Finite Group Databasebeta

← #27354943

Subgroups — #27354943

Order 13568.

584 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285271C211yesmaximal, characteristic
2#20283811C211yesmaximal, characteristic, complemented
3#20283789C211yesmaximal, characteristic
4#20273596C211yesmaximal, characteristic, complemented
5#20273590C211yesmaximal, characteristic, complemented
6#20273572C211yesmaximal, characteristic, complemented
7#20273570C211yesmaximal, characteristic, complemented
8#20273559C211yesmaximal, characteristic, complemented
9#20273557C211yesmaximal, characteristic, complemented
10#20273553C211yesmaximal, characteristic, complemented
11#20273475C211yesmaximal, characteristic, complemented
12#20273186C211yesmaximal, characteristic, complemented
13#20273182C211yesmaximal, characteristic
14#20273006C211yesmaximal, characteristic, complemented
15C53×#1582C211yesmaximal, characteristic, complemented
16#16811177C2211yescharacteristic
17#16811170C2211yescharacteristic
18#16811020C411yescharacteristic
19#16810994C411yescharacteristic
20#16810972C411yescharacteristic
21#16810968C411yescharacteristic
22#16810855C411yescharacteristic
23#16810853C411yescharacteristic
24#16810851C411yescharacteristic
25#16810849C411yescharacteristic
26#16810753C2211yescharacteristic
27#16810751—21no—
28#16810750—21no—
29Q8×#88322C2211yescharacteristic
30#16810747C2211yescharacteristic
31C4×#11719913C2211yescharacteristic
32C4×#11719924C2211yescharacteristic
33C4×#11719923C2211yescharacteristic
34#16810701C2211yescharacteristic
35#16810700C2211yescharacteristic
36#16810697—21no—
37#16810694—21no—
38C8×D212C2211yescharacteristic
39#16810685—21no—
40#16810683—21no—
41#16810679C2211yescharacteristic
42#16810679C2211yescharacteristic
43#16810674C2211yescharacteristic
44#16810674C2211yescharacteristic
45#16810589—21no—
46#16810588—21no—
47C4×C8×D53C2211yescharacteristic
48#16810375C2211yescharacteristic
49#16810375C2211yescharacteristic
50#16810374C2211yescharacteristic
51#16810355C2211yescharacteristic
52Q8⋊#88322C2211yescharacteristic
53#16810353C2211yescharacteristic
54#16810352C2211yescharacteristic
55D4⋊#88322C2211yescharacteristic
56#16810350C2211yescharacteristic
57#16810345C2211yescharacteristic
58#16810345C2211yescharacteristic
59#16810344C2211yescharacteristic
60C53×#446C2211yescharacteristic
61C4×SD16×C53C2211yescharacteristic
62C8×D4×C53C2211yescharacteristic
63C53×#334C2211yescharacteristic
64C53×Q8⋊C8C2211yescharacteristic
65C53×D4⋊C8C2211yescharacteristic
66C53×#322C2211yescharacteristic
67C2×#11719945—21no—
68C2×#11719913C2×C411yescharacteristic
69C2×#11719923C2×C411yescharacteristic
70C2×#11719924C2×C411yescharacteristic
71C2×#11719917—21no—
72C2×#11719912—21no—
73C2×C8×D53—21no—
74C2×C8×D53C2×C411yescharacteristic
75C2×C8×D53C2×C411yescharacteristic
76#13133216C2×C411yescharacteristic
77#13133216C2×C411yescharacteristic
