Finite Group Databasebeta

← #27540913

Subgroups — #27540913

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#20287060C211yesmaximal, characteristic, complemented
2#20286389C211yesmaximal, characteristic, complemented
3#20276937C211yesmaximal, characteristic, complemented
4#20276897C211yesmaximal, characteristic, complemented
5#20276851C211yesmaximal, characteristic, complemented
6#20276802C211yesmaximal, characteristic, complemented
7#20276799C211yesmaximal, characteristic, complemented
8#20276749C211yesmaximal, characteristic, complemented
9#20276629C211yesmaximal, characteristic, complemented
10#20276628C211yesmaximal, characteristic, complemented
11#20276561C211yesmaximal, characteristic, complemented
12#20276532C211yesmaximal, characteristic, complemented
13C2×#16810423C211yesmaximal, characteristic, complemented
14#357×D53C211yesmaximal, characteristic, complemented
15C53×#2232C211yesmaximal, characteristic, complemented
16C22×D424C2211yescharacteristic
17C2×D8×D53C2211yescharacteristic
18#16811274—21no—
19#16811257—21no—
20C2×D848C2211yescharacteristic
21C2×D848C2211yescharacteristic, complemented
22C2×#13133158C2211yescharacteristic, complemented
23C22×#11719908C2211yescharacteristic, complemented
24C2×C16×D53C2211yescharacteristic, complemented
25#16811027C2211yescharacteristic, complemented
26#16810998C2211yescharacteristic
27#16810938—21no—
28#16810936—21no—
29#16810902—21no—
30#16810860C2211yescharacteristic
31#16810860C2211yescharacteristic, complemented
32#16810859C2211yescharacteristic, complemented
33#16810817—21no—
34#16810758—21no—
35#16810756—21no—
36#16810705C2211yescharacteristic
37#16810705C2211yescharacteristic, complemented
38C8×C22⋊D53C2211yescharacteristic
39#107×D53C2211yescharacteristic, complemented
40#16810448C2211yescharacteristic, complemented
41#16810441C2211yescharacteristic, complemented
42#16810425C2211yescharacteristic
43#16810425C2211yescharacteristic, complemented
44#16810424C2211yescharacteristic, complemented
45#16810423C2211yescharacteristic
46#16810423C2211yescharacteristic
47#16810423C2211yescharacteristic, complemented
48#16810423C2211yescharacteristic, complemented
49#16810423C2211yescharacteristic, complemented
50#16810402C2211yescharacteristic, complemented
51C16×Dic53C2211yescharacteristic, complemented
52C53×#466C2211yescharacteristic
53C2×#107×C53C2211yescharacteristic, complemented
54C53×#366C2211yescharacteristic, complemented
55C53×#366C2211yescharacteristic, complemented
56C53×#357C2211yescharacteristic, complemented
57C53×#357C2211yescharacteristic, complemented
58C53×C22⋊C16C2211yescharacteristic, complemented
59C2×#11719945—41no—
60C22×D212—21no—
61C2×D4×D53—21no—
62C2×D4×D53—41no—
63C2×D424—21no—
64C2×D424—22no—
65C2×D424—41no—
66C2×D424C2311yescharacteristic
67C2×D424C2311yescharacteristic
68C2×D424C2311yescharacteristic
69C2×#11719922—21no—
70C2×#11719922—21no—
71D8×D53—22no—
72D8×D53—41no—
73C2×C8×D53C2311yescharacteristic
74#13133216—21no—
75#13133213—21no—
76#13133202—41no—
77#13133200—41no—
78#13133195—41no—
79#13133192—21no—
80#13133192—21no—
81#13133191—21no—
82#13133190—41no—
83#13133187—41no—
84#13133185—41no—
85#13133179—21no—
86#13133179—21no—
87C4×C22⋊D53D411yescharacteristic
88C4⋊C4×D53—21no—
89D848—22no—
90D848—41no—
91#13133158—41no—
92C2×#11719908—41no—
93C2×#11719908C2311yescharacteristic
94C2×#11719908C2311yescharacteristic
95C16×D53—41no—
96#13133151C2311yescharacteristic
97#13133150—21no—
98#13133150—21no—
99#13133146D411yescharacteristic
100#13133139—21no—
101#13133137—21no—
