Finite Group Databasebeta

← #27355040

Subgroups — #27355040

Order 13568.

616 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284739C211yesmaximal, characteristic
2#20283734C211yesmaximal, characteristic, complemented
3#20283721C211yesmaximal, characteristic, complemented
4#20273605C211yesmaximal, characteristic, complemented
5#20273599C211yesmaximal, characteristic, complemented
6#20273555C211yesmaximal, characteristic, complemented
7#20273555C211yesmaximal, characteristic, complemented
8#20273546C211yesmaximal, characteristic, complemented
9Q16×#88322C211yesmaximal, characteristic, complemented
10C8×#11719924C211yesmaximal, characteristic, complemented
11#20273494C211yesmaximal, characteristic, complemented
12#20273174C211yesmaximal, characteristic, complemented
13#20273096C211yesmaximal, characteristic, complemented
14#20272994C211yesmaximal, characteristic, complemented
15C53×#1584C211yesmaximal, characteristic, complemented
16#16811160C2211yescharacteristic
17C4×#11719949C2211yescharacteristic
18#16810963C411yescharacteristic
19#16810950C411yescharacteristic
20#16810932C411yescharacteristic
21#16810926C411yescharacteristic
22#16810847C411yescharacteristic
23#16810841C411yescharacteristic
24#16810784C411yescharacteristic
25#16810771C411yescharacteristic
26Q8×#88322C2211yescharacteristic
27Q8×#88322C2211yescharacteristic
28C4×#11719924C2211yescharacteristic
29C4×#11719924C2211yescharacteristic
30C4×D424C2211yescharacteristic
31#16810702—21no—
32#16810702C2211yescharacteristic
33#16810700—21no—
34#16810700—21no—
35#16810700C2211yescharacteristic
36#16810700C2211yescharacteristic
37#16810696—21no—
38C8×D212C2211yescharacteristic
39#16810683—21no—
40#16810679—21no—
41#16810679C2211yescharacteristic
42#16810675C2211yescharacteristic
43#16810674C2211yescharacteristic
44#16810580—21no—
45C4×#11719912—21no—
46C4×#11719912C2211yescharacteristic
47C4×C8×D53C2211yescharacteristic
48#16810376C2211yescharacteristic
49#16810373C2211yescharacteristic
50#16810373C2211yescharacteristic
51Q8⋊#88322C2211yescharacteristic
52Q8⋊#88322C2211yescharacteristic
53#16810352C2211yescharacteristic
54#16810352C2211yescharacteristic
55#16810350C2211yescharacteristic
56#16810350C2211yescharacteristic
57#16810345C2211yescharacteristic
58C8×#88322C2211yescharacteristic
59C8×#88322C2211yescharacteristic
60C53×#446C2211yescharacteristic
61C8×Q8×C53C2211yescharacteristic
62C4×Q16×C53C2211yescharacteristic
63C53×#335C2211yescharacteristic
64C53×Q8⋊C8C2211yescharacteristic
65C53×Q8⋊C8C2211yescharacteristic
66C53×#322C2211yescharacteristic
67C2×#11719949C2×C411yescharacteristic
68C2×#11719924C2×C411yescharacteristic
69C2×#11719924C2×C411yescharacteristic
70C2×D424C2×C411yescharacteristic
71C2×#11719912—21no—
72C2×#11719912—21no—
73C2×#11719912—21no—
74C2×#11719912D411yescharacteristic
75C2×#11719912D411yescharacteristic
76C2×C8×D53—21no—
