Finite Group Databasebeta

← #28316804

Subgroups — #28316804

Order 13568.

638 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291620C211yesmaximal, characteristic, complemented
2#20291563C211yesmaximal, characteristic
3#20290333C211yesmaximal, characteristic, complemented
4#20290332C211yesmaximal, characteristic, complemented
5#20290321C211yesmaximal, characteristic, complemented
6#20290320C211yesmaximal, characteristic, complemented
7#20290305C211yesmaximal, characteristic, complemented
8SD16⋊#88324C211yesmaximal, characteristic, complemented
9#20290299C211yesmaximal, characteristic, complemented
10SD16×#88324C211yesmaximal, characteristic, complemented
11#20290287C211yesmaximal, characteristic, complemented
12#20290055C211yesmaximal, characteristic
13#20290038C211yesmaximal, characteristic, complemented
14#20289933C211yesmaximal, characteristic, complemented
15#20284722C211yesmaximal, characteristic
16#16811675—21no—
17#16811675—21no—
18#16811674C2211yescharacteristic, complemented
19#16811674C2211yescharacteristic, complemented
20Q8×#88324C2211yescharacteristic, complemented
21Q8×#88324C2211yescharacteristic, complemented
22#16811651C2211yescharacteristic
23#16811651C2211yescharacteristic
24#16811650—21no—
25#16811650—21no—
26D4×#88324C2211yescharacteristic
27D4×#88324C2211yescharacteristic
28#16811648—21no—
29#16811648—21no—
30#16811645C2211yescharacteristic
31#16811642C2211yescharacteristic
32#16811641C2211yescharacteristic
33C2×#13133274—21no—
34D53⋊(C22⋊C8)—21no—
35C4×#11719936C2211yescharacteristic
36#16811487C2211yescharacteristic, complemented
37#16811486C2211yescharacteristic, complemented
38#16811485C2211yescharacteristic, complemented
39#16811484C2211yescharacteristic, complemented
40Q8⋊#88324C2211yescharacteristic, complemented
41Q8⋊#88324C2211yescharacteristic, complemented
42Q8⋊#88324C2211yescharacteristic, complemented
43Q8⋊#88324C2211yescharacteristic, complemented
44#16811471C2211yescharacteristic
45#16811471C2211yescharacteristic
46D4⋊#88324C2211yescharacteristic
47D4⋊#88324C2211yescharacteristic
48#88324⋊C8C2211yescharacteristic, complemented
49#88324⋊C8C2211yescharacteristic, complemented
50#16811464C2211yescharacteristic, complemented
51C8×#88324C2211yescharacteristic, complemented
52#16811178C2211yescharacteristic
53C4×D4×D53C2211yescharacteristic
54#16810952C411yescharacteristic
55#16810952C411yescharacteristic
56#16810914C411yescharacteristic
57#16810913C411yescharacteristic
58#16810825C411yescharacteristic
59#16810772C411yescharacteristic
60#16810772C411yescharacteristic
61SD16×Dic53C2211yescharacteristic
62SD16×Dic53C2211yescharacteristic
63#16810708C2211yescharacteristic
64#16810708C2211yescharacteristic
65#16810675C2211yescharacteristic
66C53×#461C411yescharacteristic
67C2×#11719936—21no—
68C2×D53⋊C8—21no—
69C2×D53⋊C8—21no—
70C22×#88324—21no—
71C22×#88324—21no—
