Subgroups — C₂³×C₉×C₃⋊D₃ #13049351

Order 1296.

1462 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃²×C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₉×C₃⋊D₃	C₂	1	14	yes	maximal, direct product factor
3	C₂³×C₃×C₃⋊D₃	C₃	1	1	yes	maximal, characteristic
4	C₂³×D₃×C₉	—	3	4	no	maximal
5	C₂²×C₃²×C₉	C₂²	1	7	yes	complemented
6	C₂×C₉×C₃⋊D₃	C₂²	1	28	yes	direct product factor
7	C₂³×C₃³	C₂×C₃	1	1	yes	characteristic
8	C₂²×C₃×C₃⋊D₃	C₂×C₃	1	14	yes	—
9	C₂³×C₃×C₉	—	2	4	no	—
10	C₂³×C₃×C₉	D₃	1	4	yes	complemented
11	C₂²×D₃×C₉	—	3	56	no	—
12	C₂×C₃²×C₉	C₂³	1	7	yes	complemented
13	C₉×C₃⋊D₃	C₂³	1	8	yes	direct product factor
14	C₂³×C₃⋊D₃	C₉	1	1	yes	characteristic, direct product factor
15	C₂³×C₃×D₃	—	3	4	no	—
16	C₂⁴×C₉	—	9	1	no	—
17	C₂²×C₃³	C₂²×C₃	1	7	yes	—
18	C₂×C₃×C₃⋊D₃	C₂²×C₃	1	28	yes	—
19	C₂²×C₃×C₉	—	2	28	no	—
20	C₂²×C₃×C₉	C₂×D₃	1	28	yes	complemented
21	C₂×D₃×C₉	—	3	112	no	—
22	C₃²×C₉	C₂⁴	1	1	yes	characteristic, complemented, Sylow
23	C₂³×C₃²	—	2	4	no	—
24	C₂³×C₃²	C₂×C₉	1	1	yes	characteristic, complemented
25	C₂³×C₃²	C₃×D₃	1	4	yes	—
26	C₂²×C₃⋊D₃	C₂×C₉	1	14	yes	direct product factor
27	C₂²×C₃×D₃	—	3	56	no	—
28	C₂³×C₉	—	2	4	no	—
29	C₂³×C₉	—	9	14	no	—
30	C₂³×C₉	C₃⋊D₃	1	1	yes	characteristic, direct product factor
31	C₂×C₃³	C₂³×C₃	1	7	yes	—
32	C₃×C₃⋊D₃	C₂³×C₃	1	8	yes	—
33	C₂×C₃×C₉	—	2	28	no	—
34	C₂×C₃×C₉	C₂²×D₃	1	28	yes	complemented
35	D₃×C₉	—	3	32	no	—
36	C₂⁴×C₃	—	9	1	no	—
37	C₂³×D₃	—	3	4	no	—
38	C₂²×C₃²	—	2	28	no	—
39	C₂²×C₃²	C₂²×C₉	1	7	yes	complemented
40	C₂²×C₃²	C₂×C₃×D₃	1	28	yes	—
41	C₂×C₃⋊D₃	C₂²×C₉	1	28	yes	direct product factor
42	C₂×C₃×D₃	—	3	112	no	—
43	C₂²×C₉	—	2	28	no	—
44	C₂²×C₉	—	9	28	no	—
45	C₂²×C₉	C₂×C₃⋊D₃	1	7	yes	direct product factor
46	C₃³	C₂⁴×C₃	1	1	yes	characteristic
47	C₃×C₉	—	2	4	no	—
48	C₃×C₉	C₂³×D₃	1	4	yes	complemented
49	C₂³×C₃	—	2	4	no	—
50	C₂³×C₃	—	9	14	no	—
51	C₂³×C₃	D₃×C₉	1	4	yes	complemented
52	C₂³×C₃	C₃×C₃⋊D₃	1	1	yes	characteristic
53	C₂²×D₃	—	3	56	no	—
54	C₂×C₃²	—	2	28	no	—
55	C₂×C₃²	C₂³×C₉	1	7	yes	complemented
56	C₂×C₃²	C₂²×C₃×D₃	1	28	yes	—
57	C₃⋊D₃	C₂³×C₉	1	8	yes	direct product factor
58	C₃×D₃	—	3	32	no	—
59	C₂×C₉	—	2	28	no	—
60	C₂×C₉	—	9	8	no	—
61	C₂×C₉	C₂²×C₃⋊D₃	1	7	yes	direct product factor
62	C₂⁴	—	9	1	no	Sylow
63	C₂²×C₃	—	2	28	no	—
64	C₂²×C₃	—	9	28	no	—
65	C₂²×C₃	C₂×D₃×C₉	1	28	yes	complemented
66	C₂²×C₃	C₂×C₃×C₃⋊D₃	1	7	yes	—
67	C₂×D₃	—	3	112	no	—
68	C₃²	—	2	4	no	—
69	C₃²	C₂⁴×C₉	1	1	yes	characteristic, complemented
70	C₃²	C₂³×C₃×D₃	1	4	yes	—
71	C₉	—	2	4	no	—
72	C₉	C₂³×C₃⋊D₃	1	1	yes	characteristic, direct product factor
73	C₂³	—	9	14	no	—
74	C₂³	C₉×C₃⋊D₃	1	1	yes	characteristic, direct product factor
75	C₂×C₃	—	2	28	no	—
76	C₂×C₃	—	9	8	no	—
77	C₂×C₃	C₂²×D₃×C₉	1	28	yes	complemented
78	C₂×C₃	C₂²×C₃×C₃⋊D₃	1	7	yes	—
79	D₃	—	3	32	no	—
80	C₂²	—	9	28	no	—
81	C₂²	C₂×C₉×C₃⋊D₃	1	7	yes	direct product factor
82	C₃	—	2	4	no	—
83	C₃	C₂³×D₃×C₉	1	4	yes	complemented
84	C₃	C₂³×C₃×C₃⋊D₃	1	1	yes	characteristic
85	C₂	—	9	8	no	—
86	C₂	C₂²×C₉×C₃⋊D₃	1	7	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂³×C₉	C₃⋊D₃
commutator, hypocenter	C₃²	C₂⁴×C₉
Frattini	C₃	C₂³×C₃×C₃⋊D₃
Fitting	C₂³×C₃²×C₉	C₂
socle	C₂³×C₃³	C₂×C₃
prime radical	C₂³×C₃×C₃⋊D₃	C₃
cosocle	C₃³	C₂⁴×C₃
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂³×C₉