77C2×C8×D53—21no—
78#13133212—21no—
79#13133208C2×C411yescharacteristic
80#13133206C2×C411yescharacteristic
81#13133202—21no—
82#13133193C2×C411yescharacteristic
83#13133192C2×C411yescharacteristic
84Q8×Dic53C2×C411yescharacteristic
85C4×D212—21no—
86C4×D212—21no—
87C4×D212C2311yescharacteristic
88C4×D212C2311yescharacteristic
89#13133179—21no—
90#13133179C2×C411yescharacteristic
91#13133175—21no—
92#13133175C2×C411yescharacteristic
93#13133174—21no—
94C42×D53—21no—
95C42×D53C2311yescharacteristic
96#13133151C2×C411yescharacteristic
97#13133148C2×C411yescharacteristic
98#13133148C2×C411yescharacteristic
99C4⋊#88322—21no—
100C4⋊#88322—21no—
101C4⋊#88322C2311yescharacteristic
102C4⋊#88322C2311yescharacteristic
103#13133146C2×C411yescharacteristic
104#13133146C2×C411yescharacteristic
105#13133141C2×C411yescharacteristic
106#13133141C2×C411yescharacteristic
107#13133139C2×C411yescharacteristic
108#13133139C2×C411yescharacteristic
109#13133137C2×C411yescharacteristic
110#13133137C2×C411yescharacteristic
111#13133129—21no—
112#13133129—21no—
113#13133129—21no—
114#13133129—21no—
115#13133129—21no—
116#13133129—21no—
117#13133129C2×C411yescharacteristic
118#13133129C2×C411yescharacteristic
119#13133127—21no—
120#13133127D411yescharacteristic
121#13133127D411yescharacteristic
122C4×#88322—21no—
123C4×#88322—21no—
124C4×#88322C2311yescharacteristic
125C4×#88322C2311yescharacteristic
126C4×#88322C2311yescharacteristic
127C8×Dic53—21no—
128C8×Dic53—21no—
129C2×Q16×C53C2×C411yescharacteristic
130C4×Q8×C53C2311yescharacteristic
131C4×Q8×C53C2311yescharacteristic
132#107×C53C2×C411yescharacteristic
133C4⋊C8×C53—21no—
134C4⋊C8×C53—21no—
135C4⋊C8×C53C2311yescharacteristic
136C4⋊C8×C53C2311yescharacteristic
137#103×C53C2×C411yescharacteristic
138#103×C53C2×C411yescharacteristic
139#97×C53—21no—
140C4×C8×C53—21no—
141C4×C8×C53C2311yescharacteristic
142C4×C8×C53C2311yescharacteristic
143C4×C8×C53C2311yescharacteristic
144C2×Q8×C53C22×C411yescharacteristic
145C2×Q8×C53C22×C411yescharacteristic
146#11719949—22no—
147#11719949—41no—
148C2×D212—21no—
149C2×D212—21no—
150C2×D212C22×C411yescharacteristic
151C2×D212C22×C411yescharacteristic
152C2×C4×D53—21no—
153C2×C4×D53—21no—
154C2×C4×D53—21no—
155C2×C4×D53—21no—
156C2×C4×D53C22×C411yescharacteristic
157C2×C4×D53C2×D411yescharacteristic
158C2×C4×D53#4111yescharacteristic
159Q16×C53—22no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53C22×C411yescharacteristic
164C2×C8×C53C22×C411yescharacteristic
165C2×C8×C53C2×D411yescharacteristic
166C2×C8×C53C2×D411yescharacteristic
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53C22×C411yescharacteristic
170C4⋊C4×C53C22×C411yescharacteristic
171C42×C53—21no—
172C42×C53—21no—
173C42×C53C2411yescharacteristic
174#11719924—22no—
175#11719924—41no—
176#11719921—21no—
177#11719921—21no—
178#11719921—21no—
179#11719921—21no—
180#11719921—21no—
