Subgroups — #15691657

Order 3159.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#66851	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×C₃²⋊(C₁₃⋊C₃)	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×C₃²⋊(C₁₃⋊C₃)	C₃	1	1	yes	maximal, characteristic, complemented
4	#11760436	C₃	1	9	yes	maximal, complemented
5	C₃×C₁₃×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃³×C₁₃	C₃²	1	1	yes	characteristic
7	C₃²×C₁₃⋊C₃	—	3	1	no	—
8	C₃²×C₁₃⋊C₃	C₃²	1	3	yes	complemented
9	C₁₃×3₋¹⁺²	—	3	1	no	—
10	C₁₃×3₋¹⁺²	—	3	1	no	—
11	C₁₃×3₋¹⁺²	C₃²	1	3	yes	complemented
12	C₃×C₉×C₁₃	—	3	1	no	—
13	C₃²⋊(C₁₃⋊C₃)	—	3	1	no	—
14	C₃²⋊(C₁₃⋊C₃)	—	3	1	no	—
15	C₃²⋊(C₁₃⋊C₃)	—	3	1	no	—
16	C₃²⋊(C₁₃⋊C₃)	—	3	1	no	—
17	C₃²⋊(C₁₃⋊C₃)	—	3	1	no	—
18	C₃²⋊(C₁₃⋊C₃)	C₃²	1	3	yes	—
19	#66851	—	3	1	no	—
20	#66851	—	3	1	no	—
21	#66851	C₃²	1	3	yes	complemented
22	C₃×C₁₃⋊C₉	—	3	1	no	—
23	#6909	—	13	1	no	maximal, Sylow
24	C₃²×C₁₃	—	9	1	no	—
25	C₃²×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
26	C₃²×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
27	C₃²×C₁₃	H(3,𝔽₃)	1	1	yes	characteristic
28	C₃²×C₁₃	C₃³	1	1	yes	characteristic
29	C₃×C₁₃⋊C₃	—	3	3	no	—
30	C₃×C₁₃⋊C₃	—	3	3	no	—
31	C₃×C₁₃⋊C₃	—	9	3	no	—
32	C₃×C₁₃⋊C₃	—	9	3	no	—
33	C₉×C₁₃	—	3	3	no	—
34	C₁₃⋊C₉	—	3	3	no	—
35	C₃×3₋¹⁺²	—	13	1	no	—
36	C₃×3₋¹⁺²	C₁₃⋊C₃	1	1	yes	characteristic, complemented
37	C₃×H(3,𝔽₃)	—	13	1	no	—
38	C₃×H(3,𝔽₃)	—	13	1	no	—
39	AΣL(1,𝔽₂₇)	—	13	9	no	—
40	C₃×C₁₃	—	3	1	no	—
41	C₃×C₁₃	—	3	1	no	—
42	C₃×C₁₃	—	3	1	no	—
43	C₃×C₁₃	—	3	1	no	—
44	C₃×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
45	C₁₃⋊C₃	—	9	3	no	—
46	C₁₃⋊C₃	—	9	3	no	—
47	C₃³	—	13	3	no	—
48	C₃³	—	39	1	no	—
49	C₃³	C₃×C₁₃⋊C₃	1	1	yes	characteristic
50	3₋¹⁺²	—	3	1	no	—
51	3₋¹⁺²	—	3	1	no	—
52	3₋¹⁺²	—	13	3	no	—
53	3₋¹⁺²	—	39	1	no	—
54	3₋¹⁺²	—	39	1	no	—
55	3₋¹⁺²	C₃×C₁₃⋊C₃	1	3	yes	complemented
56	H(3,𝔽₃)	—	13	3	no	—
57	H(3,𝔽₃)	—	39	1	no	—
58	H(3,𝔽₃)	—	39	1	no	—
59	H(3,𝔽₃)	—	39	1	no	—
60	H(3,𝔽₃)	—	39	1	no	—
61	H(3,𝔽₃)	—	39	1	no	—
62	C₃×C₉	—	3	1	no	—
63	C₃×C₉	—	39	1	no	—
64	C₁₃	#6909	1	1	yes	characteristic, complemented, Sylow
65	C₃²	—	9	1	no	—
66	C₃²	—	39	3	no	—
67	C₃²	—	39	3	no	—
68	C₃²	—	117	3	no	—
69	C₃²	—	117	3	no	—
70	C₃²	C₃²⋊(C₁₃⋊C₃)	1	1	yes	characteristic
71	C₃²	C₃²⋊(C₁₃⋊C₃)	1	1	yes	characteristic
72	C₃²	C₃²⋊(C₁₃⋊C₃)	1	1	yes	characteristic
73	C₃²	C₃²×C₁₃⋊C₃	1	1	yes	characteristic
74	C₉	—	3	3	no	—
75	C₉	—	39	3	no	—
76	C₃	—	3	1	no	—
77	C₃	—	3	1	no	—
78	C₃	—	3	1	no	—
79	C₃	—	3	1	no	—
80	C₃	—	117	3	no	—
81	C₃	—	117	3	no	—
82	C₃	C₃×C₃²⋊(C₁₃⋊C₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₃	C₃×C₃²⋊(C₁₃⋊C₃)
commutator, cosocle	C₃²×C₁₃	C₃³
Frattini	C₃²	C₃²×C₁₃⋊C₃
Fitting	C₃×C₁₃×3₋¹⁺²	C₃
socle	C₃×C₁₃	C₃×H(3,𝔽₃)
hypercenter	C₃×3₋¹⁺²	C₁₃⋊C₃
hypocenter	C₁₃	#6909
LCS	G ▷ C₃²×C₁₃ ▷ C₃×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
1. C₃×#66851	C₃	semidirect
2. C₃×C₃²⋊(C₁₃⋊C₃)	C₃	semidirect
3. C₃×C₃²⋊(C₁₃⋊C₃)	C₃	semidirect
4. #11760436	C₃	semidirect
5. C₃×C₁₃×3₋¹⁺²	C₃	semidirect
8. C₃²×C₁₃⋊C₃	C₃²	semidirect
11. C₁₃×3₋¹⁺²	C₃²	semidirect
21. #66851	C₃²	semidirect
36. C₃×3₋¹⁺²	C₁₃⋊C₃	semidirect
55. 3₋¹⁺²	C₃×C₁₃⋊C₃	semidirect
64. C₁₃	#6909	semidirect

Maximal subgroups

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

subgroup	index
C₃×#66851	3
C₃×C₃²⋊(C₁₃⋊C₃)	3
#11760436	3
C₃×C₁₃×3₋¹⁺²	3
#6909	13