Finite Group Databasebeta

← #27538837

Subgroups — #27538837

Order 13568.

514 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285240C211yesmaximal, characteristic, complemented
2#20283627C211yesmaximal, characteristic
3#20276892C211yesmaximal, characteristic, complemented
4#20276891C211yesmaximal, characteristic, complemented
5#20276890C211yesmaximal, characteristic, complemented
6#20276889C211yesmaximal, characteristic, complemented
7#20276885C211yesmaximal, characteristic, complemented
8#20276884C211yesmaximal, characteristic, complemented
9#20276883C211yesmaximal, characteristic, complemented
10#20276486C211yesmaximal, characteristic, complemented
11#365×D53C211yesmaximal, characteristic, complemented
12#20276479C211yesmaximal, characteristic, complemented
13#20276449C211yesmaximal, characteristic
14#360×D53C211yesmaximal, characteristic
15C53×#2230C211yesmaximal, characteristic, complemented
16#135×D53C2211yescharacteristic
17#16811388—21no—
18#16811250—21no—
19M32×D53C2211yescharacteristic
20M32×D53C2211yescharacteristic
21#16811030—21no—
22#16811029—21no—
23#16811025C2211yescharacteristic, complemented
24#16811023C2211yescharacteristic, complemented
25#16811020C2211yescharacteristic, complemented
26#16810743—21no—
27#16810742—21no—
28#16810741C2211yescharacteristic
29#16810740C2211yescharacteristic
30#16810739C2211yescharacteristic
31#108×D53C2211yescharacteristic
32#16810662—21no—
33#16810658C2211yescharacteristic
34#106×D53C2211yescharacteristic
35#16810638—21no—
36#16810577C2211yescharacteristic
37#16810447C2211yescharacteristic, complemented
38#16810447C2211yescharacteristic, complemented
39#16810446C2211yescharacteristic, complemented
40#16810446C2211yescharacteristic, complemented
41#16810445C2211yescharacteristic, complemented
42#16810445C2211yescharacteristic, complemented
43#16810444C2211yescharacteristic, complemented
44#16810443C2211yescharacteristic, complemented
45#16810443C2211yescharacteristic, complemented
46#16810434C2211yescharacteristic
47#16810434C2211yescharacteristic
48#16810433C2211yescharacteristic
49#16810433C2211yescharacteristic
50#16810432C2211yescharacteristic
51#16810432C2211yescharacteristic
52C53×#498C2211yescharacteristic, complemented
53C53×#443C2211yescharacteristic
54C53×#365C2211yescharacteristic, complemented
55C53×#365C2211yescharacteristic, complemented
56C53×#364C2211yescharacteristic, complemented
57C53×#360C2211yescharacteristic
58C53×#360C2211yescharacteristic
59#41×D53—21no—
60#13133240Q811yescharacteristic, complemented
61#13133239C2311yescharacteristic
62#13133238—21no—
63#13133237—21no—
64#13133236Q811yescharacteristic, complemented
65Q16×D53Q811yescharacteristic, complemented
66SD16×D53—21no—
67D8×D53Q811yescharacteristic, complemented
68#13133225—21no—
69#13133225—41no—
70#13133224—21no—
71#13133223—21no—
72#13133223—41no—
73C2×#11719912—41no—
74C2×#11719912D411yescharacteristic
75M16×D53—21no—
76M16×D53—41no—
77C2×C8×D53C2311yescharacteristic
78#13133216—21no—
79#13133215—41no—
80#13133209—21no—
81#13133208—21no—
82#13133207—41no—
83#13133195—41no—
84#13133193—41no—
85#13133175—21no—
86#13133174D411yescharacteristic
87C4⋊C4×D53—21no—
88#13133157C2311yescharacteristic
89#13133157C2311yescharacteristic
90#13133156—21no—
91#13133156—21no—
92C16×D53—21no—
93C16×D53—21no—
94#13133153C2311yescharacteristic
95#13133152—21no—
96#13133151—21no—
97#13133150—21no—
98#13133150—21no—
99#13133149C2311yescharacteristic
100#13133149C2311yescharacteristic
101#13133148—21no—
102#13133148—21no—
103#13133145—21no—
104#13133144—21no—
105#13133143—21no—
106#13133128D411yescharacteristic
107#13133127D411yescharacteristic
108C8×Dic53C2311yescharacteristic
109#135×C53C2311yescharacteristic
110#131×C53—21no—
111C4⋊Q8×C53—21no—
112M32×C53C2311yescharacteristic
113M32×C53C2311yescharacteristic
114#108×C53C2311yescharacteristic
115#106×C53—21no—
116#106×C53C2311yescharacteristic
117#106×C53C2311yescharacteristic
118#104×C53—21no—
119C4×C8×C53C2311yescharacteristic
120#41×C53—21no—