72#13133284—21no—
73#13133284—21no—
74#13133284D411yescharacteristic
75#13133284D411yescharacteristic
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—
86#13133274D411yescharacteristic
87#13133274D411yescharacteristic
88C22⋊#88324—21no—
89C22⋊#88324—21no—
90#13133272—21no—
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
101C2×D4×D53C2×C411yescharacteristic
102C22×C4×D53—21no—
103C2×#11719924C2×C411yescharacteristic
104C2×#11719924C2×C411yescharacteristic
105#13133217C2×C411yescharacteristic
106#13133208—21no—
107#13133197—21no—
108#13133193C2×C411yescharacteristic
109#13133193C2×C411yescharacteristic
110Q8×Dic53C2311yescharacteristic
111Q8×Dic53C2311yescharacteristic
112#13133182—21no—
113C4×D212C2×C411yescharacteristic
114D4×Dic53C2311yescharacteristic
115#13133179C2311yescharacteristic
116C4×C22⋊D53—21no—
117#13133177—21no—
118#13133175—21no—
119C4⋊C4×D53C2×C411yescharacteristic
120C22⋊C4×D53—21no—
121C42×D53C2311yescharacteristic
122#13133149C2×C411yescharacteristic
123#13133149C2×C411yescharacteristic
124#13133148C2×C411yescharacteristic
125#13133148C2×C411yescharacteristic
126C4⋊#88322C2×C411yescharacteristic
127#13133141C2×C411yescharacteristic
128#13133141C2×C411yescharacteristic
129#13133139C2×C411yescharacteristic
130#13133139C2×C411yescharacteristic
131#13133138C2×C411yescharacteristic
132#13133138C2×C411yescharacteristic
133#13133129C2×C411yescharacteristic
134#13133129C2×C411yescharacteristic
135C8×Dic53C2311yescharacteristic
136C8×Dic53C2311yescharacteristic
137C2×SD16×C53C2×C411yescharacteristic
138C2×SD16×C53C2×C411yescharacteristic
139C4⋊Q8×C53C2×C411yescharacteristic
140C4×D4×C53C2×C411yescharacteristic
141C4⋊C8×C53C2×C411yescharacteristic
142#103×C53C2×C411yescharacteristic
143#103×C53C2×C411yescharacteristic
144C23×D53—21no—
145C2×Q8×C53C22×C411yescharacteristic
146C2×Q8×C53C22×C411yescharacteristic
147C2×D4×C53C22×C411yescharacteristic
148C22×C4×C53—21no—
149C2×C22⋊D53—21no—
150C22×Dic53—21no—
151D4×D53—22no—
152D4×D53—22no—
153C2×D212C22×C411yescharacteristic
154C2×C4×D53—21no—
155C2×C4×D53—21no—
156C2×C4×D53—21no—
157C2×C4×D53—81no—
158C2×C4×D53—81no—
159C2×C4×D53C22×C411yescharacteristic
160C2×C4×D53C2×D411yescharacteristic
161C2×C4×D53#4111yescharacteristic
162C2×Dic106—21no—
163C2×#88324—21no—
164C2×#88324—21no—
165C2×#88324—21no—
166C2×#88324—21no—
167C2×#88324—21no—
168C2×#88324—21no—
169C2×#88324—81no—
170C2×#88324—81no—
171C2×#88324C2×D411yescharacteristic
172C2×#88324C2×D411yescharacteristic
173C2×#88324C2×D411yescharacteristic
174C2×#88324C2×D411yescharacteristic
175#11719936—42no—
176D53⋊C8—81no—
177D53⋊C8—81no—
178SD16×C53—22no—
179SD16×C53—22no—
180C2×C8×C53C22×C411yescharacteristic
181C2×C8×C53C22×C411yescharacteristic
182C4⋊C4×C53—21no—
183C4⋊C4×C53—21no—
184C4⋊C4×C53C22×C411yescharacteristic
185C22⋊C4×C53—21no—
186C42×C53C22×C411yescharacteristic
