Finite Group Databasebeta

← #27540924

Subgroups — #27540924

Order 13568.

622 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20287105C211yesmaximal, characteristic, complemented
2#20286389C211yesmaximal, characteristic
3#20276937C211yesmaximal, characteristic, complemented
4#20276902C211yesmaximal, characteristic, complemented
5#20276861C211yesmaximal, characteristic, complemented
6#20276812C211yesmaximal, characteristic, complemented
7#20276809C211yesmaximal, characteristic, complemented
8#20276754C211yesmaximal, characteristic, complemented
9#20276646C211yesmaximal, characteristic, complemented
10#20276645C211yesmaximal, characteristic, complemented
11#20276566C211yesmaximal, characteristic, complemented
12#20276532C211yesmaximal, characteristic, complemented
13C2×#16810423C211yesmaximal, characteristic, complemented
14#357×D53C211yesmaximal, characteristic, complemented
15C53×#2233C211yesmaximal, characteristic, complemented
16C22×D424C2211yescharacteristic
17C2×D8×D53C2211yescharacteristic
18#16811328—21no—
19#16811257—21no—
20C2×#13133162C2211yescharacteristic
21C2×#13133162C2211yescharacteristic, complemented
22C2×#13133161C2211yescharacteristic, complemented
23C22×#11719908C2211yescharacteristic, complemented
24C2×C16×D53C2211yescharacteristic, complemented
25#16811027C2211yescharacteristic, complemented
26#16810998C2211yescharacteristic
27#16810960—21no—
28#16810958—21no—
29#16810917—21no—
30#16810865C2211yescharacteristic, complemented
31#16810864C2211yescharacteristic, complemented
32#16810860C2211yescharacteristic
33#16810817—21no—
34#16810758—21no—
35#16810756—21no—
36#16810717C2211yescharacteristic, complemented
37#16810705C2211yescharacteristic
38C8×C22⋊D53C2211yescharacteristic
39#107×D53C2211yescharacteristic, complemented
40#16810450C2211yescharacteristic, complemented
41#16810441C2211yescharacteristic, complemented
42#16810426C2211yescharacteristic, complemented
43#16810425C2211yescharacteristic
44#16810425C2211yescharacteristic, complemented
45#16810424C2211yescharacteristic, complemented
46#16810423C2211yescharacteristic
47#16810423C2211yescharacteristic
48#16810423C2211yescharacteristic, complemented
49#16810423C2211yescharacteristic, complemented
50#16810402C2211yescharacteristic, complemented
51C16×Dic53C2211yescharacteristic, complemented
52C53×#466C2211yescharacteristic
53C2×#107×C53C2211yescharacteristic, complemented
54C53×#367C2211yescharacteristic, complemented
55C53×#367C2211yescharacteristic, complemented
56C53×#357C2211yescharacteristic, complemented
57C53×#357C2211yescharacteristic, complemented
58C53×C22⋊C16C2211yescharacteristic, complemented
59C22×D212—21no—
60C2×D4×D53—21no—
61C2×D4×D53—41no—
62C2×Dic212C2311yescharacteristic
63C2×D424—21no—
64C2×D424—22no—
65C2×D424—41no—
66C2×D424C2311yescharacteristic
67C2×D424C2311yescharacteristic
68C2×#11719922—21no—
69C2×#11719922—21no—
70D8×D53—22no—
71D8×D53—41no—
72C2×C8×D53C2311yescharacteristic
73#13133216—21no—
74#13133213—21no—
75#13133207—41no—
76#13133197—21no—
77#13133196—21no—
78#13133196—41no—
79#13133195—41no—
80#13133192—21no—
81#13133190—41no—
82#13133187—41no—
83#13133185—41no—
84#13133182—21no—
85#13133179—21no—
86C4×C22⋊D53—41no—
87C4×C22⋊D53D411yescharacteristic
88C4⋊C4×D53—21no—
89#13133162—22no—
90#13133162—41no—
91#13133161—41no—
92C2×#11719908—41no—
93C2×#11719908C2311yescharacteristic
94C2×#11719908C2311yescharacteristic
95C16×D53—41no—
96#13133151C2311yescharacteristic
97#13133150—21no—
98#13133150—21no—
99#13133146D411yescharacteristic
100#13133142—21no—
101#13133140—21no—
102#13133140—21no—
103#13133137—21no—
104#13133137—21no—
105C22⋊#88322D411yescharacteristic
106#13133129D411yescharacteristic
107C8×Dic53C2311yescharacteristic
108C2×D8×C53C2311yescharacteristic
109C22×C8×C53C2311yescharacteristic
110#121×C53—21no—
111C2×C4⋊C4×C53—21no—
112C2×C16×C53C2311yescharacteristic
113C2×C16×C53C2311yescharacteristic
114#107×C53—21no—
115#107×C53C2311yescharacteristic
116#107×C53C2311yescharacteristic
117#107×C53C2311yescharacteristic
118#102×C53—21no—
119C23×D53—41no—
120C23×D53—41no—
121C2×D4×C53—21no—
122C2×D4×C53—41no—