102#13133137—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⋊D53—41no—
129C2×C22⋊D53D811yescharacteristic
130D4×D53—42no—
131D4×D53—81no—
132D4×D53—84no—
133#11719945—84no—
134C2×D212—21no—
135C2×D212—21no—
136C2×D212—21no—
137C2×D212—42no—
138C2×D212—41no—
139C2×D212—41no—
140C2×D212—41no—
141C2×D212—81no—
142C2×C4×D53—41no—
143C2×C4×D53—41no—
144C2×C4×D53—41no—
145C2×C4×D53C2×D411yescharacteristic
146C2×Dic106—41no—
147D8×C53—21no—
148D8×C53—21no—
149C2×C8×C53—21no—
150C2×C8×C53—21no—
151C2×C8×C53C2×D411yescharacteristic
152C2×C8×C53C2411yescharacteristic
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—41no—
157C22⋊C4×C53—41no—
158C22⋊Dic53D811yescharacteristic
159#11719922—42no—
160#11719922—81no—
161#11719921—41no—
162#11719921—41no—
163#11719921—41no—
164#11719921—41no—
165#11719921—41no—
166#11719921D811yescharacteristic
167C4⋊Dic53—21no—
168#11719919—41no—
169#11719919—41no—
170#11719919D811yescharacteristic
171C4×Dic53C2×D411yescharacteristic
172C2×#88322C2×D411yescharacteristic
173C2×#88322C2×D411yescharacteristic
174D424—21no—
175D424—21no—
176D424—21no—
177D424—21no—
178D424—21no—
179D424—42no—
180D424#4112yescomplemented
181C8×D53—21no—
182C8×D53—21no—
183C16×C53—21no—
184C16×C53—21no—
185#11719908—21no—
186#11719908—21no—
187C23×C53—41no—
188C23×C53C2×D811yescharacteristic
189C22×D53—41no—
190C22×D53—41no—
191C22×D53—41no—
192C22×D53—82no—
193C22×D53—82no—
194C22×D53—82no—
195C22×D53—82no—
196C22×D53—161no—
197C22×D53—161no—
198C22×D53C2×D811yescharacteristic
199D4×C53—41no—
200D4×C53—41no—
201D4×C53—82no—
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—41no—
205C2×C4×C53—41no—
206C2×C4×C53—41no—
207C2×C4×C53—82no—
208C2×C4×C53C2×D811yescharacteristic
209C2×C4×C53C22×D411yescharacteristic
210C22⋊D53—82no—
211C22⋊D53—82no—
212C22⋊D53—82no—
213C22⋊D53—161no—
214C22⋊D53D1614yescomplemented
215C2×Dic53—41no—
216C2×Dic53C2×D811yescharacteristic
217C2×Dic53C2×D811yescharacteristic
218C2×Dic53C2×D811yescharacteristic
219D212—22no—
220D212—41no—
221D212—41no—
222D212—41no—
223D212—41no—
224D212—41no—
225D212—82no—
226D212—82no—
227D212—82no—
228C4×D53—21no—
229C4×D53—21no—
230C4×D53—82no—
231C4×D53—82no—
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—81no—
243C22×C53—81no—
244C22×C53—81no—
245C22×C53C2×D1612yes—
246C22×C53C22×D811yescharacteristic
247C2×D53—21no—
248C2×D53—21no—
249C2×D53—42no—
250C2×D53—81no—
251C2×D53—81no—
252C2×D53—81no—
253C2×D53—81no—
254C2×D53—81no—
255C2×D53—81no—
256C2×D53—81no—
257C2×D53—81no—
258C2×D53—164no—
259C2×D53—164no—
260C2×D53C2×D1612yes—
261C4×C53—21no—
262C4×C53—81no—
263C4×C53—81no—
264C4×C53—81no—
265C4×C53#54611yescharacteristic
266C4×C53#57711yescharacteristic
267Dic53—21no—
268Dic53—21no—
269Dic53—81no—
270Dic53C2×D1612yes—
271#3334—531no—
272D4×D8—531no—
273#2240—531no—
274#2232—531no—
275#2232D5311yescharacteristic, complemented
276#2224—531no—
277#2215—531no—
278#2215—531no—
279#2206—531no—
280#2184—531no—
281#2184—531no—
282C4×D16—531no—
283D4×C16—531no—
284C2×#357—531no—
285C2×#357—531no—
286C2×C53—22no—
287C2×C53—161no—
288C2×C53#329411yescharacteristic
289C2×C53C22×D1611yescharacteristic
290C2×C53#341611yescharacteristic
291D53—22no—
292D53—82no—
293D53—161no—
294D53—161no—
295C22×D8—531no—
296C22×D8—531no—
297#547—1061no—
298D42—1061no—
299C2×D16—531no—
300C2×D16—531no—
301C2×D16—531no—
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
313#459—1061no—
314#447—1061no—
315#447—1061no—
316C4×D8—531no—
317C4×D8—531no—
318C8×D4—531no—
319C2×#107—531no—
320C2×#107C2×D5311yescharacteristic, complemented
