Finite Group Databasebeta

← #27541116

Subgroups — #27541116

Order 13568.

610 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20287078C211yesmaximal, characteristic, complemented
2D8×C22⋊D53C211yesmaximal, characteristic
3#20276955C211yesmaximal, characteristic, complemented
4#20276907C211yesmaximal, characteristic, complemented
5#20276894C211yesmaximal, characteristic, complemented
6#20276874C211yesmaximal, characteristic, complemented
7#20276830C211yesmaximal, characteristic, complemented
8#20276808C211yesmaximal, characteristic, complemented
9#20276753C211yesmaximal, characteristic, complemented
10#20276642C211yesmaximal, characteristic, complemented
11#20276580C211yesmaximal, characteristic, complemented
12#20276543C211yesmaximal, characteristic, complemented
13#20276431C211yesmaximal, characteristic, complemented
14#357×D53C211yesmaximal, characteristic, complemented
15C53×#2187C211yesmaximal, characteristic, complemented
16C2×D8×D53C2211yescharacteristic
17#16811307—21no—
18D4×C22⋊D53—21no—
19C2×#13133160C2211yescharacteristic, complemented
20C2×#13133157C2211yescharacteristic, complemented
21C2×C16×D53C2211yescharacteristic, complemented
22#16811024C2211yescharacteristic, complemented
23#16810998C2211yescharacteristic
24#16810961—21no—
25#16810952—21no—
26#16810928—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
36#16810661C2211yescharacteristic, complemented
37#107×D53C2211yescharacteristic, complemented
38#16810451C2211yescharacteristic, complemented
39#16810449C2211yescharacteristic, complemented
40#16810448C2211yescharacteristic, 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
51#16810398C2211yescharacteristic, 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#13133213—21no—
68#13133210—41no—
69#13133208—21no—
70#13133207—41no—
71#13133199—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#13133176—21no—
82#13133175—41no—
83C4⋊C4×D53—21no—
84#13133160—41no—
85#13133157—41no—
86C2×#11719908C2311yescharacteristic
87C2×#11719908C2311yescharacteristic
88C16×D53—41no—
89#13133151C2311yescharacteristic
90#13133151C2311yescharacteristic
91#13133151C2311yescharacteristic
92#13133150—21no—
93#13133149—21no—
94#13133148—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—
136D4×D53—42no—
137D4×D53—81no—
138C2×D212—21no—
139C2×C4×D53—41no—
140C2×C4×D53C2×D411yescharacteristic
141Q16×C53—21no—
142Q16×C53—21no—
143D8×C53—21no—
144D8×C53—21no—
145D8×C53—21no—
146D8×C53—42no—
147D8×C53#4112yes—
148C2×C8×C53—21no—
149C2×C8×C53—21no—
150C2×C8×C53C2×D411yescharacteristic
151C2×C8×C53C2411yescharacteristic
152C4⋊C4×C53—21no—
153C4⋊C4×C53—41no—
154C22⋊C4×C53—41no—
155C22⋊Dic53—41no—
156C22⋊Dic53—41no—
157C22⋊Dic53—41no—
158C22⋊Dic53D811yescharacteristic
159#11719922—42no—
160#11719922—81no—
161#11719921—41no—
162#11719921—41no—
163#11719921D811yescharacteristic
164C4⋊Dic53—21no—
165C4⋊Dic53—21no—
166C4⋊Dic53—21no—
167C4⋊Dic53—41no—
168C4⋊Dic53—41no—
169#11719919—41no—
170#11719919—41no—
171#11719919—41no—
172#11719919D811yescharacteristic
173C4×Dic53—41no—
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—22no—
211C22⋊D53—82no—
212C22⋊D53—82no—
213C22⋊D53—82no—
214C22⋊D53—82no—
215C22⋊D53—161no—
216C2×Dic53—41no—
217C2×Dic53—41no—
218C2×Dic53—41no—
219C2×Dic53—82no—
220C2×Dic53C2×D811yescharacteristic
221C2×Dic53C2×D811yescharacteristic
222C2×Dic53C2×D811yescharacteristic
223D212—41no—
224D212—41no—
225C4×D53—21no—
226C4×D53—21no—
227C4×D53—82no—
228C8×C53—21no—
229C8×C53C2×#4111yescharacteristic
230C8×C53H(5,𝔽2)11yescharacteristic
231#88322—21no—
232#88322—21no—
