Subgroups — C₃×3₋¹⁺²×C₁₃⋊C₃ #15691686

Order 3159.

576 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₁₃⋊C₃	C₃	1	1	yes	maximal, characteristic
2	C₃×#66851	C₃	1	2	yes	maximal, complemented
3	C₃×#66848	C₃	1	3	yes	maximal, complemented
4	C₃×#66849	C₃	1	3	yes	maximal, complemented
5	3₋¹⁺²×C₁₃⋊C₃	C₃	1	27	yes	maximal, direct product factor
6	C₃×C₉×C₁₃⋊C₃	C₃	1	3	yes	maximal, complemented
7	C₃×C₁₃×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃³×C₁₃	C₃²	1	1	yes	characteristic
9	C₃²×C₁₃⋊C₃	—	3	9	no	—
10	C₃²×C₁₃⋊C₃	C₃²	1	9	yes	—
11	C₃²×C₁₃⋊C₃	C₃²	1	1	yes	characteristic
12	C₃²×C₁₃⋊C₃	C₃²	1	1	yes	characteristic
13	C₃²×C₁₃⋊C₃	C₃²	1	1	yes	characteristic
14	C₁₃×3₋¹⁺²	C₃²	1	9	yes	complemented
15	C₃×C₉×C₁₃	C₃²	1	3	yes	complemented
16	#66851	C₃²	1	18	yes	complemented
17	C₃×C₁₃⋊C₉	C₃²	1	6	yes	complemented
18	#66849	C₃²	1	27	yes	complemented
19	#66848	C₃²	1	27	yes	complemented
20	C₉×C₁₃⋊C₃	C₃²	1	27	yes	complemented
21	C₃²×3₋¹⁺²	—	13	1	no	maximal, Sylow
22	C₃²×C₁₃	—	3	3	no	—
23	C₃²×C₁₃	C₃³	1	3	yes	—
24	C₃²×C₁₃	C₃³	1	1	yes	characteristic
25	C₃×C₁₃⋊C₃	—	3	3	no	—
26	C₃×C₁₃⋊C₃	—	3	3	no	—
27	C₃×C₁₃⋊C₃	—	3	27	no	—
28	C₃×C₁₃⋊C₃	3₋¹⁺²	1	9	yes	direct product factor
29	C₃×C₁₃⋊C₃	C₃³	1	3	yes	—
30	C₃×C₁₃⋊C₃	C₃³	1	3	yes	—
31	C₃×C₁₃⋊C₃	C₃³	1	3	yes	—
32	C₉×C₁₃	C₃³	1	9	yes	complemented
33	C₁₃⋊C₉	C₃³	1	18	yes	complemented
34	C₃⁴	—	13	1	no	—
35	C₃×3₋¹⁺²	—	13	2	no	—
36	C₃×3₋¹⁺²	—	13	27	no	—
37	C₃×3₋¹⁺²	—	13	3	no	—
38	C₃×3₋¹⁺²	—	13	3	no	—
39	C₃×3₋¹⁺²	C₁₃⋊C₃	1	1	yes	characteristic, direct product factor
40	C₃²×C₉	—	13	3	no	—
41	C₃×C₁₃	—	3	3	no	—
42	C₃×C₁₃	C₃×3₋¹⁺²	1	3	yes	complemented
43	C₃×C₁₃	C₃⁴	1	1	yes	characteristic
44	C₁₃⋊C₃	—	3	3	no	—
45	C₁₃⋊C₃	—	3	3	no	—
46	C₁₃⋊C₃	C₃×3₋¹⁺²	1	9	yes	direct product factor
47	C₃³	—	13	1	no	—
48	C₃³	—	13	9	no	—
49	C₃³	—	13	1	no	—
50	C₃³	—	13	1	no	—
51	C₃³	—	39	9	no	—
52	C₃³	C₃×C₁₃⋊C₃	1	1	yes	characteristic
53	3₋¹⁺²	—	13	18	no	—
54	3₋¹⁺²	—	13	27	no	—
55	3₋¹⁺²	—	13	27	no	—