121C2×Q8×C53—41no—
122#11719949—21no—
123Q8×D53—21no—
124#11719947—21no—
125#11719947—41no—
126D4×D53—21no—
127D4×D53—41no—
128#11719945—21no—
129#11719945—41no—
130C2×C4×D53—41no—
131C2×C4×D53—41no—
132C2×C4×D53C2×D411yescharacteristic
133C2×Dic106—41no—
134Q16×C53C2×Q811yescharacteristic
135SD16×C53—21no—
136D8×C53C2×Q811yescharacteristic
137M16×C53—21no—
138M16×C53—41no—
139C2×C8×C53—41no—
140C2×C8×C53C2×D411yescharacteristic
141C2×C8×C53C2411yescharacteristic
142C4⋊C4×C53—21no—
143C4⋊C4×C53—21no—
144C4⋊C4×C53—41no—
145C42×C53C2×D411yescharacteristic
146#11719925—21no—
147#11719924—21no—
148#11719923—21no—
149#11719922—21no—
150#11719921—21no—
151#11719921—41no—
152C4⋊Dic53—21no—
153C4⋊Dic53—21no—
154C4⋊Dic53—41no—
155#11719919—21no—
156#11719919—41no—
157#11719919—41no—
158#11719919—41no—
159#11719919—41no—
160#11719919—41no—
161C4×Dic53—41no—
162C4×Dic53C2×D411yescharacteristic
163#11719917—21no—
164#11719917—41no—
165C2×#88322—41no—
166C2×#88322C2×D411yescharacteristic
167C2×#88322C2×D411yescharacteristic
168Dic212C2×Q811yescharacteristic
169D424C2×Q811yescharacteristic
170#11719913—21no—
171#11719912—41no—
172#11719912—41no—
173#11719912—41no—
174#11719912SD1611yescharacteristic
175#11719912SD1611yescharacteristic
176#11719912SD1611yescharacteristic
177#11719912SD1611yescharacteristic
178C8×D53—41no—
179C8×D53—41no—
180C8×D53—41no—
181C8×D53C2×Q811yescharacteristic
182C8×D53C2×Q811yescharacteristic
183C8×D53#4111yescharacteristic
184C8×D53#4111yescharacteristic
185C16×C53—21no—
186C16×C53—21no—
187#11719908—21no—
188#11719908—21no—
189C22×D53—41no—
190C22×D53AGL(1,ℤ/8)11yescharacteristic
191Q8×C53—21no—
192Q8×C53—161no—
193D4×C53—21no—
194D4×C53—41no—
195C2×C4×C53—21no—
196C2×C4×C53—41no—
197C2×C4×C53—41no—
198C2×C4×C53—41no—
199C2×C4×C53C22×D411yescharacteristic
200C22⋊D53—41no—
201C22⋊D53—41no—
202C22⋊D53—41no—
203C2×Dic53—21no—
204C2×Dic53—41no—
205C2×Dic53—41no—
206C2×Dic53—41no—
207C2×Dic53AGL(1,ℤ/8)11yescharacteristic
208D212—21no—
209D212—41no—
210C4×D53—41no—
211C4×D53—41no—
212C4×D53—41no—
213C4×D53—161no—
214C4×D53C22⋊Q811yescharacteristic
215C4×D53C22⋊Q811yescharacteristic
216C4×D53C2×SD1611yescharacteristic
217C4×D53C2×SD1611yescharacteristic
218Dic106—21no—
219Dic106—41no—
220Dic106—161no—
221C8×C53—41no—
222C8×C53C2×SD1611yescharacteristic
223C8×C53C2×SD1611yescharacteristic
224C8×C53C22×Q811yescharacteristic
225C8×C53C2×#4111yescharacteristic
226#88322—41no—
227#88322C22⋊Q811yescharacteristic
228#88322C22⋊Q811yescharacteristic
229#88322C2×SD1611yescharacteristic
230#88322C2×SD1611yescharacteristic
231#33638—531nomaximal, Sylow
232C22×C53—41no—
233C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
234C2×D53—41no—
235C2×D53—41no—
236C2×D53—161no—
237C2×D53#47611yescharacteristic
238C2×D53#47611yescharacteristic
239C4×C53—41no—
240C4×C53—41no—
241C4×C53—81no—
242C4×C53—81no—
243C4×C53C2×C22⋊Q811yescharacteristic
244C4×C53C22×SD1611yescharacteristic
245Dic53—41no—
246Dic53—41no—
247Dic53—81no—
248Dic53—81no—
249Dic53#47611yescharacteristic
250Dic53#47611yescharacteristic
251#3163—531no—
252#2957—531no—
253#2231—531no—
254#2231—531no—
255#2231—531no—
256#2231—531no—
257#2230—531no—
258#2230—531no—
259#2230—531no—
260#2230D5311yescharacteristic, complemented
261#2155—531no—
262C2×#365—531no—
263#2153—531no—
264#2145—531no—
265C2×#360—531no—
266C2×C53—21no—
267C2×C53—81no—
268C2×C53C2×#47611yescharacteristic
269D53—21no—
270D53—21no—
271D53—81no—
272C2×#135—531no—
273#568—1061no—
274#544—1061no—
275C2×M32—531no—
276C2×M32—531no—
277#501—1061no—
278#501—1061no—
279#499—531no—
280#499—531no—
281#498—531no—
282#498C2×D5311yescharacteristic, complemented
283#444—1061no—
284#444—1061no—
285#443—531no—
286#443—531no—
287#443—531no—
288#443C2×D5311yescharacteristic
289C2×#108—531no—
290#428—1061no—
291#427—531no—
292C2×#106—531no—
293#421—1061no—
294#405—531no—
295#365—531no—
296#365—531no—
297#365—531no—
298#365—531no—
299#365—531no—
300#365—531no—
301#365C2×D5311yescharacteristic, complemented