321#366—531no—
322#366C2×D5311yescharacteristic, complemented
323#366C2×D5311yescharacteristic, complemented
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, complemented
334#357C2×D5311yescharacteristic, complemented
335C22⋊C16—531no—
336C22⋊C16C2×D5311yescharacteristic, complemented
337C4×C16—531no—
338C53#3386111yescharacteristic, complemented, Sylow
339C2×#41—2121no—
340C22×D4—1061no—
341C22×D4—1061no—
342C22×D4—2121no—
343C2×D8—531no—
344C2×D8—531no—
345C2×D8—531no—
346C2×D8—1062no—
347C2×D8—1061no—
348C2×D8—1061no—
349C2×D8—1062no—
350C2×D8—1061no—
351C2×D8—2121no—
352C2×D8—2121no—
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—21no—
372C2×C4⋊C4—1061no—
373D16—1062no—
374D16—2121no—
375D16—2121no—
376C2×C16—531no—
377C2×C16—531no—
378C2×C16—2121no—
379C2×C16—2121no—
380C2×C16C22×D5311yescharacteristic
381C2×C16C22×D5311yescharacteristic
382#107—21no—
383#107—531no—
384#107—1061no—
385#107—1061no—
386#107C22×D5311yescharacteristic
387#107C22×D5311yescharacteristic
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—2121no—
400C24—2121no—
401#41—4244no—
402C2×Q8—2121no—
403C2×D4—21no—
404C2×D4—41no—
405C2×D4—531no—
406C2×D4—1061no—
407C2×D4—1061no—
408C2×D4—1061no—
409C2×D4—4241no—
410C2×D4—4241no—
411C2×D4—4244no—
412C2×D4—2122no—
413C2×D4—2121no—
414C2×D4—2121no—
415C2×D4—2121no—
416C2×D4—2121no—
417C2×D4—2121no—
418C2×D4—2122no—
419C2×D4—2121no—
420C2×D4—2121no—
421C22×C4—41no—
422C22×C4—531no—
423C22×C4—2121no—
424C22×C4—2121no—
425C22×C4—2121no—
426C22×C4D4×D5311yescharacteristic
427D8—21no—
428D8—21no—
429D8—532no—
430D8—1061no—
431D8—1061no—
432D8—1061no—
433D8—1061no—
434D8—1061no—
435D8—4241no—
436D8—2122no—
437D8—2122no—
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—21no—
448C4⋊C4—21no—
449C4⋊C4—41no—
450C4⋊C4—531no—
451C4⋊C4—1061no—
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—531no—
469C23—8481no—
470C23—8481no—
471C23—4242no—
472C23—4242no—
473C23—4242no—
474C23—4242no—
475C23—2121no—
476C23—2121no—
477C23—2121no—
478C23D8×D5311yescharacteristic
479Q8—4242no—
480D4—41no—
481D4—41no—
482D4—82no—
483D4—534no—
484D4—8481no—
485D4—1062no—
486D4—4242no—
487D4—4242no—
488D4—4242no—
489D4—4242no—
490D4—4242no—
491D4—4242no—
492D4—2121no—
493D4—2121no—
494D4—2121no—
495D4—2121no—
496D4—2121no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—82no—
503C2×C4—531no—
504C2×C4—531no—
505C2×C4—531no—
506C2×C4—1061no—
507C2×C4—1061no—
508C2×C4—4242no—
509C2×C4—4242no—
510C2×C4—4242no—
511C2×C4—2121no—
512C2×C4D8×D5311yescharacteristic
513C2×C4C2×D4×D5311yescharacteristic
514C8—21no—
515C8—1061no—
516C8—1061no—
517C8#41×D5311yescharacteristic
518C8#1313326311yescharacteristic
519C22—21no—
520C22—21no—
521C22—81no—
522C22—81no—
523C22—81no—
524C22—532no—
525C22—8484no—
526C22—8484no—
527C22—1061no—
528C22—1061no—
529C22—4241no—
530C22—4241no—
531C22—4241no—
532C22—4241no—
533C22—4241no—
534C22—4241no—
535C22—4241no—
536C22—4241no—
537C22—2122no—
538C22D16×D5312yes—
539C22C2×D8×D5311yescharacteristic
540C4—21no—
541C4—81no—
542C4—81no—
543C4—81no—
544C4—532no—
545C4—1061no—
546C4—1061no—
547C4—4241no—
548C4#1681126511yescharacteristic
549C4#1681143211yescharacteristic
550C2—22no—
551C2—161no—
552C2—8481no—
553C2—8481no—
554C2—1062no—
555C2—4242no—
556C2#2028652611yescharacteristic
557C2C2×D16×D5311yescharacteristic
558C2#2028786011yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2232C2
socleC22×C53C22×D8
hypercenter#2232D53
hypocenterC53#33861
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2232

