Finite Group Databasebeta

← #27540908

Subgroups — #27540908

Order 13568.

654 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20287059C211yesmaximal, characteristic, complemented
2D8×C22⋊D53C211yesmaximal, characteristic, complemented
3#20276937C211yesmaximal, characteristic, complemented
4#20276896C211yesmaximal, characteristic, complemented
5#20276894C211yesmaximal, characteristic, complemented
6#20276850C211yesmaximal, characteristic, complemented
7#20276799C211yesmaximal, characteristic, complemented
8#20276798C211yesmaximal, characteristic, complemented
9#20276748C211yesmaximal, characteristic, complemented
10#20276625C211yesmaximal, characteristic, complemented
11D16×Dic53C211yesmaximal, characteristic, complemented
12#20276532C211yesmaximal, characteristic, complemented
13C2×#16810424C211yesmaximal, characteristic, complemented
14#357×D53C211yesmaximal, characteristic, complemented
15C53×#2184C211yesmaximal, characteristic, complemented
16C2×D8×D53C2211yescharacteristic
17#16811273—21no—
18D4×C22⋊D53—21no—
19C2×#13133158C2211yescharacteristic, complemented
20C22×#11719908C2211yescharacteristic, complemented
21C2×C16×D53C2211yescharacteristic, complemented
22#16811027C2211yescharacteristic, complemented
23#16810998C2211yescharacteristic
24#16810935—21no—
25#16810934—21no—
26#16810901—21no—
27#16810859C2211yescharacteristic
28#16810856C2211yescharacteristic
29#16810856C2211yescharacteristic, complemented
30#16810816—21no—
31#16810757—21no—
32#16810755—21no—
33D8×Dic53C2211yescharacteristic
34D8×Dic53C2211yescharacteristic, complemented
35C8×C22⋊D53C2211yescharacteristic
36C2×#13133151C2211yescharacteristic, complemented
37#107×D53C2211yescharacteristic, complemented
38#16810449C2211yescharacteristic, complemented
39#16810449C2211yescharacteristic, complemented
40#16810448C2211yescharacteristic, complemented
41#16810441C2211yescharacteristic, complemented
42#16810425C2211yescharacteristic, complemented
43#16810425C2211yescharacteristic, complemented
44#16810424C2211yescharacteristic
45#16810424C2211yescharacteristic
46#16810424C2211yescharacteristic, complemented
47#16810424C2211yescharacteristic, complemented
48#16810424C2211yescharacteristic, complemented
49#16810423C2211yescharacteristic, complemented
50#16810402C2211yescharacteristic, complemented
51C16×Dic53C2211yescharacteristic, complemented
52C22×D8×C53C2211yescharacteristic
53C2×D16×C53C2211yescharacteristic
54C2×D16×C53C2211yescharacteristic, complemented
55C53×#466C2211yescharacteristic, complemented
56C53×#357C2211yescharacteristic
57C53×#357C2211yescharacteristic, complemented
58C53×C22⋊C16C2211yescharacteristic, complemented
59C2×#11719947—41no—
60C22×C22⋊D53—41no—
61C2×D4×D53—21no—
62C2×D424C2311yescharacteristic
63C2×#11719922—21no—
64C2×#11719922—21no—
65D8×D53—22no—
66D8×D53—41no—
67C2×C8×D53C2311yescharacteristic
68#13133216—21no—
69#13133213—21no—
70#13133199—41no—
71#13133198—41no—
72#13133194—41no—
73#13133191—21no—
74#13133189—41no—
75#13133188—21no—
76#13133188—21no—
77#13133186—41no—
78#13133185—41no—
79D4×Dic53—21no—
80D4×Dic53—21no—
81C4×C22⋊D53D411yescharacteristic
82C2×C4⋊Dic53—21no—