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃²×C₉	C₂	semidirect
2. C₂²×C₉×C₃⋊D₃	C₂	direct (N × H)
5. C₂²×C₃²×C₉	C₂²	semidirect
6. C₂×C₉×C₃⋊D₃	C₂²	direct (N × H)
10. C₂³×C₃×C₉	D₃	semidirect
12. C₂×C₃²×C₉	C₂³	semidirect
13. C₉×C₃⋊D₃	C₂³	direct (N × H)
14. C₂³×C₃⋊D₃	C₉	direct (N × H)
20. C₂²×C₃×C₉	C₂×D₃	semidirect
22. C₃²×C₉	C₂⁴	semidirect
24. C₂³×C₃²	C₂×C₉	semidirect
26. C₂²×C₃⋊D₃	C₂×C₉	direct (N × H)
30. C₂³×C₉	C₃⋊D₃	direct (N × H)
34. C₂×C₃×C₉	C₂²×D₃	semidirect
39. C₂²×C₃²	C₂²×C₉	semidirect
41. C₂×C₃⋊D₃	C₂²×C₉	direct (N × H)
45. C₂²×C₉	C₂×C₃⋊D₃	direct (N × H)
48. C₃×C₉	C₂³×D₃	semidirect
51. C₂³×C₃	D₃×C₉	semidirect
55. C₂×C₃²	C₂³×C₉	semidirect
57. C₃⋊D₃	C₂³×C₉	direct (N × H)
61. C₂×C₉	C₂²×C₃⋊D₃	direct (N × H)
65. C₂²×C₃	C₂×D₃×C₉	semidirect
69. C₃²	C₂⁴×C₉	semidirect
72. C₉	C₂³×C₃⋊D₃	direct (N × H)
74. C₂³	C₉×C₃⋊D₃	direct (N × H)
77. C₂×C₃	C₂²×D₃×C₉	semidirect
81. C₂²	C₂×C₉×C₃⋊D₃	direct (N × H)
83. C₃	C₂³×D₃×C₉	semidirect
86. C₂	C₂²×C₉×C₃⋊D₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃²×C₉	2
C₂²×C₉×C₃⋊D₃	2
C₂³×C₃×C₃⋊D₃	3
C₂³×D₃×C₉	3

Subgroups — C23×C9×C3⋊D3 #13049351

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₉×C₃⋊D₃ #13049351