Complements

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

NH ≅ G/Ntype
1. #20287060C2semidirect
2. #20286389C2semidirect
3. #20276937C2semidirect
4. #20276897C2semidirect
5. #20276851C2semidirect
6. #20276802C2semidirect
7. #20276799C2semidirect
8. #20276749C2semidirect
9. #20276629C2semidirect
10. #20276628C2semidirect
11. #20276561C2semidirect
12. #20276532C2semidirect
13. C2×#16810423C2semidirect
14. #357×D53C2semidirect
15. C53×#2232C2semidirect
21. C2×D848C22semidirect
22. C2×#13133158C22semidirect
23. C22×#11719908C22semidirect
24. C2×C16×D53C22semidirect
25. #16811027C22semidirect
31. #16810860C22semidirect
32. #16810859C22semidirect
37. #16810705C22semidirect
39. #107×D53C22semidirect
40. #16810448C22semidirect
41. #16810441C22semidirect
43. #16810425C22semidirect
44. #16810424C22semidirect
47. #16810423C22semidirect
48. #16810423C22semidirect
49. #16810423C22semidirect
50. #16810402C22semidirect
51. C16×Dic53C22semidirect
53. C2×#107×C53C22semidirect
54. C53×#366C22semidirect
55. C53×#366C22semidirect
56. C53×#357C22semidirect
57. C53×#357C22semidirect
58. C53×C22⋊C16C22semidirect
180. D424#41semidirect
214. C22⋊D53D16semidirect
275. #2232D53semidirect
320. C2×#107C2×D53semidirect
322. #366C2×D53semidirect
323. #366C2×D53semidirect
333. #357C2×D53semidirect
334. #357C2×D53semidirect
336. C22⋊C16C2×D53semidirect
338. C53#33861semidirect

Maximal subgroups

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

subgroupindex
#202870602
#202863892
#202769372
#202768972
#202768512
#202768022
#202767992
#202767492
#202766292
#202766282
#202765612
#202765322
C2×#168104232
#357×D532
C53×#22322
#3386153