8⋊#88324C2211yescharacteristic, complemented
44#16811471C2211yescharacteristic
45#16811471C2211yescharacteristic
46#16811471C2211yescharacteristic
47#16811471C2211yescharacteristic
48#88324⋊C8C2211yescharacteristic, complemented
49#88324⋊C8C2211yescharacteristic, complemented
50#16811464C2211yescharacteristic, complemented
51C8×#88324C2211yescharacteristic, complemented
52#16811175C2211yescharacteristic
53C2×#13133179C2211yescharacteristic
54#16810953C411yescharacteristic
55#16810912C411yescharacteristic
56#16810912C411yescharacteristic
57#16810827C411yescharacteristic
58#16810827C411yescharacteristic
59#16810776C411yescharacteristic
60#16810773C411yescharacteristic
61#16810710C2211yescharacteristic
62#16810710C2211yescharacteristic
63#16810708C2211yescharacteristic
64#16810708C2211yescharacteristic
65#16810673C2211yescharacteristic
66C53×#483C411yescharacteristic
67C2×#11719936—21no—
68C2×#11719936—21no—
69C2×D53⋊C8—21no—
70C2×D53⋊C8—21no—
71#13133284—21no—
72#13133284—21no—
73#13133284D411yescharacteristic
74#13133284D411yescharacteristic
75C4⋊#88324—21no—
76C4⋊#88324—21no—
77C4⋊#88324—21no—
78C4⋊#88324C2311yescharacteristic
79C4⋊#88324C2311yescharacteristic
80C4⋊#88324C2311yescharacteristic
81C4⋊#88324C2311yescharacteristic
82#13133274—21no—
83#13133274—21no—
84#13133274—21no—
85#13133274—21no—
86C22⋊#88324—21no—
87C22⋊#88324—21no—
88C22⋊#88324—21no—
89C22⋊#88324D411yescharacteristic
90C22⋊#88324D411yescharacteristic
91#13133272—21no—
92#13133272—21no—
93#13133272—21no—
94C4×#88324—21no—
95C4×#88324—21no—
96C4×#88324—21no—
97C4×#88324C2311yescharacteristic
98C4×#88324C2311yescharacteristic
99C4×#88324C2311yescharacteristic
100C4×#88324C2311yescharacteristic
101C22×D212C2×C411yescharacteristic
102C22×C4×D53—21no—
103C2×#11719913C2×C411yescharacteristic
104C2×#11719913C2×C411yescharacteristic
105C2×#11719924C2×C411yescharacteristic
106C2×#11719924C2×C411yescharacteristic
107#13133216C2×C411yescharacteristic
108#13133215C2×C411yescharacteristic
109#13133207—21no—
110#13133196C2×C411yescharacteristic
111#13133194—21no—
112Q8×Dic53C2311yescharacteristic
113#13133182—21no—
114#13133182C2311yescharacteristic
115#13133179—21no—
116#13133179—21no—
117#13133179C2×C411yescharacteristic
118#13133179C2×C411yescharacteristic
119#13133179C2311yescharacteristic
120#13133179C2311yescharacteristic
121#13133173—21no—
122C2×#11719921—21no—
123C2×C4×Dic53C2311yescharacteristic
124#13133148C2×C411yescharacteristic
125#13133148C2×C411yescharacteristic
126#13133146C2×C411yescharacteristic
127#13133146C2×C411yescharacteristic
128#13133140C2×C411yescharacteristic
129#13133140C2×C411yescharacteristic
130#13133139C2×C411yescharacteristic
131#13133139C2×C411yescharacteristic
132#13133137C2×C411yescharacteristic
133#13133137C2×C411yescharacteristic
134C22⋊#88322C2×C411yescharacteristic
135C8×Dic53C2311yescharacteristic
136C8×Dic53C2311yescharacteristic
137C22⋊Q8×C53C2×C411yescharacteristic
138C2×C4⋊C4×C53C2×C411yescharacteristic
139#106×C53C2×C411yescharacteristic
140#106×C53C2×C411yescharacteristic
141#103×C53C2×C411yescharacteristic
142#103×C53C2×C411yescharacteristic
143C22⋊C8×C53C2×C411yescharacteristic
144C23×D53—21no—
145C2×Q8×C53C22×C411yescharacteristic
146C22×C4×C53—21no—
