Finite Group Databasebeta

← #28317210

Subgroups — #28317210

Order 13568.

680 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291510C211yesmaximal, characteristic, complemented
2#20291455C211yesmaximal, characteristic
3#20290350C211yesmaximal, characteristic, complemented
4#20290349C211yesmaximal, characteristic, complemented
5#20290323C211yesmaximal, characteristic, complemented
6#20290322C211yesmaximal, characteristic, complemented
7#20290321C211yesmaximal, characteristic, complemented
8#20290319C211yesmaximal, characteristic, complemented
9#20290300C211yesmaximal, characteristic, complemented
10#20290299C211yesmaximal, characteristic, complemented
11#20290282C211yesmaximal, characteristic, complemented
12#20290063C211yesmaximal, characteristic, complemented
13C2×#16811471C211yesmaximal, characteristic
14#20289916C211yesmaximal, characteristic, complemented
15#20284720C211yesmaximal, characteristic
16C2×D53⋊M16—21no—
17#16811675—21no—
18#16811674C2211yescharacteristic, complemented
19#16811673—21no—
20#16811673C2211yescharacteristic, complemented
21#16811672C2211yescharacteristic, complemented
22Q8×#88324C2211yescharacteristic, complemented
23#16811657C2211yescharacteristic
24#16811657C2211yescharacteristic
25#16811653—21no—
26#16811653—21no—
27#16811651C2211yescharacteristic
28#16811651C2211yescharacteristic
29#16811629—21no—
30#16811626C2211yescharacteristic
31C2×C4⋊#88324C2211yescharacteristic
32#16811595—21no—
33#16811587—21no—
34C4×#11719940C2211yescharacteristic
35C2×C4×#88324C2211yescharacteristic
36#16811485C2211yescharacteristic, complemented
37#16811485C2211yescharacteristic, complemented
38#16811484C2211yescharacteristic, 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
50C8×#88324C2211yescharacteristic, 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—
69D53⋊M16—44no—
70C2×#11719940—21no—
71C2×#11719940D411yescharacteristic
72C2×#11719940D411yescharacteristic
73C2×D53⋊C8—21no—
74C2×D53⋊C8—21no—
75C22×#88324—21no—
76C22×#88324—21no—
77#13133284—21no—
78#13133284—21no—
79#13133284—21no—
80C4⋊#88324—21no—
81C4⋊#88324—21no—
82C4⋊#88324—21no—
83C4⋊#88324C2311yescharacteristic
84C4⋊#88324C2311yescharacteristic
85C4⋊#88324C2311yescharacteristic
86C4⋊#88324C2311yescharacteristic
87#13133274—21no—
88#13133274—21no—
89#13133274—21no—
90#13133274—21no—
91#13133272—21no—
92#13133272—21no—
93#13133272D411yescharacteristic
94#13133272D411yescharacteristic
95C4×#88324—21no—
96C4×#88324—21no—
97C4×#88324—21no—
98C4×#88324C2311yescharacteristic
99C4×#88324C2311yescharacteristic
100C4×#88324C2311yescharacteristic
101C4×#88324C2311yescharacteristic
102C22×D212C2×C411yescharacteristic
103C22×C4×D53—21no—
104C2×#11719913C2×C411yescharacteristic
105C2×#11719913C2×C411yescharacteristic
106C2×#11719924C2×C411yescharacteristic