181#11719921—21no—
182#11719921—21no—
183#11719921—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53C22×C411yescharacteristic
187C4⋊Dic53C22×C411yescharacteristic
188#11719919—21no—
189#11719919—21no—
190C4×Dic53—21no—
191C4×Dic53—21no—
192C4×Dic53C22×C411yescharacteristic
193C4×Dic53C2×D411yescharacteristic
194C4×Dic53#4111yescharacteristic
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322C22×C411yescharacteristic
199C2×#88322C22×C411yescharacteristic
200C2×#88322C2×D411yescharacteristic
201C2×#88322C2×D411yescharacteristic
202D424—41no—
203#11719912—22no—
204#11719912—22no—
205#11719912—42no—
206#11719912—42no—
207#11719912—42no—
208C8×D53—42no—
209C8×D53—42no—
210C22×D53—21no—
211C22×D53—21no—
212C22×D53C4×D411yescharacteristic
213Q8×C53—21no—
214Q8×C53—21no—
215Q8×C53—21no—
216Q8×C53#13112yes—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53C23×C411yescharacteristic
222C2×C4×C53C22×D411yescharacteristic
223C2×C4×C53C2×#4111yescharacteristic
224C2×Dic53—21no—
225C2×Dic53—21no—
226C2×Dic53C4×D411yescharacteristic
227C2×Dic53C4×D411yescharacteristic
228C2×Dic53C4×D411yescharacteristic
229D212—21no—
230D212—21no—
231D212—21no—
232D212—21no—
233D212—42no—
234D212—81no—
235C4×D53—21no—
236C4×D53—21no—
237C4×D53—21no—
238C4×D53—21no—
239C4×D53—41no—
240C4×D53—42no—
241C4×D53—41no—
242C4×D53—41no—
243C4×D53—41no—
244C4×D53—41no—
245C4×D53—81no—
246C8×C53—21no—
247C8×C53—21no—
248C8×C53—21no—
249C8×C53—41no—
250C8×C53—41no—
251C8×C53—41no—
252C8×C53#13512yes—
253#88322—22no—
254#88322—21no—
255#88322—21no—
256#88322—21no—
257#88322—41no—
258#88322—41no—
259#88322#13512yes—
260#17999—531nomaximal, Sylow
261C22×C53C2×C4×D411yescharacteristic
262C2×D53—21no—
263C2×D53—21no—
264C2×D53—21no—
265C2×D53—41no—
266C2×D53—41no—
267C2×D53—41no—
268C2×D53—41no—
269C2×D53—41no—
270C2×D53—41no—
271C4×C53—22no—
272C4×C53—21no—
273C4×C53—41no—
274C4×C53—41no—
275C4×C53—41no—
276C4×C53C2×#13111yescharacteristic
277C4×C53#56811yescharacteristic
278C4×C53C2×#13511yescharacteristic
279C4×C53C2×#13511yescharacteristic
280Dic53—21no—
281Dic53—21no—
282Dic53—21no—
283Dic53—41no—
284Dic53—41no—
285#3102—531no—
286#2971—531no—
287#2969—531no—
288#1590—531no—
289#1589—531no—
290#1584D5311yescharacteristic, complemented
291#1583—531no—
292#1583—531no—
293#1581—531no—
294C8×Q16—531no—
295C8×SD16—531no—
296#1572—531no—
297#1494—531no—
298#1476—531no—
299C8×M16—531no—
300C2×C53#293311yescharacteristic
301C2×C53C4×#13511yescharacteristic
302C2×C53#295611yescharacteristic
303D53—41no—
304D53—81no—
305D53—81no—
306#529—531no—
307C4×#41—531no—
308#485—531no—
309#482—531no—
310#479—531no—
311#478—531no—
312#464—531no—
313#463—531no—
314#452—531no—
315#449—531no—
316#446C2×D5311yescharacteristic
317C8×Q8—531no—
318C8×Q8—531no—
319C8×Q8C2×D5311yescharacteristic
320C4×Q16C2×D5311yescharacteristic
321C4×SD16—531no—
