Finite Group Databasebeta

← #27355859

Subgroups — #27355859

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#20285271C211yesmaximal, characteristic
2#20283657C211yesmaximal, characteristic, complemented
3#20283460C211yesmaximal, characteristic
4#20273603C211yesmaximal, characteristic, complemented
5#20273602C211yesmaximal, characteristic, complemented
6#20273557C211yesmaximal, characteristic, complemented
7#20273553C211yesmaximal, characteristic, complemented
8#20273552C211yesmaximal, characteristic, complemented
9#20273518C211yesmaximal, characteristic, complemented
10C8×#11719924C211yesmaximal, characteristic, complemented
11#334×D53C211yesmaximal, characteristic, complemented
12#20273116C211yesmaximal, characteristic, complemented
13#20273112C211yesmaximal, characteristic
14#20272994C211yesmaximal, characteristic, complemented
15C53×#1583C211yesmaximal, characteristic, complemented
16C2×#13133225—21no—
17#16811177C2211yescharacteristic
18#16811170C2211yescharacteristic
19#16811020C411yescharacteristic
20#16810994C411yescharacteristic
21#16810972C411yescharacteristic
22#16810968C411yescharacteristic
23#16810855C411yescharacteristic
24#16810853C411yescharacteristic
25#16810851C411yescharacteristic
26#16810849C411yescharacteristic
27#16810754C2211yescharacteristic
28#16810753—21no—
29#16810753C2211yescharacteristic
30#16810752—21no—
31#16810747C2211yescharacteristic
32C4×#11719913C2211yescharacteristic
33C4×#11719924C2211yescharacteristic
34C4×#11719923C2211yescharacteristic
35#16810700C2211yescharacteristic
36#16810692—21no—
37#16810691—21no—
38C8×D212C2211yescharacteristic
39D4×#88322C2211yescharacteristic
40#16810674C2211yescharacteristic
41#16810648—21no—
42C4⋊C8×D53C2211yescharacteristic
43C4⋊C8×D53C2211yescharacteristic
44#16810588—21no—
45#16810577—21no—
46C4×#11719912C2211yescharacteristic
47C4×C8×D53C2211yescharacteristic
48#16810374C2211yescharacteristic
49#16810373C2211yescharacteristic
50#16810373C2211yescharacteristic
51#16810355C2211yescharacteristic
52Q8⋊#88322C2211yescharacteristic
53#16810353C2211yescharacteristic
54#16810352C2211yescharacteristic
55D4⋊#88322C2211yescharacteristic
56#16810350C2211yescharacteristic
57#16810345C2211yescharacteristic
58C8×#88322C2211yescharacteristic
59C8×#88322C2211yescharacteristic
60C8×Q8×C53C2211yescharacteristic
61C4×SD16×C53C2211yescharacteristic
62C53×#436C2211yescharacteristic
63C53×#334C2211yescharacteristic
64C53×Q8⋊C8C2211yescharacteristic
65C53×D4⋊C8C2211yescharacteristic
66C53×#322C2211yescharacteristic
67C2×#11719945—21no—
68C2×#11719913C2×C411yescharacteristic
69C2×#11719923C2×C411yescharacteristic
70C2×#11719924C2×C411yescharacteristic
71#13133225—44no—
72C2×#11719912—21no—
73C2×#11719912D411yescharacteristic
74C2×#11719912D411yescharacteristic
75C22×#88322—21no—
76C2×C8×D53—21no—
77C2×C8×D53—21no—
78C2×C8×D53—21no—
79C2×C8×D53C2×C411yescharacteristic
80C2×C8×D53C2×C411yescharacteristic
81#13133216C2×C411yescharacteristic
82#13133216C2×C411yescharacteristic
83#13133215C2×C411yescharacteristic
84#13133213C2×C411yescharacteristic
85#13133207—21no—
86#13133203—21no—
87#13133197C2×C411yescharacteristic
88#13133195—21no—
89#13133193C2×C411yescharacteristic
90#13133192C2×C411yescharacteristic
91#13133190—21no—
92#13133188C2×C411yescharacteristic
93C4×Dic106—21no—
94C4×Dic106C2311yescharacteristic
95C4×D212C2311yescharacteristic
96C4×C22⋊D53—21no—
97#13133174—21no—
98C42×D53C2311yescharacteristic
99#13133149C2×C411yescharacteristic