187C22⋊Dic53—21no—
188#11719924—22no—
189#11719924—22no—
190#11719921—21no—
191#11719921—21no—
192#11719921—21no—
193#11719921—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53C22×C411yescharacteristic
198C4⋊Dic53C22×C411yescharacteristic
199C4⋊Dic53C22×C411yescharacteristic
200#11719919—21no—
201#11719919—21no—
202#11719919—21no—
203C4×Dic53—21no—
204C4×Dic53—21no—
205C4×Dic53C2×D411yescharacteristic
206C4×Dic53#4111yescharacteristic
207C4×Dic53C2411yescharacteristic
208C2×#88322C22×C411yescharacteristic
209C2×#88322C22×C411yescharacteristic
210C23×C53—21no—
211C22×D53—21no—
212C22×D53—24no—
213C22×D53—81no—
214C22×D53—81no—
215C22×D53C4×D411yescharacteristic
216Q8×C53—21no—
217Q8×C53—21no—
218Q8×C53—21no—
219Q8×C53—21no—
220D4×C53—21no—
221D4×C53#13112yes—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—81no—
226C2×C4×C53C4×D411yescharacteristic
227C2×C4×C53C4×D411yescharacteristic
228C2×C4×C53C23×C411yescharacteristic
229C22⋊D53—24no—
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—41no—
234C2×Dic53—41no—
235C2×Dic53C4×D411yescharacteristic
236C2×Dic53C22×D411yescharacteristic
237C2×Dic53C2×#4111yescharacteristic
238D212—21no—
239D212#13112yes—
240C4×D53—21no—
241C4×D53—21no—
242C4×D53—21no—
243C4×D53—21no—
244C4×D53—41no—
245C4×D53—41no—
246C4×D53—41no—
247C4×D53—81no—
248C4×D53—81no—
249Dic106—81no—
250#88324—22no—
251#88324—21no—
252#88324—21no—
253#88324—21no—
254#88324—21no—
255#88324—41no—
256#88324—41no—
257#88324—41no—
258#88324—41no—
259#88324—41no—
260C8×C53—21no—
261C8×C53—21no—
262#88322—21no—
263#88322—21no—
264#17924—531nomaximal, Sylow
265C22×C53—22no—
266C22×C53—41no—
267C22×C53—41no—
268C22×C53C2×C4×D411yescharacteristic
269C2×D53—21no—
270C2×D53—22no—
271C2×D53—21no—
272C2×D53—21no—
273C2×D53—41no—
274C2×D53—41no—
275C2×D53—84no—
276C4×C53—21no—
277C4×C53—21no—
278C4×C53—41no—
279C4×C53—41no—
280C4×C53—41no—
281C4×C53C2×#13111yescharacteristic
282C4×C53C2×#13111yescharacteristic
283Dic53—21no—
284Dic53—21no—
285Dic53—41no—
286Dic53—41no—
287Dic53—41no—
288Dic53#57711yescharacteristic
289Dic53#57711yescharacteristic
290#3101—531no—
291#2985—531no—
292#2972—531no—
293#1585—531no—
294#1585—531no—
295#1582—531no—
296#1582—531no—
297SD16⋊C8—531no—
298SD16⋊C8—531no—
299C8×SD16—531no—
300C8×SD16—531no—
301#1572—531no—
302#1490—531no—
303#1485—531no—
304#1453—531no—
305C2×C53—42no—
306C2×C53#292911yescharacteristic
307C2×C53#294911yescharacteristic
308C2×C53#295611yescharacteristic
309D53—41no—
310D53—42no—
311#533—531no—
312C2×C4×D4—531no—
313#483—531no—
314#483—531no—
315#476—531no—
316#476—531no—
317#461—531no—
318#461#615711yescharacteristic
319#450—531no—
320#450—531no—
321#446—531no—
322#446—531no—
323#446—1061no—
324#446—1061no—
325C8×Q8—531no—
326C8×Q8—531no—
327C4×SD16—531no—
328C4×SD16—531no—
329C4×SD16—531no—
330C4×SD16—531no—
331C8×D4—531no—
332C8×D4—531no—
333C8×D4—531no—
