Subgroups — C₂×C₃×C₄×C₅×C₆₁ #72186352

Order 7320.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×C₆₁	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
2	C₂²×C₃×C₅×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×C₅×C₆₁	C₂	1	2	yes	maximal, direct product factor
4	C₂×C₃×C₅×C₆₁	C₄	1	2	yes	direct product factor
5	C₂×C₃×C₅×C₆₁	C₂²	1	1	yes	characteristic
6	C₂×C₃×C₄×C₆₁	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₅×C₆₁	C₂×C₃	1	1	yes	characteristic
8	C₄×C₅×C₆₁	C₂×C₃	1	2	yes	direct product factor
9	C₃×C₅×C₆₁	C₂×C₄	1	1	yes	characteristic, direct product factor, Hall
10	C₂²×C₃×C₆₁	C₂×C₅	1	1	yes	characteristic
11	C₃×C₄×C₆₁	C₂×C₅	1	2	yes	direct product factor
12	C₂×C₅×C₆₁	C₃×C₄	1	2	yes	direct product factor
13	C₂×C₅×C₆₁	C₂²×C₃	1	1	yes	characteristic
14	C₂×C₄×C₆₁	C₃×C₅	1	1	yes	characteristic, direct product factor, Hall
15	C₂×C₃×C₆₁	C₄×C₅	1	2	yes	direct product factor
16	C₂×C₃×C₆₁	C₂²×C₅	1	1	yes	characteristic
17	C₅×C₆₁	C₂×C₃×C₄	1	1	yes	characteristic, direct product factor, Hall
18	C₂²×C₆₁	C₂×C₃×C₅	1	1	yes	characteristic
19	C₄×C₆₁	C₂×C₃×C₅	1	2	yes	direct product factor
20	C₃×C₆₁	C₂×C₄×C₅	1	1	yes	characteristic, direct product factor, Hall
21	C₂×C₆₁	C₃×C₄×C₅	1	2	yes	direct product factor
22	C₂×C₆₁	C₂²×C₃×C₅	1	1	yes	characteristic
23	C₂×C₃×C₄×C₅	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Hall
24	C₆₁	C₂×C₃×C₄×C₅	1	1	yes	characteristic, direct product factor, Sylow
25	C₂²×C₃×C₅	C₂×C₆₁	1	1	yes	characteristic
26	C₃×C₄×C₅	C₂×C₆₁	1	2	yes	direct product factor
27	C₂×C₄×C₅	C₃×C₆₁	1	1	yes	characteristic, direct product factor, Hall
28	C₂×C₃×C₅	C₄×C₆₁	1	2	yes	direct product factor
29	C₂×C₃×C₅	C₂²×C₆₁	1	1	yes	characteristic
30	C₂×C₃×C₄	C₅×C₆₁	1	1	yes	characteristic, direct product factor, Hall
31	C₂²×C₅	C₂×C₃×C₆₁	1	1	yes	characteristic
32	C₄×C₅	C₂×C₃×C₆₁	1	2	yes	direct product factor
33	C₃×C₅	C₂×C₄×C₆₁	1	1	yes	characteristic, direct product factor, Hall
34	C₂²×C₃	C₂×C₅×C₆₁	1	1	yes	characteristic
35	C₃×C₄	C₂×C₅×C₆₁	1	2	yes	direct product factor
36	C₂×C₅	C₃×C₄×C₆₁	1	2	yes	direct product factor
37	C₂×C₅	C₂²×C₃×C₆₁	1	1	yes	characteristic
38	C₂×C₄	C₃×C₅×C₆₁	1	1	yes	characteristic, direct product factor, Sylow
39	C₂×C₃	C₄×C₅×C₆₁	1	2	yes	direct product factor
40	C₂×C₃	C₂²×C₅×C₆₁	1	1	yes	characteristic
41	C₅	C₂×C₃×C₄×C₆₁	1	1	yes	characteristic, direct product factor, Sylow
42	C₂²	C₂×C₃×C₅×C₆₁	1	1	yes	characteristic
43	C₄	C₂×C₃×C₅×C₆₁	1	2	yes	direct product factor
44	C₃	C₂×C₄×C₅×C₆₁	1	1	yes	characteristic, direct product factor, Sylow
45	C₂	C₃×C₄×C₅×C₆₁	1	2	yes	direct product factor
46	C₂	C₂²×C₃×C₅×C₆₁	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂	C₂²×C₃×C₅×C₆₁
socle, prime radical	C₂²×C₃×C₅×C₆₁	C₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄×C₅×C₆₁	C₃	direct (N × H)
3. C₃×C₄×C₅×C₆₁	C₂	direct (N × H)
4. C₂×C₃×C₅×C₆₁	C₄	direct (N × H)
6. C₂×C₃×C₄×C₆₁	C₅	direct (N × H)
8. C₄×C₅×C₆₁	C₂×C₃	direct (N × H)
9. C₃×C₅×C₆₁	C₂×C₄	direct (N × H)
11. C₃×C₄×C₆₁	C₂×C₅	direct (N × H)
12. C₂×C₅×C₆₁	C₃×C₄	direct (N × H)
14. C₂×C₄×C₆₁	C₃×C₅	direct (N × H)
15. C₂×C₃×C₆₁	C₄×C₅	direct (N × H)
17. C₅×C₆₁	C₂×C₃×C₄	direct (N × H)
19. C₄×C₆₁	C₂×C₃×C₅	direct (N × H)
20. C₃×C₆₁	C₂×C₄×C₅	direct (N × H)
21. C₂×C₆₁	C₃×C₄×C₅	direct (N × H)
23. C₂×C₃×C₄×C₅	C₆₁	direct (N × H)
24. C₆₁	C₂×C₃×C₄×C₅	direct (N × H)
26. C₃×C₄×C₅	C₂×C₆₁	direct (N × H)
27. C₂×C₄×C₅	C₃×C₆₁	direct (N × H)
28. C₂×C₃×C₅	C₄×C₆₁	direct (N × H)
30. C₂×C₃×C₄	C₅×C₆₁	direct (N × H)
32. C₄×C₅	C₂×C₃×C₆₁	direct (N × H)
33. C₃×C₅	C₂×C₄×C₆₁	direct (N × H)
35. C₃×C₄	C₂×C₅×C₆₁	direct (N × H)
36. C₂×C₅	C₃×C₄×C₆₁	direct (N × H)
38. C₂×C₄	C₃×C₅×C₆₁	direct (N × H)
39. C₂×C₃	C₄×C₅×C₆₁	direct (N × H)
41. C₅	C₂×C₃×C₄×C₆₁	direct (N × H)
43. C₄	C₂×C₃×C₅×C₆₁	direct (N × H)
44. C₃	C₂×C₄×C₅×C₆₁	direct (N × H)
45. C₂	C₃×C₄×C₅×C₆₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×C₅×C₆₁	3
C₂²×C₃×C₅×C₆₁	2
C₃×C₄×C₅×C₆₁	2
C₂×C₃×C₄×C₆₁	5
C₂×C₃×C₄×C₅	61

Subgroups — C2×C3×C4×C5×C61 #72186352

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₄×C₅×C₆₁ #72186352