Subgroups — C₂²×C₅×D₁₂ #92076

Order 480.

472 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₅×D₃	C₂	1	2	yes	maximal, complemented
2	C₂²×C₃×C₄×C₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×D₁₂	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₅×D₄	—	3	1	no	maximal, Hall
5	C₂³×C₃×C₅	C₂²	1	1	yes	characteristic
6	C₂²×C₅×D₃	—	2	8	no	—
7	C₂²×C₅×D₃	C₂²	1	12	yes	complemented
8	C₂×C₃×C₄×C₅	C₂²	1	6	yes	complemented
9	C₅×D₁₂	C₂²	1	16	yes	direct product factor
10	C₂²×D₁₂	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂⁴×C₅	—	3	2	no	—
12	C₂×C₅×D₄	—	3	12	no	—
13	C₂²×C₄×C₅	D₃	1	1	yes	characteristic, complemented
14	C₂²×C₃×C₅	D₄	1	4	yes	complemented
15	C₂²×C₃×C₅	C₂³	1	3	yes	—
16	C₂×C₅×D₃	—	2	24	no	—
17	C₂×C₅×D₃	C₂³	1	8	yes	complemented
18	C₃×C₄×C₅	C₂³	1	4	yes	complemented
19	C₂³×D₃	C₂×C₅	1	2	yes	complemented
20	C₂²×C₃×C₄	C₂×C₅	1	1	yes	characteristic, complemented
21	C₂×D₁₂	C₂×C₅	1	12	yes	direct product factor
22	C₂³×C₅	—	3	12	no	—
23	C₂³×C₅	—	6	8	no	—
24	C₂³×C₅	C₂×D₃	1	1	yes	characteristic
25	C₅×D₄	—	3	16	no	—
26	C₂×C₄×C₅	C₂×D₃	1	6	yes	complemented
27	C₂²×D₄	—	3	1	no	Sylow
28	C₂×C₃×C₅	C₂×D₄	1	6	yes	complemented
29	C₂×C₃×C₅	C₂⁴	1	1	yes	characteristic
30	C₅×D₃	—	2	8	no	—
31	C₂³×C₃	C₂²×C₅	1	1	yes	characteristic
32	C₂²×D₃	—	2	8	no	—
33	C₂²×D₃	C₂²×C₅	1	12	yes	complemented
34	C₂×C₃×C₄	C₂²×C₅	1	6	yes	complemented
35	D₁₂	C₂²×C₅	1	16	yes	direct product factor
36	C₂²×C₅	—	3	8	no	—
37	C₂²×C₅	—	6	24	no	—
38	C₂²×C₅	D₁₂	1	4	yes	direct product factor
39	C₂²×C₅	C₂²×D₃	1	3	yes	—
40	C₄×C₅	C₂²×D₃	1	4	yes	complemented
41	C₂⁴	—	3	2	no	—
42	C₂×D₄	—	3	12	no	—
43	C₂²×C₄	C₅×D₃	1	1	yes	characteristic, complemented
44	C₃×C₅	C₂²×D₄	1	1	yes	characteristic, complemented, Hall
45	C₂²×C₃	C₅×D₄	1	4	yes	complemented
46	C₂²×C₃	C₂³×C₅	1	3	yes	—
47	C₂×D₃	—	2	24	no	—
48	C₂×D₃	C₂³×C₅	1	8	yes	complemented
49	C₃×C₄	C₂³×C₅	1	4	yes	complemented
50	C₂×C₅	—	6	8	no	—
51	C₂×C₅	C₂×D₁₂	1	6	yes	direct product factor
52	C₂×C₅	C₂³×D₃	1	1	yes	characteristic
53	C₂³	—	3	12	no	—
54	C₂³	—	6	8	no	—
55	C₂³	C₂×C₅×D₃	1	1	yes	characteristic
56	D₄	—	3	16	no	—
57	C₂×C₄	C₂×C₅×D₃	1	6	yes	complemented
58	C₂×C₃	C₂×C₅×D₄	1	6	yes	complemented
59	C₂×C₃	C₂⁴×C₅	1	1	yes	characteristic
60	D₃	—	2	8	no	—
61	C₅	C₂²×D₁₂	1	1	yes	characteristic, direct product factor, Sylow
62	C₂²	—	3	8	no	—
63	C₂²	—	6	24	no	—
64	C₂²	C₅×D₁₂	1	4	yes	direct product factor
65	C₂²	C₂²×C₅×D₃	1	3	yes	—
66	C₄	C₂²×C₅×D₃	1	4	yes	complemented
67	C₃	C₂²×C₅×D₄	1	1	yes	characteristic, complemented, Sylow
68	C₂	—	6	8	no	—
69	C₂	C₂×C₅×D₁₂	1	6	yes	direct product factor
70	C₂	C₂³×C₅×D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₅	C₂×D₃
commutator, cosocle	C₂×C₃	C₂⁴×C₅
Frattini	C₂	C₂³×C₅×D₃
Fitting	C₂²×C₃×C₄×C₅	C₂
socle	C₂³×C₃×C₅	C₂²
hypercenter	C₂²×C₄×C₅	D₃
hypocenter	C₃	C₂²×C₅×D₄
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂²×C₄×C₅

Complements

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

N	H ≅ G/N	type
1. C₂³×C₅×D₃	C₂	semidirect
2. C₂²×C₃×C₄×C₅	C₂	semidirect
3. C₂×C₅×D₁₂	C₂	direct (N × H)
7. C₂²×C₅×D₃	C₂²	semidirect
8. C₂×C₃×C₄×C₅	C₂²	semidirect
9. C₅×D₁₂	C₂²	direct (N × H)
10. C₂²×D₁₂	C₅	direct (N × H)
13. C₂²×C₄×C₅	D₃	semidirect
14. C₂²×C₃×C₅	D₄	semidirect
17. C₂×C₅×D₃	C₂³	semidirect
18. C₃×C₄×C₅	C₂³	semidirect
19. C₂³×D₃	C₂×C₅	semidirect
20. C₂²×C₃×C₄	C₂×C₅	semidirect
21. C₂×D₁₂	C₂×C₅	direct (N × H)
26. C₂×C₄×C₅	C₂×D₃	semidirect
28. C₂×C₃×C₅	C₂×D₄	semidirect
33. C₂²×D₃	C₂²×C₅	semidirect
34. C₂×C₃×C₄	C₂²×C₅	semidirect
35. D₁₂	C₂²×C₅	direct (N × H)
38. C₂²×C₅	D₁₂	direct (N × H)
40. C₄×C₅	C₂²×D₃	semidirect
43. C₂²×C₄	C₅×D₃	semidirect
44. C₃×C₅	C₂²×D₄	semidirect
45. C₂²×C₃	C₅×D₄	semidirect
48. C₂×D₃	C₂³×C₅	semidirect
49. C₃×C₄	C₂³×C₅	semidirect
51. C₂×C₅	C₂×D₁₂	direct (N × H)
57. C₂×C₄	C₂×C₅×D₃	semidirect
58. C₂×C₃	C₂×C₅×D₄	semidirect
61. C₅	C₂²×D₁₂	direct (N × H)
64. C₂²	C₅×D₁₂	direct (N × H)
66. C₄	C₂²×C₅×D₃	semidirect
67. C₃	C₂²×C₅×D₄	semidirect
69. C₂	C₂×C₅×D₁₂	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×C₅×D₃	2
C₂²×C₃×C₄×C₅	2
C₂×C₅×D₁₂	2
C₂²×C₅×D₄	3
C₂²×D₁₂	5

Subgroups — C22×C5×D12 #92076

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×D₁₂ #92076