83C4⋊C4×D53—21no—
84#13133158—41no—
85C2×#11719908—41no—
86C2×#11719908C2311yescharacteristic
87C2×#11719908C2311yescharacteristic
88C16×D53—41no—
89#13133151—21no—
90#13133151C2311yescharacteristic
91#13133151C2311yescharacteristic
92#13133151C2311yescharacteristic
93#13133150—21no—
94#13133150—21no—
95#13133146D411yescharacteristic
96#13133139—21no—
97#13133138—21no—
98#13133138—21no—
99#13133138—21no—
100#13133138—21no—
101C22⋊#88322D411yescharacteristic
102#13133129D411yescharacteristic
103C8×Dic53C2311yescharacteristic
104C22×D4×C53—21no—
105C2×D8×C53—21no—
106C2×D8×C53—22no—
107C2×D8×C53—41no—
108C2×D8×C53C2311yescharacteristic
109C2×D8×C53C2311yescharacteristic
110C2×D8×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112#121×C53—21no—
113D16×C53—22no—
114D16×C53—41no—
115C2×C16×C53C2311yescharacteristic
116C2×C16×C53C2311yescharacteristic
117#107×C53C2311yescharacteristic
118#102×C53—21no—
119C24×C53—41no—
120C23×D53—41no—
121C2×D4×C53—21no—
122C2×D4×C53—21no—
123C2×D4×C53—21no—
124C2×D4×C53—42no—
125C2×D4×C53—41no—
126C2×D4×C53—41no—
127C2×D4×C53—81no—
128C22×C4×C53C2×D411yescharacteristic
129C2×C22⋊D53—41no—
130C2×C22⋊D53—41no—
131C2×C22⋊D53—41no—
132C2×C22⋊D53—41no—
133C2×C22⋊D53—41no—
134C2×C22⋊D53—41no—
135C2×C22⋊D53—84no—
136C2×C22⋊D53D811yescharacteristic
137C22×Dic53—41no—
138C22×Dic53—41no—
139C22×Dic53—41no—
140#11719947—84no—
141D4×D53—42no—
142D4×D53—81no—
143C2×D212—21no—
144C2×C4×D53—41no—
145C2×C4×D53C2×D411yescharacteristic
146C2×Dic106—41no—
147D8×C53—21no—
148D8×C53—21no—
149D8×C53—21no—
150D8×C53—21no—
151D8×C53—21no—
152D8×C53—42no—
153D8×C53#4112yescomplemented
154C2×C8×C53—21no—
155C2×C8×C53—21no—
156C2×C8×C53C2×D411yescharacteristic
157C2×C8×C53C2411yescharacteristic
158C4⋊C4×C53—21no—
159C22⋊C4×C53—41no—
160C22⋊Dic53—41no—
161C22⋊Dic53—41no—
162C22⋊Dic53—41no—
163C22⋊Dic53—41no—
164C22⋊Dic53D811yescharacteristic
165#11719922—42no—
166#11719922—81no—
167#11719921—41no—
168#11719921D811yescharacteristic
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—41no—
173#11719919—41no—
174#11719919—41no—
175#11719919D811yescharacteristic
176C4×Dic53C2×D411yescharacteristic
177C2×#88322C2×D411yescharacteristic
178C2×#88322C2×D411yescharacteristic
179D424—21no—
180D424—21no—
181C8×D53—21no—
182C8×D53—21no—
183C16×C53—21no—
184C16×C53—21no—
185#11719908—21no—
186#11719908—21no—
187C23×C53—41no—
188C23×C53—41no—
189C23×C53—41no—
190C23×C53—82no—
191C23×C53—82no—
192C23×C53—161no—
193C23×C53C2×D811yescharacteristic
194C22×D53—41no—
195C22×D53—82no—
196C22×D53—82no—
197C22×D53—161no—
198C22×D53C2×D811yescharacteristic
199D4×C53—22no—
200D4×C53—41no—
201D4×C53—41no—
202D4×C53—41no—
203D4×C53—41no—
204D4×C53—41no—
205D4×C53—82no—
206D4×C53—82no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—41no—
210C2×C4×C53C2×D811yescharacteristic
211C2×C4×C53C22×D411yescharacteristic
212C22⋊D53—82no—
213C22⋊D53—82no—
214C22⋊D53—82no—
215C22⋊D53—82no—
216C22⋊D53—82no—
217C22⋊D53—161no—
218C22⋊D53D1614yescomplemented
219C2×Dic53—41no—
220C2×Dic53—41no—
221C2×Dic53—41no—
222C2×Dic53—82no—
223C2×Dic53—82no—
224C2×Dic53—82no—
225C2×Dic53C2×D811yescharacteristic
