Subgroups — C₃×#17987547 #28356665

Order 13851.

5588 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17987547	C₃	1	81	yes	maximal, direct product factor
2	C₃×#13095331	C₃	1	27	yes	maximal, complemented
3	C₃²×#10587025	C₃	1	12	yes	maximal, complemented
4	C₃²×C₁₉×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
5	#13095331	C₃²	1	729	yes	complemented
6	C₃×#10587025	C₃²	1	9	yes	—
7	C₃×#10587025	C₃²	1	72	yes	complemented
8	C₃×#10587025	C₃²	1	324	yes	complemented
9	C₃²×#4294	C₃²	1	36	yes	complemented
10	C₃⁴×C₁₉	C₃²	1	4	yes	—
11	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	9	yes	—
12	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	27	yes	—
13	C₃×3₋¹⁺⁴	—	19	1	no	maximal, Sylow
14	C₃³×C₁₉	—	9	12	no	—
15	C₃³×C₁₉	C₃³	1	12	yes	—
16	C₃³×C₁₉	C₃³	1	1	yes	characteristic
17	C₃³×C₁₉	C₃³	1	36	yes	—
18	C₁₉×H(3,𝔽₃)	C₃³	1	81	yes	—
19	#10587025	C₃³	1	81	yes	—
20	#10587025	C₃³	1	648	yes	complemented
21	C₃×#4294	C₃³	1	27	yes	—
22	C₃×#4294	C₃³	1	324	yes	complemented
23	3₋¹⁺⁴	—	19	81	no	—
24	C₃×#737	—	19	27	no	—
25	C₃²×3₋¹⁺²	—	19	12	no	—
26	C₃²×H(3,𝔽₃)	C₁₉⋊C₃	1	1	yes	characteristic
27	C₃²×C₁₉	—	3	3	no	—
28	C₃²×C₁₉	—	3	36	no	—
29	C₃²×C₁₉	—	9	36	no	—
30	C₃²×C₁₉	C₃⁴	1	36	yes	—
31	C₃²×C₁₉	C₃⁴	1	3	yes	—
32	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
33	#4294	C₃⁴	1	81	yes	—
34	C₃⁴	C₃×C₁₉⋊C₃	1	4	yes	—
35	#737	—	19	729	no	—
36	C₃×3₋¹⁺²	—	19	9	no	—
37	C₃×3₋¹⁺²	—	19	72	no	—
38	C₃×3₋¹⁺²	—	19	324	no	—
39	C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	9	yes	—
40	C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	27	yes	—
41	C₃²×C₉	—	19	36	no	—
42	C₃×C₁₉	—	3	3	no	—
43	C₃×C₁₉	—	3	36	no	—
44	C₃×C₁₉	3₋¹⁺⁴	1	3	yes	complemented
45	C₃×C₁₉	C₃⁵	1	1	yes	characteristic
46	C₃³	—	9	12	no	—
47	C₃³	C₃²×C₁₉⋊C₃	1	12	yes	—
48	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
49	C₃³	C₃²×C₁₉⋊C₃	1	36	yes	—
50	3₋¹⁺²	—	19	81	no	—
51	3₋¹⁺²	—	19	648	no	—
52	H(3,𝔽₃)	C₃²×C₁₉⋊C₃	1	81	yes	—
53	C₃×C₉	—	19	324	no	—
54	C₃×C₉	—	19	27	no	—
55	C₁₉	C₃×3₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
56	C₃²	—	3	3	no	—
57	C₃²	—	3	36	no	—
58	C₃²	—	9	36	no	—
59	C₃²	C₃³×C₁₉⋊C₃	1	36	yes	—
60	C₃²	C₃³×C₁₉⋊C₃	1	3	yes	—
61	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
62	C₉	—	19	81	no	—
63	C₃	—	3	3	no	—
64	C₃	—	3	36	no	—
65	C₃	#17987547	1	3	yes	direct product factor
66	C₃	C₃⁴×C₁₉⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃²	C₃³×C₁₉⋊C₃
commutator, cosocle	C₃×C₁₉	C₃⁵
Frattini	C₃	C₃⁴×C₁₉⋊C₃
Fitting, prime radical	C₃²×C₁₉×H(3,𝔽₃)	C₃
socle	C₃²×C₁₉	C₃⁴
hypercenter	C₃²×H(3,𝔽₃)	C₁₉⋊C₃
hypocenter	C₁₉	C₃×3₋¹⁺⁴
LCS	G ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₃²×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. #17987547	C₃	direct (N × H)
2. C₃×#13095331	C₃	semidirect
3. C₃²×#10587025	C₃	semidirect
5. #13095331	C₃²	semidirect
7. C₃×#10587025	C₃²	semidirect
8. C₃×#10587025	C₃²	semidirect
9. C₃²×#4294	C₃²	semidirect
20. #10587025	C₃³	semidirect
22. C₃×#4294	C₃³	semidirect
44. C₃×C₁₉	3₋¹⁺⁴	semidirect
55. C₁₉	C₃×3₋¹⁺⁴	semidirect
65. C₃	#17987547	direct (N × H)

Maximal subgroups

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

subgroup	index
#17987547	3
C₃×#13095331	3
C₃²×#10587025	3
C₃²×C₁₉×H(3,𝔽₃)	3
C₃×3₋¹⁺⁴	19

Subgroups — C3×#17987547 #28356665

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#17987547 #28356665