107C2×#11719924C2×C411yescharacteristic
108#13133216C2×C411yescharacteristic
109#13133215C2×C411yescharacteristic
110#13133207—21no—
111#13133196C2×C411yescharacteristic
112#13133194—21no—
113Q8×Dic53C2311yescharacteristic
114#13133182—21no—
115#13133182C2311yescharacteristic
116#13133179—21no—
117#13133179—21no—
118#13133179C2×C411yescharacteristic
119#13133179C2×C411yescharacteristic
120#13133179C2311yescharacteristic
121#13133179C2311yescharacteristic
122#13133173—21no—
123C2×#11719921—21no—
124C2×C4×Dic53C2311yescharacteristic
125#13133148C2×C411yescharacteristic
126#13133148C2×C411yescharacteristic
127#13133146C2×C411yescharacteristic
128#13133146C2×C411yescharacteristic
129#13133140C2×C411yescharacteristic
130#13133140C2×C411yescharacteristic
131#13133139C2×C411yescharacteristic
132#13133139C2×C411yescharacteristic
133#13133137C2×C411yescharacteristic
134#13133137C2×C411yescharacteristic
135C22⋊#88322C2×C411yescharacteristic
136C8×Dic53C2311yescharacteristic
137C8×Dic53C2311yescharacteristic
138C22⋊Q8×C53C2×C411yescharacteristic
139C2×C4⋊C4×C53C2×C411yescharacteristic
140#106×C53C2×C411yescharacteristic
141#106×C53C2×C411yescharacteristic
142#103×C53C2×C411yescharacteristic
143#103×C53C2×C411yescharacteristic
144C22⋊C8×C53C2×C411yescharacteristic
145C23×D53—21no—
146C2×Q8×C53C22×C411yescharacteristic
147C22×C4×C53—21no—
148C22×C4×C53C22×C411yescharacteristic
149C22×Dic53C2×D411yescharacteristic
150C22×Dic53#4111yescharacteristic
151C2×D212—22no—
152C2×D212—21no—
153C2×D212—41no—
154C2×D212C22×C411yescharacteristic
155C2×D212C22×C411yescharacteristic
156C2×C4×D53—21no—
157C2×C4×D53—21no—
158C2×C4×D53—21no—
159C2×C4×D53—21no—
160C2×C4×D53—42no—
161C2×C4×D53—42no—
162C2×Dic106—21no—
163C2×Dic106C22×C411yescharacteristic
164#11719940—22no—
165#11719940—42no—
166#11719940SD1614yescomplemented
167C2×#88324—21no—
168C2×#88324—21no—
169C2×#88324—21no—
170C2×#88324—21no—
171C2×#88324—21no—
172C2×#88324—21no—
173C2×#88324—42no—
174C2×#88324—42no—
175C2×#88324C2×D411yescharacteristic
176C2×#88324C2×D411yescharacteristic
177C2×#88324C2×D411yescharacteristic
178C2×#88324C2×D411yescharacteristic
179#11719936—42no—
180#11719936—42no—
181D53⋊C8—42no—
182D53⋊C8—42no—
183C2×C8×C53C22×C411yescharacteristic
184C2×C8×C53C22×C411yescharacteristic
185C4⋊C4×C53—21no—
186C4⋊C4×C53—21no—
187C4⋊C4×C53C22×C411yescharacteristic
188C4⋊C4×C53C22×C411yescharacteristic
189C4⋊C4×C53C22×C411yescharacteristic
190C22⋊C4×C53—21no—
191C22⋊Dic53—21no—
192#11719924—22no—
193#11719924—41no—
194#11719921—21no—
195#11719921—21no—
196#11719921—21no—
197#11719921—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53C22×C411yescharacteristic
200#11719919—21no—
201#11719919—21no—
202#11719919—21no—
203#11719919—21no—
204#11719919—21no—
205C4×Dic53—21no—
206C4×Dic53—21no—
