Subgroups — #15691697

Order 3159.

460 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11760454	C₃	1	18	yes	maximal, complemented
2	#11760453	C₃	1	9	yes	maximal, complemented
3	C₃×#66851	C₃	1	2	yes	maximal, complemented
4	3₋¹⁺²×C₁₃⋊C₃	C₃	1	9	yes	maximal, complemented
5	C₃×C₃²⋊(C₁₃⋊C₃)	C₃	1	1	yes	maximal, characteristic
6	C₃×C₁₃×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃³×C₁₃	C₃²	1	1	yes	characteristic
8	C₃²×C₁₃⋊C₃	C₃²	1	3	yes	—
9	C₁₃×3₋¹⁺²	C₃²	1	3	yes	—
10	C₁₃×3₋¹⁺²	C₃²	1	6	yes	complemented
11	C₃×C₉×C₁₃	C₃²	1	3	yes	complemented
12	C₃²⋊(C₁₃⋊C₃)	C₃²	1	3	yes	—
13	C₃²⋊(C₁₃⋊C₃)	C₃²	1	6	yes	—
14	#66851	C₃²	1	6	yes	—
15	#66851	C₃²	1	6	yes	complemented
16	#66851	C₃²	1	6	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	3₋¹⁺⁴	—	13	1	no	maximal, Sylow
22	C₃²×C₁₃	—	9	1	no	—
23	C₃²×C₁₃	C₃³	1	2	yes	—
24	C₃²×C₁₃	C₃³	1	1	yes	characteristic
25	C₃²×C₁₃	C₃³	1	1	yes	characteristic
26	C₃×C₁₃⋊C₃	—	9	3	no	—
27	C₃×C₁₃⋊C₃	C₃³	1	9	yes	—
28	C₉×C₁₃	C₃³	1	9	yes	—
29	C₁₃⋊C₉	C₃³	1	18	yes	—
30	#737	—	13	18	no	—
31	#737	—	13	9	no	—
32	C₃×3₋¹⁺²	—	13	2	no	—
33	C₃×3₋¹⁺²	—	13	9	no	—
34	C₃×3₋¹⁺²	C₁₃⋊C₃	1	1	yes	characteristic, complemented
35	C₃×H(3,𝔽₃)	—	13	1	no	—
36	C₃×C₁₃	—	3	1	no	—
37	C₃×C₁₃	—	3	2	no	—
38	C₃×C₁₃	—	3	1	no	—
39	C₃×C₁₃	C₃⁴	1	1	yes	characteristic
40	C₁₃⋊C₃	—	3	9	no	—
41	C₃³	—	13	3	no	—
42	C₃³	C₃×C₁₃⋊C₃	1	1	yes	characteristic
43	3₋¹⁺²	—	13	6	no	—
44	3₋¹⁺²	—	13	6	no	—
45	3₋¹⁺²	—	13	6	no	—
46	3₋¹⁺²	—	13	27	no	—
47	3₋¹⁺²	—	13	27	no	—
48	3₋¹⁺²	C₃×C₁₃⋊C₃	1	3	yes	—
49	3₋¹⁺²	C₃×C₁₃⋊C₃	1	6	yes	complemented
50	H(3,𝔽₃)	—	13	6	no	—
51	H(3,𝔽₃)	—	13	3	no	—
52	C₃×C₉	—	13	6	no	—
53	C₃×C₉	—	13	27	no	—
54	C₃×C₉	C₃×C₁₃⋊C₃	1	3	yes	complemented
55	C₁₃	3₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
56	C₃²	—	9	1	no	—
57	C₃²	—	13	9	no	—
58	C₃²	—	117	3	no	—
59	C₃²	C₃²×C₁₃⋊C₃	1	2	yes	—
60	C₃²	C₃²×C₁₃⋊C₃	1	1	yes	characteristic
61	C₃²	C₃²×C₁₃⋊C₃	1	1	yes	characteristic
62	C₉	—	13	18	no	—
63	C₉	C₃²×C₁₃⋊C₃	1	9	yes	—
64	C₃	—	3	1	no	—
65	C₃	—	3	2	no	—
66	C₃	—	3	1	no	—
67	C₃	—	39	9	no	—
68	C₃	C₃³×C₁₃⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃	C₃³×C₁₃⋊C₃
commutator, socle, cosocle	C₃×C₁₃	C₃⁴
Fitting	C₃×C₁₃×3₋¹⁺²	C₃
prime radical	C₃×C₃²⋊(C₁₃⋊C₃)	C₃
hypercenter	C₃×3₋¹⁺²	C₁₃⋊C₃
hypocenter	C₁₃	3₋¹⁺⁴
LCS	G ▷ C₃×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₃ ◁ C₃×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
1. #11760454	C₃	semidirect
2. #11760453	C₃	semidirect
3. C₃×#66851	C₃	semidirect
4. 3₋¹⁺²×C₁₃⋊C₃	C₃	semidirect
6. C₃×C₁₃×3₋¹⁺²	C₃	semidirect
10. C₁₃×3₋¹⁺²	C₃²	semidirect
11. C₃×C₉×C₁₃	C₃²	semidirect
15. #66851	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
34. C₃×3₋¹⁺²	C₁₃⋊C₃	semidirect
49. 3₋¹⁺²	C₃×C₁₃⋊C₃	semidirect
54. C₃×C₉	C₃×C₁₃⋊C₃	semidirect
55. C₁₃	3₋¹⁺⁴	semidirect

Maximal subgroups

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

subgroup	index
#11760454	3
#11760453	3
C₃×#66851	3
3₋¹⁺²×C₁₃⋊C₃	3
C₃×C₃²⋊(C₁₃⋊C₃)	3
C₃×C₁₃×3₋¹⁺²	3
3₋¹⁺⁴	13