Finite Group Databasebeta

← #27410671

Subgroups — #27410671

Order 13568.

706 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285183C211yesmaximal, characteristic
2#20275212C211yesmaximal, characteristic, complemented
3#20275206C211yesmaximal, characteristic, complemented
4#20275205C211yesmaximal, characteristic, complemented
5#20275199C211yesmaximal, characteristic, complemented
6#20275195C211yesmaximal, characteristic, complemented
7#20275193C211yesmaximal, characteristic, complemented
8#20275188C211yesmaximal, characteristic, complemented
9#20275186C211yesmaximal, characteristic, complemented
10#20275117C211yesmaximal, characteristic, complemented
11#20275102C211yesmaximal, characteristic, complemented
12#20275099C211yesmaximal, characteristic, complemented
13#20274861C211yesmaximal, characteristic, complemented
14#20274861C211yesmaximal, characteristic, complemented
15C53×#1884C211yesmaximal, characteristic, complemented
16C2×#13133247C2211yescharacteristic
17C2×#13133247C2211yescharacteristic
18C2×#13133242C2211yescharacteristic
19C2×#13133242C2211yescharacteristic
20#16811390C2211yescharacteristic
21C2×#13133205C2211yescharacteristic
22#16811100C2211yescharacteristic
23#16811097C2211yescharacteristic
24#16811015C412yescomplemented
25#16811010C412yescomplemented
26#16811005C412yescomplemented
27#16811005C412yescomplemented
28#16810640C2211yescharacteristic
29#16810640C2211yescharacteristic
30#16810633C2211yescharacteristic
31#16810631C2211yescharacteristic
32#16810629C2211yescharacteristic
33#16810627C2211yescharacteristic
34C2×#13133135C2211yescharacteristic
35C2×#13133133C2211yescharacteristic
36#16810597C2211yescharacteristic
37C4×#11719917C2211yescharacteristic
38#16810395C2211yescharacteristic
39#16810395C2211yescharacteristic
40#16810393C2211yescharacteristic
41#16810393C2211yescharacteristic
42#16810383C2211yescharacteristic
43#16810383C2211yescharacteristic
44#16810381C2211yescharacteristic
45#16810381C2211yescharacteristic
46#16810379C2211yescharacteristic
47#16810379C2211yescharacteristic
48C53×#568C2211yescharacteristic
49C53×#532C2211yescharacteristic
50C53×#421C2211yescharacteristic
51C53×#421C2211yescharacteristic
52C53×#410C2211yescharacteristic
53C53×#339C2211yescharacteristic
54C53×#339C2211yescharacteristic
55C22×Dic106C2311yescharacteristic
56C22×D212C2311yescharacteristic
57#13133247—41no—
58#13133247—41no—
59#13133247—41no—
60#13133247—41no—
61#13133242—41no—
62#13133242—41no—
63#13133242—41no—
64#13133242—41no—
65C2×#11719925C2×C412yescomplemented
66C2×#11719925C2×C412yescomplemented
67C2×#11719923C2×C412yescomplemented
68C2×#11719923C2×C412yescomplemented
69C2×#11719924C2×C412yes—
70C2×#11719924C2×C412yes—
71C2×#11719922C2×C412yes—
72C2×#11719922C2×C412yes—
73#13133224—41no—
74#13133224—41no—
75C2×#11719917—41no—
76C2×#11719917C2311yescharacteristic
77C2×#11719917C2311yescharacteristic
78C2×#11719917C2311yescharacteristic
79C2×#11719917C2311yescharacteristic
80#13133205—41no—
81#13133205C2×C412yes—
82#13133205C2×C412yes—
83C4×Dic106—41no—
84#13133182—41no—
85C4×D212—41no—
86#13133179—41no—
87#13133176C2311yescharacteristic
88#13133176C2311yescharacteristic
89#13133173—41no—
90C2×#11719921—41no—
91C22⋊C4×D53—41no—
92#13133144—41no—
93#13133144—41no—
94#13133142—41no—
95#13133140—41no—
96#13133139—41no—
97#13133137—41no—
98#13133135—41no—
99#13133133—41no—
100#13133132—41no—
101#13133128C2×C412yescomplemented
102C4×#88322C2×C412yescomplemented
103C2×#41×C53C2311yescharacteristic
104C2×#41×C53C2311yescharacteristic
105#131×C53—41no—
106#131×C53—41no—
107C2×M16×C53—41no—
108C2×M16×C53C2311yescharacteristic
109C2×M16×C53C2311yescharacteristic
110C4⋊Q8×C53C2×C411yescharacteristic, complemented
111#127×C53C2×C411yescharacteristic, complemented
112#124×C53C2×C411yescharacteristic, complemented
113#124×C53C2×C411yescharacteristic, complemented
114#121×C53—41no—
115C4×D4×C53—41no—
116#117×C53C2311yescharacteristic
117#117×C53C2311yescharacteristic
118C2×C42×C53C2311yescharacteristic
119#104×C53—41no—
120#104×C53—41no—
121#99×C53—41no—
122C23×D53—21no—
123#41×C53—21no—
124#41×C53—21no—
125#41×C53—21no—
126#41×C53—21no—
127#41×C53—21no—
128#41×C53—21no—
129#41×C53—21no—
130#41×C53—21no—
131C2×Q8×C53C22×C411yescharacteristic
132C2×Q8×C53C22×C411yescharacteristic
133C2×D4×C53—21no—
134C2×D4×C53—21no—
135C2×D4×C53—41no—
136C2×D4×C53C22×C411yescharacteristic
