Finite Group Databasebeta

← #28317717

Subgroups — #28317717

Order 13568.

654 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291510C211yesmaximal, characteristic, complemented
2#20291457C211yesmaximal, characteristic
3#20290351C211yesmaximal, characteristic, complemented
4#20290350C211yesmaximal, characteristic, complemented
5#20290325C211yesmaximal, characteristic, complemented
6#20290322C211yesmaximal, characteristic, complemented
7#20290321C211yesmaximal, characteristic, complemented
8#20290318C211yesmaximal, characteristic, complemented
9#20290300C211yesmaximal, characteristic, complemented
10SD16×#88324C211yesmaximal, characteristic, complemented
11C2×#16811484C211yesmaximal, characteristic, complemented
12#20290023C211yesmaximal, characteristic, complemented
13#20290020C211yesmaximal, characteristic
14C8×#11719940C211yesmaximal, characteristic, complemented
15#20285269C211yesmaximal, characteristic
16C2×#13133304—21no—
17#16811675—21no—
18#16811674C2211yescharacteristic
19#16811673—21no—
20#16811673C2211yescharacteristic
21#16811672C2211yescharacteristic
22Q8×#88324C2211yescharacteristic
23#16811658C2211yescharacteristic
24#16811657C2211yescharacteristic
25#16811655—21no—
26#16811652—21no—
27#16811651C2211yescharacteristic
28D4×#88324C2211yescharacteristic
29#16811640C2211yescharacteristic
30#16811629—21no—
31C2×C4⋊#88324C2211yescharacteristic
32C2×C4⋊#88324C2211yescharacteristic
33#16811596—21no—
34#16811587—21no—
35C4×#11719940C2211yescharacteristic
36#16811484C2211yescharacteristic
37#16811484C2211yescharacteristic
38#16811484C2211yescharacteristic
39#16811484C2211yescharacteristic
40#16811473C2211yescharacteristic
41#16811473C2211yescharacteristic
42Q8⋊#88324C2211yescharacteristic
43Q8⋊#88324C2211yescharacteristic
44#16811471C2211yescharacteristic
45#16811471C2211yescharacteristic
46D4⋊#88324C2211yescharacteristic
47D4⋊#88324C2211yescharacteristic
48#88324⋊C8C2211yescharacteristic
49#88324⋊C8C2211yescharacteristic
50C8×#88324C2211yescharacteristic
51C8×#88324C2211yescharacteristic
52#16811175C2211yescharacteristic
53#16811168C2211yescharacteristic
54#16811020C411yescharacteristic
55#16810994C411yescharacteristic
56#16810972C411yescharacteristic
57#16810968C411yescharacteristic
58#16810854C411yescharacteristic
59#16810852C411yescharacteristic
60#16810850C411yescharacteristic
61#16810714C2211yescharacteristic
62SD16×Dic53C2211yescharacteristic
63#16810710C2211yescharacteristic
64#16810708C2211yescharacteristic
65C2×C8×Dic53C2211yescharacteristic
66C53×#465C411yescharacteristic
67#13133304—44no—
68C2×#11719936—21no—
69C2×#11719940—21no—
70C2×#11719940D411yescharacteristic
71C2×#11719940D411yescharacteristic
72C2×D53⋊C8—21no—
73C22×#88324—21no—
74C22×#88324—21no—
75C22×#88324—21no—
76#13133284—21no—
77#13133284—21no—
78#13133284—21no—
79C4⋊#88324—21no—
80C4⋊#88324—21no—
81C4⋊#88324—21no—