56	3₋¹⁺²	C₃×C₁₃⋊C₃	1	9	yes	direct product factor
57	C₃×C₉	—	13	6	no	—
58	C₃×C₉	—	13	27	no	—
59	C₃×C₉	C₃×C₁₃⋊C₃	1	3	yes	complemented
60	C₁₃	C₃²×3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
61	C₃²	—	3	3	no	—
62	C₃²	—	13	9	no	—
63	C₃²	—	13	3	no	—
64	C₃²	—	13	3	no	—
65	C₃²	—	13	3	no	—
66	C₃²	—	39	27	no	—
67	C₃²	—	39	3	no	—
68	C₃²	—	39	3	no	—
69	C₃²	C₃²×C₁₃⋊C₃	1	3	yes	—
70	C₃²	C₃²×C₁₃⋊C₃	1	1	yes	characteristic
71	C₉	—	13	18	no	—
72	C₉	C₃²×C₁₃⋊C₃	1	9	yes	complemented
73	C₃	—	3	3	no	—
74	C₃	—	13	9	no	—
75	C₃	—	39	3	no	—
76	C₃	—	39	3	no	—
77	C₃	3₋¹⁺²×C₁₃⋊C₃	1	3	yes	direct product factor
78	C₃	C₃³×C₁₃⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃²	C₃²×C₁₃⋊C₃
commutator, cosocle	C₃×C₁₃	C₃⁴
Frattini	C₃	C₃³×C₁₃⋊C₃
Fitting	C₃×C₁₃×3₋¹⁺²	C₃
socle	C₃²×C₁₃	C₃³
prime radical	C₃³×C₁₃⋊C₃	C₃
hypercenter	C₃×3₋¹⁺²	C₁₃⋊C₃
hypocenter	C₁₃	C₃²×3₋¹⁺²
LCS	G ▷ C₃×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₃² ◁ C₃×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
2. C₃×#66851	C₃	semidirect
3. C₃×#66848	C₃	semidirect
4. C₃×#66849	C₃	semidirect
5. 3₋¹⁺²×C₁₃⋊C₃	C₃	direct (N × H)
6. C₃×C₉×C₁₃⋊C₃	C₃	semidirect
7. C₃×C₁₃×3₋¹⁺²	C₃	semidirect
14. C₁₃×3₋¹⁺²	C₃²	semidirect
15. C₃×C₉×C₁₃	C₃²	semidirect
16. #66851	C₃²	semidirect
17. C₃×C₁₃⋊C₉	C₃²	semidirect
18. #66849	C₃²	semidirect
19. #66848	C₃²	semidirect
20. C₉×C₁₃⋊C₃	C₃²	semidirect
28. C₃×C₁₃⋊C₃	3₋¹⁺²	direct (N × H)
32. C₉×C₁₃	C₃³	semidirect
33. C₁₃⋊C₉	C₃³	semidirect
39. C₃×3₋¹⁺²	C₁₃⋊C₃	direct (N × H)
42. C₃×C₁₃	C₃×3₋¹⁺²	semidirect
46. C₁₃⋊C₃	C₃×3₋¹⁺²	direct (N × H)
56. 3₋¹⁺²	C₃×C₁₃⋊C₃	direct (N × H)
59. C₃×C₉	C₃×C₁₃⋊C₃	semidirect
60. C₁₃	C₃²×3₋¹⁺²	semidirect
72. C₉	C₃²×C₁₃⋊C₃	semidirect
77. C₃	3₋¹⁺²×C₁₃⋊C₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×C₁₃⋊C₃	3
C₃×#66851	3
C₃×#66848	3
C₃×#66849	3
3₋¹⁺²×C₁₃⋊C₃	3
C₃×C₉×C₁₃⋊C₃	3
C₃×C₁₃×3₋¹⁺²	3
C₃²×3₋¹⁺²	13

Subgroups — C3×3−1+2×C13⋊C3 #15691686

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×3₋¹⁺²×C₁₃⋊C₃ #15691686