137C2×D4×C53C22×C411yescharacteristic
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53C2411yescharacteristic
142C22×Dic53—21no—
143C2×D212—21no—
144C2×D212—21no—
145C2×D212—21no—
146C2×D212—21no—
147C2×D212—21no—
148C2×D212C22×C412yes—
149C2×C4×D53—42no—
150C2×Dic106—21no—
151C2×Dic106—21no—
152C2×Dic106—21no—
153C2×Dic106—21no—
154C2×Dic106—21no—
155C2×Dic106C22×C412yes—
156M16×C53—21no—
157M16×C53—21no—
158M16×C53—21no—
159M16×C53—21no—
160M16×C53—42no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53—42no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—41no—
167C22⋊C4×C53—21no—
168C22⋊C4×C53—21no—
169C22⋊C4×C53—41no—
170C22⋊C4×C53—41no—
171C42×C53—21no—
172C42×C53—21no—
173C42×C53—21no—
174C42×C53C22×C411yescharacteristic
175C42×C53C22×C411yescharacteristic
176C22⋊Dic53—21no—
177C22⋊Dic53—21no—
178#11719925—42no—
179#11719925—42no—
180#11719924—42no—
181#11719924—42no—
182#11719923—42no—
183#11719923—42no—
184#11719922—42no—
185#11719922—42no—
186#11719921—42no—
187#11719921—42no—
188#11719921—42no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191#11719919—42no—
192C4×Dic53—21no—
193C4×Dic53—21no—
194C4×Dic53—42no—
195#11719917—21no—
196#11719917—21no—
197#11719917—21no—
198#11719917—21no—
199#11719917—21no—
200#11719917—21no—
201#11719917—21no—
202#11719917—21no—
203#11719917—42no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—42no—
207C2×#88322C22×C412yes—
208C2×#88322C22×C412yes—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53H(5,𝔽2)11yescharacteristic
212C22×D53—22no—
213C22×D53—41no—
214C22×D53—41no—
215C22×D53—82no—
216Q8×C53—21no—
217Q8×C53—21no—
218Q8×C53—21no—
219Q8×C53—21no—
220D4×C53—21no—
221D4×C53—21no—
222D4×C53—21no—
223D4×C53—21no—
224D4×C53—41no—
225D4×C53—41no—
226D4×C53—41no—
227D4×C53—41no—
228D4×C53—82no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—81no—
240C2×C4×C53C23×C411yescharacteristic
241C2×C4×C53H(5,𝔽2)11yescharacteristic
242C2×C4×C53H(5,𝔽2)11yescharacteristic
243C2×C4×C53H(5,𝔽2)11yescharacteristic
244C2×C4×C53H(5,𝔽2)11yescharacteristic
245C2×C4×C53H(5,𝔽2)11yescharacteristic
246C2×Dic53—22no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251D212—22no—
252D212—22no—
253D212—41no—
254D212—41no—
255C4×D53—162no—
256Dic106—22no—
257Dic106—22no—
258Dic106—41no—
259Dic106—41no—
260C8×C53—41no—
261C8×C53—41no—
262#88322—22no—
263#88322—22no—
264#88322—41no—
265#88322—41no—
266#23362—531nomaximal, Sylow
267C22×C53—41no—
268C22×C53—41no—
269C22×C53—41no—
270C22×C53—81no—
271C22×C53—81no—
272C22×C53#51811yescharacteristic
273C22×C53#56011yescharacteristic
274C22×C53#56011yescharacteristic
275C2×D53—42no—
276C2×D53—82no—
277C2×D53—82no—
278C2×D53—161no—
279C4×C53—41no—
280C4×C53—41no—
281C4×C53—41no—
282C4×C53—41no—
283C4×C53—81no—
284C4×C53—81no—
285C4×C53#51811yescharacteristic
286C4×C53#51811yescharacteristic
287C4×C53#56011yescharacteristic
288C4×C53#56011yescharacteristic
289Dic53—42no—
290Dic53—81no—
291Dic53—81no—
292#3153—531no—
293#1905—531no—
294#1905—531no—
295#1904—531no—
296#1904—531no—
297#1903—531no—
298#1903—531no—
299#1902—531no—
300#1902—531no—
301#1888—531no—
302#1885—531no—
303#1884—531no—
304#1884D5311yescharacteristic, complemented
305#1841—531no—
306#1841—531no—
307C2×C53—21no—
308C2×C53—21no—
309C2×C53—21no—
310C2×C53—81no—
311C2×C53—81no—
312C2×C53#238011yescharacteristic
313D53—82no—
314C2×#137—531no—
315C2×#137—531no—
316C2×AGL(1,ℤ/8)—531no—
317C2×AGL(1,ℤ/8)—531no—
318#568—531no—
319#568C2×D5311yescharacteristic
320#532C2×D5311yescharacteristic
321C2×#124—531no—
322#519—531no—
323#518—531no—
324#497—532no—
325#496—532no—
326#495—532no—
327#495—532no—
328#421—531no—
329#421—531no—
330#421C2×D5311yescharacteristic
331#421C2×D5311yescharacteristic
332#419—531no—
333#419—531no—
334#418—531no—
335#418—531no—
336C2×#101—531no—
337C2×#100—531no—
338#410—531no—
339#410C2×D5311yescharacteristic
340C4×M16—531no—
341#344—531no—
342#344—531no—
343#344—531no—
344#344—531no—
345#339—531no—
346#339—531no—
347#339C2×D5311yescharacteristic
348#339C2×D5311yescharacteristic
349#338—531no—
350#338—531no—
351H(3,ℤ/4)—531no—
352H(3,ℤ/4)—531no—
353C53#2336211yescharacteristic, complemented, Sylow
354C2×#41C22×D5311yescharacteristic
355C2×#41C22×D5311yescharacteristic
356C22×Q8—531no—
357C22×D4—531no—
358#137—2121no—
359#137—2121no—
360#137—2121no—
361#137—2121no—
362AGL(1,ℤ/8)—2121no—
363AGL(1,ℤ/8)—2121no—
364AGL(1,ℤ/8)—2121no—
