Subgroups — C₃²×#4020 #13087799

Order 1440.

648 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×D₂₀	C₂	1	6	yes	maximal, complemented
2	C₃²×C₄²×C₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#4020	C₃	1	4	yes	maximal, direct product factor
4	C₂²×C₃²×D₅	C₂²	1	4	yes	—
5	C₂×C₃²×C₄×C₅	C₂²	1	3	yes	—
6	C₃²×D₂₀	—	2	12	no	—
7	C₃²×#127	—	5	1	no	maximal, Hall
8	C₂×C₃×D₂₀	C₂×C₃	1	24	yes	complemented
9	C₃×C₄²×C₅	C₂×C₃	1	4	yes	complemented
10	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
11	C₂×C₃²×D₅	—	2	12	no	—
12	C₃²×C₄×C₅	D₄	1	6	yes	complemented
13	#4020	C₃²	1	1	yes	characteristic, direct product factor, Hall
14	C₂×C₃²×D₄	—	5	6	no	—
15	C₃²×C₄²	D₅	1	1	yes	characteristic, complemented
16	C₂²×C₃×D₅	C₂²×C₃	1	16	yes	—
17	C₂×C₃×C₄×C₅	C₂²×C₃	1	12	yes	—
18	C₃×D₂₀	—	2	48	no	—
19	C₃×#127	—	5	4	no	—
20	C₂×C₃²×C₅	C₂×D₄	1	3	yes	—
21	C₃²×D₅	—	4	4	no	—
22	C₂×D₂₀	C₂×C₃²	1	6	yes	complemented
23	C₄²×C₅	C₂×C₃²	1	1	yes	characteristic, complemented
24	C₂³×C₃²	—	5	4	no	—
25	C₃²×D₄	—	10	12	no	—
26	C₂×C₃²×C₄	C₂×D₅	1	3	yes	—
27	C₂²×C₃×C₅	C₂³×C₃	1	4	yes	—
28	C₂×C₃×D₅	—	2	48	no	—
29	C₃×C₄×C₅	C₃×D₄	1	24	yes	complemented
30	C₂×C₃×D₄	—	5	24	no	—
31	C₃×C₄²	C₃×D₅	1	4	yes	complemented
32	C₃²×C₅	#127	1	1	yes	characteristic, complemented, Hall
33	C₂²×D₅	C₂²×C₃²	1	4	yes	—
34	C₂×C₄×C₅	C₂²×C₃²	1	3	yes	—
35	D₂₀	—	2	12	no	—
36	C₂²×C₃²	—	10	12	no	—
37	C₂²×C₃²	C₂²×D₅	1	1	yes	characteristic
38	C₃²×C₄	D₂₀	1	6	yes	complemented
39	#127	—	5	1	no	Sylow
40	C₂×C₃×C₅	C₂×C₃×D₄	1	12	yes	—
41	C₃×D₅	—	4	16	no	—
42	C₂³×C₃	—	5	16	no	—
43	C₃×D₄	—	10	48	no	—
44	C₂×C₃×C₄	C₂×C₃×D₅	1	12	yes	—
45	C₂²×C₅	C₂³×C₃²	1	1	yes	characteristic
46	C₂×D₅	—	2	12	no	—
47	C₄×C₅	C₃²×D₄	1	6	yes	complemented
48	C₂×C₃²	—	20	4	no	—
49	C₂×C₃²	C₂×D₂₀	1	3	yes	—
50	C₂×D₄	—	5	6	no	—
51	C₄²	C₃²×D₅	1	1	yes	characteristic, complemented
52	C₃×C₅	C₃×#127	1	4	yes	complemented
53	C₂²×C₃	—	10	48	no	—
54	C₂²×C₃	C₂²×C₃×D₅	1	4	yes	—
55	C₃×C₄	C₃×D₂₀	1	24	yes	complemented
56	C₂×C₅	C₂×C₃²×D₄	1	3	yes	—
57	D₅	—	4	4	no	—
58	C₃²	#4020	1	1	yes	characteristic, direct product factor, Sylow
59	C₂³	—	5	4	no	—
60	D₄	—	10	12	no	—
61	C₂×C₄	C₂×C₃²×D₅	1	3	yes	—
62	C₂×C₃	—	20	16	no	—
63	C₂×C₃	C₂×C₃×D₂₀	1	12	yes	—
64	C₅	C₃²×#127	1	1	yes	characteristic, complemented, Sylow
65	C₂²	—	10	12	no	—
66	C₂²	C₂²×C₃²×D₅	1	1	yes	characteristic
67	C₄	C₃²×D₂₀	1	6	yes	complemented
68	C₃	C₃×#4020	1	4	yes	direct product factor
69	C₂	—	20	4	no	—
70	C₂	C₂×C₃²×D₂₀	1	3	yes	—

	subgroup	G/H
center	C₂²×C₃²	C₂²×D₅
commutator, cosocle	C₂²×C₅	C₂³×C₃²
Frattini	C₂²	C₂²×C₃²×D₅
Fitting	C₃²×C₄²×C₅	C₂
socle	C₂²×C₃²×C₅	C₂³
hypercenter	C₃²×C₄²	D₅
hypocenter	C₅	C₃²×#127
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄²

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×D₂₀	C₂	semidirect
2. C₃²×C₄²×C₅	C₂	semidirect
3. C₃×#4020	C₃	direct (N × H)
8. C₂×C₃×D₂₀	C₂×C₃	semidirect
9. C₃×C₄²×C₅	C₂×C₃	semidirect
12. C₃²×C₄×C₅	D₄	semidirect
13. #4020	C₃²	direct (N × H)
15. C₃²×C₄²	D₅	semidirect
22. C₂×D₂₀	C₂×C₃²	semidirect
23. C₄²×C₅	C₂×C₃²	semidirect
29. C₃×C₄×C₅	C₃×D₄	semidirect
31. C₃×C₄²	C₃×D₅	semidirect
32. C₃²×C₅	#127	semidirect
38. C₃²×C₄	D₂₀	semidirect
47. C₄×C₅	C₃²×D₄	semidirect
51. C₄²	C₃²×D₅	semidirect
52. C₃×C₅	C₃×#127	semidirect
55. C₃×C₄	C₃×D₂₀	semidirect
58. C₃²	#4020	direct (N × H)
64. C₅	C₃²×#127	semidirect
67. C₄	C₃²×D₂₀	semidirect
68. C₃	C₃×#4020	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃²×D₂₀	2
C₃²×C₄²×C₅	2
C₃×#4020	3
C₃²×#127	5

Subgroups — C32×#4020 #13087799

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#4020 #13087799