Finite Group Databasebeta

← #28317734

Subgroups — #28317734

Order 13568.

600 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291548C211yesmaximal, characteristic, complemented
2#20291546C211yesmaximal, characteristic, complemented
3#20290351C211yesmaximal, characteristic, complemented
4#20290350C211yesmaximal, characteristic, complemented
5#20290329C211yesmaximal, characteristic, complemented
6#20290326C211yesmaximal, characteristic, complemented
7#20290317C211yesmaximal, characteristic, complemented
8#20290314C211yesmaximal, characteristic, complemented
9#20290300C211yesmaximal, characteristic, complemented
10SD16×#88324C211yesmaximal, characteristic, complemented
11#20290262C211yesmaximal, characteristic, complemented
12#20290028C211yesmaximal, characteristic, complemented
13#20290026C211yesmaximal, characteristic, complemented
14C8×#11719940C211yesmaximal, characteristic, complemented
15#20285278C211yesmaximal, characteristic
16#16811674C2211yescharacteristic
17#16811673C2211yescharacteristic
18#16811672C2211yescharacteristic
19Q8×#88324C2211yescharacteristic
20#16811658—21no—
21#16811658C2211yescharacteristic
22#16811657C2211yescharacteristic
23#16811656—21no—
24#16811655—21no—
25#16811654—21no—
26#16811651C2211yescharacteristic
27D4×#88324C2211yescharacteristic
28#16811648—21no—
29#16811647—21no—
30#16811640C2211yescharacteristic
31#16811639C2211yescharacteristic
32#16811639C2211yescharacteristic
33#16811588—21no—
34#16811587—21no—
35C4×#11719940C2211yescharacteristic
36#16811486C2211yescharacteristic
37#16811486C2211yescharacteristic
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#16811159C2211yescharacteristic
53#16811154C2211yescharacteristic
54#16811023C411yescharacteristic
55#16811004C411yescharacteristic
56#16810979C411yescharacteristic
57#16810976C411yescharacteristic
58#16810862C411yescharacteristic
59#16810857C411yescharacteristic
60#16810852C411yescharacteristic
61#16810715C2211yescharacteristic
62SD16×Dic53C2211yescharacteristic
63#16810710C2211yescharacteristic
64#16810703C2211yescharacteristic
65C2×C8×Dic53C2211yescharacteristic
66C53×#465C411yescharacteristic
67C2×#11719936—21no—
68C2×#11719940—21no—
69C2×#11719940D411yescharacteristic
70C2×#11719940D411yescharacteristic
71C2×D53⋊C8—21no—
72C22×#88324—21no—
73#13133284—21no—
74#13133284—21no—
75#13133284—21no—
76C4⋊#88324—21no—
77C4⋊#88324—21no—
78C4⋊#88324C2311yescharacteristic
79C4⋊#88324C2311yescharacteristic
80C4⋊#88324C2311yescharacteristic
81C4⋊#88324C2311yescharacteristic
82#13133274—21no—
83#13133274—21no—
84C22⋊#88324—21no—
85C22⋊#88324—21no—
86C22⋊#88324—21no—
87C22⋊#88324—21no—
88C22⋊#88324—21no—
89#13133272—21no—
90#13133272—21no—
91#13133272D411yescharacteristic
92#13133272D411yescharacteristic
93C4×#88324—21no—
94C4×#88324—21no—
95C4×#88324—21no—
96C4×#88324—21no—
97C4×#88324C2311yescharacteristic
98C4×#88324C2311yescharacteristic
99C4×#88324C2311yescharacteristic
100C4×#88324C2311yescharacteristic
101C2×#11719925C2×C411yescharacteristic
102C2×#11719913C2×C411yescharacteristic
103C2×#11719922C2×C411yescharacteristic
104C22×#88322C2×C411yescharacteristic
105#13133212—21no—
106#13133210—21no—
107#13133208C2×C411yescharacteristic
108#13133208C2×C411yescharacteristic
109#13133206C2×C411yescharacteristic
110#13133204C2×C411yescharacteristic
111#13133201—21no—
