Finite Group Databasebeta

← #27540919

Subgroups — #27540919

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#20287104C211yesmaximal, characteristic, complemented
2D8×C22⋊D53C211yesmaximal, characteristic
3#20276937C211yesmaximal, characteristic, complemented
4#20276901C211yesmaximal, characteristic, complemented
5#20276899C211yesmaximal, characteristic, complemented
6#20276860C211yesmaximal, characteristic, complemented
7#20276813C211yesmaximal, characteristic, complemented
8#20276808C211yesmaximal, characteristic, complemented
9#20276753C211yesmaximal, characteristic, complemented
10#20276642C211yesmaximal, characteristic, complemented
11SD32×Dic53C211yesmaximal, characteristic, complemented
12#20276532C211yesmaximal, characteristic, complemented
13C2×#16810424C211yesmaximal, characteristic, complemented
14#357×D53C211yesmaximal, characteristic, complemented
15C53×#2187C211yesmaximal, characteristic, complemented
16C2×D8×D53C2211yescharacteristic
17#16811331—21no—
18D4×C22⋊D53—21no—
19C2×#13133160C2211yescharacteristic, complemented
20C22×#11719908C2211yescharacteristic, complemented
21C2×C16×D53C2211yescharacteristic, complemented
22#16811027C2211yescharacteristic, complemented
23#16810998C2211yescharacteristic
24#16810957—21no—
25#16810956—21no—
26#16810916—21no—
27#16810861C2211yescharacteristic, complemented
28#16810859C2211yescharacteristic
29#16810856C2211yescharacteristic
30#16810816—21no—
31#16810757—21no—
32#16810755—21no—
33Q16×Dic53C2211yescharacteristic, complemented
34D8×Dic53C2211yescharacteristic
35C8×C22⋊D53C2211yescharacteristic
36C2×#13133151C2211yescharacteristic, complemented
37#107×D53C2211yescharacteristic, complemented
38#16810451C2211yescharacteristic, complemented
39#16810451C2211yescharacteristic, complemented
40#16810450C2211yescharacteristic, complemented
41#16810441C2211yescharacteristic, complemented
42#16810427C2211yescharacteristic, complemented
43#16810425C2211yescharacteristic, complemented
44#16810425C2211yescharacteristic, complemented
45#16810424C2211yescharacteristic
46#16810424C2211yescharacteristic
47#16810424C2211yescharacteristic, complemented
48#16810424C2211yescharacteristic, complemented
49#16810423C2211yescharacteristic, complemented
50#16810402C2211yescharacteristic, complemented
51C16×Dic53C2211yescharacteristic, complemented
52C22×D8×C53C2211yescharacteristic
53C2×SD32×C53C2211yescharacteristic
54C2×SD32×C53C2211yescharacteristic, complemented
55C53×#467C2211yescharacteristic, complemented
56C53×#357C2211yescharacteristic
57C53×#357C2211yescharacteristic, complemented
58C53×C22⋊C16C2211yescharacteristic, complemented
59C22×C22⋊D53—41no—
60C2×D4×D53—21no—
61C2×D424C2311yescharacteristic
62C2×#11719922—21no—
63C2×#11719922—21no—
64D8×D53—22no—
65D8×D53—41no—
66C2×C8×D53C2311yescharacteristic
67#13133216—21no—
68#13133213—21no—
69#13133207—41no—
70#13133197—41no—
71#13133194—41no—
72#13133193—21no—
73#13133191—21no—
74#13133189—41no—
75#13133188—21no—
76#13133186—41no—
77#13133185—41no—
78Q8×Dic53—21no—
79D4×Dic53—21no—
80C4×C22⋊D53D411yescharacteristic
81#13133177—41no—
82C2×C4⋊Dic53—21no—
83C4⋊C4×D53—21no—