78#13133215C2×C411yescharacteristic
79#13133213C2×C411yescharacteristic
80#13133207—21no—
81#13133203—21no—
82#13133197C2×C411yescharacteristic
83#13133195—21no—
84#13133193C2×C411yescharacteristic
85#13133192C2×C411yescharacteristic
86#13133190—21no—
87#13133188C2×C411yescharacteristic
88C4×Dic106—21no—
89C4×Dic106C2311yescharacteristic
90C4×D212C2311yescharacteristic
91C4×C22⋊D53—21no—
92#13133174—21no—
93C42×D53C2311yescharacteristic
94#13133149C2×C411yescharacteristic
95#13133148C2×C411yescharacteristic
96#13133148C2×C411yescharacteristic
97C4⋊#88322—21no—
98C4⋊#88322C2311yescharacteristic
99C4⋊#88322C2311yescharacteristic
100#13133146—21no—
101#13133146—21no—
102#13133146—21no—
103#13133146D411yescharacteristic
104#13133146D411yescharacteristic
105#13133142C2×C411yescharacteristic
106#13133141C2×C411yescharacteristic
107#13133140C2×C411yescharacteristic
108#13133139C2×C411yescharacteristic
109#13133138C2×C411yescharacteristic
110#13133137C2×C411yescharacteristic
111C22⋊#88322—21no—
112#13133129—21no—
113#13133129—21no—
114#13133129—21no—
115#13133129—21no—
116#13133129—21no—
117#13133129D411yescharacteristic
118#13133129D411yescharacteristic
119#13133127—21no—
120#13133127—21no—
121#13133127—21no—
122C4×#88322—21no—
123C4×#88322C2311yescharacteristic
124C4×#88322C2311yescharacteristic
125C4×#88322C2311yescharacteristic
126C8×Dic53—21no—
127C8×Dic53C2×C411yescharacteristic
128C8×Dic53C2×C411yescharacteristic
129C2×SD16×C53C2×C411yescharacteristic
130C22×C8×C53—21no—
131C4×Q8×C53C2311yescharacteristic
132C4×D4×C53C2311yescharacteristic
133#106×C53C2×C411yescharacteristic
134C4⋊C8×C53—21no—
135C4⋊C8×C53C2311yescharacteristic
136C4⋊C8×C53C2311yescharacteristic
137#103×C53C2×C411yescharacteristic
138#102×C53C2×C411yescharacteristic
139C22⋊C8×C53—21no—
140#97×C53—21no—
141C4×C8×C53C2311yescharacteristic
142C4×C8×C53C2311yescharacteristic
143C4×C8×C53C2311yescharacteristic
144C2×Q8×C53C22×C411yescharacteristic
145C2×D4×C53C22×C411yescharacteristic
146C22×C4×C53—21no—
147C2×C22⋊D53—21no—
148C2×C22⋊D53—21no—
149C2×C22⋊D53—21no—
150#11719945—81no—
151#11719945—81no—
152C2×D212—21no—
153C2×D212C22×C411yescharacteristic
154C2×C4×D53—21no—
155C2×C4×D53C22×C411yescharacteristic
156C2×C4×D53C2×D411yescharacteristic
157C2×C4×D53#4111yescharacteristic
158C2×Dic106—21no—
159C2×Dic106—21no—
160C2×Dic106C22×C411yescharacteristic
161SD16×C53—41no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53—81no—
166C2×C8×C53C22×C411yescharacteristic
167C2×C8×C53C22×C411yescharacteristic
168C2×C8×C53C2×D411yescharacteristic
169C2×C8×C53C2×D411yescharacteristic
170C4⋊C4×C53—21no—
171C4⋊C4×C53C22×C411yescharacteristic
172C4⋊C4×C53C22×C411yescharacteristic
173C22⋊C4×C53—21no—
174C42×C53—21no—
175C42×C53C2411yescharacteristic
176C22⋊Dic53—21no—
177#11719924—41no—
178#11719923—41no—
179#11719921—21no—