112#13133198—21no—
113#13133196C2×C411yescharacteristic
114#13133191C2×C411yescharacteristic
115Q8×Dic53C2311yescharacteristic
116#13133182C2311yescharacteristic
117D4×Dic53—21no—
118D4×Dic53C2311yescharacteristic
119#13133179C2311yescharacteristic
120#13133177—21no—
121#13133175—21no—
122#13133173—21no—
123C2×C4×Dic53C2311yescharacteristic
124#13133150C2×C411yescharacteristic
125#13133150C2×C411yescharacteristic
126#13133149C2×C411yescharacteristic
127#13133142C2×C411yescharacteristic
128#13133141C2×C411yescharacteristic
129#13133140C2×C411yescharacteristic
130#13133139C2×C411yescharacteristic
131#13133138C2×C411yescharacteristic
132#13133137C2×C411yescharacteristic
133C8×Dic53C2×C411yescharacteristic
134C8×Dic53C2×C411yescharacteristic
135C8×Dic53C2311yescharacteristic
136C8×Dic53C2311yescharacteristic
137C2×SD16×C53C2×C411yescharacteristic
138C22×C8×C53C2×C411yescharacteristic
139C22⋊Q8×C53C2×C411yescharacteristic
140#121×C53C2×C411yescharacteristic
141#106×C53C2×C411yescharacteristic
142#103×C53C2×C411yescharacteristic
143#102×C53C2×C411yescharacteristic
144C2×Q8×C53C22×C411yescharacteristic
145C2×D4×C53—21no—
146C2×D4×C53C22×C411yescharacteristic
147C22×C4×C53C22×C411yescharacteristic
148C2×C22⋊D53—21no—
149C22×Dic53—21no—
150C22×Dic53C2×D411yescharacteristic
151C22×Dic53#4111yescharacteristic
152C2×D212C22×C411yescharacteristic
153C2×C4×D53—21no—
154C2×Dic106C22×C411yescharacteristic
155#11719940—22no—
156#11719940—22no—
157#11719940—42no—
158C2×#88324—21no—
159C2×#88324—21no—
160C2×#88324—21no—
161C2×#88324—21no—
162C2×#88324—21no—
163C2×#88324—21no—
164C2×#88324—42no—
165C2×#88324C2×D411yescharacteristic
166C2×#88324C2×D411yescharacteristic
167C2×#88324C2×D411yescharacteristic
168C2×#88324C2×D411yescharacteristic
169#11719936—42no—
170D53⋊C8—42no—
171SD16×C53—22no—
172C2×C8×C53—22no—
173C2×C8×C53C22×C411yescharacteristic
174C2×C8×C53C22×C411yescharacteristic
175C4⋊C4×C53—21no—
176C4⋊C4×C53C22×C411yescharacteristic
177C4⋊C4×C53C22×C411yescharacteristic
178C22⋊C4×C53—21no—
179C22⋊C4×C53—21no—
180C22⋊Dic53—21no—
181C22⋊Dic53—21no—
182C22⋊Dic53—21no—
183C22⋊Dic53—21no—
184#11719925—41no—
185#11719922—41no—
186#11719921—21no—
187#11719921—21no—
188#11719921—21no—
189#11719921—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53C22×C411yescharacteristic
194C4⋊Dic53C22×C411yescharacteristic
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×#88322—41no—
206C2×#88322C22×C411yescharacteristic
207C2×#88322C22×C411yescharacteristic
208#11719913—22no—
209C23×C53—21no—
210C23×C53C4×D411yescharacteristic
211C22×D53—21no—
212Q8×C53—21no—
213Q8×C53—21no—
214D4×C53—21no—
215D4×C53—21no—
216D4×C53—81no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53C4×D411yescharacteristic
223C2×C4×C53C23×C411yescharacteristic
224C22⋊D53—81no—
225C2×Dic53—21no—
226C2×Dic53—21no—
227C2×Dic53—21no—
228C2×Dic53—21no—
229C2×Dic53—21no—
230C2×Dic53—41no—
231C2×Dic53—41no—
232C2×Dic53—41no—
233C2×Dic53C4×D411yescharacteristic
234C2×Dic53C4×D411yescharacteristic
235C2×Dic53C22×D411yescharacteristic
236C2×Dic53C2×#4111yescharacteristic
237D212—21no—
238D212—21no—
239C4×D53—41no—
240C4×D53—41no—
241Dic106—21no—
242Dic106—21no—
243#88324—21no—
244#88324—21no—
245#88324—41no—
246#88324—41no—
247#88324—41no—
248#88324—41no—
249#88324—41no—
250#88324—41no—
251#88324#13512yes—