100#13133148C2×C411yescharacteristic
101#13133148C2×C411yescharacteristic
102C4⋊#88322—21no—
103C4⋊#88322C2311yescharacteristic
104C4⋊#88322C2311yescharacteristic
105#13133146—21no—
106#13133146—21no—
107#13133146—21no—
108#13133146—21no—
109#13133146—21no—
110#13133142C2×C411yescharacteristic
111#13133141C2×C411yescharacteristic
112#13133140C2×C411yescharacteristic
113#13133139C2×C411yescharacteristic
114#13133138C2×C411yescharacteristic
115#13133137C2×C411yescharacteristic
116C22⋊#88322—21no—
117#13133129—21no—
118#13133129—21no—
119#13133129—21no—
120#13133128—21no—
121#13133127—21no—
122#13133127D411yescharacteristic
123#13133127D411yescharacteristic
124C4×#88322C2311yescharacteristic
125C4×#88322C2311yescharacteristic
126C4×#88322C2311yescharacteristic
127C8×Dic53—21no—
128C8×Dic53C2×C411yescharacteristic
129C8×Dic53C2×C411yescharacteristic
130C2×SD16×C53C2×C411yescharacteristic
131C2×M16×C53—21no—
132C4×Q8×C53C2311yescharacteristic
133C4×D4×C53C2311yescharacteristic
134#106×C53C2×C411yescharacteristic
135C4⋊C8×C53—21no—
136C4⋊C8×C53C2311yescharacteristic
137C4⋊C8×C53C2311yescharacteristic
138#103×C53C2×C411yescharacteristic
139#102×C53C2×C411yescharacteristic
140C22⋊C8×C53—21no—
141C4×C8×C53—21no—
142C4×C8×C53C2311yescharacteristic
143C4×C8×C53C2311yescharacteristic
144C4×C8×C53C2311yescharacteristic
145C2×Q8×C53C22×C411yescharacteristic
146C2×D4×C53C22×C411yescharacteristic
147C22×C4×C53—21no—
148C2×C22⋊D53—21no—
149C2×C22⋊D53—21no—
150C2×C22⋊D53—21no—
151#11719945—42no—
152#11719945—42no—
153C2×D212—21no—
154C2×D212C22×C411yescharacteristic
155C2×C4×D53—21no—
156C2×C4×D53C22×C411yescharacteristic
157C2×C4×D53C2×D411yescharacteristic
158C2×C4×D53#4111yescharacteristic
159C2×Dic106—21no—
160C2×Dic106—21no—
161C2×Dic106C22×C411yescharacteristic
162SD16×C53—41no—
163M16×C53—42no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×C53C22×C411yescharacteristic
168C2×C8×C53C22×C411yescharacteristic
169C2×C8×C53C2×D411yescharacteristic
170C2×C8×C53C2×D411yescharacteristic
171C4⋊C4×C53—21no—
172C4⋊C4×C53C22×C411yescharacteristic
173C4⋊C4×C53C22×C411yescharacteristic
174C22⋊C4×C53—21no—
175C42×C53—21no—
176C42×C53C2411yescharacteristic
177C22⋊Dic53—21no—
178#11719924—22no—
179#11719923—22no—
180#11719921—21no—
181#11719921—21no—
182#11719921—21no—
183#11719921—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53C22×C411yescharacteristic
186C4⋊Dic53C22×C411yescharacteristic
187#11719919—21no—
188#11719919—21no—
189#11719919—21no—
190#11719919—21no—
191#11719919—21no—
192#11719919—21no—
193C4×Dic53—21no—
194C4×Dic53C22×C411yescharacteristic
195C4×Dic53C2×D411yescharacteristic
196C4×Dic53#4111yescharacteristic
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—42no—
201C2×#88322C22×C411yescharacteristic
202C2×#88322C22×C411yescharacteristic
203C2×#88322C2×D411yescharacteristic
204C2×#88322C2×D411yescharacteristic
205#11719913—41no—
206#11719912—22no—
207#11719912—42no—
208#11719912SD1614yescomplemented
209C8×D53—22no—
210C8×D53—41no—
211C8×D53—42no—
212C8×D53—42no—
213C8×D53—42no—
214C23×C53—21no—
215C22×D53—21no—
216C22×D53C4×D411yescharacteristic
217Q8×C53—21no—
218Q8×C53—21no—
219D4×C53—21no—
220D4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53C23×C411yescharacteristic
227C2×C4×C53C22×D411yescharacteristic
228C2×C4×C53C2×#4111yescharacteristic
229C22⋊D53—42no—
230C22⋊D53—42no—
231C22⋊D53—81no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53C4×D411yescharacteristic
236C2×Dic53C4×D411yescharacteristic
