Subgroups — #15691662

Order 3159.

136 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×#66851	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×C₃²⋊(C₁₃⋊C₃)	C₃	1	1	yes	maximal, characteristic
4	#11760439	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	C₁₃×3₋¹⁺²	—	3	1	no	—
9	C₁₃×3₋¹⁺²	—	3	1	no	—
10	C₁₃×3₋¹⁺²	C₃²	1	3	yes	complemented
11	C₃×C₉×C₁₃	—	3	1	no	—
12	C₃²⋊(C₁₃⋊C₃)	—	3	1	no	—
13	C₃²⋊(C₁₃⋊C₃)	—	3	1	no	—
14	C₃²⋊(C₁₃⋊C₃)	C₃²	1	3	yes	—
15	#66851	—	3	1	no	—
16	#66851	—	3	1	no	—
17	#66851	—	3	1	no	—
18	#66851	—	3	1	no	—
19	#66851	—	3	1	no	—
20	#66851	C₃²	1	3	yes	complemented
21	C₃×C₁₃⋊C₉	—	3	1	no	—
22	C₃×C₁₃⋊C₉	C₃²	1	3	yes	complemented
23	#6910	—	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	—	9	3	no	—
31	C₉×C₁₃	—	3	3	no	—
32	C₁₃⋊C₉	—	3	3	no	—
33	C₁₃⋊C₉	—	3	3	no	—
34	C₃×3₋¹⁺²	—	13	1	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	#731	—	13	9	no	—
39	C₃×C₁₃	—	3	1	no	—
40	C₃×C₁₃	—	3	1	no	—
41	C₃×C₁₃	—	3	1	no	—
42	C₃×C₁₃	—	3	1	no	—
43	C₃×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
44	C₁₃⋊C₃	—	9	3	no	—
45	C₃³	—	39	1	no	—
46	C₃³	C₃×C₁₃⋊C₃	1	1	yes	characteristic
47	3₋¹⁺²	—	3	1	no	—
48	3₋¹⁺²	—	3	1	no	—
49	3₋¹⁺²	—	13	3	no	—
50	3₋¹⁺²	—	39	1	no	—
51	3₋¹⁺²	—	39	1	no	—
52	3₋¹⁺²	—	39	1	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	C₃×C₉	—	3	1	no	—
60	C₃×C₉	—	13	3	no	—
61	C₃×C₉	—	39	1	no	—
62	C₁₃	#6910	1	1	yes	characteristic, complemented, Sylow
63	C₃²	—	9	1	no	—
64	C₃²	—	39	3	no	—
65	C₃²	—	117	3	no	—
66	C₃²	C₃²⋊(C₁₃⋊C₃)	1	1	yes	characteristic
67	C₃²	C₃²⋊(C₁₃⋊C₃)	1	1	yes	characteristic
68	C₃²	C₃²⋊(C₁₃⋊C₃)	1	1	yes	characteristic
69	C₃²	C₃²×C₁₃⋊C₃	1	1	yes	characteristic
70	C₉	—	3	3	no	—
71	C₉	—	39	3	no	—
72	C₉	—	39	3	no	—
73	C₃	—	3	1	no	—
74	C₃	—	3	1	no	—
75	C₃	—	3	1	no	—
76	C₃	—	3	1	no	—
77	C₃	—	117	3	no	—
78	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)
prime radical	C₃×C₃²⋊(C₁₃⋊C₃)	C₃
hypercenter	C₃×3₋¹⁺²	C₁₃⋊C₃
hypocenter	C₁₃	#6910
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×#66851	C₃	semidirect
4. #11760439	C₃	semidirect
5. C₃×C₁₃×3₋¹⁺²	C₃	semidirect
10. C₁₃×3₋¹⁺²	C₃²	semidirect
20. #66851	C₃²	semidirect
22. C₃×C₁₃⋊C₉	C₃²	semidirect
36. C₃×3₋¹⁺²	C₁₃⋊C₃	semidirect
55. 3₋¹⁺²	C₃×C₁₃⋊C₃	semidirect
62. 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×#66851	3
C₃×C₃²⋊(C₁₃⋊C₃)	3
#11760439	3
C₃×C₁₃×3₋¹⁺²	3
#6910	13