82C4⋊#88324—21no—
83C4⋊#88324C2311yescharacteristic
84C4⋊#88324C2311yescharacteristic
85C4⋊#88324C2311yescharacteristic
86C4⋊#88324C2311yescharacteristic
87#13133274—21no—
88#13133274—21no—
89C22⋊#88324—21no—
90C22⋊#88324—21no—
91C22⋊#88324—21no—
92#13133272—21no—
93#13133272—21no—
94#13133272D411yescharacteristic
95#13133272D411yescharacteristic
96C4×#88324—21no—
97C4×#88324—21no—
98C4×#88324C2311yescharacteristic
99C4×#88324C2311yescharacteristic
100C4×#88324C2311yescharacteristic
101C4×#88324C2311yescharacteristic
102C2×#11719947—21no—
103C2×#11719913C2×C411yescharacteristic
104C2×#11719923C2×C411yescharacteristic
105C2×#11719924C2×C411yescharacteristic
106C22×#88322C2×C411yescharacteristic
107#13133216C2×C411yescharacteristic
108#13133216C2×C411yescharacteristic
109#13133215C2×C411yescharacteristic
110#13133213C2×C411yescharacteristic
111#13133207—21no—
112#13133203—21no—
113#13133196C2×C411yescharacteristic
114#13133194—21no—
115#13133191C2×C411yescharacteristic
116#13133189—21no—
117Q8×Dic53C2311yescharacteristic
118#13133182—21no—
119#13133182C2311yescharacteristic
120D4×Dic53C2311yescharacteristic
121#13133179C2311yescharacteristic
122#13133177—21no—
123#13133173—21no—
124C2×C4×Dic53C2311yescharacteristic
125#13133149C2×C411yescharacteristic
126#13133148C2×C411yescharacteristic
127#13133148C2×C411yescharacteristic
128#13133142C2×C411yescharacteristic
129#13133141C2×C411yescharacteristic
130#13133140C2×C411yescharacteristic
131#13133139C2×C411yescharacteristic
132#13133138C2×C411yescharacteristic
133#13133137C2×C411yescharacteristic
134C8×Dic53C2×C411yescharacteristic
135C8×Dic53C2×C411yescharacteristic
136C8×Dic53C2311yescharacteristic
137C8×Dic53C2311yescharacteristic
138C2×SD16×C53C2×C411yescharacteristic
139C22×C8×C53C2×C411yescharacteristic
140C22⋊Q8×C53C2×C411yescharacteristic
141#121×C53C2×C411yescharacteristic
142#106×C53C2×C411yescharacteristic
143#103×C53C2×C411yescharacteristic
144#102×C53C2×C411yescharacteristic
145C2×Q8×C53C22×C411yescharacteristic
146C2×D4×C53—21no—
147C2×D4×C53C22×C411yescharacteristic
148C22×C4×C53C22×C411yescharacteristic
149C2×C22⋊D53—21no—
150C2×C22⋊D53—21no—
151C2×C22⋊D53—21no—
152C22×Dic53—21no—
153C22×Dic53C2×D411yescharacteristic
154C22×Dic53#4111yescharacteristic
155#11719947—42no—
156#11719947—42no—
157C2×D212C22×C411yescharacteristic
158C2×C4×D53—21no—
159C2×Dic106—21no—
160C2×Dic106—21no—
161C2×Dic106C22×C411yescharacteristic
162#11719940—22no—
163#11719940—42no—
164#11719940SD1614yescomplemented
165C2×#88324—21no—
166C2×#88324—21no—
167C2×#88324—21no—
168C2×#88324—21no—
169C2×#88324—21no—
170C2×#88324—21no—
171C2×#88324—42no—
172C2×#88324—42no—
173C2×#88324—42no—
174C2×#88324C2×D411yescharacteristic
175C2×#88324C2×D411yescharacteristic
176C2×#88324C2×D411yescharacteristic
177C2×#88324C2×D411yescharacteristic
178#11719936—42no—
179D53⋊C8—42no—
180SD16×C53—22no—