237C2×Dic53C4×D411yescharacteristic
238D212—21no—
239D212—21no—
240D212—42no—
241C4×D53—21no—
242C4×D53—21no—
243C4×D53—41no—
244C4×D53—41no—
245C4×D53—41no—
246C4×D53C2×SD1612yes—
247C4×D53#13512yes—
248Dic106—21no—
249Dic106—21no—
250Dic106—42no—
251Dic106—81no—
252C8×C53—21no—
253C8×C53—21no—
254C8×C53—21no—
255C8×C53—41no—
256C8×C53—41no—
257C8×C53—41no—
258C8×C53C2×SD1612yes—
259#88322—21no—
260#88322—21no—
261#88322—41no—
262#88322—41no—
263#88322—41no—
264#88322#13112yescomplemented
265#88322C2×SD1612yes—
266#18067—531nomaximal, Sylow
267C22×C53—41no—
268C22×C53—41no—
269C22×C53—41no—
270C22×C53C2×C4×D411yescharacteristic
271C2×D53—21no—
272C2×D53—21no—
273C2×D53—41no—
274C2×D53—41no—
275C2×D53—41no—
276C2×D53C4×SD1612yes—
277C4×C53—21no—
278C4×C53—41no—
279C4×C53—41no—
280C4×C53—41no—
281C4×C53C2×#13111yescharacteristic
282C4×C53#56811yescharacteristic
283C4×C53C22×SD1611yescharacteristic
284C4×C53C2×#13511yescharacteristic
285Dic53—21no—
286Dic53—21no—
287Dic53—41no—
288Dic53—41no—
289Dic53—41no—
290Dic53C4×SD1612yes—
291#3167—531no—
292#2962—531no—
293#2935—531no—
294#1590—531no—
295#1590—531no—
296#1583—531no—
297#1583D5311yescharacteristic, complemented
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—81no—
307C2×C53#293511yescharacteristic
308C2×C53C2×C4×SD1611yescharacteristic
309C2×C53#295711yescharacteristic
310D53—22no—
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—531no—
323#446—531no—
324#446—531no—
325#446—1061no—
326#446—1061no—
327C8×Q8—531no—
328C8×Q8C2×D5311yescharacteristic
329C4×SD16—531no—
330C4×SD16—531no—
331C4×SD16—531no—
332C4×SD16C2×D5311yescharacteristic
333#436—531no—
334#436C2×D5311yescharacteristic
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#334C2×D5311yescharacteristic
351Q8⋊C8—531no—
352Q8⋊C8—531no—
353Q8⋊C8—531no—
354Q8⋊C8C2×D5311yescharacteristic
355D4⋊C8—531no—
356D4⋊C8—531no—
357D4⋊C8—531no—
358D4⋊C8C2×D5311yescharacteristic
359#322—531no—
360#322C2×D5311yescharacteristic
361C82—531no—
362C82—531no—
363C53#1806711yescharacteristic, complemented, Sylow
364C2×#41—1061no—
365C2×SD16—531no—
366C2×SD16—531no—
367C2×SD16—531no—
368C2×SD16C4×D5311yescharacteristic
369#131—2124no—
370C2×M16—21no—
371C2×M16—531no—
372C2×M16—531no—
373C2×M16—1061no—
374C22×C8—531no—
375C22×C8—531no—
376C22×C8—1061no—
377C22×C8—1061no—
378C22×C8—1061no—
379C22×C8—1061no—
380C4⋊Q8—531no—
381C4⋊Q8—531no—
382C4⋊Q8—531no—
383#127—531no—
384#125—1061no—
385#124—1061no—
386C22⋊Q8—531no—
387C22⋊Q8—531no—
388C22⋊Q8—1061no—
389#121—531no—
390#121—531no—
391#121—1061no—
392C4×Q8—531no—
393C4×Q8—1061no—
394C4×Q8C22×D5311yescharacteristic
395C4×D4—531no—
396C4×D4—1061no—
397C4×D4C22×D5311yescharacteristic
398#117—1061no—
399C2×C42—531no—
400#106—531no—
401#106—531no—
402#106—531no—
403#106C4×D5311yescharacteristic
404C4⋊C8—21no—
405C4⋊C8—531no—
406C4⋊C8—531no—
407C4⋊C8—1061no—
408C4⋊C8—1061no—
409C4⋊C8—1061no—
410C4⋊C8—1061no—
411C4⋊C8—1061no—
412C4⋊C8—1061no—
413C4⋊C8C22×D5311yescharacteristic
414C4⋊C8C22×D5311yescharacteristic
415#103—531no—
416#103—531no—
417#103—531no—
418#103C4×D5311yescharacteristic
419#102—531no—
420#102—531no—
421#102—531no—
422#102C4×D5311yescharacteristic
423C22⋊C8—21no—
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—21no—
433C4×C8—531no—
434C4×C8—531no—
435C4×C8—531no—
436C4×C8—531no—
437C4×C8—531no—
438C4×C8—1061no—
439C4×C8C22×D5311yescharacteristic
440C4×C8C22×D5311yescharacteristic
441C4×C8C22×D5311yescharacteristic
442#41—2122no—
443#41—2122no—
444C2×Q8—531no—
