Subgroups — C₃₁×Dic₁₆₅ #35205608

Order 20460.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₁₁×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×Dic₅₅	—	3	1	no	maximal, Hall
3	C₃×C₅×C₁₁×C₃₁	C₄	1	1	yes	characteristic, complemented, Hall
4	C₃₁×Dic₃₃	—	5	1	no	maximal, Hall
5	C₂×C₅×C₁₁×C₃₁	D₃	1	1	yes	characteristic
6	C₂×C₃×C₁₁×C₃₁	D₅	1	1	yes	characteristic
7	C₃₁×Dic₁₅	—	11	1	no	maximal, Hall
8	C₅×C₁₁×C₃₁	Dic₃	1	1	yes	characteristic, complemented, Hall
9	C₃₁×Dic₁₁	—	15	1	no	Hall
10	C₃×C₁₁×C₃₁	Dic₅	1	1	yes	characteristic, complemented, Hall
11	C₂×C₃×C₅×C₃₁	D₁₁	1	1	yes	characteristic
12	C₂×C₁₁×C₃₁	D₁₅	1	1	yes	characteristic
13	Dic₁₆₅	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
14	Dic₅×C₃₁	—	33	1	no	Hall
15	C₃×C₅×C₃₁	Dic₁₁	1	1	yes	characteristic, complemented, Hall
16	Dic₃×C₃₁	—	55	1	no	Hall
17	C₁₁×C₃₁	Dic₁₅	1	1	yes	characteristic, complemented, Hall
18	C₂×C₃×C₅×C₁₁	C₂×C₃₁	1	1	yes	characteristic
19	C₂×C₅×C₃₁	D₃₃	1	1	yes	characteristic
20	Dic₅₅	—	3	1	no	Hall
21	C₂×C₃×C₃₁	D₅₅	1	1	yes	characteristic
22	C₃×C₅×C₁₁	C₄×C₃₁	1	1	yes	characteristic, complemented, Hall
23	C₅×C₃₁	Dic₃₃	1	1	yes	characteristic, complemented, Hall
24	Dic₃₃	—	5	1	no	Hall
25	C₄×C₃₁	—	165	1	no	Hall
26	C₂×C₅×C₁₁	D₃×C₃₁	1	1	yes	characteristic
27	C₃×C₃₁	Dic₅₅	1	1	yes	characteristic, complemented, Hall
28	C₂×C₃×C₁₁	D₅×C₃₁	1	1	yes	characteristic
29	C₂×C₃₁	D₁₆₅	1	1	yes	characteristic
30	Dic₁₅	—	11	1	no	Hall
31	C₅×C₁₁	Dic₃×C₃₁	1	1	yes	characteristic, complemented, Hall
32	Dic₁₁	—	15	1	no	Hall
33	C₃×C₁₁	Dic₅×C₃₁	1	1	yes	characteristic, complemented, Hall
34	C₃₁	Dic₁₆₅	1	1	yes	characteristic, direct product factor, Sylow
35	C₂×C₃×C₅	D₁₁×C₃₁	1	1	yes	characteristic
36	C₂×C₁₁	D₁₅×C₃₁	1	1	yes	characteristic
37	Dic₅	—	33	1	no	Hall
38	C₃×C₅	C₃₁×Dic₁₁	1	1	yes	characteristic, complemented, Hall
39	Dic₃	—	55	1	no	Hall
40	C₁₁	C₃₁×Dic₁₅	1	1	yes	characteristic, complemented, Sylow
41	C₂×C₅	C₃₁×D₃₃	1	1	yes	characteristic
42	C₂×C₃	C₃₁×D₅₅	1	1	yes	characteristic
43	C₅	C₃₁×Dic₃₃	1	1	yes	characteristic, complemented, Sylow
44	C₄	—	165	1	no	Sylow
45	C₃	C₃₁×Dic₅₅	1	1	yes	characteristic, complemented, Sylow
46	C₂	C₃₁×D₁₆₅	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃₁	D₁₆₅
commutator, hypocenter	C₃×C₅×C₁₁	C₄×C₃₁
Frattini	C₂	C₃₁×D₁₆₅
Fitting, socle, prime radical	C₂×C₃×C₅×C₁₁×C₃₁	C₂
cosocle	C₂×C₃×C₅×C₁₁	C₂×C₃₁
LCS	G ▷ C₃×C₅×C₁₁
UCS	C₁ ◁ C₂×C₃₁

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×C₁₁×C₃₁	C₄	semidirect
8. C₅×C₁₁×C₃₁	Dic₃	semidirect
10. C₃×C₁₁×C₃₁	Dic₅	semidirect
13. Dic₁₆₅	C₃₁	direct (N × H)
15. C₃×C₅×C₃₁	Dic₁₁	semidirect
17. C₁₁×C₃₁	Dic₁₅	semidirect
22. C₃×C₅×C₁₁	C₄×C₃₁	semidirect
23. C₅×C₃₁	Dic₃₃	semidirect
27. C₃×C₃₁	Dic₅₅	semidirect
31. C₅×C₁₁	Dic₃×C₃₁	semidirect
33. C₃×C₁₁	Dic₅×C₃₁	semidirect
34. C₃₁	Dic₁₆₅	direct (N × H)
38. C₃×C₅	C₃₁×Dic₁₁	semidirect
40. C₁₁	C₃₁×Dic₁₅	semidirect
43. C₅	C₃₁×Dic₃₃	semidirect
45. C₃	C₃₁×Dic₅₅	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×C₁₁×C₃₁	2
C₃₁×Dic₅₅	3
C₃₁×Dic₃₃	5
C₃₁×Dic₁₅	11
Dic₁₆₅	31

Subgroups — C31×Dic165 #35205608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×Dic₁₆₅ #35205608