226C2×Dic53C2×D811yescharacteristic
227C2×Dic53C2×D811yescharacteristic
228D212—41no—
229D212—41no—
230C4×D53—21no—
231C4×D53—21no—
232C4×D53—82no—
233Dic106—82no—
234C8×C53—21no—
235C8×C53C2×#4111yescharacteristic
236C8×C53H(5,𝔽2)11yescharacteristic
237#88322—21no—
238#88322—21no—
239#33861—531nomaximal, Sylow
240C22×C53—21no—
241C22×C53—21no—
242C22×C53—42no—
243C22×C53—81no—
244C22×C53—81no—
245C22×C53—81no—
246C22×C53—81no—
247C22×C53—81no—
248C22×C53—81no—
249C22×C53—81no—
250C22×C53—81no—
251C22×C53—164no—
252C22×C53C2×D1612yes—
253C22×C53C22×D811yescharacteristic
254C2×D53—21no—
255C2×D53—21no—
256C2×D53—81no—
257C2×D53—81no—
258C2×D53—81no—
259C2×D53—164no—
260C2×D53C2×D1612yes—
261C4×C53—21no—
262C4×C53—81no—
263C4×C53#54611yescharacteristic
264C4×C53#57711yescharacteristic
265Dic53—21no—
266Dic53—21no—
267Dic53—81no—
268Dic53—81no—
269Dic53—81no—
270Dic53C2×D1612yes—
271#3334—531no—
272D4×D8—531no—
273#2240—531no—
274#2232—531no—
275#2232—531no—
276#2224—531no—
277#2215—531no—
278#2215—531no—
279#2206—531no—
280#2184—531no—
281#2184D5311yescharacteristic, complemented
282C4×D16—531no—
283D4×C16—531no—
284C2×#357—531no—
285C2×#357—531no—
286C2×C53—22no—
287C2×C53—82no—
288C2×C53—161no—
289C2×C53—161no—
290C2×C53#329411yescharacteristic
291C2×C53C22×D1611yescharacteristic
292C2×C53#341611yescharacteristic
293D53—22no—
294D53—161no—
295C22×D8—531no—
296C22×D8C2×D5311yescharacteristic
297#547—1061no—
298D42—1061no—
299C2×D16—531no—
300C2×D16C2×D5311yescharacteristic
301C2×D16C2×D5311yescharacteristic, complemented
302C22×C16—531no—
303C22×C16—531no—
304#500—531no—
305#493—531no—
306#480—1061no—
307#480—1061no—
308#474—1061no—
309#466—531no—
310#466—531no—
311#466—531no—
312#466C2×D5311yescharacteristic, complemented
313#459—1061no—
314#447—1061no—
315#447—1061no—
316C4×D8—531no—
317C4×D8—531no—
318C8×D4—531no—
319C2×#107—531no—
320C2×#107—531no—
321#366—531no—
322#366—531no—
323#366—531no—
324C4⋊C16—531no—
325#357—531no—
326#357—531no—
327#357—531no—
328#357—531no—
329#357—531no—
330#357—531no—
331#357—531no—
332#357—531no—
333#357C2×D5311yescharacteristic
334#357C2×D5311yescharacteristic, complemented
335C22⋊C16—531no—
336C22⋊C16C2×D5311yescharacteristic, complemented
337C4×C16—531no—
338C53#3386111yescharacteristic, complemented, Sylow
339C2×#41—2121no—
340C22×D4—21no—
341C22×D4—1061no—
342C22×D4—2121no—
343C2×D8—22no—
344C2×D8—21no—
345C2×D8—41no—
346C2×D8—531no—
347C2×D8—1061no—
348C2×D8—1061no—
349C2×D8—1062no—
350C2×D8—2121no—
351C2×D8C22×D5311yescharacteristic
352C2×D8C22×D5311yescharacteristic
353C2×D8C22×D5311yescharacteristic
354C22×C8—531no—
355C22×C8C22×D5311yescharacteristic
356C4⋊Q8—1061no—
357#127—1061no—
358#123—2121no—
359#123—2121no—
360C22⋊Q8—2121no—
361#121—21no—
362#121—1061no—
363#121—1061no—
364#121—1061no—
365#121—2121no—
366#120—2121no—
367#120—2121no—
368C4×D4—531no—
369C4×D4—1061no—
370C4×D4—1061no—
371C2×C4⋊C4—1061no—
372C2×C4⋊C4—1061no—
373D16—22no—
374D16—41no—
375D16—2121no—
376C2×C16—531no—
377C2×C16—531no—
378C2×C16—2121no—
379C2×C16—2121no—
380C2×C16C22×D5311yescharacteristic
381C2×C16C22×D5311yescharacteristic
382#107—531no—