181C2×C8×C53—22no—
182C2×C8×C53C22×C411yescharacteristic
183C2×C8×C53C22×C411yescharacteristic
184C4⋊C4×C53—21no—
185C4⋊C4×C53C22×C411yescharacteristic
186C4⋊C4×C53C22×C411yescharacteristic
187C22⋊C4×C53—21no—
188C22⋊C4×C53—21no—
189C22⋊Dic53—21no—
190C22⋊Dic53—21no—
191#11719924—41no—
192#11719923—41no—
193#11719921—21no—
194#11719921—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53C22×C411yescharacteristic
197C4⋊Dic53C22×C411yescharacteristic
198#11719919—21no—
199#11719919—21no—
200#11719919—21no—
201#11719919—21no—
202#11719919—21no—
203#11719919—21no—
204C4×Dic53—21no—
205C4×Dic53—21no—
206C4×Dic53C22×C411yescharacteristic
207C4×Dic53C2×D411yescharacteristic
208C4×Dic53#4111yescharacteristic
209C4×Dic53C2411yescharacteristic
210C2×#88322—41no—
211C2×#88322C22×C411yescharacteristic
212C2×#88322C22×C411yescharacteristic
213#11719913—22no—
214C23×C53—21no—
215C23×C53C4×D411yescharacteristic
216C22×D53—21no—
217Q8×C53—21no—
218Q8×C53—21no—
219D4×C53—21no—
220D4×C53—21no—
221D4×C53—42no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53C4×D411yescharacteristic
228C2×C4×C53C23×C411yescharacteristic
229C22⋊D53—42no—
230C22⋊D53—42no—
231C22⋊D53—81no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—41no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53C4×D411yescharacteristic
239C2×Dic53C4×D411yescharacteristic
240C2×Dic53C2×SD1612yes—
241C2×Dic53#13512yes—
242C2×Dic53C22×D411yescharacteristic
243C2×Dic53C2×#4111yescharacteristic
244D212—21no—
245D212—21no—
246C4×D53—41no—
247C4×D53—41no—
248Dic106—21no—
249Dic106—21no—
250Dic106—42no—
251Dic106—81no—
252#88324—21no—
253#88324—21no—
254#88324—41no—
255#88324—41no—
256#88324—41no—
257#88324—41no—
258#88324—41no—
259#88324—41no—
260#88324C2×SD1612yes—
261#88324C2×SD1612yes—
262C8×C53—21no—
263C8×C53#13112yescomplemented
264#88322—21no—
265#88322—21no—
266#18067—531nomaximal, Sylow
267C22×C53—21no—
268C22×C53—21no—
269C22×C53—41no—
270C22×C53—41no—
271C22×C53—41no—
272C22×C53C4×SD1612yes—
273C22×C53C2×C4×D411yescharacteristic
274C2×D53—41no—
275C2×D53—41no—
276C2×D53—41no—
277C4×C53—21no—
278C4×C53—41no—
279C4×C53—41no—
280C4×C53—41no—
281C4×C53C2×#13111yescharacteristic
282C4×C53#56811yescharacteristic
283Dic53—21no—
284Dic53—21no—
285Dic53—41no—
286Dic53—41no—
287Dic53—41no—
288Dic53C4×SD1612yes—
289Dic53C22×SD1611yescharacteristic
290Dic53C2×#13511yescharacteristic
291#3167—531no—
292#2962—531no—
293#2935—531no—
294#1590—531no—
295#1590—531no—
296#1583—531no—
297#1583—531no—
298#1582—531no—
299#1582—531no—
300C8×SD16—531no—
301C8×SD16—531no—
302C2×#334—531no—
303#1481—531no—
304#1480—531no—
305C8×M16—531no—
306C2×C53—22no—
307C2×C53—81no—
308C2×C53#293511yescharacteristic
309C2×C53C2×C4×SD1611yescharacteristic
310C2×C53#295711yescharacteristic
311D53—81no—
312C2×#131—1061no—
313#533—531no—
314#532—531no—
315#498—531no—
316#492—531no—
317#487—531no—