302#365C2×D5311yescharacteristic, complemented
303#364—531no—
304#364—531no—
305#364—531no—
306#364C2×D5311yescharacteristic, complemented
307#360—531no—
308#360—531no—
309#360—531no—
310#360—531no—
311#360—531no—
312#360—531no—
313#360C2×D5311yescharacteristic
314#360C2×D5311yescharacteristic
315C53#3363811yescharacteristic, complemented, Sylow
316C2×#41—1061no—
317#135—531no—
318#135—531no—
319#135—531no—
320#135—1061no—
321#135—1061no—
322#135C22×D5311yescharacteristic
323C2×Q16—531no—
324C2×SD16—1061no—
325C2×D8—531no—
326#131—21no—
327#131—1061no—
328#131—1061no—
329#131—1061no—
330#131—2121no—
331#131—2121no—
332C2×M16—531no—
333C2×M16—1061no—
334C2×M16—2121no—
335C2×M16—2121no—
336C22×C8—531no—
337C4⋊Q8—21no—
338C4⋊Q8—1061no—
339C4⋊Q8—2121no—
340#125—1061no—
341#125—1061no—
342#125—2121no—
343C22⋊Q8—2121no—
344C22⋊Q8—2121no—
345#117—531no—
346#117—1061no—
347C2×C4⋊C4—1061no—
348M32—531no—
349M32—531no—
350M32—1061no—
351M32—1061no—
352M32C22×D5311yescharacteristic
353M32C22×D5311yescharacteristic
354C2×C16—1061no—
355C2×C16—1061no—
356#108—531no—
357#108—1061no—
358#108C22×D5311yescharacteristic
359#107—1061no—
360#107—1061no—
361#107—1061no—
362#106—21no—
363#106—531no—
364#106—531no—
365#106—1061no—
366#106—1061no—
367#106C22×D5311yescharacteristic
368#106C22×D5311yescharacteristic
369#104—21no—
370#104—1061no—
371#104—1061no—
372#104—1061no—
373#97—531no—
374#97—531no—
375C4×C8—531no—
376C4×C8C22×D5311yescharacteristic
377#41—21no—
378#41—1061no—
379#41—1061no—
380#41—1061no—
381#41—2121no—
382#41—2121no—
383C2×Q8—41no—
384C2×Q8—1061no—
385C2×Q8—2121no—
386C2×D4—1061no—
387C2×D4—2121no—
388C22×C4—531no—
389C22×C4—2121no—
390C22×C4—2121no—
391Q16—531no—
392Q16—1061no—
393Q16Q8×D5311yescharacteristic
394SD16—21no—
395SD16—1061no—
396SD16—1061no—
397SD16—1061no—
398D8—531no—
399D8—1061no—
400D8Q8×D5311yescharacteristic
401M16—21no—
402M16—41no—
403M16—531no—
404M16—531no—
405M16—531no—
406M16—531no—
407M16—1061no—
408M16—2121no—
409M16—2121no—
410M16—2121no—
411M16—2121no—
412C2×C8—41no—
413C2×C8—531no—
414C2×C8—531no—
415C2×C8—531no—
416C2×C8—531no—
417C2×C8—531no—
418C2×C8—531no—
419C2×C8—2121no—
420C2×C8—2121no—
421C2×C8—2121no—
422C2×C8—2121no—
423C2×C8D4×D5311yescharacteristic
424C2×C8C23×D5311yescharacteristic
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—41no—
428C4⋊C4—1061no—
429C4⋊C4—1061no—
430C4⋊C4—1061no—
431C4⋊C4—2121no—
432C4⋊C4—2121no—
433C4⋊C4—2121no—
434C4⋊C4—2121no—
435C4⋊C4—2121no—
436C4⋊C4—2121no—
437C22⋊C4—1061no—
438C22⋊C4—2121no—
439C42—531no—
440C42—2121no—
441C42D4×D5311yescharacteristic
442C16—21no—
443C16—21no—
444C16—1061no—
445C16—1061no—
446C23—531no—
447C23—2121no—
448Q8—21no—
449Q8—161no—
450Q8—8481no—
451Q8—1061no—
452Q8—2121no—
453D4—21no—
454D4—41no—
455D4—1061no—
456D4—2121no—
457D4—2121no—
458D4—2121no—
459D4—2121no—
460C2×C4—21no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—531no—
465C2×C4—531no—
466C2×C4—531no—
467C2×C4—531no—
468C2×C4—531no—
469C2×C4—8481no—
470C2×C4—1061no—
471C2×C4—2121no—
472C2×C4—2121no—
473C2×C4—2121no—
474C2×C4—2121no—
475C2×C4—2121no—
476C2×C4—2121no—
477C2×C4C2×D4×D5311yescharacteristic
478C8—41no—
479C8—531no—
480C8—531no—
481C8—531no—
482C8—531no—
483C8—2121no—
484C8SD16×D5311yescharacteristic
485C8SD16×D5311yescharacteristic
486C8C2×Q8×D5311yescharacteristic
487C8#41×D5311yescharacteristic
488C22—41no—
489C22—531no—
490C22—531no—
491C22—8481no—
492C22—2121no—
493C22—2121no—
494C22AGL(1,ℤ/8)×D5311yescharacteristic
495C4—41no—
496C4—41no—
497C4—81no—
498C4—81no—
499C4—531no—
500C4—531no—
501C4—4241no—
502C4—4241no—
503C4—2121no—
504C4—2121no—
505C4C22⋊Q8×D5311yescharacteristic
506C4C2×SD16×D5311yescharacteristic
507C2—21no—
508C2—81no—
509C2—1061no—
510C2—1061no—
511C2—4241no—
512C2#476×D5311yescharacteristic
subgroupG/H
centerC2#476×D53
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2230C2
socleC2×C53C2×#476
prime radical#135×D53C22
hypercenter#2230D53
hypocenterC53#33638
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2230