383#107—531no—
384#107—531no—
385#107—1061no—
386#107—1061no—
387#107—1061no—
388#107C22×D5311yescharacteristic
389C4⋊C8—531no—
390#102—21no—
391#102—1061no—
392#102—1061no—
393#102—1061no—
394#102—1061no—
395#102—1061no—
396C22⋊C8—531no—
397C22⋊C8—531no—
398C4×C8—531no—
399C24—41no—
400C24—2121no—
401#41—4244no—
402C2×Q8—2121no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—42no—
407C2×D4—41no—
408C2×D4—41no—
409C2×D4—81no—
410C2×D4—531no—
411C2×D4—1061no—
412C2×D4—4241no—
413C2×D4—4244no—
414C2×D4—2121no—
415C2×D4—2121no—
416C2×D4—2121no—
417C2×D4—2122no—
418C2×D4—2121no—
419C2×D4—2121no—
420C2×D4—2121no—
421C22×C4—531no—
422C22×C4—2121no—
423C22×C4—2121no—
424C22×C4—2121no—
425C22×C4—2121no—
426C22×C4D4×D5311yescharacteristic
427D8—21no—
428D8—21no—
429D8—21no—
430D8—21no—
431D8—21no—
432D8—42no—
433D8—1061no—
434D8—1061no—
435D8—4241no—
436D8—2122no—
437D8#1171994712yescomplemented
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—531no—
441C2×C8—531no—
442C2×C8—1061no—
443C2×C8—1061no—
444C2×C8D4×D5311yescharacteristic
445C2×C8C23×D5311yescharacteristic
446C4⋊C4—21no—
447C4⋊C4—531no—
448C4⋊C4—1061no—
449C4⋊C4—1061no—
450C4⋊C4—1061no—
451C4⋊C4—2121no—
452C4⋊C4—2121no—
453C4⋊C4—2121no—
454C22⋊C4—41no—
455C22⋊C4—531no—
456C22⋊C4—531no—
457C22⋊C4—2121no—
458C22⋊C4—2121no—
459C22⋊C4—2121no—
460C22⋊C4—2121no—
461C22⋊C4—2121no—
462C42—531no—
463C16—21no—
464C16—21no—
465C16—1061no—
466C16—1061no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—82no—
471C23—82no—
472C23—161no—
473C23—531no—
474C23—8481no—
475C23—4242no—
476C23—4242no—
477C23—2121no—
478C23D8×D5311yescharacteristic
479Q8—4242no—
480D4—22no—
481D4—41no—
482D4—41no—
483D4—41no—
484D4—41no—
485D4—41no—
486D4—82no—
487D4—82no—
488D4—534no—
489D4—8481no—
490D4—4242no—
491D4—4242no—
492D4—4242no—
493D4—4242no—
494D4—4242no—
495D4—2121no—
496D4—2121no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—41no—
500C2×C4—531no—
501C2×C4—531no—
502C2×C4—531no—
503C2×C4—1061no—
504C2×C4—1061no—
505C2×C4—4242no—
506C2×C4—4242no—
507C2×C4—4242no—
508C2×C4—4242no—
509C2×C4—2121no—
510C2×C4—2121no—
511C2×C4—2121no—
512C2×C4D8×D5311yescharacteristic
513C2×C4C2×D4×D5311yescharacteristic
514C8—21no—
515C8—1061no—
516C8—1061no—
517C8C2×#1171994711yescharacteristic
518C8#1313326211yescharacteristic
519C22—21no—
520C22—21no—
521C22—42no—
522C22—81no—
523C22—81no—
524C22—81no—
525C22—81no—
526C22—81no—
527C22—81no—
528C22—81no—
529C22—81no—
530C22—164no—
531C22—532no—
532C22—8484no—
533C22—1061no—
534C22—1061no—
535C22—4241no—
536C22—4241no—
537C22—4241no—
538C22D16×D5312yes—
539C22C2×D8×D5311yescharacteristic
540C4—21no—
541C4—81no—
542C4—532no—
543C4—1061no—
544C4—1061no—
545C4—4241no—
546C4—4241no—
547C4—4241no—
548C4#1681126011yescharacteristic
549C4#1681143311yescharacteristic
550C2—22no—
551C2—82no—
552C2—161no—
553C2—161no—
554C2—8481no—
555C2—1062no—
556C2#2028652111yescharacteristic
557C2C2×D16×D5311yescharacteristic
558C2#2028786111yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2184C2
socleC22×C53C22×D8
hypercenter#2184D53
hypocenterC53#33861
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2184

