Subgroups — C₂²×C₃⋊D₄₅ #11762012

Order 1080.

320 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃⋊D₄₅	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₃⋊D₁₅	—	3	1	no	maximal
4	C₂²×D₄₅	—	3	3	no	maximal
5	C₂×C₃×C₅×C₉	C₂²	1	3	yes	complemented
6	C₃⋊D₄₅	C₂²	1	4	yes	direct product factor
7	C₂²×C₃⋊D₉	—	5	1	no	maximal, Hall
8	C₂²×C₃²×C₅	D₃	1	1	yes	characteristic
9	C₂×C₃⋊D₁₅	—	3	6	no	—
10	C₂²×C₅×C₉	D₃	1	3	yes	complemented
11	C₂×D₄₅	—	3	18	no	—
12	C₃×C₅×C₉	C₂³	1	1	yes	characteristic, complemented, Hall
13	C₂²×D₁₅	—	9	3	no	—
14	C₂²×D₁₅	—	9	1	no	—
15	C₂²×C₃×C₉	D₅	1	1	yes	characteristic, complemented
16	C₂×C₃⋊D₉	—	5	6	no	—
17	C₂×C₃²×C₅	C₂×D₃	1	3	yes	—
18	C₃⋊D₁₅	—	3	4	no	—
19	C₂×C₅×C₉	C₂×D₃	1	9	yes	complemented
20	D₄₅	—	3	12	no	—
21	C₂²×C₃⋊D₃	—	15	1	no	—
22	C₂²×D₉	—	15	3	no	—
23	C₂²×C₃×C₅	D₉	1	3	yes	complemented
24	C₂²×C₃×C₅	C₃⋊D₃	1	1	yes	characteristic
25	C₂×D₁₅	—	9	18	no	—
26	C₂×D₁₅	—	9	6	no	—
27	C₂×C₃×C₉	C₂×D₅	1	3	yes	complemented
28	C₃⋊D₉	—	5	4	no	—
29	C₃²×C₅	C₂²×D₃	1	1	yes	characteristic
30	C₅×C₉	C₂²×D₃	1	3	yes	complemented
31	C₂²×D₅	—	27	1	no	Hall
32	C₂²×C₃²	D₁₅	1	1	yes	characteristic
33	C₂×C₃⋊D₃	—	15	6	no	—
34	C₂²×C₉	D₁₅	1	3	yes	complemented
35	C₂×D₉	—	15	18	no	—
36	C₂×C₃×C₅	C₂×D₉	1	9	yes	complemented
37	C₂×C₃×C₅	C₂×C₃⋊D₃	1	3	yes	—
38	D₁₅	—	9	12	no	—
39	D₁₅	—	9	4	no	—
40	C₃×C₉	C₂²×D₅	1	1	yes	characteristic, complemented, Sylow
41	C₂²×D₃	—	45	3	no	—
42	C₂²×D₃	—	45	1	no	—
43	C₂²×C₅	C₃⋊D₉	1	1	yes	characteristic, complemented
44	C₂×D₅	—	27	6	no	—
45	C₂×C₃²	C₂×D₁₅	1	3	yes	—
46	C₃⋊D₃	—	15	4	no	—
47	C₂×C₉	C₂×D₁₅	1	9	yes	complemented
48	D₉	—	15	12	no	—
49	C₃×C₅	C₂²×D₉	1	3	yes	complemented
50	C₃×C₅	C₂²×C₃⋊D₃	1	1	yes	characteristic
51	C₂²×C₃	D₄₅	1	3	yes	complemented
52	C₂²×C₃	C₃⋊D₁₅	1	1	yes	characteristic
53	C₂×D₃	—	45	18	no	—
54	C₂×D₃	—	45	6	no	—
55	C₂×C₅	C₂×C₃⋊D₉	1	3	yes	complemented
56	D₅	—	27	4	no	—
57	C₃²	C₂²×D₁₅	1	1	yes	characteristic
58	C₉	C₂²×D₁₅	1	3	yes	complemented
59	C₂³	—	135	1	no	Sylow
60	C₂×C₃	C₂×D₄₅	1	9	yes	complemented
61	C₂×C₃	C₂×C₃⋊D₁₅	1	3	yes	—
62	D₃	—	45	12	no	—
63	D₃	—	45	4	no	—
64	C₅	C₂²×C₃⋊D₉	1	1	yes	characteristic, complemented, Sylow
65	C₂²	—	135	6	no	—
66	C₂²	C₃⋊D₄₅	1	1	yes	characteristic, direct product factor
67	C₃	C₂²×D₄₅	1	3	yes	complemented
68	C₃	C₂²×C₃⋊D₁₅	1	1	yes	characteristic
69	C₂	—	135	4	no	—
70	C₂	C₂×C₃⋊D₄₅	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²	C₃⋊D₄₅
commutator, cosocle, hypocenter	C₃×C₅×C₉	C₂³
Frattini	C₃	C₂²×C₃⋊D₁₅
Fitting	C₂²×C₃×C₅×C₉	C₂
socle	C₂²×C₃²×C₅	D₃
LCS	G ▷ C₃×C₅×C₉
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₅×C₉	C₂	semidirect
2. C₂×C₃⋊D₄₅	C₂	direct (N × H)
5. C₂×C₃×C₅×C₉	C₂²	semidirect
6. C₃⋊D₄₅	C₂²	direct (N × H)
10. C₂²×C₅×C₉	D₃	semidirect
12. C₃×C₅×C₉	C₂³	semidirect
15. C₂²×C₃×C₉	D₅	semidirect
19. C₂×C₅×C₉	C₂×D₃	semidirect
23. C₂²×C₃×C₅	D₉	semidirect
27. C₂×C₃×C₉	C₂×D₅	semidirect
30. C₅×C₉	C₂²×D₃	semidirect
34. C₂²×C₉	D₁₅	semidirect
36. C₂×C₃×C₅	C₂×D₉	semidirect
40. C₃×C₉	C₂²×D₅	semidirect
43. C₂²×C₅	C₃⋊D₉	semidirect
47. C₂×C₉	C₂×D₁₅	semidirect
49. C₃×C₅	C₂²×D₉	semidirect
51. C₂²×C₃	D₄₅	semidirect
55. C₂×C₅	C₂×C₃⋊D₉	semidirect
58. C₉	C₂²×D₁₅	semidirect
60. C₂×C₃	C₂×D₄₅	semidirect
64. C₅	C₂²×C₃⋊D₉	semidirect
66. C₂²	C₃⋊D₄₅	direct (N × H)
67. C₃	C₂²×D₄₅	semidirect
70. C₂	C₂×C₃⋊D₄₅	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×C₃×C₅×C₉	2
C₂×C₃⋊D₄₅	2
C₂²×C₃⋊D₁₅	3
C₂²×D₄₅	3
C₂²×C₃⋊D₉	5

Subgroups — C22×C3⋊D45 #11762012

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃⋊D₄₅ #11762012