252#88324#13512yes—
253C8×C53—21no—
254C8×C53#13112yes—
255#88322—21no—
256#88322—21no—
257#18070—531nomaximal, Sylow
258C22×C53—21no—
259C22×C53—21no—
260C22×C53—21no—
261C22×C53—41no—
262C22×C53—41no—
263C22×C53—41no—
264C22×C53C2×C4×D411yescharacteristic
265C2×D53—41no—
266C2×D53—41no—
267C2×D53—41no—
268C4×C53—21no—
269C4×C53—41no—
270C4×C53—41no—
271C4×C53—41no—
272C4×C53C2×#13111yescharacteristic
273C4×C53#56811yescharacteristic
274Dic53—21no—
275Dic53—21no—
276Dic53—21no—
277Dic53—41no—
278Dic53—41no—
279Dic53—41no—
280Dic53C2×#13511yescharacteristic
281Dic53C2×#13511yescharacteristic
282#3168—531no—
283#2969—531no—
284#2969—531no—
285#1590—531no—
286#1590—531no—
287#1584—531no—
288#1584—531no—
289#1581—531no—
290#1581—531no—
291C8×SD16—531no—
292C8×SD16—531no—
293#1564—531no—
294#1482—531no—
295#1482—531no—
296C8×M16—531no—
297C2×C53—41no—
298C2×C53—81no—
299C2×C53#293511yescharacteristic
300C2×C53C4×#13511yescharacteristic
301C2×C53#295711yescharacteristic
302D53—81no—
303#529—531no—
304#529—531no—
305#499—531no—
306#495—531no—
307#488—531no—
308#488—531no—
309#467—531no—
310#466—531no—
311#465—531no—
312#465#615711yescharacteristic
313#446—531no—
314#446—531no—
315C8×Q8—531no—
316C8×Q8—531no—
317C4×Q16—531no—
318C4×SD16—531no—
319C4×SD16—531no—
320C4×D8—531no—
321#436—531no—
322#436—531no—
323#436—1061no—
324#436—1061no—
325#435—1061no—
326#435—1061no—
327C8×D4—531no—
328C8×D4—531no—
329#433—1061no—
330#433—1061no—
331#431—531no—
332#431—531no—
333#431—531no—
334#405—1061no—
335#405—1061no—
336C4×M16—531no—
337C2×C4×C8—531no—
338#335—531no—
339#335—531no—
340#334—531no—
341#334—531no—
342Q8⋊C8—531no—
343Q8⋊C8—531no—
344Q8⋊C8—531no—
345Q8⋊C8—531no—
346D4⋊C8—531no—
347D4⋊C8—531no—
348D4⋊C8—531no—
349D4⋊C8—531no—
350#322—531no—
351#322—531no—
352C82—531no—
353C82—531no—
354C53#1807011yescharacteristic, complemented, Sylow
355C2×Q16—531no—
356C2×SD16—531no—
357C2×SD16C2×#615711yescharacteristic
358C2×D8—531no—
359C2×M16—531no—
360C2×M16—531no—
361C2×M16—1061no—
362C2×M16—1061no—
363C22×C8—531no—
364C22×C8—1061no—
365C22×C8—1061no—
366C22×C8C2×#615711yescharacteristic
367#126—1061no—
368#126—1061no—
369#125—531no—
370#125—531no—
371#124—531no—
372#124—531no—
373#123—1061no—
374#123—1061no—
375C22⋊Q8—531no—
376C22⋊Q8C2×#615711yescharacteristic
377#121—531no—
378#121C2×#615711yescharacteristic
379C4×Q8—531no—
380C4×Q8—531no—
381C4×D4—531no—
382C4×D4—531no—
383C4×D4—1061no—
384C4×D4—1061no—
385#117—1061no—
386#117—1061no—
387C2×C42—531no—
388#107—531no—
389#107—531no—
390#106—531no—
391#106C2×#615711yescharacteristic
392C4⋊C8—531no—
393C4⋊C8—531no—
394C4⋊C8—531no—
395C4⋊C8—531no—
396C4⋊C8—1061no—
397C4⋊C8—1061no—
398C4⋊C8—1061no—
399C4⋊C8—1061no—
400C4⋊C8—1061no—
401#103—531no—
402#103—531no—
403#103—531no—
404#103C2×#615711yescharacteristic
405#102—531no—
406#102—531no—
407#102—531no—
408#102C2×#615711yescharacteristic
409C22⋊C8—1061no—
410C22⋊C8—1061no—
411C22⋊C8—1061no—
412C22⋊C8—1061no—
413C22⋊C8—1061no—
414C22⋊C8—1061no—
415C22⋊C8—1061no—
416#97—531no—
417#97—531no—
418#97—1061no—
419#97—1061no—
420C4×C8—531no—
421C4×C8—531no—
422C4×C8—531no—
423C4×C8—531no—
424C4×C8—531no—
425C4×C8—531no—
426C4×C8—531no—
427C4×C8—531no—
428C4×C8—1061no—
429C4×C8—1061no—
430C4×C8—1061no—
431C4×C8—1061no—
432C2×Q8—531no—
433C2×Q8C22×#615711yescharacteristic