365AGL(1,ℤ/8)—2121no—
366C2×Q16—532no—
367C2×Q16—532no—
368C2×SD16—532no—
369C2×SD16—532no—
370C2×SD16—532no—
371C2×SD16—532no—
372C2×D8—532no—
373C2×D8—532no—
374#131—41no—
375#131—41no—
376#131—2121no—
377#131—2121no—
378C2×M16—41no—
379C2×M16—531no—
380C2×M16—531no—
381C2×M16—531no—
382C2×M16—531no—
383C2×M16—2121no—
384C2×M16C22×D5311yescharacteristic
385C2×M16C22×D5311yescharacteristic
386C4⋊Q8C4×D5311yescharacteristic, complemented
387#127C4×D5311yescharacteristic, complemented
388#124—532no—
389#124—532no—
390#124—2121no—
391#124C4×D5311yescharacteristic, complemented
392#124C4×D5311yescharacteristic, complemented
393#121—41no—
394C4×Q8—2121no—
395C4×Q8—2121no—
396C4×D4—41no—
397C4×D4—2121no—
398C4×D4—2121no—
399#117—531no—
400#117—531no—
401#117—2121no—
402#117C22×D5311yescharacteristic
403#117C22×D5311yescharacteristic
404C2×C22⋊C4—2121no—
405C2×C22⋊C4—2121no—
406C2×C42C22×D5311yescharacteristic
407#104—41no—
408#104—41no—
409#104—2121no—
410#104—2121no—
411#103—2121no—
412#103—2121no—
413#102—2121no—
414#102—2121no—
415#101—2121no—
416#100—2121no—
417#99—41no—
418#99—2121no—
419#97—532no—
420C4×C8—532no—
421C24—1061no—
422#41—21no—
423#41—21no—
424#41—21no—
425#41—21no—
426#41—21no—
427#41—21no—
428#41—21no—
429#41—21no—
430C2×Q8—532no—
431C2×Q8—1061no—
432C2×Q8—1061no—
433C2×Q8—1061no—
434C2×Q8—1061no—
435C2×Q8—1061no—
436C2×Q8C2×C4×D5311yescharacteristic
437C2×Q8C2×C4×D5311yescharacteristic
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—41no—
441C2×D4—532no—
442C2×D4—1061no—
443C2×D4—1061no—
444C2×D4—1061no—
445C2×D4—1061no—
446C2×D4—1061no—
447C2×D4C2×C4×D5311yescharacteristic
448C2×D4C2×C4×D5311yescharacteristic
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—1061no—
453C22×C4—2122no—
454C22×C4C23×D5311yescharacteristic
455Q16—2122no—
456Q16—2122no—
457SD16—2122no—
458SD16—2122no—
459SD16—2122no—
460SD16—2122no—
461D8—2122no—
462D8—2122no—
463M16—21no—
464M16—21no—
465M16—21no—
466M16—21no—
467M16—42no—
468M16—1061no—
469M16—1061no—
470M16—1061no—
471M16—1061no—
472M16—1061no—
473M16—1061no—
474M16—1061no—
475M16—1061no—
476M16—2122no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—42no—
480C2×C8—532no—
481C2×C8—532no—
482C2×C8—1061no—
483C2×C8—1061no—
484C2×C8—2122no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—41no—
488C4⋊C4—1061no—
489C4⋊C4—1061no—
490C4⋊C4—2122no—
491C22⋊C4—21no—
492C22⋊C4—21no—
493C22⋊C4—41no—
494C22⋊C4—41no—
495C22⋊C4—1061no—
496C22⋊C4—1061no—
497C22⋊C4—2122no—
498C22⋊C4—2122no—
499C22⋊C4—2122no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—1061no—
504C42—1061no—
505C42—2122no—
506C42C2×C4×D5311yescharacteristic
507C42C2×C4×D5311yescharacteristic
508C23—21no—
509C23—21no—
510C23—1062no—
511C23—4242no—
512C23—2121no—
513C23—2121no—
514C23#1313326211yescharacteristic
515Q8—21no—
516Q8—21no—
517Q8—21no—
518Q8—21no—
519Q8—1062no—
520Q8—1062no—
521Q8—2121no—
522Q8—2121no—
523D4—21no—
524D4—21no—
525D4—21no—
526D4—21no—
527D4—41no—
528D4—41no—
529D4—41no—
530D4—41no—
531D4—82no—
532D4—1062no—
533D4—1062no—
534D4—2121no—
535D4—2121no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—81no—
547C2×C4—8482no—
548C2×C4—1062no—
549C2×C4—2121no—
550C2×C4—2121no—
551C2×C4—2121no—
552C2×C4—2121no—
553C2×C4C22×C4×D5311yescharacteristic
554C2×C4#1313326211yescharacteristic
555C2×C4#1313326211yescharacteristic
556C2×C4#1313326211yescharacteristic
557C2×C4#1313326311yescharacteristic
558C2×C4#1313326311yescharacteristic
559C8—41no—
560C8—41no—
561C8—1062no—
562C8—1062no—
563C8—2121no—
564C8—2121no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—81no—
569C22—81no—
570C22—8481no—
571C22—4242no—
572C22—4242no—
573C22—2122no—
574C22#1681109611yescharacteristic
575C22#1681136111yescharacteristic
576C22#1681136111yescharacteristic
577C4—41no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—81no—
582C4—81no—
583C4—4241no—
584C4—4241no—
585C4—2122no—
586C4#1681109511yescharacteristic
587C4#1681109511yescharacteristic
588C4#1681136211yescharacteristic
589C4#1681136311yescharacteristic
590C2—21no—
591C2—21no—
592C2—21no—
593C2—81no—
594C2—81no—
595C2—4242no—
596C2#2027789011yescharacteristic
subgroupG/H
centerC2#20277890
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1884C2
socleC2×C53#2380
prime radical#20285183C2
cosocleC22×C4×C53C24
hypercenter#1884D53
hypocenterC53#23362
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1884