147C22×C4×C53C22×C411yescharacteristic
148C22×Dic53C2×D411yescharacteristic
149C22×Dic53#4111yescharacteristic
150C2×D212—22no—
151C2×D212—21no—
152C2×D212—41no—
153C2×D212C22×C411yescharacteristic
154C2×D212C22×C411yescharacteristic
155C2×C4×D53—21no—
156C2×C4×D53—21no—
157C2×C4×D53—21no—
158C2×C4×D53—21no—
159C2×C4×D53—81no—
160C2×C4×D53—81no—
161C2×Dic106—21no—
162C2×Dic106C22×C411yescharacteristic
163C2×#88324—21no—
164C2×#88324—21no—
165C2×#88324—21no—
166C2×#88324—21no—
167C2×#88324—21no—
168C2×#88324—21no—
169C2×#88324C2×D411yescharacteristic
170C2×#88324C2×D411yescharacteristic
171C2×#88324C2×D411yescharacteristic
172C2×#88324C2×D411yescharacteristic
173#11719936—81no—
174#11719936—81no—
175D53⋊C8—81no—
176D53⋊C8—81no—
177C2×C8×C53C22×C411yescharacteristic
178C2×C8×C53C22×C411yescharacteristic
179C4⋊C4×C53—21no—
180C4⋊C4×C53—21no—
181C4⋊C4×C53C22×C411yescharacteristic
182C4⋊C4×C53C22×C411yescharacteristic
183C4⋊C4×C53C22×C411yescharacteristic
184C22⋊C4×C53—21no—
185C22⋊Dic53—21no—
186#11719924—22no—
187#11719924—41no—
188#11719921—21no—
189#11719921—21no—
190#11719921—21no—
191#11719921—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53C22×C411yescharacteristic
194#11719919—21no—
195#11719919—21no—
196#11719919—21no—
197#11719919—21no—
198#11719919—21no—
199C4×Dic53—21no—
200C4×Dic53—21no—
201C4×Dic53C22×C411yescharacteristic
202C4×Dic53C2×D411yescharacteristic
203C4×Dic53#4111yescharacteristic
204C4×Dic53C2411yescharacteristic
205C2×#88322C22×C411yescharacteristic
206C2×#88322C22×C411yescharacteristic
207#11719913—22no—
208#11719913—41no—
209C23×C53C4×D411yescharacteristic
210C22×D53—21no—
211C22×D53—21no—
212C22×D53—42no—
213C22×D53—42no—
214C22×D53—81no—
215Q8×C53—21no—
216Q8×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—81no—
224C2×C4×C53C4×D411yescharacteristic
225C2×C4×C53C23×C411yescharacteristic
226C2×Dic53—21no—
227C2×Dic53—21no—
228C2×Dic53—21no—
229C2×Dic53—21no—
230C2×Dic53—41no—
231C2×Dic53C4×D411yescharacteristic
232C2×Dic53C4×D411yescharacteristic
233C2×Dic53C22×D411yescharacteristic
234C2×Dic53C2×#4111yescharacteristic
235D212—21no—
236D212—21no—
237D212—21no—
238D212—42no—
239D212#13112yes—
240C4×D53—41no—
241C4×D53—41no—
242C4×D53—41no—
243C4×D53—41no—
244C4×D53—81no—
245C4×D53—81no—
246Dic106—21no—
247Dic106—21no—
248Dic106—42no—
249#88324—21no—
250#88324—21no—
251#88324—21no—
252#88324—21no—
253#88324—41no—
254#88324—41no—
255#88324—41no—
256#88324—41no—
257#88324—41no—
258#88324—41no—
259C8×C53—21no—
260C8×C53—21no—
261#88322—21no—
262#88322—21no—
263#18028—531nomaximal, Sylow
264C22×C53—21no—
265C22×C53—21no—
266C22×C53—21no—
267C22×C53C2×C4×D411yescharacteristic
268C2×D53—22no—
269C2×D53—41no—
270C2×D53—41no—
271C2×D53—41no—
272C2×D53—41no—
273C2×D53—41no—
274C2×D53—84no—
275C4×C53—21no—
276C4×C53—41no—
277C4×C53—41no—
278C4×C53—41no—
279C4×C53—41no—
280C4×C53C2×#13111yescharacteristic
281C4×C53#56811yescharacteristic
282Dic53—21no—
283Dic53—21no—
284Dic53—21no—
285Dic53—41no—
286Dic53—41no—
287Dic53#57711yescharacteristic
288Dic53#57711yescharacteristic
289#3101—531no—
290#2986—531no—
291#2974—531no—
292#1592—531no—
293#1590—531no—
294#1588—531no—
295#1585—531no—
296#1583—531no—
297#1582—531no—
298SD16⋊C8—531no—