207C4×Dic53C22×C411yescharacteristic
208C4×Dic53C2×D411yescharacteristic
209C4×Dic53#4111yescharacteristic
210C4×Dic53C2411yescharacteristic
211C2×#88322C22×C411yescharacteristic
212C2×#88322C22×C411yescharacteristic
213#11719913—22no—
214#11719913—41no—
215C23×C53C4×D411yescharacteristic
216C22×D53—21no—
217C22×D53—21no—
218C22×D53—42no—
219C22×D53—42no—
220C22×D53—81no—
221Q8×C53—21no—
222Q8×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—42no—
230C2×C4×C53C4×D411yescharacteristic
231C2×C4×C53C23×C411yescharacteristic
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—41no—
235C2×Dic53C4×D411yescharacteristic
236C2×Dic53C4×D411yescharacteristic
237C2×Dic53C2×SD1612yes—
238C2×Dic53#13512yes—
239C2×Dic53C22×D411yescharacteristic
240C2×Dic53C2×#4111yescharacteristic
241D212—21no—
242D212—21no—
243D212—21no—
244D212—42no—
245D212#13112yes—
246C4×D53—41no—
247C4×D53—41no—
248C4×D53—41no—
249C4×D53—41no—
250C4×D53—42no—
251C4×D53—42no—
252Dic106—21no—
253Dic106—21no—
254Dic106—81no—
255#88324—22no—
256#88324—21no—
257#88324—21no—
258#88324—41no—
259#88324—41no—
260#88324—41no—
261#88324—41no—
262#88324—41no—
263#88324C2×SD1612yes—
264#88324C2×SD1612yes—
265C8×C53—21no—
266C8×C53—21no—
267#88322—21no—
268#88322—21no—
269#17995—531nomaximal, Sylow
270C22×C53—21no—
271C22×C53—21no—
272C22×C53C4×SD1612yes—
273C22×C53C2×C4×D411yescharacteristic
274C2×D53—22no—
275C2×D53—41no—
276C2×D53—41no—
277C2×D53—41no—
278C2×D53—41no—
279C2×D53—41no—
280C2×D53—84no—
281C4×C53—21no—
282C4×C53—41no—
283C4×C53—41no—
284C4×C53—41no—
285C4×C53—41no—
286C4×C53C2×#13111yescharacteristic
287C4×C53#56811yescharacteristic
288Dic53—21no—
289Dic53—21no—
290Dic53—41no—
291Dic53—41no—
292Dic53C4×SD1612yes—
293Dic53C22×SD1611yescharacteristic
294Dic53C2×#13511yescharacteristic
295#3101—531no—
296#2962—531no—
297D4×M16—531no—
298#1590—531no—
299#1590—531no—
300#1583—531no—
301#1583—531no—
302#1582—531no—
303#1582—531no—
304C8×SD16—531no—
305C8×SD16—531no—
306#1570—531no—
307#1493—531no—
308C2×D4⋊C8—531no—
309C8×M16—531no—
310C2×C53—22no—
311C2×C53#293311yescharacteristic
312C2×C53C2×C4×SD1611yescharacteristic
313C2×C53#295611yescharacteristic
314D53—42no—
315D53—81no—
316C22×M16—1061no—
317#533—531no—
318C2×C4×D4—531no—
319#483—531no—
320#483#615711yescharacteristic
321#476—531no—
322#476—531no—
323#461—531no—
324#461—531no—
325#450—531no—
326#450—531no—
327#446—531no—
328#446—531no—
329#446—1061no—
330#446—1061no—
331C8×Q8—531no—
332C8×Q8—531no—
333C4×SD16—531no—
334C4×SD16—531no—
335C4×SD16—531no—
336C4×SD16—531no—
337#436—531no—
338#436—531no—
339#435—1061no—
340#435—1061no—
341C8×D4—531no—
342C8×D4—531no—
343#431—531no—
344#424—1061no—
345#423—531no—
346C2×C4⋊C8—531no—
347#407—1061no—
348#405—1061no—
349C4×M16—531no—
350C2×C4×C8—531no—
351#334—531no—
352#334—531no—