180#11719921—21no—
181#11719921—21no—
182#11719921—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53C22×C411yescharacteristic
185C4⋊Dic53C22×C411yescharacteristic
186#11719919—21no—
187#11719919—21no—
188#11719919—21no—
189#11719919—21no—
190#11719919—21no—
191#11719919—21no—
192C4×Dic53—21no—
193C4×Dic53C22×C411yescharacteristic
194C4×Dic53C2×D411yescharacteristic
195C4×Dic53#4111yescharacteristic
196#11719917—81no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322C22×C411yescharacteristic
201C2×#88322C22×C411yescharacteristic
202C2×#88322C2×D411yescharacteristic
203C2×#88322C2×D411yescharacteristic
204#11719913—41no—
205#11719912—81no—
206C8×D53—41no—
207C8×D53—41no—
208C8×D53—81no—
209C23×C53—21no—
210C22×D53—21no—
211C22×D53C4×D411yescharacteristic
212Q8×C53—21no—
213Q8×C53—21no—
214D4×C53—21no—
215D4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53C23×C411yescharacteristic
222C2×C4×C53C22×D411yescharacteristic
223C2×C4×C53C2×#4111yescharacteristic
224C22⋊D53—81no—
225C22⋊D53—81no—
226C22⋊D53—81no—
227C2×Dic53—21no—
228C2×Dic53—21no—
229C2×Dic53—21no—
230C2×Dic53C4×D411yescharacteristic
231C2×Dic53C4×D411yescharacteristic
232C2×Dic53C4×D411yescharacteristic
233D212—21no—
234D212—21no—
235D212—81no—
236C4×D53—21no—
237C4×D53—21no—
238C4×D53—21no—
239C4×D53—21no—
240C4×D53—41no—
241C4×D53—41no—
242C4×D53—41no—
243Dic106—21no—
244Dic106—21no—
245Dic106—81no—
246Dic106—81no—
247C8×C53—21no—
248C8×C53—21no—
249C8×C53—21no—
250C8×C53—21no—
251C8×C53—41no—
252C8×C53—41no—
253C8×C53—41no—
254#88322—21no—
255#88322—21no—
256#88322—21no—
257#88322—21no—
258#88322—41no—
259#88322—41no—
260#88322—41no—
261#17990—531nomaximal, Sylow
262C22×C53—41no—
263C22×C53—41no—
264C22×C53—41no—
265C22×C53C2×C4×D411yescharacteristic
266C2×D53—21no—
267C2×D53—21no—
268C2×D53—21no—
269C2×D53—41no—
270C2×D53—41no—
271C2×D53—41no—
272C4×C53—21no—
273C4×C53—41no—
274C4×C53—41no—
275C4×C53—41no—
276C4×C53#56811yescharacteristic
277C4×C53#56811yescharacteristic
278C4×C53#57711yescharacteristic
279C4×C53#57711yescharacteristic
280Dic53—21no—
281Dic53—21no—
282Dic53—21no—
283Dic53—41no—
284Dic53—41no—
285Dic53—41no—
286#3167—531no—
287#2985—531no—
288#2981—531no—
289#1588—531no—
290#1588—531no—
291#1585—531no—
292#1585—531no—
293#1583—531no—
294#1583—531no—
295#1582—531no—
296#1582D5311yescharacteristic, complemented
297#1568—531no—
298#1497—531no—
299#1496—531no—
300#1452—531no—
301C2×C53—81no—
302C2×C53#293211yescharacteristic
303C2×C53#294911yescharacteristic
304C2×C53#295711yescharacteristic
305D53—41no—
306D53—81no—
307#533—531no—
308#532—531no—
309#498—531no—
310#492—531no—
311#487—531no—
312#486—531no—
313#465—531no—
314#465—531no—
315#465—531no—
316#465—531no—
317#446—531no—
318#446—1061no—
319#446—1061no—
320#446C2×D5311yescharacteristic
321C8×Q8—531no—
322C8×Q8—531no—
323C4×SD16—531no—
324C4×SD16—531no—