Complements

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

NH ≅ G/Ntype
2. #20275212C2semidirect
3. #20275206C2semidirect
4. #20275205C2semidirect
5. #20275199C2semidirect
6. #20275195C2semidirect
7. #20275193C2semidirect
8. #20275188C2semidirect
9. #20275186C2semidirect
10. #20275117C2semidirect
11. #20275102C2semidirect
12. #20275099C2semidirect
13. #20274861C2semidirect
14. #20274861C2semidirect
15. C53×#1884C2semidirect
24. #16811015C4semidirect
25. #16811010C4semidirect
26. #16811005C4semidirect
27. #16811005C4semidirect
65. C2×#11719925C2×C4semidirect
66. C2×#11719925C2×C4semidirect
67. C2×#11719923C2×C4semidirect
68. C2×#11719923C2×C4semidirect
101. #13133128C2×C4semidirect
102. C4×#88322C2×C4semidirect
110. C4⋊Q8×C53C2×C4semidirect
111. #127×C53C2×C4semidirect
112. #124×C53C2×C4semidirect
113. #124×C53C2×C4semidirect
304. #1884D53semidirect
353. C53#23362semidirect
386. C4⋊Q8C4×D53semidirect
387. #127C4×D53semidirect
391. #124C4×D53semidirect
392. #124C4×D53semidirect

Maximal subgroups

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

subgroupindex
#202851832
#202752122
#202752062
#202752052
#202751992
#202751952
#202751932
#202751882
#202751862
#202751172
#202751022
#202750992
#202748612
C53×#18842
#2336253