334C8×D4—531no—
335C8×D4—1061no—
336C8×D4—1061no—
337#433—1061no—
338#433—1061no—
339#432—531no—
340#431—531no—
341#431—531no—
342#431—531no—
343C2×C22⋊C8—1061no—
344C2×C22⋊C8—1061no—
345C4×M16—531no—
346#335—531no—
347#335—531no—
348#334—531no—
349#334—531no—
350Q8⋊C8—531no—
351Q8⋊C8—531no—
352Q8⋊C8—531no—
353Q8⋊C8—531no—
354D4⋊C8—531no—
355D4⋊C8—531no—
356D4⋊C8—531no—
357D4⋊C8—531no—
358#322—531no—
359#322—531no—
360#322—531no—
361C82—531no—
362C53#1792411yescharacteristic, complemented, Sylow
363C22×D4—531no—
364C23×C4—1061no—
365C2×SD16—531no—
366C2×SD16—531no—
367C2×SD16C2×#615711yescharacteristic
368C2×SD16C2×#615711yescharacteristic
369C2×M16—1061no—
370C22×C8—1061no—
371C22×C8—1061no—
372C22×C8—1061no—
373C22×C8—1061no—
374C4⋊Q8—531no—
375C4⋊Q8C2×#615711yescharacteristic
376#125—1061no—
377C22⋊Q8—531no—
378C22⋊Q8—531no—
379C22⋊Q8—1061no—
380C4×Q8—531no—
381C4×Q8—531no—
382C4×Q8—1061no—
383C4×D4—531no—
384C4×D4—531no—
385C4×D4—531no—
386C4×D4—1061no—
387C4×D4—1061no—
388C4×D4C2×#615711yescharacteristic
389#117—1061no—
390C2×C4⋊C4—531no—
391C2×C22⋊C4—1061no—
392C2×C42—531no—
393#106—531no—
394#106—531no—
395#106—531no—
396#106—531no—
397C4⋊C8—531no—
398C4⋊C8—531no—
399C4⋊C8—531no—
400C4⋊C8—531no—
401C4⋊C8—531no—
402C4⋊C8—531no—
403C4⋊C8—531no—
404C4⋊C8—1061no—
405C4⋊C8—1061no—
406C4⋊C8—1061no—
407C4⋊C8—1061no—
408C4⋊C8C2×#615711yescharacteristic
409#103—531no—
410#103—531no—
411#103C2×#615711yescharacteristic
412#103C2×#615711yescharacteristic
413#102—531no—
414#102—531no—
415#102—531no—
416#102—531no—
417C22⋊C8—531no—
418C22⋊C8—531no—
419C22⋊C8—531no—
420C22⋊C8—531no—
421C22⋊C8—1061no—
422C22⋊C8—1061no—
423C22⋊C8—1061no—
424C22⋊C8—1061no—
425C22⋊C8—1061no—
426C22⋊C8—1061no—
427#97—1061no—
428#97—1061no—
429#97—1061no—
430#97—1061no—
431C4×C8—531no—
432C4×C8—531no—
433C4×C8—531no—
434C4×C8—531no—
435C4×C8—531no—
436C4×C8—531no—
437C4×C8—1061no—
438C4×C8—1061no—
439C4×C8—1061no—
440C24—1061no—
441C2×Q8—1061no—
442C2×Q8C22×#615711yescharacteristic
443C2×Q8C22×#615711yescharacteristic
444C2×D4—531no—
445C2×D4—1061no—
446C2×D4—1062no—
447C2×D4—1062no—
448C2×D4C22×#615711yescharacteristic
449C22×C4—21no—
450C22×C4—531no—
451C22×C4—531no—
452C22×C4—531no—
453C22×C4—1061no—
454C22×C4—1061no—
455C22×C4—1061no—
456C22×C4—1061no—
457C22×C4—4241no—
458C22×C4—4241no—
459SD16—22no—
460SD16—22no—
461SD16—1062no—
462SD16—1062no—
463M16—2122no—
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×C8—4241no—
479C2×C8—4241no—
480C2×C8C22×#615711yescharacteristic
481C2×C8C22×#615711yescharacteristic
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—531no—
485C4⋊C4—531no—
486C4⋊C4—531no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
489C4⋊C4—1061no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4C22×#615711yescharacteristic
494C22⋊C4—21no—
495C22⋊C4—1061no—
496C22⋊C4—1061no—
497C22⋊C4—1061no—
