Subgroups — C₂²×C₃×C₅×D₄ #92105

Order 480.

472 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, socle	C₂³×C₃×C₅	C₂²
commutator, Frattini, cosocle	C₂	C₂⁴×C₃×C₅
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₃×C₅	C₂	semidirect
2. C₂×C₃×C₅×D₄	C₂	direct (N × H)
3. C₂²×C₃×C₄×C₅	C₂	semidirect
4. C₂²×C₅×D₄	C₃	direct (N × H)
7. C₂³×C₃×C₅	C₂²	semidirect
8. C₃×C₅×D₄	C₂²	direct (N × H)
9. C₂×C₃×C₄×C₅	C₂²	semidirect
10. C₂²×C₃×D₄	C₅	direct (N × H)
11. C₂⁴×C₅	C₂×C₃	semidirect
12. C₂×C₅×D₄	C₂×C₃	direct (N × H)
13. C₂²×C₄×C₅	C₂×C₃	semidirect
15. C₂²×C₃×C₅	D₄	direct (N × H)
17. C₂²×C₃×C₅	C₂³	semidirect
18. C₃×C₄×C₅	C₂³	semidirect
19. C₂⁴×C₃	C₂×C₅	semidirect
20. C₂×C₃×D₄	C₂×C₅	direct (N × H)
21. C₂²×C₃×C₄	C₂×C₅	semidirect
24. C₂³×C₅	C₂²×C₃	semidirect
25. C₅×D₄	C₂²×C₃	direct (N × H)
26. C₂×C₄×C₅	C₂²×C₃	semidirect
27. C₂²×D₄	C₃×C₅	direct (N × H)
29. C₂×C₃×C₅	C₂×D₄	direct (N × H)
33. C₂³×C₃	C₂²×C₅	semidirect
34. C₃×D₄	C₂²×C₅	direct (N × H)
35. C₂×C₃×C₄	C₂²×C₅	semidirect
37. C₂²×C₅	C₃×D₄	direct (N × H)
39. C₂²×C₅	C₂³×C₃	semidirect
40. C₄×C₅	C₂³×C₃	semidirect
41. C₂⁴	C₂×C₃×C₅	semidirect
42. C₂×D₄	C₂×C₃×C₅	direct (N × H)
43. C₂²×C₄	C₂×C₃×C₅	semidirect
44. C₃×C₅	C₂²×D₄	direct (N × H)
46. C₂²×C₃	C₅×D₄	direct (N × H)
48. C₂²×C₃	C₂³×C₅	semidirect
49. C₃×C₄	C₂³×C₅	semidirect
51. C₂×C₅	C₂×C₃×D₄	direct (N × H)
55. C₂³	C₂²×C₃×C₅	semidirect
56. D₄	C₂²×C₃×C₅	direct (N × H)
57. C₂×C₄	C₂²×C₃×C₅	semidirect
59. C₂×C₃	C₂×C₅×D₄	direct (N × H)
61. C₅	C₂²×C₃×D₄	direct (N × H)
63. C₂²	C₃×C₅×D₄	direct (N × H)
65. C₂²	C₂³×C₃×C₅	semidirect
66. C₄	C₂³×C₃×C₅	semidirect
67. C₃	C₂²×C₅×D₄	direct (N × H)
69. C₂	C₂×C₃×C₅×D₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₃×C₅	2
C₂×C₃×C₅×D₄	2
C₂²×C₃×C₄×C₅	2
C₂²×C₅×D₄	3
C₂²×C₃×D₄	5

Subgroups — C22×C3×C5×D4 #92105

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₅×D₄ #92105