Subgroups — C₅×#64083 #13088057

Order 1440.

648 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₄⋊(C₃⋊D₃)	C₂	1	6	yes	maximal, complemented
2	C₃²×C₄²×C₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#874	—	3	4	no	maximal
4	C₂²×C₅×C₃⋊D₃	C₂²	1	4	yes	—
5	C₂×C₃²×C₄×C₅	C₂²	1	3	yes	—
6	C₅×C₄⋊(C₃⋊D₃)	—	2	12	no	—
7	#64083	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂×C₅×D₁₂	—	3	24	no	—
9	C₃×C₄²×C₅	D₃	1	4	yes	complemented
10	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
11	C₂×C₅×C₃⋊D₃	—	2	12	no	—
12	C₃²×C₄×C₅	D₄	1	6	yes	complemented
13	C₅×#127	—	9	1	no	Hall
14	C₂×C₄⋊(C₃⋊D₃)	C₂×C₅	1	6	yes	complemented
15	C₃²×C₄²	C₂×C₅	1	1	yes	characteristic, complemented
16	C₂²×C₅×D₃	—	3	16	no	—
17	C₂×C₃×C₄×C₅	C₂×D₃	1	12	yes	—
18	C₅×D₁₂	—	6	48	no	—
19	#874	—	3	4	no	—
20	C₂×C₃²×C₅	C₂×D₄	1	3	yes	—
21	C₅×C₃⋊D₃	—	4	4	no	—
22	C₂×C₅×D₄	—	9	6	no	—
23	C₄²×C₅	C₃⋊D₃	1	1	yes	characteristic, complemented
24	C₂²×C₃⋊D₃	C₂²×C₅	1	4	yes	—
25	C₂×C₃²×C₄	C₂²×C₅	1	3	yes	—
26	C₄⋊(C₃⋊D₃)	—	2	12	no	—
27	C₂²×C₃×C₅	C₂²×D₃	1	4	yes	—
28	C₂×C₅×D₃	—	6	48	no	—
29	C₃×C₄×C₅	D₁₂	1	24	yes	complemented
30	C₂×D₁₂	—	3	24	no	—
31	C₃×C₄²	C₅×D₃	1	4	yes	complemented
32	C₃²×C₅	#127	1	1	yes	characteristic, complemented, Hall
33	C₂³×C₅	—	9	4	no	—
34	C₅×D₄	—	18	12	no	—
35	C₂×C₄×C₅	C₂×C₃⋊D₃	1	3	yes	—
36	C₂²×C₃²	C₂³×C₅	1	1	yes	characteristic
37	C₂×C₃⋊D₃	—	2	12	no	—
38	C₃²×C₄	C₅×D₄	1	6	yes	complemented
39	#127	—	9	1	no	Sylow
40	C₂×C₃×C₅	C₂×D₁₂	1	12	yes	—
41	C₅×D₃	—	12	16	no	—
42	C₂²×D₃	—	3	16	no	—
43	C₂×C₃×C₄	C₂×C₅×D₃	1	12	yes	—
44	D₁₂	—	6	48	no	—
45	C₂²×C₅	—	18	12	no	—
46	C₂²×C₅	C₂²×C₃⋊D₃	1	1	yes	characteristic
47	C₄×C₅	C₄⋊(C₃⋊D₃)	1	6	yes	complemented
48	C₂×C₃²	C₂×C₅×D₄	1	3	yes	—
49	C₃⋊D₃	—	4	4	no	—
50	C₂×D₄	—	9	6	no	—
51	C₄²	C₅×C₃⋊D₃	1	1	yes	characteristic, complemented
52	C₃×C₅	#874	1	4	yes	complemented
53	C₂²×C₃	C₂²×C₅×D₃	1	4	yes	—
54	C₂×D₃	—	6	48	no	—
55	C₃×C₄	C₅×D₁₂	1	24	yes	complemented
56	C₂×C₅	—	36	4	no	—
57	C₂×C₅	C₂×C₄⋊(C₃⋊D₃)	1	3	yes	—
58	C₃²	C₅×#127	1	1	yes	characteristic, complemented, Sylow
59	C₂³	—	9	4	no	—
60	D₄	—	18	12	no	—
61	C₂×C₄	C₂×C₅×C₃⋊D₃	1	3	yes	—
62	C₂×C₃	C₂×C₅×D₁₂	1	12	yes	—
63	D₃	—	12	16	no	—
64	C₅	#64083	1	1	yes	characteristic, direct product factor, Sylow
65	C₂²	—	18	12	no	—
66	C₂²	C₂²×C₅×C₃⋊D₃	1	1	yes	characteristic
67	C₄	C₅×C₄⋊(C₃⋊D₃)	1	6	yes	complemented
68	C₃	C₅×#874	1	4	yes	complemented
69	C₂	—	36	4	no	—
70	C₂	C₂×C₅×C₄⋊(C₃⋊D₃)	1	3	yes	—

	subgroup	G/H
center	C₂²×C₅	C₂²×C₃⋊D₃
commutator, cosocle	C₂²×C₃²	C₂³×C₅
Frattini	C₂²	C₂²×C₅×C₃⋊D₃
Fitting	C₃²×C₄²×C₅	C₂
socle	C₂²×C₃²×C₅	C₂³
hypercenter	C₄²×C₅	C₃⋊D₃
hypocenter	C₃²	C₅×#127
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×C₄⋊(C₃⋊D₃)	C₂	semidirect
2. C₃²×C₄²×C₅	C₂	semidirect
7. #64083	C₅	direct (N × H)
9. C₃×C₄²×C₅	D₃	semidirect
12. C₃²×C₄×C₅	D₄	semidirect
14. C₂×C₄⋊(C₃⋊D₃)	C₂×C₅	semidirect
15. C₃²×C₄²	C₂×C₅	semidirect
23. C₄²×C₅	C₃⋊D₃	semidirect
29. C₃×C₄×C₅	D₁₂	semidirect
31. C₃×C₄²	C₅×D₃	semidirect
32. C₃²×C₅	#127	semidirect
38. C₃²×C₄	C₅×D₄	semidirect
47. C₄×C₅	C₄⋊(C₃⋊D₃)	semidirect
51. C₄²	C₅×C₃⋊D₃	semidirect
52. C₃×C₅	#874	semidirect
55. C₃×C₄	C₅×D₁₂	semidirect
58. C₃²	C₅×#127	semidirect
64. C₅	#64083	direct (N × H)
67. C₄	C₅×C₄⋊(C₃⋊D₃)	semidirect
68. C₃	C₅×#874	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×C₄⋊(C₃⋊D₃)	2
C₃²×C₄²×C₅	2
C₅×#874	3
#64083	5

Subgroups — C5×#64083 #13088057

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#64083 #13088057