Subgroups — #15691663

Order 3159.

160 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	H(3,𝔽₃)×C₁₃⋊C₃	C₃	1	3	yes	maximal, complemented
3	#11760441	C₃	1	2	yes	maximal, complemented
4	#11760440	C₃	1	6	yes	maximal, complemented
5	C₁₃×#732	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃²×C₁₃⋊C₃	—	3	3	no	—
7	C₃²×C₁₃⋊C₃	C₃²	1	1	yes	characteristic
8	C₁₃×H(3,𝔽₃)	C₃²	1	3	yes	complemented
9	C₃×C₉×C₁₃	C₃²	1	1	yes	characteristic, complemented
10	C₃²⋊(C₁₃⋊C₃)	—	3	6	no	—
11	C₃²⋊(C₁₃⋊C₃)	C₃²	1	6	yes	complemented
12	C₃×C₁₃⋊C₉	C₃²	1	2	yes	complemented
13	#66848	—	3	3	no	—
14	#6911	—	13	1	no	maximal, Sylow
15	C₃²×C₁₃	—	3	3	no	—
16	C₃²×C₁₃	C₃³	1	1	yes	characteristic
17	C₃×C₁₃⋊C₃	—	3	6	no	—
18	C₃×C₁₃⋊C₃	—	9	1	no	—
19	C₃×C₁₃⋊C₃	—	9	9	no	—
20	C₃×C₁₃⋊C₃	H(3,𝔽₃)	1	3	yes	—
21	C₉×C₁₃	—	3	1	no	—
22	C₁₃⋊C₉	—	3	2	no	—
23	C₃×3₋¹⁺²	—	13	1	no	—
24	C₃×H(3,𝔽₃)	—	13	3	no	—
25	#732	—	13	6	no	—
26	#732	—	13	2	no	—
27	#732	C₁₃⋊C₃	1	1	yes	characteristic, complemented
28	C₃×C₁₃	—	3	1	no	—
29	C₃×C₁₃	—	9	3	no	—
30	C₃×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
31	C₁₃⋊C₃	—	3	3	no	—
32	C₁₃⋊C₃	—	9	6	no	—
33	C₃³	—	13	1	no	—
34	C₃³	—	39	3	no	—
35	3₋¹⁺²	—	39	3	no	—
36	H(3,𝔽₃)	—	13	6	no	—
37	H(3,𝔽₃)	—	39	6	no	—
38	H(3,𝔽₃)	C₃×C₁₃⋊C₃	1	3	yes	complemented
39	C₃×C₉	—	13	2	no	—
40	C₃×C₉	C₃×C₁₃⋊C₃	1	1	yes	characteristic, complemented
41	C₁₃	#6911	1	1	yes	characteristic, complemented, Sylow
42	C₃²	—	3	3	no	—
43	C₃²	—	13	3	no	—
44	C₃²	—	39	6	no	—
45	C₃²	—	117	1	no	—
46	C₃²	—	117	9	no	—
47	C₃²	C₃²×C₁₃⋊C₃	1	1	yes	characteristic
48	C₉	—	3	1	no	—
49	C₉	—	39	2	no	—
50	C₃	—	3	1	no	—
51	C₃	—	9	3	no	—
52	C₃	—	39	3	no	—
53	C₃	—	117	6	no	—
54	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×#732	C₃
socle	C₃×C₁₃	C₃×H(3,𝔽₃)
hypercenter	#732	C₁₃⋊C₃
hypocenter	C₁₃	#6911
LCS	G ▷ C₃²×C₁₃ ▷ C₃×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₃ ◁ C₃² ◁ #732

Complements

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

N	H ≅ G/N	type
1. C₃×#66848	C₃	semidirect
2. H(3,𝔽₃)×C₁₃⋊C₃	C₃	semidirect
3. #11760441	C₃	semidirect
4. #11760440	C₃	semidirect
5. C₁₃×#732	C₃	semidirect
8. C₁₃×H(3,𝔽₃)	C₃²	semidirect
9. C₃×C₉×C₁₃	C₃²	semidirect
11. C₃²⋊(C₁₃⋊C₃)	C₃²	semidirect
12. C₃×C₁₃⋊C₉	C₃²	semidirect
27. #732	C₁₃⋊C₃	semidirect
38. H(3,𝔽₃)	C₃×C₁₃⋊C₃	semidirect
40. C₃×C₉	C₃×C₁₃⋊C₃	semidirect
41. C₁₃	#6911	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
H(3,𝔽₃)×C₁₃⋊C₃	3
#11760441	3
#11760440	3
C₁₃×#732	3
#6911	13