84#13133160—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#13133141—21no—
97#13133141—21no—
98#13133139—21no—
99#13133138—21no—
100#13133138—21no—
101C22⋊#88322D411yescharacteristic
102#13133129D411yescharacteristic
103C8×Dic53C2311yescharacteristic
104C22×D4×C53—21no—
105C2×Q16×C53C2311yescharacteristic
106C2×D8×C53—21no—
107C2×D8×C53—22no—
108C2×D8×C53—41no—
109C2×D8×C53C2311yescharacteristic
110C2×D8×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112C22⋊Q8×C53—21no—
113SD32×C53—22no—
114SD32×C53—41no—
115C2×C16×C53C2311yescharacteristic
116C2×C16×C53C2311yescharacteristic
117#107×C53C2311yescharacteristic
118#103×C53—21no—
119C24×C53—41no—
120C23×D53—41no—
121C2×Q8×C53—21no—
122C2×D4×C53—21no—
123C2×D4×C53—21no—
124C2×D4×C53—42no—
125C2×D4×C53—41no—
126C2×D4×C53—81no—
127C22×C4×C53C2×D411yescharacteristic
128C2×C22⋊D53—41no—
129C2×C22⋊D53—41no—
130C2×C22⋊D53—41no—
131C2×C22⋊D53—41no—
132C2×C22⋊D53—41no—
133C2×C22⋊D53—84no—
134C2×C22⋊D53D811yescharacteristic
135C22×Dic53—41no—
136C22×Dic53—41no—
137D4×D53—42no—
138D4×D53—81no—
139C2×D212—21no—
140C2×C4×D53—41no—
141C2×C4×D53C2×D411yescharacteristic
142C2×Dic106—41no—
143Q16×C53—21no—
144Q16×C53—21no—
145D8×C53—21no—
146D8×C53—21no—
147D8×C53—21no—
148D8×C53—42no—
149D8×C53#4112yes—
150C2×C8×C53—21no—
151C2×C8×C53—21no—
152C2×C8×C53C2×D411yescharacteristic
153C2×C8×C53C2411yescharacteristic
154C4⋊C4×C53—21no—
155C4⋊C4×C53—41no—
156C22⋊C4×C53—41no—
157C22⋊Dic53—41no—
158C22⋊Dic53—41no—
159C22⋊Dic53D811yescharacteristic
160#11719922—42no—
161#11719922—81no—
162#11719921—41no—
163#11719921—41no—
164#11719921D811yescharacteristic
165C4⋊Dic53—21no—
166C4⋊Dic53—21no—
167C4⋊Dic53—21no—
168C4⋊Dic53—41no—
169C4⋊Dic53—41no—
170#11719919—41no—
171#11719919—41no—
172#11719919—41no—
173#11719919D811yescharacteristic
174C4×Dic53—41no—
175C4×Dic53C2×D411yescharacteristic
176C2×#88322C2×D411yescharacteristic
177C2×#88322C2×D411yescharacteristic
178D424—21no—
179D424—21no—
180C8×D53—21no—
181C8×D53—21no—
182C16×C53—21no—
183C16×C53—21no—
184#11719908—21no—
185#11719908—21no—
186C23×C53—41no—
187C23×C53—41no—
188C23×C53—82no—
189C23×C53—82no—
190C23×C53—161no—
191C23×C53C2×D811yescharacteristic
192C22×D53—41no—
193C22×D53—82no—
194C22×D53—82no—
195C22×D53—161no—
196C22×D53C2×D811yescharacteristic
197Q8×C53—41no—
198Q8×C53—41no—
199D4×C53—22no—
200D4×C53—41no—
201D4×C53—41no—
202D4×C53—41no—
203D4×C53—82no—
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—41no—
207C2×C4×C53—41no—
208C2×C4×C53C2×D811yescharacteristic
209C2×C4×C53C22×D411yescharacteristic
210C22⋊D53—82no—
211C22⋊D53—82no—
212C22⋊D53—82no—
213C22⋊D53—82no—
214C22⋊D53—161no—
215C22⋊D53SD3214yescomplemented
216C2×Dic53—41no—
217C2×Dic53—41no—
218C2×Dic53—41no—
219C2×Dic53—82no—
220C2×Dic53—161no—
221C2×Dic53C2×D811yescharacteristic
222C2×Dic53C2×D811yescharacteristic
223C2×Dic53C2×D811yescharacteristic
224D212—41no—
225D212—41no—
226C4×D53—21no—
227C4×D53—21no—
228C4×D53—161no—
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—42no—
239C22×C53—81no—
240C22×C53—81no—
241C22×C53—81no—
242C22×C53—81no—
243C22×C53—81no—
244C22×C53—164no—
245C22×C53C2×SD3212yes—
246C22×C53C22×D811yescharacteristic