445C2×Q8—1061no—
446C2×Q8—1061no—
447C2×Q8C2×C4×D5311yescharacteristic
448C2×D4—531no—
449C2×D4—1061no—
450C2×D4—1061no—
451C2×D4—1061no—
452C2×D4—1061no—
453C2×D4C2×C4×D5311yescharacteristic
454C22×C4—21no—
455C22×C4—531no—
456C22×C4—531no—
457C22×C4—531no—
458C22×C4—1061no—
459SD16—41no—
460SD16—1062no—
461SD16—1062no—
462SD16—2121no—
463M16—42no—
464M16—534no—
465M16—1062no—
466M16—2122no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—531no—
471C2×C8—531no—
472C2×C8—531no—
473C2×C8—531no—
474C2×C8—1062no—
475C2×C8—1061no—
476C2×C8—1061no—
477C2×C8—1061no—
478C2×C8—2121no—
479C2×C8—2122no—
480C2×C8—2122no—
481C2×C8—2122no—
482C2×C8—2122no—
483C2×C8C2×C4×D5311yescharacteristic
484C2×C8C2×C4×D5311yescharacteristic
485C2×C8D4×D5311yescharacteristic
486C2×C8D4×D5311yescharacteristic
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⋊C4C2×C4×D5311yescharacteristic
498C4⋊C4C2×C4×D5311yescharacteristic
499C22⋊C4—21no—
500C22⋊C4—1061no—
501C22⋊C4—1061no—
502C22⋊C4—1061no—
503C22⋊C4—1061no—
504C22⋊C4—1061no—
505C42—21no—
506C42—531no—
507C42—531no—
508C42—531no—
509C42—1061no—
510C42C23×D5311yescharacteristic
511C23—21no—
512C23—531no—
513C23—1061no—
514Q8—21no—
515Q8—21no—
516Q8—1061no—
517Q8—1061no—
518Q8—4241no—
519Q8—2122no—
520D4—21no—
521D4—21no—
522D4—1061no—
523D4—1061no—
524D4—4241no—
525D4—2122no—
526D4—2122no—
527D4—2122no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—531no—
534C2×C4—532no—
535C2×C4—532no—
536C2×C4—531no—
537C2×C4—531no—
538C2×C4—1061no—
539C2×C4—1061no—
540C2×C4—1061no—
541C2×C4—1061no—
542C2×C4—1061no—
543C2×C4—2121no—
544C2×C4—2121no—
545C2×C4—2121no—
546C2×C4C22×C4×D5311yescharacteristic
547C2×C4C2×D4×D5311yescharacteristic
548C2×C4#41×D5311yescharacteristic
549C8—21no—
550C8—21no—
551C8—21no—
552C8—41no—
553C8—41no—
554C8—41no—
555C8—532no—
556C8—532no—
557C8—1061no—
558C8—1061no—
559C8—2121no—
560C8—2121no—
561C8—2121no—
562C8SD16×D5312yes—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—532no—
567C22—1061no—
568C22—1061no—
569C22—2121no—
570C22—2121no—
571C22—2121no—
572C22C4×D4×D5311yescharacteristic
573C4—21no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—532no—
578C4—1061no—
579C4—1061no—
580C4—2121no—
581C4—2121no—
582C4—2121no—
583C4#131×D5311yescharacteristic
584C4#1681138911yescharacteristic
585C4C2×SD16×D5311yescharacteristic
586C4#135×D5311yescharacteristic
587C2—81no—
588C2—1062no—
589C2—4241no—
590C2#2028345611yescharacteristic
591C2C4×SD16×D5311yescharacteristic
592C2#2028362911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC53×#1583C2
socleC22×C53C2×C4×D4
prime radical#16811170C22
cosocleC42×C53C24
hypercenter#1583D53
hypocenterC53#18067
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1583

Complements

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

NH ≅ G/Ntype
2. #20283657C2semidirect
4. #20273603C2semidirect
5. #20273602C2semidirect
6. #20273557C2semidirect
7. #20273553C2semidirect
8. #20273552C2semidirect
9. #20273518C2semidirect
10. C8×#11719924C2semidirect
11. #334×D53C2semidirect
12. #20273116C2semidirect
14. #20272994C2semidirect
15. C53×#1583C2semidirect
208. #11719912SD16semidirect
264. #88322#131semidirect
297. #1583D53semidirect
363. C53#18067semidirect

Maximal subgroups

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

subgroupindex
#202852712
#202836572
#202834602
#202736032
#202736022
#202735572
#202735532
#202735522
#202735182
C8×#117199242
#334×D532
#202731162
#202731122
#202729942
C53×#15832
#1806753