Complements

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

NH ≅ G/Ntype
1. #20285240C2semidirect
3. #20276892C2semidirect
4. #20276891C2semidirect
5. #20276890C2semidirect
6. #20276889C2semidirect
7. #20276885C2semidirect
8. #20276884C2semidirect
9. #20276883C2semidirect
10. #20276486C2semidirect
11. #365×D53C2semidirect
12. #20276479C2semidirect
15. C53×#2230C2semidirect
23. #16811025C22semidirect
24. #16811023C22semidirect
25. #16811020C22semidirect
37. #16810447C22semidirect
38. #16810447C22semidirect
39. #16810446C22semidirect
40. #16810446C22semidirect
41. #16810445C22semidirect
42. #16810445C22semidirect
43. #16810444C22semidirect
44. #16810443C22semidirect
45. #16810443C22semidirect
52. C53×#498C22semidirect
54. C53×#365C22semidirect
55. C53×#365C22semidirect
56. C53×#364C22semidirect
60. #13133240Q8semidirect
64. #13133236Q8semidirect
65. Q16×D53Q8semidirect
67. D8×D53Q8semidirect
260. #2230D53semidirect
282. #498C2×D53semidirect
301. #365C2×D53semidirect
302. #365C2×D53semidirect
306. #364C2×D53semidirect
315. C53#33638semidirect

Maximal subgroups

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

subgroupindex
#202852402
#202836272
#202768922
#202768912
#202768902
#202768892
#202768852
#202768842
#202768832
#202764862
#365×D532
#202764792
#202764492
#360×D532
C53×#22302
#3363853