Subgroups — C₅×C₆₁×D₅₁ #44527441

Order 31110.

32 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×C₁₇×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₅×D₁₇×C₆₁	—	3	1	no	maximal, Hall
3	C₆₁×D₅₁	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₅×C₁₇×C₆₁	D₃	1	1	yes	characteristic, complemented, Hall
5	C₃×C₁₇×C₆₁	C₂×C₅	1	1	yes	characteristic, complemented, Hall
6	D₁₇×C₆₁	—	3	1	no	Hall
7	C₅×D₃×C₆₁	—	17	1	no	maximal, Hall
8	C₁₇×C₆₁	C₅×D₃	1	1	yes	characteristic, complemented, Hall
9	C₃×C₅×C₆₁	D₁₇	1	1	yes	characteristic, complemented, Hall
10	C₂×C₅×C₆₁	—	51	1	no	Hall
11	C₅×D₅₁	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Hall
12	D₃×C₆₁	—	17	1	no	Hall
13	C₅×C₆₁	D₅₁	1	1	yes	characteristic, direct product factor, Hall
14	C₃×C₅×C₁₇	C₂×C₆₁	1	1	yes	characteristic, complemented, Hall
15	C₃×C₆₁	C₅×D₁₇	1	1	yes	characteristic, complemented, Hall
16	C₅×D₁₇	—	3	1	no	Hall
17	C₂×C₆₁	—	51	1	no	Hall
18	D₅₁	C₅×C₆₁	1	1	yes	characteristic, direct product factor, Hall
19	C₅×C₁₇	D₃×C₆₁	1	1	yes	characteristic, complemented, Hall
20	C₆₁	C₅×D₅₁	1	1	yes	characteristic, direct product factor, Sylow
21	C₃×C₁₇	C₂×C₅×C₆₁	1	1	yes	characteristic, complemented, Hall
22	D₁₇	—	3	1	no	Hall
23	C₅×D₃	—	17	1	no	Hall
24	C₁₇	C₅×D₃×C₆₁	1	1	yes	characteristic, complemented, Sylow
25	C₃×C₅	D₁₇×C₆₁	1	1	yes	characteristic, complemented, Hall
26	C₂×C₅	—	51	1	no	Hall
27	D₃	—	17	1	no	Hall
28	C₅	C₆₁×D₅₁	1	1	yes	characteristic, direct product factor, Sylow
29	C₃	C₅×D₁₇×C₆₁	1	1	yes	characteristic, complemented, Sylow
30	C₂	—	51	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₅×C₆₁	D₅₁
commutator, cosocle, hypocenter	C₃×C₁₇	C₂×C₅×C₆₁
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₁₇×C₆₁	C₂
LCS	G ▷ C₃×C₁₇
UCS	C₁ ◁ C₅×C₆₁

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×C₁₇×C₆₁	C₂	semidirect
3. C₆₁×D₅₁	C₅	direct (N × H)
4. C₅×C₁₇×C₆₁	D₃	semidirect
5. C₃×C₁₇×C₆₁	C₂×C₅	semidirect
8. C₁₇×C₆₁	C₅×D₃	semidirect
9. C₃×C₅×C₆₁	D₁₇	semidirect
11. C₅×D₅₁	C₆₁	direct (N × H)
13. C₅×C₆₁	D₅₁	direct (N × H)
14. C₃×C₅×C₁₇	C₂×C₆₁	semidirect
15. C₃×C₆₁	C₅×D₁₇	semidirect
18. D₅₁	C₅×C₆₁	direct (N × H)
19. C₅×C₁₇	D₃×C₆₁	semidirect
20. C₆₁	C₅×D₅₁	direct (N × H)
21. C₃×C₁₇	C₂×C₅×C₆₁	semidirect
24. C₁₇	C₅×D₃×C₆₁	semidirect
25. C₃×C₅	D₁₇×C₆₁	semidirect
28. C₅	C₆₁×D₅₁	direct (N × H)
29. C₃	C₅×D₁₇×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×C₁₇×C₆₁	2
C₅×D₁₇×C₆₁	3
C₆₁×D₅₁	5
C₅×D₃×C₆₁	17
C₅×D₅₁	61

Subgroups — C5×C61×D51 #44527441

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₆₁×D₅₁ #44527441