247C2×D53—21no—
248C2×D53—21no—
249C2×D53—81no—
250C2×D53—81no—
251C2×D53—81no—
252C2×D53—164no—
253C2×D53C2×SD3212yes—
254C4×C53—21no—
255C4×C53—81no—
256C4×C53—81no—
257C4×C53#54611yescharacteristic
258C4×C53#57711yescharacteristic
259Dic53—21no—
260Dic53—21no—
261Dic53—81no—
262Dic53—81no—
263Dic53—81no—
264Dic53C2×SD3212yes—
265#3338—531no—
266D4×D8—531no—
267#2240—531no—
268#2233—531no—
269#2233—531no—
270#2226—531no—
271#2217—531no—
272#2217—531no—
273#2207—531no—
274#2187—531no—
275#2187D5311yescharacteristic, complemented
276C4×SD32—531no—
277D4×C16—531no—
278C2×#357—531no—
279C2×#357—531no—
280C2×C53—22no—
281C2×C53—82no—
282C2×C53—161no—
283C2×C53#329411yescharacteristic
284C2×C53C22×SD3211yescharacteristic
285C2×C53#341511yescharacteristic
286D53—22no—
287D53—161no—
288C22×D8—531no—
289C22×D8C2×D5311yescharacteristic
290#554—1061no—
291D42—1061no—
292C2×SD32—531no—
293C2×SD32C2×D5311yescharacteristic
294C2×SD32C2×D5311yescharacteristic, complemented
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#467C2×D5311yescharacteristic, complemented
304#466—531no—
305#466—531no—
306#459—1061no—
307#447—1061no—
308#447—1061no—
309C4×Q16—531no—
310C4×D8—531no—
311C8×D4—531no—
312C2×#107—531no—
313C2×#107—531no—
314#367—531no—
315#367—531no—
316#367—531no—
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
327#357C2×D5311yescharacteristic, complemented
328C22⋊C16—531no—
329C22⋊C16C2×D5311yescharacteristic, complemented
330C4×C16—531no—
331C53#3386211yescharacteristic, complemented, Sylow
332C22×D4—21no—
333C22×D4—1061no—
334C22×D4—2121no—
335C2×Q16C22×D5311yescharacteristic
336C2×D8—22no—
337C2×D8—21no—
338C2×D8—41no—
339C2×D8—531no—
340C2×D8—1061no—
341C2×D8—1061no—
342C2×D8—1062no—
343C2×D8—2121no—
344C2×D8C22×D5311yescharacteristic
345C2×D8C22×D5311yescharacteristic
346C22×C8—531no—
347C22×C8C22×D5311yescharacteristic
348C4⋊Q8—1061no—
349#127—1061no—
350#125—2121no—
351C22⋊Q8—21no—
352C22⋊Q8—1061no—
353C22⋊Q8—2121no—
354C22⋊Q8—2121no—
355#121—1061no—
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—1061no—
365C2×C4⋊C4—1061no—
366SD32—22no—
367SD32—41no—
368SD32—2121no—
369C2×C16—531no—
370C2×C16—531no—
371C2×C16—2121no—
372C2×C16—2121no—
373C2×C16C22×D5311yescharacteristic
374C2×C16C22×D5311yescharacteristic
375#107—531no—
376#107—531no—
377#107—531no—
378#107—1061no—
379#107—1061no—
380#107—1061no—
381#107C22×D5311yescharacteristic
382C4⋊C8—531no—
383#103—21no—
384#103—1061no—
385#103—1061no—
386#102—1061no—
387#102—1061no—
388#102—1061no—
389C22⋊C8—531no—
390C22⋊C8—531no—
391C4×C8—531no—
392C24—41no—
393C24—2121no—
394C2×Q8—21no—
395C2×Q8—2121no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—42no—
399C2×D4—41no—
400C2×D4—81no—
401C2×D4—531no—
402C2×D4—1061no—
403C2×D4—4241no—
404C2×D4—4244no—
405C2×D4—2121no—
406C2×D4—2121no—
407C2×D4—2121no—
408C2×D4—2122no—
409C2×D4—2121no—
410C2×D4—2121no—
411C22×C4—531no—
412C22×C4—2121no—
413C22×C4—2121no—
414C22×C4—2121no—
415C22×C4D4×D5311yescharacteristic
416Q16—21no—
417Q16—21no—
418D8—21no—
419D8—21no—
420D8—21no—
421D8—42no—
422D8—1061no—
423D8—1061no—
424D8—4241no—
425D8—2122no—
426D8#1171994712yes—
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—41no—