318#486—531no—
319#465—531no—
320#465—531no—
321#465—531no—
322#465#615711yescharacteristic
323#446—531no—
324#446—531no—
325#446—1061no—
326#446—1061no—
327C8×Q8—531no—
328C8×Q8—531no—
329C4×SD16—531no—
330C4×SD16—531no—
331C4×SD16—531no—
332C4×SD16—531no—
333#436—531no—
334#436—531no—
335#435—1061no—
336#435—1061no—
337C8×D4—531no—
338C8×D4—531no—
339#431—531no—
340#424—1061no—
341C2×C4⋊C8—531no—
342C2×C4⋊C8—531no—
343#408—1061no—
344#405—1061no—
345C4×M16—531no—
346C2×C4×C8—531no—
347#334—531no—
348#334—531no—
349#334—531no—
350#334—531no—
351Q8⋊C8—531no—
352Q8⋊C8—531no—
353Q8⋊C8—531no—
354Q8⋊C8—531no—
355D4⋊C8—531no—
356D4⋊C8—531no—
357D4⋊C8—531no—
358D4⋊C8—531no—
359#322—531no—
360#322—531no—
361C82—531no—
362C82—531no—
363C53#1806711yescharacteristic, complemented, Sylow
364C2×#41—1061no—
365C2×SD16—531no—
366C2×SD16—531no—
367C2×SD16—531no—
368C2×SD16C2×#615711yescharacteristic
369#131—2124no—
370C2×M16—531no—
371C2×M16—531no—
372C2×M16—1061no—
373C2×M16—1061no—
374C22×C8—531no—
375C22×C8—1061no—
376C22×C8—1061no—
377C22×C8—1061no—
378C22×C8—1061no—
379C22×C8C2×#615711yescharacteristic
380C4⋊Q8—531no—
381C4⋊Q8—531no—
382C4⋊Q8—531no—
383#127—531no—
384#125—1061no—
385#124—1061no—
386C22⋊Q8—531no—
387C22⋊Q8—1061no—
388C22⋊Q8C2×#615711yescharacteristic
389#121—531no—
390#121—1061no—
391#121C2×#615711yescharacteristic
392C4×Q8—531no—
393C4×Q8—531no—
394C4×Q8—1061no—
395C4×D4—531no—
396C4×D4—531no—
397C4×D4—1061no—
398#117—1061no—
399C2×C42—531no—
400#106—531no—
401#106—531no—
402#106—531no—
403#106C2×#615711yescharacteristic
404C4⋊C8—531no—
405C4⋊C8—531no—
406C4⋊C8—531no—
407C4⋊C8—531no—
408C4⋊C8—1061no—
409C4⋊C8—1061no—
410C4⋊C8—1061no—
411C4⋊C8—1061no—
412C4⋊C8—1061no—
413C4⋊C8—1061no—
414C4⋊C8—1061no—
415#103—531no—
416#103—531no—
417#103—531no—
418#103C2×#615711yescharacteristic
419#102—531no—
420#102—531no—
421#102—531no—
422#102C2×#615711yescharacteristic
423C22⋊C8—1061no—
424C22⋊C8—1061no—
425C22⋊C8—1061no—
426C22⋊C8—1061no—
427C22⋊C8—1061no—
428#97—531no—
429#97—531no—
430#97—1061no—
431#97—1061no—
432C4×C8—531no—
433C4×C8—531no—
434C4×C8—531no—
435C4×C8—531no—
436C4×C8—531no—
437C4×C8—531no—
438C4×C8—531no—
439C4×C8—531no—
440C4×C8—1061no—
441C4×C8—1061no—
442#41—2122no—
443#41—2122no—
444C2×Q8—531no—
445C2×Q8—1061no—
446C2×Q8—1061no—
447C2×Q8C22×#615711yescharacteristic
448C2×D4—21no—
449C2×D4—531no—
450C2×D4—1061no—
451C2×D4—1061no—
452C2×D4—1061no—
453C2×D4C22×#615711yescharacteristic
454C22×C4—531no—
455C22×C4—531no—
456C22×C4—1061no—
457C22×C4—1061no—
458C22×C4C22×#615711yescharacteristic
459SD16—22no—
460SD16—1062no—
461SD16—2121no—
462SD16—2121no—
463M16—534no—
464M16—1062no—
465M16—2122no—
466M16—2122no—
467C2×C8—22no—
468C2×C8—531no—
469C2×C8—531no—
470C2×C8—531no—
471C2×C8—531no—
472C2×C8—531no—
473C2×C8—531no—
474C2×C8—1061no—
475C2×C8—1061no—
476C2×C8—1061no—
477C2×C8—1061no—