233#33877—531nomaximal, Sylow
234C22×C53—21no—
235C22×C53—21no—
236C22×C53—21no—
237C22×C53—42no—
238C22×C53—81no—
239C22×C53—81no—
240C22×C53—81no—
241C22×C53—81no—
242C22×C53—81no—
243C22×C53—164no—
244C22×C53C22×D811yescharacteristic
245C2×D53—21no—
246C2×D53—21no—
247C2×D53—81no—
248C2×D53—81no—
249C2×D53—81no—
250C2×D53—164no—
251C2×D53#50912yes—
252C4×C53—21no—
253C4×C53—81no—
254C4×C53—81no—
255C4×C53#54611yescharacteristic
256C4×C53#57711yescharacteristic
257Dic53—21no—
258Dic53—21no—
259Dic53—21no—
260Dic53—81no—
261Dic53—81no—
262Dic53—81no—
263#3336—531no—
264D4×D8—531no—
265#2244—531no—
266#2234—531no—
267#2232—531no—
268#2228—531no—
269#2220—531no—
270#2217—531no—
271#2207—531no—
272#2187—531no—
273#2187D5311yescharacteristic, complemented
274#2175—531no—
275#2168—531no—
276#2140—531no—
277C2×#357—531no—
278C2×C53—41no—
279C2×C53—82no—
280C2×C53—161no—
281C2×C53#329411yescharacteristic
282C2×C53C2×#50911yescharacteristic
283C2×C53#341611yescharacteristic
284D53—22no—
285D53—161no—
286C22×D8—531no—
287C22×D8C2×D5311yescharacteristic
288#552—1061no—
289D42—1061no—
290C2×SD32—531no—
291C2×SD32C2×D5311yescharacteristic
292C2×SD32C2×D5311yescharacteristic, complemented
293C2×M32—531no—
294C22×C16—531no—
295#499—531no—
296#493—531no—
297#485—1061no—
298#483—1061no—
299#479—1061no—
300#467—531no—
301#467C2×D5311yescharacteristic, complemented
302#466—531no—
303#466—531no—
304#459—1061no—
305#447—1061no—
306#447—1061no—
307C4×Q16—531no—
308C4×D8—531no—
309C8×D4—531no—
310#427—531no—
311C2×#107—531no—
312#367—531no—
313#366—531no—
314#366—531no—
315C4⋊C16—531no—
316#358—531no—
317#357—531no—
318#357—531no—
319#357—531no—
320#357—531no—
321#357—531no—
322#357—531no—
323#357—531no—
324#357C2×D5311yescharacteristic
325#357C2×D5311yescharacteristic, complemented
326C22⋊C16—531no—
327C22⋊C16C2×D5311yescharacteristic, complemented
328#346—531no—
329C53#3387711yescharacteristic, complemented, Sylow
330C22×D4—21no—
331C22×D4—1061no—
332C22×D4—2121no—
333C2×Q16C22×D5311yescharacteristic
334C2×D8—22no—
335C2×D8—21no—
336C2×D8—41no—
337C2×D8—531no—
338C2×D8—1061no—
339C2×D8—1061no—
340C2×D8—1062no—
341C2×D8—2121no—
342C2×D8C22×D5311yescharacteristic
343C2×D8C22×D5311yescharacteristic
344C22×C8—531no—
345C22×C8C22×D5311yescharacteristic
346#127—1061no—
347#126—2121no—
348#125—1061no—
349#125—2121no—
350#123—2121no—
351C22⋊Q8—21no—
352C22⋊Q8—1061no—
353#121—1061no—
354#121—1061no—
355#121—2121no—
356#120—2121no—
357#120—2121no—
358C4×Q8—1061no—
359C4×D4—531no—
360C4×D4—1061no—
361#117—1061no—
362#117—2121no—
363C2×C4⋊C4—1061no—
364SD32—22no—
365SD32—41no—
366SD32—2121no—
367M32—2121no—
368C2×C16—531no—
369C2×C16—531no—
370C2×C16—2121no—
371C2×C16C22×D5311yescharacteristic
372C2×C16C22×D5311yescharacteristic
373#107—531no—
374#107—531no—
375#107—531no—
376#107—1061no—
377#107C22×D5311yescharacteristic
378#106—1061no—
379#106—1061no—
380C4⋊C8—531no—
381#103—21no—
382#103—1061no—
383#103—1061no—
384#102—1061no—
385#102—1061no—
386#102—1061no—
387C22⋊C8—531no—
388C22⋊C8—531no—
389C4×C8—531no—
390C24—41no—
391C24—2121no—
392C2×Q8—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—42no—
396C2×D4—41no—
397C2×D4—81no—
398C2×D4—531no—
399C2×D4—1061no—
400C2×D4—4241no—
401C2×D4—4244no—
402C2×D4—2121no—
403C2×D4—2121no—
404C2×D4—2121no—
405C2×D4—2122no—
406C2×D4—2121no—
407C2×D4—2121no—
408C22×C4—531no—
409C22×C4—2121no—
410C22×C4—2121no—
411C22×C4D4×D5311yescharacteristic
412Q16—21no—
413Q16—21no—
414D8—21no—
415D8—21no—
416D8—21no—
417D8—42no—
418D8—1061no—
419D8—1061no—
420D8—4241no—
421D8—2122no—