123C22×C4×C53—41no—
124C22×C4×C53C2×D411yescharacteristic
125C2×C22⋊D53—41no—
126C2×C22⋊D53—41no—
127C2×C22⋊D53—41no—
128C2×C22⋊D53D811yescharacteristic
129D4×D53—42no—
130D4×D53—81no—
131D4×D53—84no—
132C2×D212—21no—
133C2×D212—21no—
134C2×D212—42no—
135C2×D212—41no—
136C2×D212—41no—
137C2×D212—81no—
138C2×C4×D53—41no—
139C2×C4×D53—41no—
140C2×C4×D53C2×D411yescharacteristic
141C2×Dic106—21no—
142C2×Dic106—41no—
143D8×C53—21no—
144D8×C53—21no—
145C2×C8×C53—21no—
146C2×C8×C53—21no—
147C2×C8×C53C2×D411yescharacteristic
148C2×C8×C53C2411yescharacteristic
149C4⋊C4×C53—21no—
150C4⋊C4×C53—21no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—41no—
153C22⋊C4×C53—41no—
154C22⋊Dic53—41no—
155C22⋊Dic53D811yescharacteristic
156#11719922—42no—
157#11719922—81no—
158#11719921—41no—
159#11719921—41no—
160#11719921—41no—
161#11719921D811yescharacteristic
162C4⋊Dic53—21no—
163C4⋊Dic53—41no—
164#11719919—41no—
165#11719919—41no—
166#11719919—41no—
167#11719919—41no—
168#11719919D811yescharacteristic
169C4×Dic53—41no—
170C4×Dic53C2×D411yescharacteristic
171C2×#88322C2×D411yescharacteristic
172C2×#88322C2×D411yescharacteristic
173Dic212—21no—
174Dic212—21no—
175D424—21no—
176D424—21no—
177D424—21no—
178D424—42no—
179D424#4112yes—
180C8×D53—21no—
181C8×D53—21no—
182C16×C53—21no—
183C16×C53—21no—
184#11719908—21no—
185#11719908—21no—
186C23×C53—41no—
187C23×C53C2×D811yescharacteristic
188C22×D53—41no—
189C22×D53—41no—
190C22×D53—82no—
191C22×D53—82no—
192C22×D53—82no—
193C22×D53—82no—
194C22×D53—161no—
195C22×D53—161no—
196C22×D53C2×D811yescharacteristic
197D4×C53—41no—
198D4×C53—41no—
199D4×C53—82no—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—41no—
203C2×C4×C53—41no—
204C2×C4×C53—41no—
205C2×C4×C53—161no—
206C2×C4×C53C2×D811yescharacteristic
207C2×C4×C53C22×D411yescharacteristic
208C22⋊D53—82no—
209C22⋊D53—82no—
210C22⋊D53—161no—
211C22⋊D53SD3214yescomplemented
212C2×Dic53—41no—
213C2×Dic53—41no—
214C2×Dic53C2×D811yescharacteristic
215C2×Dic53C2×D811yescharacteristic
216C2×Dic53C2×D811yescharacteristic
217D212—22no—
218D212—41no—
219D212—41no—
220D212—41no—
221D212—82no—
222D212—82no—
223C4×D53—21no—
224C4×D53—21no—
225C4×D53—82no—
226C4×D53—161no—
227Dic106—41no—
228Dic106—41no—
229Dic106—161no—
230C8×C53—21no—
231C8×C53C2×#4111yescharacteristic
232C8×C53H(5,𝔽2)11yescharacteristic
233#88322—21no—
234#88322—21no—
235#33862—531nomaximal, Sylow
236C22×C53—21no—
237C22×C53—21no—
238C22×C53—81no—
239C22×C53—81no—
240C22×C53—81no—
241C22×C53C2×SD3212yes—
242C22×C53C22×D811yescharacteristic
243C2×D53—21no—
244C2×D53—21no—
245C2×D53—42no—
246C2×D53—81no—
247C2×D53—81no—
248C2×D53—81no—
249C2×D53—81no—
250C2×D53—81no—
251C2×D53—164no—
252C2×D53—164no—
253C2×D53C2×SD3212yes—
254C4×C53—21no—
255C4×C53—81no—
256C4×C53—81no—
257C4×C53—81no—
258C4×C53#54611yescharacteristic
259C4×C53#57711yescharacteristic
260Dic53—21no—
261Dic53—21no—
262Dic53—81no—
263Dic53—81no—
264Dic53C2×SD3212yes—
265#3338—531no—
266D4×D8—531no—
267#2240—531no—
268#2233—531no—
269#2233D5311yescharacteristic, complemented
270#2226—531no—
271#2217—531no—
272#2217—531no—
273#2207—531no—
274#2187—531no—
275#2187—531no—
276C4×SD32—531no—
277D4×C16—531no—
278C2×#357—531no—
279C2×#357—531no—
280C2×C53—22no—
281C2×C53—161no—
282C2×C53#329411yescharacteristic
283C2×C53C22×SD3211yescharacteristic
284C2×C53#341511yescharacteristic
285D53—22no—
286D53—82no—
287D53—161no—
288C22×D8—531no—
289C22×D8—531no—
290#554—1061no—
291D42—1061no—
292C2×SD32—531no—
293C2×SD32—531no—
294C2×SD32—531no—
295C22×C16—531no—
296C22×C16—531no—
297#500—531no—
298#493—531no—
299#484—1061no—
300#484—1061no—
301#477—1061no—
302#467—531no—
303#467—531no—
304#466—531no—
305#466C2×D5311yescharacteristic
306#459—1061no—
307#447—1061no—
308#447—1061no—
309C4×Q16—531no—
310C4×D8—531no—
311C8×D4—531no—
312C2×#107—531no—
313C2×#107C2×D5311yescharacteristic, complemented
314#367—531no—
315#367C2×D5311yescharacteristic, complemented
316#367C2×D5311yescharacteristic, complemented
317C4⋊C16—531no—
318#358—531no—
319#357—531no—
320#357—531no—
321#357—531no—
322#357—531no—
323#357—531no—
324#357—531no—
325#357—531no—
