Finite Group Databasebeta

← #28317443

Subgroups — #28317443

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#20291626C211yesmaximal, characteristic, complemented
2#20291573C211yesmaximal, characteristic
3#20290358C211yesmaximal, characteristic, complemented
4#20290351C211yesmaximal, characteristic, complemented
5#20290344C211yesmaximal, characteristic, complemented
6#20290333C211yesmaximal, characteristic, complemented
7#20290324C211yesmaximal, characteristic, complemented
8#20290321C211yesmaximal, characteristic, complemented
9#20290305C211yesmaximal, characteristic, complemented
10#20290299C211yesmaximal, characteristic, complemented
11#20290287C211yesmaximal, characteristic, complemented
12#20290078C211yesmaximal, characteristic, complemented
13#20290015C211yesmaximal, characteristic
14#20289937C211yesmaximal, characteristic, complemented
15#20284722C211yesmaximal, characteristic
16#16811674C2211yescharacteristic, complemented
17#16811674C2211yescharacteristic, complemented
18#16811673—21no—
19#16811672—21no—
20Q8×#88324C2211yescharacteristic, complemented
21Q8×#88324C2211yescharacteristic, complemented
22#16811658C2211yescharacteristic
23#16811658C2211yescharacteristic
24#16811655—21no—
25#16811655—21no—
26#16811651C2211yescharacteristic
27#16811651C2211yescharacteristic
28#16811648—21no—
29#16811648—21no—
30#16811645C2211yescharacteristic
31#16811642C2211yescharacteristic
32#16811642C2211yescharacteristic
33#16811641C2211yescharacteristic
34#16811595—21no—
35D53⋊(C22⋊C8)—21no—
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×#11719940—21no—
68C2×#11719940—21no—
69C2×D53⋊C8—21no—
70C2×D53⋊C8—21no—
71#13133284—21no—
72#13133284—21no—
73#13133284—21no—
74C4⋊#88324—21no—
75C4⋊#88324—21no—
76C4⋊#88324D411yescharacteristic
77C4⋊#88324D411yescharacteristic
78C4⋊#88324C2311yescharacteristic
79C4⋊#88324C2311yescharacteristic
80C4⋊#88324C2311yescharacteristic
81C4⋊#88324C2311yescharacteristic
82#13133274—21no—
83#13133274—21no—
84#13133274—21no—
85#13133274—21no—
86#13133274—21no—
87#13133274D411yescharacteristic
88#13133274D411yescharacteristic
89C22⋊#88324—21no—
90C22⋊#88324—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—41no—
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—
163#11719940—81no—
164#11719940—81no—
165C2×#88324—21no—
166C2×#88324—21no—
167C2×#88324—21no—
168C2×#88324—21no—
169C2×#88324—21no—
170C2×#88324—21no—
171C2×#88324C2×D411yescharacteristic
172C2×#88324C2×D411yescharacteristic
173C2×#88324C2×D411yescharacteristic
174C2×#88324C2×D411yescharacteristic
175D53⋊C8—81no—
176D53⋊C8—81no—
177SD16×C53—41no—
178SD16×C53—41no—
179C2×C8×C53C22×C411yescharacteristic
180C2×C8×C53C22×C411yescharacteristic
181C4⋊C4×C53—21no—
182C4⋊C4×C53—21no—
183C4⋊C4×C53C22×C411yescharacteristic
184C22⋊C4×C53—21no—
185C42×C53C22×C411yescharacteristic
186C22⋊Dic53—21no—
187#11719924—22no—
188#11719924—22no—
189#11719921—21no—
190#11719921—21no—
191#11719921—21no—
192#11719921—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53C22×C411yescharacteristic
197C4⋊Dic53C22×C411yescharacteristic
198C4⋊Dic53C22×C411yescharacteristic
199#11719919—21no—
200#11719919—21no—
201#11719919—21no—
202C4×Dic53—21no—
203C4×Dic53—21no—
204C4×Dic53C2×D411yescharacteristic
205C4×Dic53#4111yescharacteristic
206C4×Dic53C2411yescharacteristic
207C2×#88322C22×C411yescharacteristic
208C2×#88322C22×C411yescharacteristic
209C23×C53—21no—
210C22×D53—21no—
211C22×D53—42no—
212C22×D53—42no—
213C22×D53—81no—
214C22×D53C4×D411yescharacteristic
215Q8×C53—21no—
216Q8×C53—21no—
217Q8×C53—21no—
218Q8×C53—21no—
219D4×C53—21no—
220D4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—81no—
225C2×C4×C53C4×D411yescharacteristic
226C2×C4×C53C4×D411yescharacteristic
227C2×C4×C53C23×C411yescharacteristic
228C22⋊D53—42no—
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—41no—
233C2×Dic53—41no—
234C2×Dic53C4×D411yescharacteristic
235C2×Dic53C22×D411yescharacteristic
236C2×Dic53C2×#4111yescharacteristic
237D212—21no—
238D212#13112yes—
239C4×D53—21no—
240C4×D53—21no—
241C4×D53—21no—
242C4×D53—21no—
243C4×D53—41no—
244C4×D53—41no—
245C4×D53—41no—
246C4×D53—81no—
247C4×D53—81no—
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—41no—
265C22×C53—41no—
266C22×C53—41no—
267C22×C53C2×C4×D411yescharacteristic
268C2×D53—21no—
269C2×D53—22no—
270C2×D53—21no—
271C2×D53—21no—