325C4×SD16—531no—
326C4×SD16C2×D5311yescharacteristic
327#436—531no—
328#436—531no—
329#435—1061no—
330#435—1061no—
331C8×D4—531no—
332C8×D4C2×D5311yescharacteristic
333#433—1061no—
334#433—1061no—
335#432—531no—
336#432—531no—
337#431—531no—
338#431—531no—
339#408—1061no—
340#408—1061no—
341C2×C4×C8—531no—
342#335—531no—
343#335—531no—
344#334—531no—
345#334C2×D5311yescharacteristic
346Q8⋊C8—531no—
347Q8⋊C8—531no—
348Q8⋊C8—531no—
349Q8⋊C8C2×D5311yescharacteristic
350D4⋊C8—531no—
351D4⋊C8—531no—
352D4⋊C8—531no—
353D4⋊C8C2×D5311yescharacteristic
354#322—531no—
355#322—531no—
356#322—531no—
357#322C2×D5311yescharacteristic
358C53#1799011yescharacteristic, complemented, Sylow
359C2×#41—1061no—
360C2×SD16—531no—
361C2×SD16—531no—
362C2×SD16—531no—
363C2×SD16C4×D5311yescharacteristic
364C2×M16—1061no—
365C2×M16—1061no—
366C22×C8—21no—
367C22×C8—531no—
368C22×C8—531no—
369C22×C8—1061no—
370C4⋊Q8—531no—
371C4⋊Q8—531no—
372C4⋊Q8—531no—
373#127—531no—
374#125—1061no—
375#124—1061no—
376C22⋊Q8—531no—
377C22⋊Q8—531no—
378C22⋊Q8—1061no—
379#121—531no—
380#121—531no—
381#121—1061no—
382C4×Q8—531no—
383C4×Q8—1061no—
384C4×Q8C22×D5311yescharacteristic
385C4×D4—531no—
386C4×D4—1061no—
387C4×D4C22×D5311yescharacteristic
388#117—1061no—
389C2×C42—531no—
390#106—531no—
391#106—531no—
392#106—531no—
393#106C4×D5311yescharacteristic
394C4⋊C8—21no—
395C4⋊C8—531no—
396C4⋊C8—531no—
397C4⋊C8—531no—
398C4⋊C8—531no—
399C4⋊C8—1061no—
400C4⋊C8—1061no—
401C4⋊C8—1061no—
402C4⋊C8—1061no—
403C4⋊C8C22×D5311yescharacteristic
404C4⋊C8C22×D5311yescharacteristic
405#103—531no—
406#103—531no—
407#103—531no—
408#103C4×D5311yescharacteristic
409#102—531no—
410#102—531no—
411#102—531no—
412#102C4×D5311yescharacteristic
413C22⋊C8—21no—
414C22⋊C8—531no—
415C22⋊C8—531no—
416C22⋊C8—1061no—
417C22⋊C8—1061no—
418C22⋊C8—1061no—
419C22⋊C8—1061no—
420C22⋊C8—1061no—
421C22⋊C8—1061no—
422#97—21no—
423#97—1061no—
424#97—1061no—
425#97—1061no—
426C4×C8—531no—
427C4×C8—531no—
428C4×C8—531no—
429C4×C8—531no—
430C4×C8—531no—
431C4×C8—1061no—
432C4×C8—1061no—
433C4×C8C22×D5311yescharacteristic
434C4×C8C22×D5311yescharacteristic
435C4×C8C22×D5311yescharacteristic
436#41—4241no—
437#41—4241no—
438C2×Q8—531no—
439C2×Q8—1061no—
440C2×Q8—1061no—
441C2×Q8C2×C4×D5311yescharacteristic
442C2×D4—531no—
443C2×D4—1061no—
444C2×D4—1061no—
445C2×D4—1061no—
446C2×D4—1061no—
447C2×D4C2×C4×D5311yescharacteristic
448C22×C4—21no—
449C22×C4—531no—
450C22×C4—531no—
451C22×C4—531no—
452C22×C4—1061no—
453SD16—41no—
454SD16—2121no—
455SD16—2121no—
456SD16—2121no—
457M16—4241no—
458M16—4241no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—81no—
463C2×C8—531no—
464C2×C8—531no—
465C2×C8—531no—
466C2×C8—531no—
467C2×C8—1061no—
468C2×C8—1061no—
469C2×C8—1061no—
470C2×C8—4241no—
471C2×C8—2121no—
472C2×C8—2121no—
473C2×C8C2×C4×D5311yescharacteristic
474C2×C8C2×C4×D5311yescharacteristic