326#357C2×D5311yescharacteristic, complemented
327#357C2×D5311yescharacteristic, complemented
328C22⋊C16—531no—
329C22⋊C16C2×D5311yescharacteristic, complemented
330C4×C16—531no—
331C53#3386211yescharacteristic, complemented, Sylow
332C22×D4—1061no—
333C22×D4—1061no—
334C22×D4—2121no—
335C2×Q16—531no—
336C2×D8—531no—
337C2×D8—531no—
338C2×D8—1062no—
339C2×D8—1061no—
340C2×D8—1061no—
341C2×D8—1062no—
342C2×D8—1061no—
343C2×D8—2121no—
344C2×D8—2121no—
345C2×D8C22×D5311yescharacteristic
346C22×C8—531no—
347C22×C8C22×D5311yescharacteristic
348C4⋊Q8—1061no—
349#127—1061no—
350#125—2121no—
351C22⋊Q8—1061no—
352C22⋊Q8—1061no—
353C22⋊Q8—2121no—
354C22⋊Q8—2121no—
355#121—21no—
356#121—1061no—
357#121—2121no—
358#120—2121no—
359#120—2121no—
360C4×Q8—1061no—
361C4×D4—531no—
362C4×D4—1061no—
363C4×D4—2121no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—1061no—
366SD32—1062no—
367SD32—2121no—
368SD32—2121no—
369C2×C16—531no—
370C2×C16—531no—
371C2×C16—2121no—
372C2×C16—2121no—
373C2×C16C22×D5311yescharacteristic
374C2×C16C22×D5311yescharacteristic
375#107—21no—
376#107—531no—
377#107—1061no—
378#107—1061no—
379#107C22×D5311yescharacteristic
380#107C22×D5311yescharacteristic
381#107C22×D5311yescharacteristic
382C4⋊C8—531no—
383#103—1061no—
384#103—1061no—
385#103—1061no—
386#102—21no—
387#102—1061no—
388#102—1061no—
389C22⋊C8—531no—
390C22⋊C8—531no—
391C4×C8—531no—
392C24—2121no—
393C24—2121no—
394C2×Q8—1061no—
395C2×Q8—2121no—
396C2×D4—21no—
397C2×D4—41no—
398C2×D4—531no—
399C2×D4—1061no—
400C2×D4—1061no—
401C2×D4—4241no—
402C2×D4—4241no—
403C2×D4—4244no—
404C2×D4—2122no—
405C2×D4—2121no—
406C2×D4—2121no—
407C2×D4—2121no—
408C2×D4—2121no—
409C2×D4—2122no—
410C2×D4—2121no—
411C22×C4—41no—
412C22×C4—531no—
413C22×C4—2121no—
414C22×C4—2121no—
415C22×C4D4×D5311yescharacteristic
416Q16—1061no—
417Q16—1061no—
418D8—21no—
419D8—21no—
420D8—532no—
421D8—1061no—
422D8—1061no—
423D8—1061no—
424D8—4241no—
425D8—2122no—
426D8—2122no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—531no—
430C2×C8—531no—
431C2×C8—1061no—
432C2×C8—1061no—
433C2×C8D4×D5311yescharacteristic
434C2×C8C23×D5311yescharacteristic
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—41no—
439C4⋊C4—531no—
440C4⋊C4—1061no—
441C4⋊C4—2121no—
442C4⋊C4—2121no—
443C4⋊C4—2121no—
444C4⋊C4—2121no—
445C4⋊C4—2121no—
446C22⋊C4—41no—
447C22⋊C4—531no—
448C22⋊C4—531no—
449C22⋊C4—2121no—
450C22⋊C4—2121no—
451C22⋊C4—2121no—
452C22⋊C4—2121no—
453C42—531no—
454C42—2121no—
455C16—21no—
456C16—21no—
457C16—1061no—
458C16—1061no—
459C23—41no—
460C23—531no—
461C23—8481no—
462C23—8481no—
463C23—4242no—
464C23—4242no—
465C23—4242no—
466C23—4242no—
467C23—2121no—
468C23—2121no—
469C23D8×D5311yescharacteristic
470Q8—8481no—
471Q8—2121no—
472Q8—2121no—
473D4—41no—
474D4—41no—
475D4—82no—
476D4—534no—
477D4—8481no—
478D4—1062no—
479D4—4242no—
480D4—4242no—
481D4—4242no—
482D4—4242no—
483D4—2121no—
484D4—2121no—
485D4—2121no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—161no—
492C2×C4—531no—
493C2×C4—531no—
494C2×C4—531no—
495C2×C4—8481no—
496C2×C4—1061no—
497C2×C4—1061no—
498C2×C4—4242no—
499C2×C4—2121no—
500C2×C4—2121no—
501C2×C4D8×D5311yescharacteristic
502C2×C4C2×D4×D5311yescharacteristic
503C8—21no—
504C8—1061no—
505C8—1061no—
506C8#41×D5311yescharacteristic
507C8#1313326311yescharacteristic
508C22—21no—
509C22—21no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22—532no—
514C22—8484no—
515C22—8484no—
516C22—1061no—
517C22—1061no—
518C22—4241no—
519C22—4241no—
520C22—4241no—
521C22—4241no—
522C22—4241no—
523C22—2122no—
524C22SD32×D5312yes—
525C22C2×D8×D5311yescharacteristic
526C4—21no—
527C4—81no—
528C4—81no—
529C4—81no—
530C4—532no—
531C4—1061no—
532C4—1061no—
533C4—4241no—
534C4—4241no—
535C4#1681126511yescharacteristic
536C4#1681143211yescharacteristic
537C2—22no—
538C2—161no—
539C2—8481no—
540C2—1062no—
541C2—4242no—
542C2#2028652611yescharacteristic
543C2C2×SD32×D5311yescharacteristic
544C2#2028785511yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2233C2
socleC22×C53C22×D8
prime radical#20286389C2
hypercenter#2233D53
hypocenterC53#33862
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2233

