Subgroups — #15691659

Order 3159.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#66848	C₃	1	1	yes	maximal, characteristic, complemented
2	3₋¹⁺²×C₁₃⋊C₃	C₃	1	2	yes	maximal, complemented
3	H(3,𝔽₃)×C₁₃⋊C₃	C₃	1	1	yes	maximal, characteristic
4	#11760439	C₃	1	4	yes	maximal, complemented
5	#11760438	C₃	1	2	yes	maximal, complemented
6	#11760437	C₃	1	2	yes	maximal, complemented
7	C₁₃×#731	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃²×C₁₃⋊C₃	—	3	1	no	—
9	C₃²×C₁₃⋊C₃	C₃²	1	1	yes	characteristic
10	C₁₃×3₋¹⁺²	C₃²	1	2	yes	complemented
11	C₁₃×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
12	C₃×C₉×C₁₃	C₃²	1	1	yes	characteristic, complemented
13	C₃²⋊(C₁₃⋊C₃)	—	3	2	no	—
14	C₃²⋊(C₁₃⋊C₃)	C₃²	1	2	yes	—
15	#66851	C₃²	1	4	yes	complemented
16	C₃×C₁₃⋊C₉	C₃²	1	2	yes	complemented
17	#66849	—	3	2	no	—
18	#66848	—	3	2	no	—
19	#66848	—	3	2	no	—
20	#66848	—	3	1	no	—
21	C₉×C₁₃⋊C₃	—	3	2	no	—
22	#6910	—	13	1	no	maximal, Sylow
23	C₃²×C₁₃	—	3	1	no	—
24	C₃²×C₁₃	C₃³	1	1	yes	characteristic
25	C₃×C₁₃⋊C₃	—	3	2	no	—
26	C₃×C₁₃⋊C₃	—	9	1	no	—
27	C₃×C₁₃⋊C₃	—	9	3	no	—
28	C₃×C₁₃⋊C₃	H(3,𝔽₃)	1	2	yes	—
29	C₃×C₁₃⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
30	C₉×C₁₃	—	3	2	no	—
31	C₉×C₁₃	—	3	1	no	—
32	C₁₃⋊C₉	—	3	4	no	—
33	C₁₃⋊C₉	—	3	2	no	—
34	C₃×3₋¹⁺²	—	13	1	no	—
35	C₃×3₋¹⁺²	—	13	2	no	—
36	C₃×H(3,𝔽₃)	—	13	1	no	—
37	#731	—	13	4	no	—
38	#731	—	13	2	no	—
39	#731	—	13	2	no	—
40	#731	C₁₃⋊C₃	1	1	yes	characteristic, complemented
41	C₃×C₁₃	—	3	1	no	—
42	C₃×C₁₃	—	9	1	no	—
43	C₃×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
44	C₁₃⋊C₃	—	3	1	no	—
45	C₁₃⋊C₃	—	3	2	no	—
46	C₁₃⋊C₃	—	9	2	no	—
47	C₃³	—	13	1	no	—
48	C₃³	—	39	1	no	—
49	3₋¹⁺²	—	13	4	no	—
50	3₋¹⁺²	—	39	2	no	—
51	3₋¹⁺²	—	39	2	no	—
52	3₋¹⁺²	—	39	1	no	—
53	3₋¹⁺²	—	39	2	no	—
54	3₋¹⁺²	C₃×C₁₃⋊C₃	1	2	yes	complemented
55	H(3,𝔽₃)	—	13	2	no	—
56	H(3,𝔽₃)	—	39	2	no	—
57	H(3,𝔽₃)	C₃×C₁₃⋊C₃	1	1	yes	characteristic
58	C₃×C₉	—	13	2	no	—
59	C₃×C₉	—	39	2	no	—
60	C₃×C₉	C₃×C₁₃⋊C₃	1	1	yes	characteristic, complemented
61	C₁₃	#6910	1	1	yes	characteristic, complemented, Sylow
62	C₃²	—	3	1	no	—
63	C₃²	—	13	2	no	—
64	C₃²	—	13	1	no	—
65	C₃²	—	39	2	no	—
66	C₃²	—	117	1	no	—
67	C₃²	—	117	3	no	—
68	C₃²	C₃²×C₁₃⋊C₃	1	1	yes	characteristic
69	C₉	—	3	2	no	—
70	C₉	—	3	1	no	—
71	C₉	—	39	2	no	—
72	C₉	—	39	4	no	—
73	C₃	—	3	1	no	—
74	C₃	—	9	1	no	—
75	C₃	—	39	1	no	—
76	C₃	—	39	2	no	—
77	C₃	—	117	2	no	—
78	C₃	H(3,𝔽₃)×C₁₃⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃	H(3,𝔽₃)×C₁₃⋊C₃
commutator, cosocle	C₃²×C₁₃	C₃³
Frattini	C₃²	C₃²×C₁₃⋊C₃
Fitting	C₁₃×#731	C₃
socle	C₃×C₁₃	C₃×H(3,𝔽₃)
prime radical	H(3,𝔽₃)×C₁₃⋊C₃	C₃
hypercenter	#731	C₁₃⋊C₃
hypocenter	C₁₃	#6910
LCS	G ▷ C₃²×C₁₃ ▷ C₃×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₃ ◁ C₃² ◁ #731

Complements

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

N	H ≅ G/N	type
1. C₃×#66848	C₃	semidirect
2. 3₋¹⁺²×C₁₃⋊C₃	C₃	semidirect
4. #11760439	C₃	semidirect
5. #11760438	C₃	semidirect
6. #11760437	C₃	semidirect
7. C₁₃×#731	C₃	semidirect
10. C₁₃×3₋¹⁺²	C₃²	semidirect
12. C₃×C₉×C₁₃	C₃²	semidirect
15. #66851	C₃²	semidirect
16. C₃×C₁₃⋊C₉	C₃²	semidirect
40. #731	C₁₃⋊C₃	semidirect
54. 3₋¹⁺²	C₃×C₁₃⋊C₃	semidirect
60. C₃×C₉	C₃×C₁₃⋊C₃	semidirect
61. C₁₃	#6910	semidirect

Maximal subgroups

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

subgroup	index
C₃×#66848	3
3₋¹⁺²×C₁₃⋊C₃	3
H(3,𝔽₃)×C₁₃⋊C₃	3
#11760439	3
#11760438	3
#11760437	3
C₁₃×#731	3
#6910	13