475C2×C8D4×D5311yescharacteristic
476C2×C8D4×D5311yescharacteristic
477C4⋊C4—21no—
478C4⋊C4—531no—
479C4⋊C4—531no—
480C4⋊C4—1061no—
481C4⋊C4—1061no—
482C4⋊C4—1061no—
483C4⋊C4—1061no—
484C4⋊C4—1061no—
485C4⋊C4—1061no—
486C4⋊C4—1061no—
487C4⋊C4C2×C4×D5311yescharacteristic
488C4⋊C4C2×C4×D5311yescharacteristic
489C22⋊C4—21no—
490C22⋊C4—1061no—
491C22⋊C4—1061no—
492C22⋊C4—1061no—
493C22⋊C4—1061no—
494C22⋊C4—1061no—
495C42—21no—
496C42—531no—
497C42—531no—
498C42—531no—
499C42—1061no—
500C42C23×D5311yescharacteristic
501C23—21no—
502C23—531no—
503C23—1061no—
504Q8—21no—
505Q8—21no—
506Q8—1061no—
507Q8—1061no—
508Q8—4241no—
509Q8—4241no—
510D4—21no—
511D4—21no—
512D4—1061no—
513D4—1061no—
514D4—4241no—
515D4—4241no—
516D4—4241no—
517D4—4241no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—531no—
524C2×C4—531no—
525C2×C4—531no—
526C2×C4—1061no—
527C2×C4—1061no—
528C2×C4—1061no—
529C2×C4—1061no—
530C2×C4—1061no—
531C2×C4—1061no—
532C2×C4—1061no—
533C2×C4—2121no—
534C2×C4—2121no—
535C2×C4—2121no—
536C2×C4C22×C4×D5311yescharacteristic
537C2×C4C2×D4×D5311yescharacteristic
538C2×C4#41×D5311yescharacteristic
539C8—21no—
540C8—21no—
541C8—21no—
542C8—21no—
543C8—41no—
544C8—41no—
545C8—41no—
546C8—1061no—
547C8—1061no—
548C8—1061no—
549C8—1061no—
550C8—2121no—
551C8—2121no—
552C8—2121no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—1061no—
557C22—1061no—
558C22—1061no—
559C22—2121no—
560C22—2121no—
561C22—2121no—
562C22C4×D4×D5311yescharacteristic
563C4—21no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—1061no—
568C4—1061no—
569C4—1061no—
570C4—2121no—
571C4—2121no—
572C4—2121no—
573C4#1681138811yescharacteristic
574C4#1681138911yescharacteristic
575C4#1681142911yescharacteristic
576C4#1681143011yescharacteristic
577C2—81no—
578C2—4241no—
579C2—2121no—
580C2#2028342911yescharacteristic
581C2#2028356911yescharacteristic
582C2#2028362911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC53×#1582C2
socleC22×C53C2×C4×D4
prime radical#16811170C22
cosocleC42×C53C24
hypercenter#1582D53
hypocenterC53#17990
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1582

Complements

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

NH ≅ G/Ntype
2. #20283811C2semidirect
4. #20273596C2semidirect
5. #20273590C2semidirect
6. #20273572C2semidirect
7. #20273570C2semidirect
8. #20273559C2semidirect
9. #20273557C2semidirect
10. #20273553C2semidirect
11. #20273475C2semidirect
12. #20273186C2semidirect
14. #20273006C2semidirect
15. C53×#1582C2semidirect
296. #1582D53semidirect
358. C53#17990semidirect

Maximal subgroups

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

subgroupindex
#202852712
#202838112
#202837892
#202735962
#202735902
#202735722
#202735702
#202735592
#202735572
#202735532
#202734752
#202731862
#202731822
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C53×#15822
#1799053