Complements

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

NH ≅ G/Ntype
1. #20287105C2semidirect
3. #20276937C2semidirect
4. #20276902C2semidirect
5. #20276861C2semidirect
6. #20276812C2semidirect
7. #20276809C2semidirect
8. #20276754C2semidirect
9. #20276646C2semidirect
10. #20276645C2semidirect
11. #20276566C2semidirect
12. #20276532C2semidirect
13. C2×#16810423C2semidirect
14. #357×D53C2semidirect
15. C53×#2233C2semidirect
21. C2×#13133162C22semidirect
22. C2×#13133161C22semidirect
23. C22×#11719908C22semidirect
24. C2×C16×D53C22semidirect
25. #16811027C22semidirect
30. #16810865C22semidirect
31. #16810864C22semidirect
36. #16810717C22semidirect
39. #107×D53C22semidirect
40. #16810450C22semidirect
41. #16810441C22semidirect
42. #16810426C22semidirect
44. #16810425C22semidirect
45. #16810424C22semidirect
48. #16810423C22semidirect
49. #16810423C22semidirect
50. #16810402C22semidirect
51. C16×Dic53C22semidirect
53. C2×#107×C53C22semidirect
54. C53×#367C22semidirect
55. C53×#367C22semidirect
56. C53×#357C22semidirect
57. C53×#357C22semidirect
58. C53×C22⋊C16C22semidirect
211. C22⋊D53SD32semidirect
269. #2233D53semidirect
313. C2×#107C2×D53semidirect
315. #367C2×D53semidirect
316. #367C2×D53semidirect
326. #357C2×D53semidirect
327. #357C2×D53semidirect
329. C22⋊C16C2×D53semidirect
331. C53#33862semidirect

Maximal subgroups

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

subgroupindex
#202871052
#202863892
#202769372
#202769022
#202768612
#202768122
#202768092
#202767542
#202766462
#202766452
#202765662
#202765322
C2×#168104232
#357×D532
C53×#22332
#3386253