498C22⋊C4—1061no—
499C22⋊C4—1061no—
500C42—531no—
501C42—531no—
502C42—531no—
503C42—1061no—
504C42—1061no—
505C42C22×#615711yescharacteristic
506C23—21no—
507C23—531no—
508C23—1064no—
509C23—1061no—
510C23—4241no—
511C23—4241no—
512Q8—21no—
513Q8—21no—
514Q8—21no—
515Q8—21no—
516Q8—4241no—
517D4—21no—
518D4—532no—
519D4—1064no—
520D4—1061no—
521D4#1313330412yes—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—81no—
526C2×C4—531no—
527C2×C4—531no—
528C2×C4—531no—
529C2×C4—1061no—
530C2×C4—1061no—
531C2×C4—1061no—
532C2×C4—1061no—
533C2×C4—1061no—
534C2×C4—1061no—
535C2×C4—1061no—
536C2×C4—4241no—
537C2×C4—4241no—
538C2×C4—2121no—
539C2×C4—2121no—
540C2×C4—2121no—
541C2×C4—2121no—
542C2×C4—2121no—
543C2×C4D4×#615711yescharacteristic
544C2×C4D4×#615711yescharacteristic
545C2×C4C23×#615711yescharacteristic
546C8—21no—
547C8—21no—
548C8—1061no—
549C8—1061no—
550C8—1061no—
551C8—1061no—
552C8—1062no—
553C8—1061no—
554C8—1061no—
555C8—2121no—
556C8—2121no—
557C8—2121no—
558C8—2121no—
559C8—2121no—
560C22—22no—
561C22—41no—
562C22—41no—
563C22—1061no—
564C22—1062no—
565C22—1061no—
566C22—1061no—
567C22—4244no—
568C22—2121no—
569C22—2121no—
570C22C2×D4×#615711yescharacteristic
571C4—21no—
572C4—21no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—531no—
577C4—531no—
578C4—1061no—
579C4—1061no—
580C4—2121no—
581C4—2121no—
582C4—2121no—
583C4C2×#1313330411yescharacteristic
584C4D53⋊#13111yescharacteristic
585C2—42no—
586C2—2121no—
587C2—2122no—
588C2#2029142911yescharacteristic
589C2#2029147511yescharacteristic
590C2#2029149011yescharacteristic
subgroupG/H
centerC22C2×D4×#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC53×#461C4
socleC22×C53C2×C4×D4
prime radicalC2×D4×D53C2×C4
cosocleC4×Dic53C24
hypercenter#461#6157
hypocenterC53#17924
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #20291620C2semidirect
3. #20290333C2semidirect
4. #20290332C2semidirect
5. #20290321C2semidirect
6. #20290320C2semidirect
7. #20290305C2semidirect
8. SD16⋊#88324C2semidirect
9. #20290299C2semidirect
10. SD16×#88324C2semidirect
11. #20290287C2semidirect
13. #20290038C2semidirect
14. #20289933C2semidirect
18. #16811674C22semidirect
19. #16811674C22semidirect
20. Q8×#88324C22semidirect
21. Q8×#88324C22semidirect
36. #16811487C22semidirect
37. #16811486C22semidirect
38. #16811485C22semidirect
39. #16811484C22semidirect
40. Q8⋊#88324C22semidirect
41. Q8⋊#88324C22semidirect
42. Q8⋊#88324C22semidirect
43. Q8⋊#88324C22semidirect
48. #88324⋊C8C22semidirect
49. #88324⋊C8C22semidirect
50. #16811464C22semidirect
51. C8×#88324C22semidirect
362. C53#17924semidirect

Maximal subgroups

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

subgroupindex
#202916202
#202915632
#202903332
#202903322
#202903212
#202903202
#202903052
SD16⋊#883242
#202902992
SD16×#883242
#202902872
#202900552
#202900382
#202899332
#202847222
#1792453