299C8×SD16—531no—
300#1572—531no—
301#1497—531no—
302#1478—531no—
303#1454—531no—
304C2×C53—41no—
305C2×C53#293311yescharacteristic
306C2×C53#294911yescharacteristic
307C2×C53#295511yescharacteristic
308D53—42no—
309D53—81no—
310#533—531no—
311C2×C4×D4—531no—
312#483—531no—
313#483#615711yescharacteristic
314#476—531no—
315#476—531no—
316#461—531no—
317#461—531no—
318#450—531no—
319#450—531no—
320#446—531no—
321#446—531no—
322#446—1061no—
323C8×Q8—531no—
324C8×Q8—531no—
325C8×Q8—1061no—
326C4×SD16—531no—
327C4×SD16—531no—
328C4×SD16—531no—
329C4×SD16—531no—
330#436—531no—
331#436—531no—
332#435—1061no—
333#435—1061no—
334C8×D4—531no—
335C8×D4—531no—
336#433—1061no—
337#433—1061no—
338#432—531no—
339#431—531no—
340#431—531no—
341#431—531no—
342#431—531no—
343#407—1061no—
344C2×C22⋊C8—1061no—
345#335—531no—
346#335—531no—
347#334—531no—
348#334—531no—
349Q8⋊C8—531no—
350Q8⋊C8—531no—
351Q8⋊C8—531no—
352Q8⋊C8—531no—
353D4⋊C8—531no—
354D4⋊C8—531no—
355D4⋊C8—531no—
356D4⋊C8—531no—
357#322—531no—
358#322—531no—
359#322—531no—
360C82—531no—
361C53#1802811yescharacteristic, complemented, Sylow
362C22×D4—531no—
363C23×C4—1061no—
364C2×SD16—531no—
365C2×SD16—531no—
366C2×SD16—531no—
367C2×SD16—531no—
368C2×M16—1061no—
369C2×M16—1061no—
370C22×C8—1061no—
371C22×C8—1061no—
372C4⋊Q8—531no—
373C4⋊Q8—531no—
374#125—1061no—
375C22⋊Q8—531no—
376C22⋊Q8—1061no—
377C22⋊Q8C2×#615711yescharacteristic
378C4×Q8—531no—
379C4×Q8—531no—
380C4×Q8—1061no—
381C4×D4—531no—
382C4×D4—531no—
383C4×D4—531no—
384C4×D4—531no—
385C4×D4—1061no—
386C4×D4—1061no—
387#117—1061no—
388C2×C4⋊C4C2×#615711yescharacteristic
389C2×C22⋊C4—1061no—
390C2×C42—531no—
391#106—531no—
392#106—531no—
393#106C2×#615711yescharacteristic
394#106C2×#615711yescharacteristic
395C4⋊C8—531no—
396C4⋊C8—531no—
397C4⋊C8—531no—
398C4⋊C8—531no—
399C4⋊C8—531no—
400C4⋊C8—531no—
401C4⋊C8—531no—
402C4⋊C8—531no—
403C4⋊C8—1061no—
404C4⋊C8—1061no—
405C4⋊C8—1061no—
406C4⋊C8—1061no—
407C4⋊C8—1061no—
408#103—531no—
409#103—531no—
410#103C2×#615711yescharacteristic
411#103C2×#615711yescharacteristic
412#102—531no—
413#102—531no—
414#102—531no—
415#102—531no—
416C22⋊C8—531no—
417C22⋊C8—531no—
418C22⋊C8—531no—
419C22⋊C8—1061no—
420C22⋊C8—1061no—
421C22⋊C8—1061no—
422C22⋊C8—1061no—
423C22⋊C8—1061no—
424C22⋊C8—1061no—
425C22⋊C8—1061no—
426C22⋊C8C2×#615711yescharacteristic
427#97—1061no—
428#97—1061no—
429#97—1061no—
430C4×C8—531no—
431C4×C8—531no—
432C4×C8—531no—
433C4×C8—531no—
434C4×C8—531no—
435C4×C8—531no—
436C4×C8—1061no—
437C4×C8—1061no—
438C4×C8—1061no—
439C24—1061no—
440C2×Q8—531no—
441C2×Q8—1061no—
442C2×Q8C22×#615711yescharacteristic
443C2×D4—531no—
444C2×D4—531no—
445C2×D4—1062no—
446C2×D4—1061no—
447C2×D4—2121no—
448C22×C4—21no—
449C22×C4—531no—
450C22×C4—531no—
451C22×C4—1061no—
452C22×C4—1061no—
453C22×C4—1061no—
454C22×C4—1061no—
455C22×C4—4241no—
456C22×C4—4241no—
457C22×C4C22×#615711yescharacteristic
458SD16—1062no—
459SD16—1062no—
460SD16—2121no—
461SD16—2121no—
462M16—4241no—
463M16—4241no—
464C2×C8—531no—
465C2×C8—531no—
466C2×C8—531no—
467C2×C8—531no—
468C2×C8—531no—
469C2×C8—531no—
470C2×C8—1061no—
471C2×C8—1061no—
472C2×C8—1061no—
473C2×C8—1061no—
474C2×C8—1061no—
475C2×C8—1061no—
476C2×C8—4241no—
477C2×C8—4241no—
