Subgroups — C₂²×C₃×C₅×D₉ #11762002

Order 1080.

212 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×C₅×D₉	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₃×C₅×D₃	—	3	1	no	maximal
4	C₂²×C₅×D₉	C₃	1	1	yes	maximal, characteristic, direct product factor
5	C₂×C₃×C₅×C₉	C₂²	1	3	yes	complemented
6	C₃×C₅×D₉	C₂²	1	4	yes	direct product factor
7	C₂²×C₃×D₉	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂²×C₃²×C₅	D₃	1	1	yes	characteristic
9	C₂×C₃×C₅×D₃	—	3	6	no	—
10	C₂²×C₅×C₉	—	2	1	no	—
11	C₂²×C₅×C₉	C₂×C₃	1	1	yes	characteristic, complemented
12	C₂×C₅×D₉	C₂×C₃	1	6	yes	direct product factor
13	C₃×C₅×C₉	C₂³	1	1	yes	characteristic, complemented, Hall
14	C₂³×C₃×C₅	—	9	1	no	—
15	C₂²×C₅×D₃	—	3	1	no	—
16	C₂²×C₃×C₉	C₂×C₅	1	1	yes	characteristic, complemented
17	C₂×C₃×D₉	C₂×C₅	1	6	yes	direct product factor
18	C₂×C₃²×C₅	C₂×D₃	1	3	yes	—
19	C₃×C₅×D₃	—	3	4	no	—
20	C₂×C₅×C₉	—	2	3	no	—
21	C₂×C₅×C₉	C₂²×C₃	1	3	yes	complemented
22	C₅×D₉	C₂²×C₃	1	4	yes	direct product factor
23	C₂²×C₃×D₃	—	3	1	no	—
24	C₂²×D₉	C₃×C₅	1	1	yes	characteristic, direct product factor
25	C₂²×C₃×C₅	—	2	1	no	—
26	C₂²×C₃×C₅	—	9	6	no	—
27	C₂²×C₃×C₅	D₉	1	1	yes	characteristic, direct product factor
28	C₂²×C₃×C₅	C₃×D₃	1	1	yes	characteristic
29	C₂×C₅×D₃	—	3	6	no	—
30	C₂×C₃×C₉	C₂²×C₅	1	3	yes	complemented
31	C₃×D₉	C₂²×C₅	1	4	yes	direct product factor
32	C₃²×C₅	C₂²×D₃	1	1	yes	characteristic
33	C₅×C₉	—	2	1	no	—
34	C₅×C₉	C₂³×C₃	1	1	yes	characteristic, complemented
35	C₂³×C₅	—	9	1	no	Hall
36	C₂²×C₃²	C₅×D₃	1	1	yes	characteristic
37	C₂×C₃×D₃	—	3	6	no	—
38	C₂²×C₉	—	2	1	no	—
39	C₂²×C₉	C₂×C₃×C₅	1	1	yes	characteristic, complemented
40	C₂×D₉	C₂×C₃×C₅	1	6	yes	direct product factor
41	C₂×C₃×C₅	—	2	3	no	—
42	C₂×C₃×C₅	—	9	4	no	—
43	C₂×C₃×C₅	C₂×D₉	1	3	yes	direct product factor
44	C₂×C₃×C₅	C₂×C₃×D₃	1	3	yes	—
45	C₅×D₃	—	3	4	no	—
46	C₃×C₉	C₂³×C₅	1	1	yes	characteristic, complemented, Sylow
47	C₂³×C₃	—	9	1	no	—
48	C₂²×D₃	—	3	1	no	—
49	C₂²×C₅	—	9	6	no	—
50	C₂²×C₅	C₃×D₉	1	1	yes	characteristic, direct product factor
51	C₂×C₃²	C₂×C₅×D₃	1	3	yes	—
52	C₃×D₃	—	3	4	no	—
53	C₂×C₉	—	2	3	no	—
54	C₂×C₉	C₂²×C₃×C₅	1	3	yes	complemented
55	D₉	C₂²×C₃×C₅	1	4	yes	direct product factor
56	C₃×C₅	—	2	1	no	—
57	C₃×C₅	C₂²×D₉	1	1	yes	characteristic, direct product factor
58	C₃×C₅	C₂²×C₃×D₃	1	1	yes	characteristic
59	C₂²×C₃	—	2	1	no	—
60	C₂²×C₃	—	9	6	no	—
61	C₂²×C₃	C₅×D₉	1	1	yes	characteristic, direct product factor
62	C₂²×C₃	C₃×C₅×D₃	1	1	yes	characteristic
63	C₂×D₃	—	3	6	no	—
64	C₂×C₅	—	9	4	no	—
65	C₂×C₅	C₂×C₃×D₉	1	3	yes	direct product factor
66	C₃²	C₂²×C₅×D₃	1	1	yes	characteristic
67	C₉	—	2	1	no	—
68	C₉	C₂³×C₃×C₅	1	1	yes	characteristic, complemented
69	C₂³	—	9	1	no	Sylow
70	C₂×C₃	—	2	3	no	—
71	C₂×C₃	—	9	4	no	—
72	C₂×C₃	C₂×C₅×D₉	1	3	yes	direct product factor
73	C₂×C₃	C₂×C₃×C₅×D₃	1	3	yes	—
74	D₃	—	3	4	no	—
75	C₅	C₂²×C₃×D₉	1	1	yes	characteristic, direct product factor, Sylow
76	C₂²	—	9	6	no	—
77	C₂²	C₃×C₅×D₉	1	1	yes	characteristic, direct product factor
78	C₃	—	2	1	no	—
79	C₃	C₂²×C₅×D₉	1	1	yes	characteristic, direct product factor
80	C₃	C₂²×C₃×C₅×D₃	1	1	yes	characteristic
81	C₂	—	9	4	no	—
82	C₂	C₂×C₃×C₅×D₉	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₃×C₅	D₉
commutator, cosocle, hypocenter	C₉	C₂³×C₃×C₅
Frattini	C₃	C₂²×C₃×C₅×D₃
Fitting	C₂²×C₃×C₅×C₉	C₂
socle	C₂²×C₃²×C₅	D₃
LCS	G ▷ C₉
UCS	C₁ ◁ C₂²×C₃×C₅