478C2×C8—1061no—
479C2×C8—1061no—
480C2×C8—2122no—
481C2×C8—2122no—
482C2×C8—2121no—
483C2×C8—2122no—
484C2×C8—2122no—
485C2×C8C22×#615711yescharacteristic
486C2×C8C22×#615711yescharacteristic
487C4⋊C4—21no—
488C4⋊C4—531no—
489C4⋊C4—531no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—1061no—
497C4⋊C4C22×#615711yescharacteristic
498C4⋊C4C22×#615711yescharacteristic
499C22⋊C4—21no—
500C22⋊C4—21no—
501C22⋊C4—1061no—
502C22⋊C4—1061no—
503C22⋊C4—1061no—
504C22⋊C4—1061no—
505C42—531no—
506C42—531no—
507C42—531no—
508C42—531no—
509C42—1061no—
510C42—1061no—
511C23—21no—
512C23—1061no—
513C23D4×#615711yescharacteristic
514Q8—21no—
515Q8—21no—
516Q8—1061no—
517Q8—1061no—
518Q8—4241no—
519Q8—2122no—
520D4—21no—
521D4—21no—
522D4—42no—
523D4—1061no—
524D4—1061no—
525D4—4241no—
526D4—2122no—
527D4—2122no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—532no—
534C2×C4—532no—
535C2×C4—531no—
536C2×C4—531no—
537C2×C4—531no—
538C2×C4—531no—
539C2×C4—1061no—
540C2×C4—1061no—
541C2×C4—1061no—
542C2×C4—2121no—
543C2×C4—2121no—
544C2×C4—2121no—
545C2×C4—2121no—
546C2×C4—2121no—
547C2×C4D4×#615711yescharacteristic
548C2×C4C23×#615711yescharacteristic
549C8—21no—
550C8—532no—
551C8—532no—
552C8—1061no—
553C8—1061no—
554C8—1061no—
555C8—1061no—
556C8—2121no—
557C8—2121no—
558C8—2121no—
559C8—2121no—
560C8—2121no—
561C8—2121no—
562C8#1313330412yescomplemented
563C22—21no—
564C22—21no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—2121no—
569C22—2121no—
570C22—2121no—
571C22SD16×#615712yes—
572C22C2×D4×#615711yescharacteristic
573C4—21no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—531no—
578C4—531no—
579C4—532no—
580C4—1061no—
581C4—1061no—
582C4—2121no—
583C4—2121no—
584C4—2121no—
585C4C2×#1313330411yescharacteristic
586C4#1681170211yescharacteristic
587C2—22no—
588C2—81no—
589C2—4241no—
590C2#2029145711yescharacteristic
591C2C2×SD16×#615711yescharacteristic
592C2#2029149411yescharacteristic
subgroupG/H
centerC22C2×D4×#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC53×#465C4
socleC22×C53C2×C4×D4
prime radical#16811168C22
cosocleC4×Dic53C24
hypercenter#465#6157
hypocenterC53#18067
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #20291510C2semidirect
3. #20290351C2semidirect
4. #20290350C2semidirect
5. #20290325C2semidirect
6. #20290322C2semidirect
7. #20290321C2semidirect
8. #20290318C2semidirect
9. #20290300C2semidirect
10. SD16×#88324C2semidirect
11. C2×#16811484C2semidirect
12. #20290023C2semidirect
14. C8×#11719940C2semidirect
164. #11719940SD16semidirect
263. C8×C53#131semidirect
363. C53#18067semidirect
562. C8#13133304semidirect

Maximal subgroups

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

subgroupindex
#202915102
#202914572
#202903512
#202903502
#202903252
#202903222
#202903212
#202903182
#202903002
SD16×#883242
C2×#168114842
#202900232
#202900202
C8×#117199402
#202852692
#1806753