Complements

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

NH ≅ G/Ntype
1. #20287059C2semidirect
2. D8×C22⋊D53C2semidirect
3. #20276937C2semidirect
4. #20276896C2semidirect
5. #20276894C2semidirect
6. #20276850C2semidirect
7. #20276799C2semidirect
8. #20276798C2semidirect
9. #20276748C2semidirect
10. #20276625C2semidirect
11. D16×Dic53C2semidirect
12. #20276532C2semidirect
13. C2×#16810424C2semidirect
14. #357×D53C2semidirect
15. C53×#2184C2semidirect
19. C2×#13133158C22semidirect
20. C22×#11719908C22semidirect
21. C2×C16×D53C22semidirect
22. #16811027C22semidirect
29. #16810856C22semidirect
34. D8×Dic53C22semidirect
36. C2×#13133151C22semidirect
37. #107×D53C22semidirect
38. #16810449C22semidirect
39. #16810449C22semidirect
40. #16810448C22semidirect
41. #16810441C22semidirect
42. #16810425C22semidirect
43. #16810425C22semidirect
46. #16810424C22semidirect
47. #16810424C22semidirect
48. #16810424C22semidirect
49. #16810423C22semidirect
50. #16810402C22semidirect
51. C16×Dic53C22semidirect
54. C2×D16×C53C22semidirect
55. C53×#466C22semidirect
57. C53×#357C22semidirect
58. C53×C22⋊C16C22semidirect
153. D8×C53#41semidirect
218. C22⋊D53D16semidirect
281. #2184D53semidirect
301. C2×D16C2×D53semidirect
312. #466C2×D53semidirect
334. #357C2×D53semidirect
336. C22⋊C16C2×D53semidirect
338. C53#33861semidirect
437. D8#11719947semidirect

Maximal subgroups

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

subgroupindex
#202870592
D8×C22⋊D532
#202769372
#202768962
#202768942
#202768502
#202767992
#202767982
#202767482
#202766252
D16×Dic532
#202765322
C2×#168104242
#357×D532
C53×#21842
#3386153