Subgroups — C₃×C₅×#120 #91849

Order 480.

260 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃×C₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₅×D₄	C₂	1	3	yes	maximal, complemented
3	C₃×C₅×C₂²⋊C₄	C₂	1	3	yes	maximal, complemented
4	C₅×#120	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂³×C₃×C₅	—	2	6	no	—
6	C₂³×C₃×C₅	C₂²	1	3	yes	—
7	C₂³×C₃×C₅	C₂²	1	1	yes	characteristic, complemented
8	C₃×C₅×D₄	—	2	6	no	—
9	C₂×C₃×C₄×C₅	C₂²	1	3	yes	complemented
10	C₃×#120	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂⁴×C₅	C₂×C₃	1	1	yes	characteristic, complemented
12	C₂×C₅×D₄	C₂×C₃	1	3	yes	complemented
13	C₅×C₂²⋊C₄	C₂×C₃	1	3	yes	complemented
14	C₂²×C₃×C₅	—	2	8	no	—
15	C₂²×C₃×C₅	—	2	6	no	—
16	C₂²×C₃×C₅	—	2	3	no	—
17	C₂²×C₃×C₅	D₄	1	6	yes	complemented
18	C₂²×C₃×C₅	C₂³	1	1	yes	characteristic
19	C₃×C₄×C₅	—	2	3	no	—
20	C₂⁴×C₃	C₂×C₅	1	1	yes	characteristic, complemented
21	C₂×C₃×D₄	C₂×C₅	1	3	yes	complemented
22	C₃×C₂²⋊C₄	C₂×C₅	1	3	yes	complemented
23	C₂³×C₅	—	2	6	no	—
24	C₂³×C₅	C₂²×C₃	1	3	yes	—
25	C₂³×C₅	C₂²×C₃	1	1	yes	characteristic, complemented
26	C₅×D₄	—	2	6	no	—
27	C₂×C₄×C₅	C₂²×C₃	1	3	yes	complemented
28	#120	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
29	C₂×C₃×C₅	—	2	6	no	—
30	C₂×C₃×C₅	—	4	1	no	—
31	C₂×C₃×C₅	C₂×D₄	1	3	yes	—
32	C₂³×C₃	—	2	6	no	—
33	C₂³×C₃	C₂²×C₅	1	3	yes	—
34	C₂³×C₃	C₂²×C₅	1	1	yes	characteristic, complemented
35	C₃×D₄	—	2	6	no	—
36	C₂×C₃×C₄	C₂²×C₅	1	3	yes	complemented
37	C₂²×C₅	—	2	8	no	—
38	C₂²×C₅	—	2	6	no	—
39	C₂²×C₅	—	2	3	no	—
40	C₂²×C₅	C₃×D₄	1	6	yes	complemented
41	C₂²×C₅	C₂³×C₃	1	1	yes	characteristic
42	C₄×C₅	—	2	3	no	—
43	C₂⁴	C₂×C₃×C₅	1	1	yes	characteristic, complemented
44	C₂×D₄	C₂×C₃×C₅	1	3	yes	complemented
45	C₂²⋊C₄	C₂×C₃×C₅	1	3	yes	complemented
46	C₃×C₅	#120	1	1	yes	characteristic, direct product factor, Hall
47	C₂²×C₃	—	2	8	no	—
48	C₂²×C₃	—	2	6	no	—
49	C₂²×C₃	—	2	3	no	—
50	C₂²×C₃	C₅×D₄	1	6	yes	complemented
51	C₂²×C₃	C₂³×C₅	1	1	yes	characteristic
52	C₃×C₄	—	2	3	no	—
53	C₂×C₅	—	2	6	no	—
54	C₂×C₅	—	4	1	no	—
55	C₂×C₅	C₂×C₃×D₄	1	3	yes	—
56	C₂³	—	2	6	no	—
57	C₂³	C₂²×C₃×C₅	1	3	yes	—
58	C₂³	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
59	D₄	—	2	6	no	—
60	C₂×C₄	C₂²×C₃×C₅	1	3	yes	complemented
61	C₂×C₃	—	2	6	no	—
62	C₂×C₃	—	4	1	no	—
63	C₂×C₃	C₂×C₅×D₄	1	3	yes	—
64	C₅	C₃×#120	1	1	yes	characteristic, direct product factor, Sylow
65	C₂²	—	2	8	no	—
66	C₂²	—	2	6	no	—
67	C₂²	—	2	3	no	—
68	C₂²	C₃×C₅×D₄	1	6	yes	complemented
69	C₂²	C₂³×C₃×C₅	1	1	yes	characteristic
70	C₄	—	2	3	no	—
71	C₃	C₅×#120	1	1	yes	characteristic, direct product factor, Sylow
72	C₂	—	2	6	no	—
73	C₂	—	4	1	no	—
74	C₂	C₂×C₃×C₅×D₄	1	3	yes	—

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂³
commutator, Frattini, cosocle	C₂²	C₂³×C₃×C₅
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×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	semidirect
3. C₃×C₅×C₂²⋊C₄	C₂	semidirect
4. C₅×#120	C₃	direct (N × H)
7. C₂³×C₃×C₅	C₂²	semidirect
9. C₂×C₃×C₄×C₅	C₂²	semidirect
10. C₃×#120	C₅	direct (N × H)
11. C₂⁴×C₅	C₂×C₃	semidirect
12. C₂×C₅×D₄	C₂×C₃	semidirect
13. C₅×C₂²⋊C₄	C₂×C₃	semidirect
17. C₂²×C₃×C₅	D₄	semidirect
20. C₂⁴×C₃	C₂×C₅	semidirect
21. C₂×C₃×D₄	C₂×C₅	semidirect
22. C₃×C₂²⋊C₄	C₂×C₅	semidirect
25. C₂³×C₅	C₂²×C₃	semidirect
27. C₂×C₄×C₅	C₂²×C₃	semidirect
28. #120	C₃×C₅	direct (N × H)
34. C₂³×C₃	C₂²×C₅	semidirect
36. C₂×C₃×C₄	C₂²×C₅	semidirect
40. C₂²×C₅	C₃×D₄	semidirect
43. C₂⁴	C₂×C₃×C₅	semidirect
44. C₂×D₄	C₂×C₃×C₅	semidirect
45. C₂²⋊C₄	C₂×C₃×C₅	semidirect
46. C₃×C₅	#120	direct (N × H)
50. C₂²×C₃	C₅×D₄	semidirect
58. C₂³	C₂²×C₃×C₅	semidirect
60. C₂×C₄	C₂²×C₃×C₅	semidirect
64. C₅	C₃×#120	direct (N × H)
68. C₂²	C₃×C₅×D₄	semidirect
71. C₃	C₅×#120	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
C₃×C₅×C₂²⋊C₄	2
C₅×#120	3
C₃×#120	5

Subgroups — C3×C5×#120 #91849

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#120 #91849