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×C₅×D₉	C₂	direct (N × H)
4. C₂²×C₅×D₉	C₃	direct (N × H)
5. C₂×C₃×C₅×C₉	C₂²	semidirect
6. C₃×C₅×D₉	C₂²	direct (N × H)
7. C₂²×C₃×D₉	C₅	direct (N × H)
11. C₂²×C₅×C₉	C₂×C₃	semidirect
12. C₂×C₅×D₉	C₂×C₃	direct (N × H)
13. C₃×C₅×C₉	C₂³	semidirect
16. C₂²×C₃×C₉	C₂×C₅	semidirect
17. C₂×C₃×D₉	C₂×C₅	direct (N × H)
21. C₂×C₅×C₉	C₂²×C₃	semidirect
22. C₅×D₉	C₂²×C₃	direct (N × H)
24. C₂²×D₉	C₃×C₅	direct (N × H)
27. C₂²×C₃×C₅	D₉	direct (N × H)
30. C₂×C₃×C₉	C₂²×C₅	semidirect
31. C₃×D₉	C₂²×C₅	direct (N × H)
34. C₅×C₉	C₂³×C₃	semidirect
39. C₂²×C₉	C₂×C₃×C₅	semidirect
40. C₂×D₉	C₂×C₃×C₅	direct (N × H)
43. C₂×C₃×C₅	C₂×D₉	direct (N × H)
46. C₃×C₉	C₂³×C₅	semidirect
50. C₂²×C₅	C₃×D₉	direct (N × H)
54. C₂×C₉	C₂²×C₃×C₅	semidirect
55. D₉	C₂²×C₃×C₅	direct (N × H)
57. C₃×C₅	C₂²×D₉	direct (N × H)
61. C₂²×C₃	C₅×D₉	direct (N × H)
65. C₂×C₅	C₂×C₃×D₉	direct (N × H)
68. C₉	C₂³×C₃×C₅	semidirect
72. C₂×C₃	C₂×C₅×D₉	direct (N × H)
75. C₅	C₂²×C₃×D₉	direct (N × H)
77. C₂²	C₃×C₅×D₉	direct (N × H)
79. C₃	C₂²×C₅×D₉	direct (N × H)
82. C₂	C₂×C₃×C₅×D₉	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×C₅×D₉	2
C₂²×C₃×C₅×D₃	3
C₂²×C₅×D₉	3
C₂²×C₃×D₉	5

Subgroups — C22×C3×C5×D9 #11762002

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₅×D₉ #11762002