Subgroups — C₂³×C₃²×3₋¹⁺² #14519384

Order 1944.

4608 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×3₋¹⁺²	C₂	1	7	yes	maximal, direct product factor
2	C₂³×C₃⁴	C₃	1	1	yes	maximal, characteristic
3	C₂³×C₃×3₋¹⁺²	C₃	1	36	yes	maximal, direct product factor
4	C₂³×C₃²×C₉	C₃	1	3	yes	maximal, complemented
5	C₂×C₃²×3₋¹⁺²	C₂²	1	7	yes	direct product factor
6	C₂²×C₃⁴	C₂×C₃	1	7	yes	—
7	C₂²×C₃×3₋¹⁺²	C₂×C₃	1	252	yes	direct product factor
8	C₂²×C₃²×C₉	C₂×C₃	1	21	yes	complemented
9	C₃²×3₋¹⁺²	C₂³	1	1	yes	characteristic, direct product factor, Sylow
10	C₂³×C₃³	—	3	9	no	—
11	C₂³×C₃³	C₃²	1	12	yes	—
12	C₂³×C₃³	C₃²	1	1	yes	characteristic
13	C₂³×3₋¹⁺²	C₃²	1	81	yes	direct product factor
14	C₂³×C₃×C₉	C₃²	1	36	yes	complemented
15	C₂×C₃⁴	C₂²×C₃	1	7	yes	—
16	C₂×C₃×3₋¹⁺²	C₂²×C₃	1	252	yes	direct product factor
17	C₂×C₃²×C₉	C₂²×C₃	1	21	yes	complemented
18	C₂²×C₃³	—	3	63	no	—
19	C₂²×C₃³	C₂×C₃²	1	84	yes	—
20	C₂²×C₃³	C₂×C₃²	1	7	yes	—
21	C₂²×3₋¹⁺²	C₂×C₃²	1	567	yes	direct product factor
22	C₂²×C₃×C₉	C₂×C₃²	1	252	yes	complemented
23	C₃⁴	C₂³×C₃	1	1	yes	characteristic
24	C₃×3₋¹⁺²	C₂³×C₃	1	36	yes	direct product factor
25	C₃²×C₉	C₂³×C₃	1	3	yes	complemented
26	C₂³×C₃²	—	3	36	no	—
27	C₂³×C₃²	3₋¹⁺²	1	9	yes	direct product factor
28	C₂³×C₃²	C₃³	1	9	yes	—
29	C₂³×C₃²	C₃³	1	4	yes	—
30	C₂³×C₉	C₃³	1	27	yes	complemented
31	C₂×C₃³	—	3	63	no	—
32	C₂×C₃³	C₂²×C₃²	1	84	yes	—
33	C₂×C₃³	C₂²×C₃²	1	7	yes	—
34	C₂×3₋¹⁺²	C₂²×C₃²	1	567	yes	direct product factor
35	C₂×C₃×C₉	C₂²×C₃²	1	252	yes	complemented
36	C₂²×C₃²	—	3	252	no	—
37	C₂²×C₃²	C₂×3₋¹⁺²	1	63	yes	direct product factor
38	C₂²×C₃²	C₂×C₃³	1	63	yes	—
39	C₂²×C₃²	C₂×C₃³	1	28	yes	—
40	C₂²×C₉	C₂×C₃³	1	189	yes	complemented
41	C₃³	—	3	9	no	—
42	C₃³	C₂³×C₃²	1	12	yes	—
43	C₃³	C₂³×C₃²	1	1	yes	characteristic
44	3₋¹⁺²	C₂³×C₃²	1	81	yes	direct product factor
45	C₃×C₉	C₂³×C₃²	1	36	yes	complemented
46	C₂³×C₃	—	3	9	no	—
47	C₂³×C₃	C₃×3₋¹⁺²	1	12	yes	direct product factor
48	C₂³×C₃	C₃⁴	1	1	yes	characteristic
49	C₂×C₃²	—	3	252	no	—
50	C₂×C₃²	C₂²×3₋¹⁺²	1	63	yes	direct product factor
51	C₂×C₃²	C₂²×C₃³	1	63	yes	—
52	C₂×C₃²	C₂²×C₃³	1	28	yes	—
53	C₂×C₉	C₂²×C₃³	1	189	yes	complemented
54	C₂²×C₃	—	3	63	no	—
55	C₂²×C₃	C₂×C₃×3₋¹⁺²	1	84	yes	direct product factor
56	C₂²×C₃	C₂×C₃⁴	1	7	yes	—
57	C₃²	—	3	36	no	—
58	C₃²	C₂³×3₋¹⁺²	1	9	yes	direct product factor
59	C₃²	C₂³×C₃³	1	9	yes	—
60	C₃²	C₂³×C₃³	1	4	yes	—
61	C₉	C₂³×C₃³	1	27	yes	complemented
62	C₂³	C₃²×3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
63	C₂×C₃	—	3	63	no	—
64	C₂×C₃	C₂²×C₃×3₋¹⁺²	1	84	yes	direct product factor
65	C₂×C₃	C₂²×C₃⁴	1	7	yes	—
66	C₂²	C₂×C₃²×3₋¹⁺²	1	7	yes	direct product factor
67	C₃	—	3	9	no	—
68	C₃	C₂³×C₃×3₋¹⁺²	1	12	yes	direct product factor
69	C₃	C₂³×C₃⁴	1	1	yes	characteristic
70	C₂	C₂²×C₃²×3₋¹⁺²	1	7	yes	direct product factor