353#334—531no—
354#334—531no—
355Q8⋊C8—531no—
356Q8⋊C8—531no—
357Q8⋊C8—531no—
358Q8⋊C8—531no—
359D4⋊C8—531no—
360D4⋊C8—531no—
361D4⋊C8—531no—
362D4⋊C8—531no—
363#322—531no—
364#322—531no—
365C82—531no—
366C82—531no—
367C53#1799511yescharacteristic, complemented, Sylow
368C22×D4—531no—
369C23×C4—1061no—
370C2×SD16—531no—
371C2×SD16—531no—
372C2×SD16—531no—
373C2×SD16—531no—
374C2×M16—531no—
375C2×M16—531no—
376C2×M16—1061no—
377C2×M16—1061no—
378C2×M16—1061no—
379C2×M16—2124no—
380C22×C8—1061no—
381C22×C8—1061no—
382C22×C8—1061no—
383C22×C8—1061no—
384C4⋊Q8—531no—
385C4⋊Q8—531no—
386#125—1061no—
387C22⋊Q8—531no—
388C22⋊Q8—1061no—
389C22⋊Q8C2×#615711yescharacteristic
390C4×Q8—531no—
391C4×Q8—531no—
392C4×Q8—1061no—
393C4×D4—531no—
394C4×D4—531no—
395C4×D4—531no—
396C4×D4—531no—
397C4×D4—1061no—
398C4×D4—1061no—
399#117—1061no—
400C2×C4⋊C4C2×#615711yescharacteristic
401C2×C22⋊C4—1061no—
402C2×C42—531no—
403#106—531no—
404#106—531no—
405#106C2×#615711yescharacteristic
406#106C2×#615711yescharacteristic
407C4⋊C8—531no—
408C4⋊C8—531no—
409C4⋊C8—531no—
410C4⋊C8—531no—
411C4⋊C8—531no—
412C4⋊C8—531no—
413C4⋊C8—1061no—
414C4⋊C8—1061no—
415C4⋊C8—1061no—
416C4⋊C8—1061no—
417C4⋊C8—1061no—
418C4⋊C8—1061no—
419#103—531no—
420#103—531no—
421#103C2×#615711yescharacteristic
422#103C2×#615711yescharacteristic
423#102—531no—
424#102—531no—
425#102—531no—
426#102—531no—
427C22⋊C8—531no—
428C22⋊C8—1061no—
429C22⋊C8—1061no—
430C22⋊C8—1061no—
431C22⋊C8—1061no—
432C22⋊C8C2×#615711yescharacteristic
433#97—531no—
434#97—531no—
435#97—1061no—
436#97—1061no—
437C4×C8—531no—
438C4×C8—531no—
439C4×C8—531no—
440C4×C8—531no—
441C4×C8—531no—
442C4×C8—531no—
443C4×C8—1061no—
444C4×C8—1061no—
445C4×C8—1061no—
446C24—1061no—
447C2×Q8—531no—
448C2×Q8—1061no—
449C2×Q8C22×#615711yescharacteristic
450C2×D4—531no—
451C2×D4—531no—
452C2×D4—1062no—
453C2×D4—1061no—
454C2×D4—2121no—
455C22×C4—21no—
456C22×C4—531no—
457C22×C4—531no—
458C22×C4—1061no—
459C22×C4—1061no—
460C22×C4—1061no—
461C22×C4—1061no—
462C22×C4—2122no—
463C22×C4—2122no—
464C22×C4C22×#615711yescharacteristic
465SD16—1062no—
466SD16—1062no—
467SD16—2121no—
468SD16—2121no—
469M16—534no—
470M16—1062no—
471M16—2122no—
472M16—2122no—
473M16—2122no—
474C2×C8—531no—
475C2×C8—531no—
476C2×C8—531no—
477C2×C8—531no—
478C2×C8—531no—
479C2×C8—531no—
480C2×C8—1061no—
481C2×C8—1061no—
482C2×C8—1061no—
483C2×C8—1061no—
484C2×C8—1061no—
485C2×C8—1061no—
486C2×C8—2122no—
487C2×C8—2122no—
488C2×C8—2122no—
489C2×C8—2122no—
490C2×C8C22×#615711yescharacteristic
491C2×C8C22×#615711yescharacteristic
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—531no—
495C4⋊C4—1061no—
496C4⋊C4—1061no—
497C4⋊C4—1061no—
498C4⋊C4—1061no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4C22×#615711yescharacteristic
502C4⋊C4C22×#615711yescharacteristic