272C2×D53—41no—
273C2×D53—41no—
274C2×D53—84no—
275C4×C53—21no—
276C4×C53—21no—
277C4×C53—41no—
278C4×C53—41no—
279C4×C53—41no—
280C4×C53C2×#13111yescharacteristic
281C4×C53#56811yescharacteristic
282Dic53—21no—
283Dic53—21no—
284Dic53—41no—
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—81no—
305C2×C53#293311yescharacteristic
306C2×C53#294911yescharacteristic
307C2×C53#295511yescharacteristic
308D53—41no—
309D53—42no—
310#533—531no—
311C2×C4×D4—531no—
312#483—531no—
313#483—531no—
314#476—531no—
315#476—531no—
316#461—531no—
317#461#615711yescharacteristic
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×SD16C2×#615711yescharacteristic
367C2×SD16C2×#615711yescharacteristic
368C2×M16—1061no—
369C2×M16—1061no—
370C22×C8—1061no—
371C22×C8—1061no—
372C4⋊Q8—531no—
373C4⋊Q8C2×#615711yescharacteristic
374#125—1061no—
375C22⋊Q8—531no—
376C22⋊Q8—531no—
377C22⋊Q8—1061no—
378C4×Q8—531no—
379C4×Q8—531no—
380C4×Q8—1061no—
381C4×D4—531no—
382C4×D4—531no—
383C4×D4—531no—
384C4×D4—1061no—
385C4×D4—1061no—
386C4×D4C2×#615711yescharacteristic
387#117—1061no—
388C2×C4⋊C4—531no—
389C2×C22⋊C4—1061no—
390C2×C42—531no—
391#106—531no—
392#106—531no—
393#106—531no—
394#106—531no—
395C4⋊C8—531no—
396C4⋊C8—531no—
397C4⋊C8—531no—
398C4⋊C8—531no—
399C4⋊C8—531no—
400C4⋊C8—531no—
401C4⋊C8—531no—
402C4⋊C8—1061no—
403C4⋊C8—1061no—
404C4⋊C8—1061no—
405C4⋊C8—1061no—
406C4⋊C8—1061no—
407C4⋊C8C2×#615711yescharacteristic
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—531no—
420C22⋊C8—1061no—
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—
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—1061no—
441C2×Q8C22×#615711yescharacteristic
442C2×Q8C22×#615711yescharacteristic
443C2×D4—531no—
444C2×D4—1061no—
445C2×D4—1062no—
446C2×D4—2121no—
447C2×D4C22×#615711yescharacteristic
448C22×C4—21no—
449C22×C4—531no—
450C22×C4—531no—
451C22×C4—531no—
452C22×C4—1061no—
453C22×C4—1061no—
454C22×C4—1061no—
455C22×C4—1061no—
456C22×C4—4241no—
457C22×C4—4241no—
458SD16—41no—
459SD16—41no—
460SD16—1062no—
461SD16—1062no—
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—531no—
484C4⋊C4—531no—
485C4⋊C4—1061no—
486C4⋊C4—1061no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
489C4⋊C4—1061no—
490C4⋊C4—1061no—
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—1061no—
502C42—1061no—
503C42C22×#615711yescharacteristic
504C23—21no—
505C23—531no—
506C23—1061no—
507C23—4241no—
508C23—2122no—
509C23—2122no—
510Q8—21no—
511Q8—21no—
512Q8—21no—
513Q8—21no—
514Q8—2122no—
515D4—21no—
516D4—21no—
517D4—532no—
518D4—1061no—
519D4—2122no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—81no—
524C2×C4—531no—
525C2×C4—531no—
526C2×C4—531no—
527C2×C4—1061no—
528C2×C4—1061no—
529C2×C4—1061no—
530C2×C4—1061no—
531C2×C4—1061no—
532C2×C4—1061no—
533C2×C4—1061no—
534C2×C4—4241no—
535C2×C4—4241no—
536C2×C4—2121no—
537C2×C4—2121no—
538C2×C4—2121no—
539C2×C4—2121no—
540C2×C4—2121no—
541C2×C4D4×#615711yescharacteristic
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—41no—
559C22—41no—
560C22—41no—
561C22—1061no—
562C22—1062no—
563C22—1061no—
564C22—1061no—
565C22—4244no—
566C22—2121no—
567C22—2121no—
568C22C2×D4×#615711yescharacteristic
569C4—21no—
570C4—21no—
571C4—41no—
572C4—41no—
573C4—41no—
574C4—531no—
575C4—531no—
576C4—1061no—
577C4—1061no—
578C4—2121no—
579C4—2121no—
580C4—2121no—
581C4D53⋊#13111yescharacteristic
582C4#1681170111yescharacteristic
583C2—81no—
584C2—2121no—
585C2—2122no—
586C2#2029144611yescharacteristic
587C2#2029147511yescharacteristic
588C2#2029148811yescharacteristic
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#18028
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. #20291626C2semidirect
3. #20290358C2semidirect
4. #20290351C2semidirect
5. #20290344C2semidirect
6. #20290333C2semidirect
7. #20290324C2semidirect
8. #20290321C2semidirect
9. #20290305C2semidirect
10. #20290299C2semidirect
11. #20290287C2semidirect
12. #20290078C2semidirect
14. #20289937C2semidirect
16. #16811674C22semidirect
17. #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
361. C53#18028semidirect

Maximal subgroups

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

subgroupindex
#202916262
#202915732
#202903582
#202903512
#202903442
#202903332
#202903242
#202903212
#202903052
#202902992
#202902872
#202900782
#202900152
#202899372
#202847222
#1802853