434C2×D4—21no—
435C2×D4—531no—
436C2×D4—1061no—
437C2×D4C22×#615711yescharacteristic
438C22×C4—531no—
439C22×C4—531no—
440C22×C4—1061no—
441C22×C4—1061no—
442C22×C4C22×#615711yescharacteristic
443Q16—2121no—
444SD16—22no—
445SD16—1062no—
446D8—2121no—
447M16—1062no—
448M16—1062no—
449M16—2122no—
450M16—2122no—
451C2×C8—22no—
452C2×C8—531no—
453C2×C8—531no—
454C2×C8—531no—
455C2×C8—531no—
456C2×C8—531no—
457C2×C8—531no—
458C2×C8—1061no—
459C2×C8—1061no—
460C2×C8—1061no—
461C2×C8—1061no—
462C2×C8—1061no—
463C2×C8—1061no—
464C2×C8—2122no—
465C2×C8—2122no—
466C2×C8—2121no—
467C2×C8C22×#615711yescharacteristic
468C2×C8C22×#615711yescharacteristic
469C4⋊C4—21no—
470C4⋊C4—531no—
471C4⋊C4—531no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4—1061no—
475C4⋊C4—1061no—
476C4⋊C4—1061no—
477C4⋊C4—1061no—
478C4⋊C4—1061no—
479C4⋊C4C22×#615711yescharacteristic
480C4⋊C4C22×#615711yescharacteristic
481C22⋊C4—21no—
482C22⋊C4—21no—
483C22⋊C4—1061no—
484C22⋊C4—1061no—
485C22⋊C4—1061no—
486C22⋊C4—1061no—
487C22⋊C4—1061no—
488C22⋊C4—1061no—
489C22⋊C4—1061no—
490C22⋊C4—1061no—
491C42—531no—
492C42—531no—
493C42—531no—
494C42—531no—
495C42—1061no—
496C42—1061no—
497C23—21no—
498C23—1061no—
499C23D4×#615711yescharacteristic
500Q8—21no—
501Q8—21no—
502Q8—1061no—
503Q8—1061no—
504D4—21no—
505D4—21no—
506D4—81no—
507D4—1061no—
508D4—1061no—
509D4—4241no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—531no—
516C2×C4—531no—
517C2×C4—531no—
518C2×C4—531no—
519C2×C4—1061no—
520C2×C4—1061no—
521C2×C4—1061no—
522C2×C4—1061no—
523C2×C4—1061no—
524C2×C4—2121no—
525C2×C4—2121no—
526C2×C4—2121no—
527C2×C4—2121no—
528C2×C4—2121no—
529C2×C4D4×#615711yescharacteristic
530C2×C4C23×#615711yescharacteristic
531C8—21no—
532C8—532no—
533C8—532no—
534C8—1061no—
535C8—1061no—
536C8—1061no—
537C8—1061no—
538C8—2121no—
539C8—2121no—
540C8—2121no—
541C8—2121no—
542C8—2121no—
543C8—2121no—
544C8#1313330412yes—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—2121no—
552C22—2121no—
553C22—2121no—
554C22C2×D4×#615711yescharacteristic
555C4—21no—
556C4—41no—
557C4—41no—
558C4—41no—
559C4—531no—
560C4—531no—
561C4—1061no—
562C4—1061no—
563C4—1061no—
564C4—2121no—
565C4—2121no—
566C4—2121no—
567C4C2×#1313330411yescharacteristic
568C4#1681170211yescharacteristic
569C2—41no—
570C2—81no—
571C2—4241no—
572C2#2029145711yescharacteristic
573C2#135×#615711yescharacteristic
574C2#2029149311yescharacteristic
subgroupG/H
centerC22C2×D4×#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC53×#465C4
socleC22×C53C2×C4×D4
prime radical#16810857C4
cosocleC4×Dic53C24
hypercenter#465#6157
hypocenterC53#18070
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. #20291548C2semidirect
2. #20291546C2semidirect
3. #20290351C2semidirect
4. #20290350C2semidirect
5. #20290329C2semidirect
6. #20290326C2semidirect
7. #20290317C2semidirect
8. #20290314C2semidirect
9. #20290300C2semidirect
10. SD16×#88324C2semidirect
11. #20290262C2semidirect
12. #20290028C2semidirect
13. #20290026C2semidirect
14. C8×#11719940C2semidirect
354. C53#18070semidirect

Maximal subgroups

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

subgroupindex
#202915482
#202915462
#202903512
#202903502
#202903292
#202903262
#202903172
#202903142
#202903002
SD16×#883242
#202902622
#202900282
#202900262
C8×#117199402
#202852782
#1807053