322C4×SD16—531no—
323C4×D8—531no—
324#436—531no—
325#436—531no—
326#436—531no—
327#436—1061no—
328#436—1061no—
329#436—1061no—
330#435—1061no—
331C8×D4—531no—
332#433—1061no—
333#432—531no—
334#432—1061no—
335#431—531no—
336#431—531no—
337#405—1061no—
338C4×M16—531no—
339C4×M16—1061no—
340C2×C4×C8—531no—
341#335—531no—
342#335C2×D5311yescharacteristic
343#334—531no—
344#334—531no—
345Q8⋊C8—531no—
346Q8⋊C8—531no—
347Q8⋊C8C2×D5311yescharacteristic
348Q8⋊C8C2×D5311yescharacteristic
349D4⋊C8—531no—
350D4⋊C8—531no—
351D4⋊C8—531no—
352D4⋊C8—531no—
353#322—531no—
354#322C2×D5311yescharacteristic
355C82—531no—
356C82—531no—
357C53#1799911yescharacteristic, complemented, Sylow
358C2×#41—531no—
359C2×Q16C4×D5311yescharacteristic
360C2×SD16—531no—
361C2×SD16—531no—
362C2×D8—531no—
363C2×M16—531no—
364C2×M16—531no—
365C2×M16—1061no—
366C2×M16—1061no—
367C2×M16—1061no—
368C22×C8—1061no—
369C22×C8—1061no—
370#126—1061no—
371#125—531no—
372#124—531no—
373#123—1061no—
374C22⋊Q8—531no—
375#121—531no—
376C4×Q8—531no—
377C4×Q8C22×D5311yescharacteristic
378C4×Q8C22×D5311yescharacteristic
379C4×D4—531no—
380C4×D4—531no—
381C4×D4—531no—
382C4×D4—1061no—
383C4×D4—1061no—
384C4×D4—1061no—
385#117—531no—
386#117—1061no—
387#117—1061no—
388C2×C42—531no—
389C2×C42—1061no—
390#107—531no—
391#107C4×D5311yescharacteristic
392#106—531no—
393#106—531no—
394C4⋊C8—21no—
395C4⋊C8—21no—
396C4⋊C8—531no—
397C4⋊C8—531no—
398C4⋊C8—531no—
399C4⋊C8—531no—
400C4⋊C8—1061no—
401C4⋊C8—1061no—
402C4⋊C8C22×D5311yescharacteristic
403C4⋊C8C22×D5311yescharacteristic
404#103—531no—
405#103—531no—
406#103C4×D5311yescharacteristic
407#103C4×D5311yescharacteristic
408#102—531no—
409#102—531no—
410#102—531no—
411#102—531no—
412C22⋊C8—531no—
413C22⋊C8—531no—
414C22⋊C8—1061no—
415C22⋊C8—1061no—
416C22⋊C8—1061no—
417C22⋊C8—1061no—
418C22⋊C8—1061no—
419C22⋊C8—1061no—
420#97—21no—
421#97—531no—
422#97—531no—
423#97—1061no—
424C4×C8—21no—
425C4×C8—531no—
426C4×C8—531no—
427C4×C8—531no—
428C4×C8—1061no—
429C4×C8—1061no—
430C4×C8—1061no—
431C4×C8—1061no—
432C4×C8C22×D5311yescharacteristic
433C4×C8C22×D5311yescharacteristic
434C4×C8C22×D5311yescharacteristic
435#41—1062no—
436#41—2121no—
437C2×Q8C2×C4×D5311yescharacteristic
438C2×Q8C2×C4×D5311yescharacteristic
439C2×D4—531no—
440C2×D4—531no—
441C2×D4—1061no—
442C2×D4—1061no—
443C22×C4—531no—
444C22×C4—531no—
445C22×C4—531no—
446C22×C4—1061no—
447C22×C4—1061no—
448C22×C4—1061no—
449C22×C4—1061no—
450Q16—22no—
451SD16—1062no—
452SD16—2121no—
453D8—2121no—
454M16—1062no—
455M16—1062no—
456M16—2122no—
457M16—2122no—
458M16—2122no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—531no—
463C2×C8—531no—
464C2×C8—531no—
465C2×C8—531no—
466C2×C8—1061no—
467C2×C8—1061no—
468C2×C8—1061no—
469C2×C8—2122no—
470C2×C8—2122no—
471C2×C8C2×C4×D5311yescharacteristic
472C2×C8C2×C4×D5311yescharacteristic
473C2×C8D4×D5311yescharacteristic
