Subgroups — C₂²×D₅×3₋¹⁺² #11762001

Order 1080.

256 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₅×3₋¹⁺²	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₃²×D₅	C₃	1	1	yes	maximal, characteristic
4	C₂²×C₉×D₅	C₃	1	3	yes	maximal, complemented
5	C₂×C₅×3₋¹⁺²	C₂²	1	3	yes	complemented
6	D₅×3₋¹⁺²	C₂²	1	4	yes	direct product factor
7	C₂³×3₋¹⁺²	—	5	1	no	maximal, Hall
8	C₂²×C₃²×C₅	C₂×C₃	1	1	yes	characteristic
9	C₂×C₃²×D₅	C₂×C₃	1	6	yes	—
10	C₂²×C₅×C₉	C₂×C₃	1	3	yes	complemented
11	C₂×C₉×D₅	C₂×C₃	1	18	yes	complemented
12	C₅×3₋¹⁺²	C₂³	1	1	yes	characteristic, complemented, Hall
13	C₂²×C₃×D₅	—	3	1	no	—
14	C₂²×C₃×D₅	C₃²	1	1	yes	characteristic
15	C₂²×3₋¹⁺²	—	5	6	no	—
16	C₂²×3₋¹⁺²	D₅	1	1	yes	characteristic, direct product factor
17	C₂×C₃²×C₅	C₂²×C₃	1	3	yes	—
18	C₃²×D₅	C₂²×C₃	1	4	yes	—
19	C₂×C₅×C₉	C₂²×C₃	1	9	yes	complemented
20	C₉×D₅	C₂²×C₃	1	12	yes	complemented
21	C₂³×C₃²	—	5	1	no	—
22	C₂³×C₉	—	5	3	no	—
23	C₂²×C₃×C₅	—	3	1	no	—
24	C₂²×C₃×C₅	C₂×C₃²	1	1	yes	characteristic
25	C₂×C₃×D₅	—	3	6	no	—
26	C₂×C₃×D₅	C₂×C₃²	1	6	yes	—
27	C₂×3₋¹⁺²	—	5	4	no	—
28	C₂×3₋¹⁺²	C₂×D₅	1	3	yes	direct product factor
29	C₃²×C₅	C₂³×C₃	1	1	yes	characteristic
30	C₅×C₉	C₂³×C₃	1	3	yes	complemented
31	C₂²×D₅	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Hall
32	C₂²×C₃²	—	5	6	no	—
33	C₂²×C₃²	C₃×D₅	1	1	yes	characteristic
34	C₂²×C₉	—	5	18	no	—
35	C₂²×C₉	C₃×D₅	1	3	yes	complemented
36	C₂×C₃×C₅	—	3	3	no	—
37	C₂×C₃×C₅	C₂²×C₃²	1	3	yes	—
38	C₃×D₅	—	3	4	no	—
39	C₃×D₅	C₂²×C₃²	1	4	yes	—
40	3₋¹⁺²	C₂²×D₅	1	1	yes	characteristic, direct product factor, Sylow
41	C₂³×C₃	—	5	1	no	—
42	C₂³×C₃	—	15	1	no	—
43	C₂²×C₅	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
44	C₂×D₅	C₂×3₋¹⁺²	1	6	yes	direct product factor
45	C₂×C₃²	—	5	4	no	—
46	C₂×C₃²	C₂×C₃×D₅	1	3	yes	—
47	C₂×C₉	—	5	12	no	—
48	C₂×C₉	C₂×C₃×D₅	1	9	yes	complemented
49	C₃×C₅	—	3	1	no	—
50	C₃×C₅	C₂³×C₃²	1	1	yes	characteristic
51	C₂²×C₃	—	3	1	no	—
52	C₂²×C₃	—	5	6	no	—
53	C₂²×C₃	—	15	6	no	—
54	C₂²×C₃	C₃²×D₅	1	1	yes	characteristic
55	C₂×C₅	C₂²×3₋¹⁺²	1	3	yes	complemented
56	D₅	C₂²×3₋¹⁺²	1	4	yes	direct product factor
57	C₃²	C₂²×C₃×D₅	1	1	yes	characteristic
58	C₉	C₂²×C₃×D₅	1	3	yes	complemented
59	C₂³	—	5	1	no	Sylow
60	C₂×C₃	—	3	3	no	—
61	C₂×C₃	—	5	4	no	—
62	C₂×C₃	—	15	4	no	—
63	C₂×C₃	C₂×C₃²×D₅	1	3	yes	—
64	C₅	C₂³×3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
65	C₂²	—	5	6	no	—
66	C₂²	D₅×3₋¹⁺²	1	1	yes	characteristic, direct product factor
67	C₃	—	3	1	no	—
68	C₃	C₂²×C₃²×D₅	1	1	yes	characteristic
69	C₂	—	5	4	no	—
70	C₂	C₂×D₅×3₋¹⁺²	1	3	yes	direct product factor

	subgroup	G/H
center	C₂²×C₃	C₃²×D₅
commutator, cosocle	C₃×C₅	C₂³×C₃²
Frattini	C₃	C₂²×C₃²×D₅
Fitting	C₂²×C₅×3₋¹⁺²	C₂
socle	C₂²×C₃×C₅	C₂×C₃²
prime radical	C₂²×C₃²×D₅	C₃
hypercenter	C₂²×3₋¹⁺²	D₅
hypocenter	C₅	C₂³×3₋¹⁺²
LCS	G ▷ C₃×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×3₋¹⁺²	C₂	semidirect
2. C₂×D₅×3₋¹⁺²	C₂	direct (N × H)
4. C₂²×C₉×D₅	C₃	semidirect
5. C₂×C₅×3₋¹⁺²	C₂²	semidirect
6. D₅×3₋¹⁺²	C₂²	direct (N × H)
10. C₂²×C₅×C₉	C₂×C₃	semidirect
11. C₂×C₉×D₅	C₂×C₃	semidirect
12. C₅×3₋¹⁺²	C₂³	semidirect
16. C₂²×3₋¹⁺²	D₅	direct (N × H)
19. C₂×C₅×C₉	C₂²×C₃	semidirect
20. C₉×D₅	C₂²×C₃	semidirect
28. C₂×3₋¹⁺²	C₂×D₅	direct (N × H)
30. C₅×C₉	C₂³×C₃	semidirect
31. C₂²×D₅	3₋¹⁺²	direct (N × H)
35. C₂²×C₉	C₃×D₅	semidirect
40. 3₋¹⁺²	C₂²×D₅	direct (N × H)
43. C₂²×C₅	C₂×3₋¹⁺²	semidirect
44. C₂×D₅	C₂×3₋¹⁺²	direct (N × H)
48. C₂×C₉	C₂×C₃×D₅	semidirect
55. C₂×C₅	C₂²×3₋¹⁺²	semidirect
56. D₅	C₂²×3₋¹⁺²	direct (N × H)
58. C₉	C₂²×C₃×D₅	semidirect
64. C₅	C₂³×3₋¹⁺²	semidirect
66. C₂²	D₅×3₋¹⁺²	direct (N × H)
70. C₂	C₂×D₅×3₋¹⁺²	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×3₋¹⁺²	2
C₂×D₅×3₋¹⁺²	2
C₂²×C₃²×D₅	3
C₂²×C₉×D₅	3
C₂³×3₋¹⁺²	5

Subgroups — C22×D5×3−1+2 #11762001

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₅×3₋¹⁺² #11762001