503C4⋊C4C22×#615711yescharacteristic
504C22⋊C4—21no—
505C22⋊C4—1061no—
506C22⋊C4—1061no—
507C22⋊C4—1061no—
508C22⋊C4—1061no—
509C22⋊C4—1061no—
510C42—531no—
511C42—531no—
512C42—531no—
513C42—531no—
514C42—1061no—
515C42—1061no—
516C23—1061no—
517C23—1061no—
518C23—4241no—
519C23—2122no—
520C23—2122no—
521C23D4×#615711yescharacteristic
522Q8—21no—
523Q8—21no—
524Q8—1061no—
525Q8—1061no—
526Q8—4241no—
527D4—532no—
528D4—1061no—
529D4—1061no—
530D4—1061no—
531D4—2122no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—42no—
539C2×C4—532no—
540C2×C4—532no—
541C2×C4—531no—
542C2×C4—531no—
543C2×C4—531no—
544C2×C4—531no—
545C2×C4—1061no—
546C2×C4—1061no—
547C2×C4—2121no—
548C2×C4—2121no—
549C2×C4—2121no—
550C2×C4—2121no—
551C2×C4—2122no—
552C2×C4—2122no—
553C2×C4—2121no—
554C2×C4D4×#615711yescharacteristic
555C2×C4C23×#615711yescharacteristic
556C8—21no—
557C8—21no—
558C8—532no—
559C8—532no—
560C8—1061no—
561C8—1061no—
562C8—1062no—
563C8—1061no—
564C8—1061no—
565C8—2121no—
566C8—2121no—
567C8—2121no—
568C8—2121no—
569C8—2121no—
570C22—21no—
571C22—21no—
572C22—1062no—
573C22—4244no—
574C22—2121no—
575C22—2121no—
576C22—2121no—
577C22—2121no—
578C22—2121no—
579C22SD16×#615712yes—
580C22C2×D4×#615711yescharacteristic
581C4—21no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—41no—
586C4—531no—
587C4—531no—
588C4—532no—
589C4—1061no—
590C4—1061no—
591C4—2121no—
592C4—2121no—
593C4D53⋊#13111yescharacteristic
594C4#1681170211yescharacteristic
595C2—22no—
596C2—4241no—
597C2—2122no—
598C2#2029144911yescharacteristic
599C2C2×SD16×#615711yescharacteristic
600C2#2029149011yescharacteristic
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#17995
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. #20291510C2semidirect
3. #20290350C2semidirect
4. #20290349C2semidirect
5. #20290323C2semidirect
6. #20290322C2semidirect
7. #20290321C2semidirect
8. #20290319C2semidirect
9. #20290300C2semidirect
10. #20290299C2semidirect
11. #20290282C2semidirect
12. #20290063C2semidirect
14. #20289916C2semidirect
18. #16811674C22semidirect
20. #16811673C22semidirect
21. #16811672C22semidirect
22. Q8×#88324C22semidirect
36. #16811485C22semidirect
37. #16811485C22semidirect
38. #16811484C22semidirect
39. #16811484C22semidirect
40. #16811473C22semidirect
41. #16811473C22semidirect
42. Q8⋊#88324C22semidirect
43. Q8⋊#88324C22semidirect
48. #88324⋊C8C22semidirect
49. #88324⋊C8C22semidirect
50. C8×#88324C22semidirect
51. C8×#88324C22semidirect
166. #11719940SD16semidirect
367. C53#17995semidirect

Maximal subgroups

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

subgroupindex
#202915102
#202914552
#202903502
#202903492
#202903232
#202903222
#202903212
#202903192
#202903002
#202902992
#202902822
#202900632
C2×#168114712
#202899162
#202847202
#1799553