437C4⋊C4—531no—
438C4⋊C4—1061no—
439C4⋊C4—1061no—
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—41no—
461C23—82no—
462C23—82no—
463C23—161no—
464C23—531no—
465C23—8481no—
466C23—4242no—
467C23—4242no—
468C23—2121no—
469C23D8×D5311yescharacteristic
470Q8—41no—
471Q8—41no—
472Q8—8481no—
473D4—22no—
474D4—41no—
475D4—41no—
476D4—41no—
477D4—82no—
478D4—534no—
479D4—8481no—
480D4—4242no—
481D4—4242no—
482D4—4242no—
483D4—4242no—
484D4—2121no—
485D4—2121no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—531no—
491C2×C4—531no—
492C2×C4—531no—
493C2×C4—8481no—
494C2×C4—8481no—
495C2×C4—1061no—
496C2×C4—1061no—
497C2×C4—4242no—
498C2×C4—2121no—
499C2×C4—2121no—
500C2×C4—2121no—
501C2×C4D8×D5311yescharacteristic
502C2×C4C2×D4×D5311yescharacteristic
503C8—21no—
504C8—1061no—
505C8—1061no—
506C8C2×#1171994711yescharacteristic
507C8#1313326211yescharacteristic
508C22—21no—
509C22—21no—
510C22—42no—
511C22—81no—
512C22—81no—
513C22—81no—
514C22—81no—
515C22—81no—
516C22—164no—
517C22—532no—
518C22—8484no—
519C22—1061no—
520C22—1061no—
521C22—4241no—
522C22—4241no—
523C22—4241no—
524C22SD32×D5312yes—
525C22C2×D8×D5311yescharacteristic
526C4—21no—
527C4—81no—
528C4—81no—
529C4—532no—
530C4—1061no—
531C4—1061no—
532C4—4241no—
533C4—4241no—
534C4—4241no—
535C4#1681126011yescharacteristic
536C4#1681143311yescharacteristic
537C2—22no—
538C2—82no—
539C2—161no—
540C2—8481no—
541C2—1062no—
542C2#2028652111yescharacteristic
543C2C2×SD32×D5311yescharacteristic
544C2#2028785511yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2187C2
socleC22×C53C22×D8
prime radicalD8×C22⋊D53C2
hypercenter#2187D53
hypocenterC53#33862
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2187

Complements

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

NH ≅ G/Ntype
1. #20287104C2semidirect
3. #20276937C2semidirect
4. #20276901C2semidirect
5. #20276899C2semidirect
6. #20276860C2semidirect
7. #20276813C2semidirect
8. #20276808C2semidirect
9. #20276753C2semidirect
10. #20276642C2semidirect
11. SD32×Dic53C2semidirect
12. #20276532C2semidirect
13. C2×#16810424C2semidirect
14. #357×D53C2semidirect
15. C53×#2187C2semidirect
19. C2×#13133160C22semidirect
20. C22×#11719908C22semidirect
21. C2×C16×D53C22semidirect
22. #16811027C22semidirect
27. #16810861C22semidirect
33. Q16×Dic53C22semidirect
36. C2×#13133151C22semidirect
37. #107×D53C22semidirect
38. #16810451C22semidirect
39. #16810451C22semidirect
40. #16810450C22semidirect
41. #16810441C22semidirect
42. #16810427C22semidirect
43. #16810425C22semidirect
44. #16810425C22semidirect
47. #16810424C22semidirect
48. #16810424C22semidirect
49. #16810423C22semidirect
50. #16810402C22semidirect
51. C16×Dic53C22semidirect
54. C2×SD32×C53C22semidirect
55. C53×#467C22semidirect
57. C53×#357C22semidirect
58. C53×C22⋊C16C22semidirect
215. C22⋊D53SD32semidirect
275. #2187D53semidirect
294. C2×SD32C2×D53semidirect
303. #467C2×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
#202871042
D8×C22⋊D532
#202769372
#202769012
#202768992
#202768602
#202768132
#202768082
#202767532
#202766422
SD32×Dic532
#202765322
C2×#168104242
#357×D532
C53×#21872
#3386253