Subgroups — C₂³×C₅×D₂₁ #13132982

Order 1680.

536 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₅×C₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₅×D₂₁	C₂	1	14	yes	maximal, direct product factor
3	C₂³×C₅×D₇	—	3	1	no	maximal, Hall
4	C₂²×C₃×C₅×C₇	C₂²	1	7	yes	complemented
5	C₂×C₅×D₂₁	C₂²	1	28	yes	direct product factor
6	C₂³×D₂₁	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂³×C₅×C₇	D₃	1	1	yes	characteristic, complemented
8	C₂²×C₅×D₇	—	3	14	no	—
9	C₂³×C₅×D₃	—	7	1	no	maximal, Hall
10	C₂×C₃×C₅×C₇	C₂³	1	7	yes	complemented
11	C₅×D₂₁	C₂³	1	8	yes	direct product factor
12	C₂³×C₃×C₇	C₂×C₅	1	1	yes	characteristic, complemented
13	C₂²×D₂₁	C₂×C₅	1	14	yes	direct product factor
14	C₂²×C₅×C₇	C₂×D₃	1	7	yes	complemented
15	C₂×C₅×D₇	—	3	28	no	—
16	C₂³×C₃×C₅	D₇	1	1	yes	characteristic, complemented
17	C₂²×C₅×D₃	—	7	14	no	—
18	C₂³×D₇	—	3	1	no	Hall
19	C₃×C₅×C₇	C₂⁴	1	1	yes	characteristic, complemented, Hall
20	C₂²×C₃×C₇	C₂²×C₅	1	7	yes	complemented
21	C₂×D₂₁	C₂²×C₅	1	28	yes	direct product factor
22	C₂⁴×C₅	—	21	1	no	Hall
23	C₂×C₅×C₇	C₂²×D₃	1	7	yes	complemented
24	C₅×D₇	—	3	8	no	—
25	C₂²×C₃×C₅	C₂×D₇	1	7	yes	complemented
26	C₂×C₅×D₃	—	7	28	no	—
27	C₂³×C₇	C₅×D₃	1	1	yes	characteristic, complemented
28	C₂²×D₇	—	3	14	no	—
29	C₂³×D₃	—	7	1	no	Hall
30	C₂×C₃×C₇	C₂³×C₅	1	7	yes	complemented
31	D₂₁	C₂³×C₅	1	8	yes	direct product factor
32	C₂³×C₅	—	21	14	no	—
33	C₂³×C₅	D₂₁	1	1	yes	characteristic, direct product factor
34	C₅×C₇	C₂³×D₃	1	1	yes	characteristic, complemented, Hall
35	C₂×C₃×C₅	C₂²×D₇	1	7	yes	complemented
36	C₅×D₃	—	7	8	no	—
37	C₂²×C₇	C₂×C₅×D₃	1	7	yes	complemented
38	C₂×D₇	—	3	28	no	—
39	C₂³×C₃	C₅×D₇	1	1	yes	characteristic, complemented
40	C₂²×D₃	—	7	14	no	—
41	C₃×C₇	C₂⁴×C₅	1	1	yes	characteristic, complemented, Hall
42	C₂²×C₅	—	21	28	no	—
43	C₂²×C₅	C₂×D₂₁	1	7	yes	direct product factor
44	C₂⁴	—	21	1	no	Sylow
45	C₃×C₅	C₂³×D₇	1	1	yes	characteristic, complemented, Hall
46	C₂×C₇	C₂²×C₅×D₃	1	7	yes	complemented
47	D₇	—	3	8	no	—
48	C₂²×C₃	C₂×C₅×D₇	1	7	yes	complemented
49	C₂×D₃	—	7	28	no	—
50	C₂×C₅	—	21	8	no	—
51	C₂×C₅	C₂²×D₂₁	1	7	yes	direct product factor
52	C₂³	—	21	14	no	—
53	C₂³	C₅×D₂₁	1	1	yes	characteristic, direct product factor
54	C₇	C₂³×C₅×D₃	1	1	yes	characteristic, complemented, Sylow
55	C₂×C₃	C₂²×C₅×D₇	1	7	yes	complemented
56	D₃	—	7	8	no	—
57	C₅	C₂³×D₂₁	1	1	yes	characteristic, direct product factor, Sylow
58	C₂²	—	21	28	no	—
59	C₂²	C₂×C₅×D₂₁	1	7	yes	direct product factor
60	C₃	C₂³×C₅×D₇	1	1	yes	characteristic, complemented, Sylow
61	C₂	—	21	8	no	—
62	C₂	C₂²×C₅×D₂₁	1	7	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂³×C₅	D₂₁
commutator, cosocle, hypocenter	C₃×C₇	C₂⁴×C₅
Frattini	C₁	G
Fitting, socle	C₂³×C₃×C₅×C₇	C₂
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₂³×C₅

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃×C₅×C₇	C₂	semidirect
2. C₂²×C₅×D₂₁	C₂	direct (N × H)
4. C₂²×C₃×C₅×C₇	C₂²	semidirect
5. C₂×C₅×D₂₁	C₂²	direct (N × H)
6. C₂³×D₂₁	C₅	direct (N × H)
7. C₂³×C₅×C₇	D₃	semidirect
10. C₂×C₃×C₅×C₇	C₂³	semidirect
11. C₅×D₂₁	C₂³	direct (N × H)
12. C₂³×C₃×C₇	C₂×C₅	semidirect
13. C₂²×D₂₁	C₂×C₅	direct (N × H)
14. C₂²×C₅×C₇	C₂×D₃	semidirect
16. C₂³×C₃×C₅	D₇	semidirect
19. C₃×C₅×C₇	C₂⁴	semidirect
20. C₂²×C₃×C₇	C₂²×C₅	semidirect
21. C₂×D₂₁	C₂²×C₅	direct (N × H)
23. C₂×C₅×C₇	C₂²×D₃	semidirect
25. C₂²×C₃×C₅	C₂×D₇	semidirect
27. C₂³×C₇	C₅×D₃	semidirect
30. C₂×C₃×C₇	C₂³×C₅	semidirect
31. D₂₁	C₂³×C₅	direct (N × H)
33. C₂³×C₅	D₂₁	direct (N × H)
34. C₅×C₇	C₂³×D₃	semidirect
35. C₂×C₃×C₅	C₂²×D₇	semidirect
37. C₂²×C₇	C₂×C₅×D₃	semidirect
39. C₂³×C₃	C₅×D₇	semidirect
41. C₃×C₇	C₂⁴×C₅	semidirect
43. C₂²×C₅	C₂×D₂₁	direct (N × H)
45. C₃×C₅	C₂³×D₇	semidirect
46. C₂×C₇	C₂²×C₅×D₃	semidirect
48. C₂²×C₃	C₂×C₅×D₇	semidirect
51. C₂×C₅	C₂²×D₂₁	direct (N × H)
53. C₂³	C₅×D₂₁	direct (N × H)
54. C₇	C₂³×C₅×D₃	semidirect
55. C₂×C₃	C₂²×C₅×D₇	semidirect
57. C₅	C₂³×D₂₁	direct (N × H)
59. C₂²	C₂×C₅×D₂₁	direct (N × H)
60. C₃	C₂³×C₅×D₇	semidirect
62. C₂	C₂²×C₅×D₂₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₅×C₇	2
C₂²×C₅×D₂₁	2
C₂³×C₅×D₇	3
C₂³×D₂₁	5
C₂³×C₅×D₃	7

Subgroups — C23×C5×D21 #13132982

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₅×D₂₁ #13132982