474C2×C8D4×D5311yescharacteristic
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—531no—
478C4⋊C4—531no—
479C4⋊C4—1061no—
480C4⋊C4—1061no—
481C4⋊C4—1061no—
482C4⋊C4—1061no—
483C4⋊C4C2×C4×D5311yescharacteristic
484C4⋊C4C2×C4×D5311yescharacteristic
485C22⋊C4—1061no—
486C22⋊C4—1061no—
487C22⋊C4—1061no—
488C22⋊C4—1061no—
489C22⋊C4—1061no—
490C22⋊C4—1061no—
491C22⋊C4—1061no—
492C22⋊C4—1061no—
493C42—21no—
494C42—21no—
495C42—531no—
496C42—531no—
497C42—531no—
498C42—1061no—
499C42—1061no—
500C42C23×D5311yescharacteristic
501C23—531no—
502C23—1061no—
503C23—1061no—
504Q8—21no—
505Q8—21no—
506Q8—21no—
507Q8#1313322512yes—
508D4—1061no—
509D4—1061no—
510D4—1061no—
511D4—1061no—
512D4—4241no—
513D4—2122no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—531no—
519C2×C4—531no—
520C2×C4—531no—
521C2×C4—1061no—
522C2×C4—1061no—
523C2×C4—1061no—
524C2×C4—1061no—
525C2×C4—1061no—
526C2×C4—1061no—
527C2×C4—4241no—
528C2×C4—2121no—
529C2×C4—2122no—
530C2×C4—2121no—
531C2×C4—2121no—
532C2×C4—2121no—
533C2×C4—2121no—
534C2×C4C22×C4×D5311yescharacteristic
535C2×C4C2×D4×D5311yescharacteristic
536C2×C4#41×D5311yescharacteristic
537C8—21no—
538C8—21no—
539C8—21no—
540C8—41no—
541C8—41no—
542C8—41no—
543C8—532no—
544C8—1062no—
545C8—1061no—
546C8—1061no—
547C8—1061no—
548C8—2121no—
549C8—2121no—
550C8#1313324012yes—
551C22—1061no—
552C22—1061no—
553C22—1061no—
554C22—2121no—
555C22—2121no—
556C22—2121no—
557C22—2121no—
558C22—2121no—
559C22—2121no—
560C22C4×D4×D5311yescharacteristic
561C4—22no—
562C4—21no—
563C4—41no—
564C4—41no—
565C4—41no—
566C4—1061no—
567C4—1061no—
568C4—1061no—
569C4—2121no—
570C4—2121no—
571C4C2×#1313322511yescharacteristic
572C4#1681138811yescharacteristic
573C4#135×D5311yescharacteristic
574C4C2×#1313324011yescharacteristic
575C2—4241no—
576C2—4241no—
577C2—2121no—
578C2#2028344111yescharacteristic
579C2C4×#1313324011yescharacteristic
580C2#2028362311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC53×#1584C2
socleC22×C53C2×C4×D4
prime radical#16810771C4
cosocleC42×C53C24
hypercenter#1584D53
hypocenterC53#17999
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1584

Complements

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

NH ≅ G/Ntype
2. #20283734C2semidirect
3. #20283721C2semidirect
4. #20273605C2semidirect
5. #20273599C2semidirect
6. #20273555C2semidirect
7. #20273555C2semidirect
8. #20273546C2semidirect
9. Q16×#88322C2semidirect
10. C8×#11719924C2semidirect
11. #20273494C2semidirect
12. #20273174C2semidirect
13. #20273096C2semidirect
14. #20272994C2semidirect
15. C53×#1584C2semidirect
290. #1584D53semidirect
357. C53#17999semidirect

Maximal subgroups

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

subgroupindex
#202847392
#202837342
#202837212
#202736052
#202735992
#202735552
#202735462
Q16×#883222
C8×#117199242
#202734942
#202731742
#202730962
#202729942
C53×#15842
#1799953