422D8#1171994712yes—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—531no—
426C2×C8—531no—
427C2×C8—1061no—
428C2×C8—1061no—
429C2×C8D4×D5311yescharacteristic
430C2×C8C23×D5311yescharacteristic
431C4⋊C4—21no—
432C4⋊C4—41no—
433C4⋊C4—531no—
434C4⋊C4—1061no—
435C4⋊C4—1061no—
436C4⋊C4—1061no—
437C4⋊C4—2121no—
438C4⋊C4—2121no—
439C4⋊C4—2121no—
440C4⋊C4—2121no—
441C4⋊C4—2121no—
442C22⋊C4—41no—
443C22⋊C4—531no—
444C22⋊C4—531no—
445C22⋊C4—2121no—
446C22⋊C4—2121no—
447C22⋊C4—2121no—
448C22⋊C4—2121no—
449C22⋊C4—2121no—
450C42—531no—
451C42—2121no—
452C42—2121no—
453C16—21no—
454C16—21no—
455C16—1061no—
456C16—1061no—
457C23—41no—
458C23—41no—
459C23—82no—
460C23—82no—
461C23—161no—
462C23—531no—
463C23—8481no—
464C23—4242no—
465C23—4242no—
466C23—2121no—
467C23D8×D5311yescharacteristic
468Q8—41no—
469Q8—41no—
470D4—22no—
471D4—41no—
472D4—41no—
473D4—41no—
474D4—82no—
475D4—8481no—
476D4—1062no—
477D4—4242no—
478D4—4242no—
479D4—4242no—
480D4—4242no—
481D4—2121no—
482D4—2121no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—531no—
488C2×C4—531no—
489C2×C4—531no—
490C2×C4—1061no—
491C2×C4—1061no—
492C2×C4—4242no—
493C2×C4—4242no—
494C2×C4—2121no—
495C2×C4—2121no—
496C2×C4—2121no—
497C2×C4D8×D5311yescharacteristic
498C2×C4C2×D4×D5311yescharacteristic
499C8—21no—
500C8—1061no—
501C8—1061no—
502C8C2×#1171994711yescharacteristic
503C8#1313326211yescharacteristic
504C22—21no—
505C22—21no—
506C22—21no—
507C22—42no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22—164no—
514C22—532no—
515C22—8484no—
516C22—1061no—
517C22—1061no—
518C22—4241no—
519C22—4241no—
520C22—4241no—
521C22C2×D8×D5311yescharacteristic
522C4—21no—
523C4—81no—
524C4—81no—
525C4—1061no—
526C4—1061no—
527C4—1061no—
528C4—4241no—
529C4—4241no—
530C4—4241no—
531C4#1681126011yescharacteristic
532C4#1681143311yescharacteristic
533C2—41no—
534C2—82no—
535C2—161no—
536C2—8481no—
537C2—1062no—
538C2#2028652111yescharacteristic
539C2#509×D5311yescharacteristic
540C2#2028785711yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2187C2
socleC22×C53C22×D8
prime radicalD8×C22⋊D53C2
hypercenter#2187D53
hypocenterC53#33877
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. #20287078C2semidirect
3. #20276955C2semidirect
4. #20276907C2semidirect
5. #20276894C2semidirect
6. #20276874C2semidirect
7. #20276830C2semidirect
8. #20276808C2semidirect
9. #20276753C2semidirect
10. #20276642C2semidirect
11. #20276580C2semidirect
12. #20276543C2semidirect
13. #20276431C2semidirect
14. #357×D53C2semidirect
15. C53×#2187C2semidirect
19. C2×#13133160C22semidirect
20. C2×#13133157C22semidirect
21. C2×C16×D53C22semidirect
22. #16811024C22semidirect
27. #16810861C22semidirect
33. Q16×Dic53C22semidirect
36. #16810661C22semidirect
37. #107×D53C22semidirect
38. #16810451C22semidirect
39. #16810449C22semidirect
40. #16810448C22semidirect
41. #16810441C22semidirect
42. #16810427C22semidirect
43. #16810425C22semidirect
44. #16810425C22semidirect
47. #16810424C22semidirect
48. #16810424C22semidirect
49. #16810423C22semidirect
50. #16810402C22semidirect
51. #16810398C22semidirect
54. C2×SD32×C53C22semidirect
55. C53×#467C22semidirect
57. C53×#357C22semidirect
58. C53×C22⋊C16C22semidirect
273. #2187D53semidirect
292. C2×SD32C2×D53semidirect
301. #467C2×D53semidirect
325. #357C2×D53semidirect
327. C22⋊C16C2×D53semidirect
329. C53#33877semidirect

Maximal subgroups

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

subgroupindex
#202870782
D8×C22⋊D532
#202769552
#202769072
#202768942
#202768742
#202768302
#202768082
#202767532
#202766422
#202765802
#202765432
#202764312
#357×D532
C53×#21872
#3387753