Subgroups — C₂²×C₃×D₂₀ #92061

Order 480.

472 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×D₅	C₂	1	2	yes	maximal, complemented
2	C₂²×C₃×C₄×C₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×D₂₀	C₂	1	12	yes	maximal, direct product factor
4	C₂²×D₂₀	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂³×C₃×C₅	C₂²	1	1	yes	characteristic
6	C₂²×C₃×D₅	—	2	8	no	—
7	C₂²×C₃×D₅	C₂²	1	12	yes	complemented
8	C₂×C₃×C₄×C₅	C₂²	1	6	yes	complemented
9	C₃×D₂₀	C₂²	1	16	yes	direct product factor
10	C₂²×C₃×D₄	—	5	1	no	maximal, Hall
11	C₂³×D₅	C₂×C₃	1	2	yes	complemented
12	C₂²×C₄×C₅	C₂×C₃	1	1	yes	characteristic, complemented
13	C₂×D₂₀	C₂×C₃	1	12	yes	direct product factor
14	C₂²×C₃×C₅	D₄	1	4	yes	complemented
15	C₂²×C₃×C₅	C₂³	1	3	yes	—
16	C₂×C₃×D₅	—	2	24	no	—
17	C₂×C₃×D₅	C₂³	1	8	yes	complemented
18	C₃×C₄×C₅	C₂³	1	4	yes	complemented
19	C₂⁴×C₃	—	5	2	no	—
20	C₂×C₃×D₄	—	5	12	no	—
21	C₂²×C₃×C₄	D₅	1	1	yes	characteristic, complemented
22	C₂³×C₅	C₂²×C₃	1	1	yes	characteristic
23	C₂²×D₅	—	2	8	no	—
24	C₂²×D₅	C₂²×C₃	1	12	yes	complemented
25	C₂×C₄×C₅	C₂²×C₃	1	6	yes	complemented
26	D₂₀	C₂²×C₃	1	16	yes	direct product factor
27	C₂²×D₄	—	5	1	no	Sylow
28	C₂×C₃×C₅	C₂×D₄	1	6	yes	complemented
29	C₂×C₃×C₅	C₂⁴	1	1	yes	characteristic
30	C₃×D₅	—	2	8	no	—
31	C₂³×C₃	—	5	12	no	—
32	C₂³×C₃	—	10	8	no	—
33	C₂³×C₃	C₂×D₅	1	1	yes	characteristic
34	C₃×D₄	—	5	16	no	—
35	C₂×C₃×C₄	C₂×D₅	1	6	yes	complemented
36	C₂²×C₅	C₃×D₄	1	4	yes	complemented
37	C₂²×C₅	C₂³×C₃	1	3	yes	—
38	C₂×D₅	—	2	24	no	—
39	C₂×D₅	C₂³×C₃	1	8	yes	complemented
40	C₄×C₅	C₂³×C₃	1	4	yes	complemented
41	C₂⁴	—	5	2	no	—
42	C₂×D₄	—	5	12	no	—
43	C₂²×C₄	C₃×D₅	1	1	yes	characteristic, complemented
44	C₃×C₅	C₂²×D₄	1	1	yes	characteristic, complemented, Hall
45	C₂²×C₃	—	5	8	no	—
46	C₂²×C₃	—	10	24	no	—
47	C₂²×C₃	D₂₀	1	4	yes	direct product factor
48	C₂²×C₃	C₂²×D₅	1	3	yes	—
49	C₃×C₄	C₂²×D₅	1	4	yes	complemented
50	C₂×C₅	C₂×C₃×D₄	1	6	yes	complemented
51	C₂×C₅	C₂⁴×C₃	1	1	yes	characteristic
52	D₅	—	2	8	no	—
53	C₂³	—	5	12	no	—
54	C₂³	—	10	8	no	—
55	C₂³	C₂×C₃×D₅	1	1	yes	characteristic
56	D₄	—	5	16	no	—
57	C₂×C₄	C₂×C₃×D₅	1	6	yes	complemented
58	C₂×C₃	—	10	8	no	—
59	C₂×C₃	C₂×D₂₀	1	6	yes	direct product factor
60	C₂×C₃	C₂³×D₅	1	1	yes	characteristic
61	C₅	C₂²×C₃×D₄	1	1	yes	characteristic, complemented, Sylow
62	C₂²	—	5	8	no	—
63	C₂²	—	10	24	no	—
64	C₂²	C₃×D₂₀	1	4	yes	direct product factor
65	C₂²	C₂²×C₃×D₅	1	3	yes	—
66	C₄	C₂²×C₃×D₅	1	4	yes	complemented
67	C₃	C₂²×D₂₀	1	1	yes	characteristic, direct product factor, Sylow
68	C₂	—	10	8	no	—
69	C₂	C₂×C₃×D₂₀	1	6	yes	direct product factor
70	C₂	C₂³×C₃×D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	C₂×D₅
commutator, cosocle	C₂×C₅	C₂⁴×C₃
Frattini	C₂	C₂³×C₃×D₅
Fitting	C₂²×C₃×C₄×C₅	C₂
socle	C₂³×C₃×C₅	C₂²
hypercenter	C₂²×C₃×C₄	D₅
hypocenter	C₅	C₂²×C₃×D₄
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×C₄

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃×D₅	C₂	semidirect
2. C₂²×C₃×C₄×C₅	C₂	semidirect
3. C₂×C₃×D₂₀	C₂	direct (N × H)
4. C₂²×D₂₀	C₃	direct (N × H)
7. C₂²×C₃×D₅	C₂²	semidirect
8. C₂×C₃×C₄×C₅	C₂²	semidirect
9. C₃×D₂₀	C₂²	direct (N × H)
11. C₂³×D₅	C₂×C₃	semidirect
12. C₂²×C₄×C₅	C₂×C₃	semidirect
13. C₂×D₂₀	C₂×C₃	direct (N × H)
14. C₂²×C₃×C₅	D₄	semidirect
17. C₂×C₃×D₅	C₂³	semidirect
18. C₃×C₄×C₅	C₂³	semidirect
21. C₂²×C₃×C₄	D₅	semidirect
24. C₂²×D₅	C₂²×C₃	semidirect
25. C₂×C₄×C₅	C₂²×C₃	semidirect
26. D₂₀	C₂²×C₃	direct (N × H)
28. C₂×C₃×C₅	C₂×D₄	semidirect
35. C₂×C₃×C₄	C₂×D₅	semidirect
36. C₂²×C₅	C₃×D₄	semidirect
39. C₂×D₅	C₂³×C₃	semidirect
40. C₄×C₅	C₂³×C₃	semidirect
43. C₂²×C₄	C₃×D₅	semidirect
44. C₃×C₅	C₂²×D₄	semidirect
47. C₂²×C₃	D₂₀	direct (N × H)
49. C₃×C₄	C₂²×D₅	semidirect
50. C₂×C₅	C₂×C₃×D₄	semidirect
57. C₂×C₄	C₂×C₃×D₅	semidirect
59. C₂×C₃	C₂×D₂₀	direct (N × H)
61. C₅	C₂²×C₃×D₄	semidirect
64. C₂²	C₃×D₂₀	direct (N × H)
66. C₄	C₂²×C₃×D₅	semidirect
67. C₃	C₂²×D₂₀	direct (N × H)
69. C₂	C₂×C₃×D₂₀	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×D₅	2
C₂²×C₃×C₄×C₅	2
C₂×C₃×D₂₀	2
C₂²×D₂₀	3
C₂²×C₃×D₄	5

Subgroups — C22×C3×D20 #92061

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×D₂₀ #92061