478C2×C8C22×#615711yescharacteristic
479C2×C8C22×#615711yescharacteristic
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—531no—
483C4⋊C4—1061no—
484C4⋊C4—1061no—
485C4⋊C4—1061no—
486C4⋊C4—1061no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
489C4⋊C4C22×#615711yescharacteristic
490C4⋊C4C22×#615711yescharacteristic
491C4⋊C4C22×#615711yescharacteristic
492C22⋊C4—21no—
493C22⋊C4—1061no—
494C22⋊C4—1061no—
495C22⋊C4—1061no—
496C22⋊C4—1061no—
497C22⋊C4—1061no—
498C42—531no—
499C42—531no—
500C42—531no—
501C42—531no—
502C42—1061no—
503C42—1061no—
504C23—1061no—
505C23—1061no—
506C23—4241no—
507C23—2122no—
508C23—2122no—
509C23D4×#615711yescharacteristic
510Q8—21no—
511Q8—21no—
512Q8—1061no—
513Q8—1061no—
514Q8—2122no—
515D4—532no—
516D4—1061no—
517D4—1061no—
518D4—1061no—
519D4—2122no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—81no—
527C2×C4—531no—
528C2×C4—531no—
529C2×C4—531no—
530C2×C4—531no—
531C2×C4—1061no—
532C2×C4—1061no—
533C2×C4—1061no—
534C2×C4—1061no—
535C2×C4—4241no—
536C2×C4—4241no—
537C2×C4—2121no—
538C2×C4—2121no—
539C2×C4—2121no—
540C2×C4—2121no—
541C2×C4—2121no—
542C2×C4D4×#615711yescharacteristic
543C2×C4C23×#615711yescharacteristic
544C8—21no—
545C8—21no—
546C8—1061no—
547C8—1061no—
548C8—1061no—
549C8—1061no—
550C8—1061no—
551C8—1061no—
552C8—2121no—
553C8—2121no—
554C8—2121no—
555C8—2121no—
556C8—2121no—
557C8—2121no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—1062no—
562C22—4244no—
563C22—2121no—
564C22—2121no—
565C22—2121no—
566C22—2121no—
567C22—2121no—
568C22C2×D4×#615711yescharacteristic
569C4—21no—
570C4—41no—
571C4—41no—
572C4—41no—
573C4—41no—
574C4—531no—
575C4—531no—
576C4—1061no—
577C4—1061no—
578C4—1061no—
579C4—2121no—
580C4—2121no—
581C4D53⋊#13111yescharacteristic
582C4#1681170211yescharacteristic
583C2—41no—
584C2—4241no—
585C2—2122no—
586C2#2029144911yescharacteristic
587C2#2029147511yescharacteristic
588C2#2029148811yescharacteristic
subgroupG/H
centerC22C2×D4×#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC53×#483C4
socleC22×C53C2×C4×D4
prime radicalC22×D212C2×C4
cosocleC4×Dic53C24
hypercenter#483#6157
hypocenterC53#18028
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
1. #20291624C2semidirect
3. #20290359C2semidirect
4. #20290350C2semidirect
5. #20290342C2semidirect
6. #20290331C2semidirect
7. #20290323C2semidirect
8. #20290321C2semidirect
9. #20290307C2semidirect
10. #20290299C2semidirect
11. #20290286C2semidirect
12. #20290077C2semidirect
14. #20289936C2semidirect
17. #16811674C22semidirect
18. #16811673C22semidirect
20. #16811672C22semidirect
21. Q8×#88324C22semidirect
36. #16811487C22semidirect
37. #16811486C22semidirect
38. #16811485C22semidirect
39. #16811484C22semidirect
40. #16811473C22semidirect
41. #16811473C22semidirect
42. Q8⋊#88324C22semidirect
43. Q8⋊#88324C22semidirect
48. #88324⋊C8C22semidirect
49. #88324⋊C8C22semidirect
50. #16811464C22semidirect
51. C8×#88324C22semidirect
361. C53#18028semidirect

Maximal subgroups

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

subgroupindex
#202916242
#202915762
#202903592
#202903502
#202903422
#202903312
#202903232
#202903212
#202903072
#202902992
#202902862
#202900772
#202900162
#202899362
#202847202
#1802853