	subgroup	G/H
center, socle	C₂³×C₃³	C₃²
commutator, Frattini, cosocle	C₃	C₂³×C₃⁴
prime radical	C₂³×C₃⁴	C₃
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂³×C₃³ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃²×3₋¹⁺²	C₂	direct (N × H)
3. C₂³×C₃×3₋¹⁺²	C₃	direct (N × H)
4. C₂³×C₃²×C₉	C₃	semidirect
5. C₂×C₃²×3₋¹⁺²	C₂²	direct (N × H)
7. C₂²×C₃×3₋¹⁺²	C₂×C₃	direct (N × H)
8. C₂²×C₃²×C₉	C₂×C₃	semidirect
9. C₃²×3₋¹⁺²	C₂³	direct (N × H)
13. C₂³×3₋¹⁺²	C₃²	direct (N × H)
14. C₂³×C₃×C₉	C₃²	semidirect
16. C₂×C₃×3₋¹⁺²	C₂²×C₃	direct (N × H)
17. C₂×C₃²×C₉	C₂²×C₃	semidirect
21. C₂²×3₋¹⁺²	C₂×C₃²	direct (N × H)
22. C₂²×C₃×C₉	C₂×C₃²	semidirect
24. C₃×3₋¹⁺²	C₂³×C₃	direct (N × H)
25. C₃²×C₉	C₂³×C₃	semidirect
27. C₂³×C₃²	3₋¹⁺²	direct (N × H)
30. C₂³×C₉	C₃³	semidirect
34. C₂×3₋¹⁺²	C₂²×C₃²	direct (N × H)
35. C₂×C₃×C₉	C₂²×C₃²	semidirect
37. C₂²×C₃²	C₂×3₋¹⁺²	direct (N × H)
40. C₂²×C₉	C₂×C₃³	semidirect
44. 3₋¹⁺²	C₂³×C₃²	direct (N × H)
45. C₃×C₉	C₂³×C₃²	semidirect
47. C₂³×C₃	C₃×3₋¹⁺²	direct (N × H)
50. C₂×C₃²	C₂²×3₋¹⁺²	direct (N × H)
53. C₂×C₉	C₂²×C₃³	semidirect
55. C₂²×C₃	C₂×C₃×3₋¹⁺²	direct (N × H)
58. C₃²	C₂³×3₋¹⁺²	direct (N × H)
61. C₉	C₂³×C₃³	semidirect
62. C₂³	C₃²×3₋¹⁺²	direct (N × H)
64. C₂×C₃	C₂²×C₃×3₋¹⁺²	direct (N × H)
66. C₂²	C₂×C₃²×3₋¹⁺²	direct (N × H)
68. C₃	C₂³×C₃×3₋¹⁺²	direct (N × H)
70. C₂	C₂²×C₃²×3₋¹⁺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×C₃²×3₋¹⁺²	2
C₂³×C₃⁴	3
C₂³×C₃×3₋¹⁺²	3
C₂³×C₃²×C₉	3

Subgroups — C23×C32×3−1+2 #14519384

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃²×3₋¹⁺² #14519384