Subgroups — C₃²×C₁₉×#737 #28353136

Order 13851.

5804 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₁₉×#737	C₃	1	108	yes	maximal, direct product factor
2	C₃²×C₁₉×3₋¹⁺²	C₃	1	8	yes	maximal, complemented
3	C₃²×C₁₉×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
4	C₃³×C₉×C₁₉	C₃	1	4	yes	maximal, complemented
5	C₃⁴×C₁₉	C₃²	1	4	yes	—
6	C₁₉×#737	C₃²	1	729	yes	direct product factor
7	C₃×C₁₉×3₋¹⁺²	C₃²	1	288	yes	complemented
8	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	36	yes	—
9	C₃²×C₉×C₁₉	C₃²	1	8	yes	—
10	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
11	C₃²×C₉×C₁₉	C₃²	1	144	yes	complemented
12	C₃²×#737	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₃³×C₁₉	—	3	36	no	—
14	C₃³×C₁₉	C₃³	1	48	yes	—
15	C₃³×C₁₉	C₃³	1	1	yes	characteristic
16	C₁₉×3₋¹⁺²	C₃³	1	648	yes	complemented
17	C₁₉×H(3,𝔽₃)	C₃³	1	81	yes	—
18	C₃×C₉×C₁₉	C₃³	1	96	yes	—
19	C₃×C₉×C₁₉	C₃³	1	12	yes	—
20	C₃×C₉×C₁₉	C₃³	1	324	yes	complemented
21	C₃×#737	C₃×C₁₉	1	108	yes	direct product factor
22	C₃²×3₋¹⁺²	C₃×C₁₉	1	8	yes	complemented
23	C₃²×H(3,𝔽₃)	C₃×C₁₉	1	1	yes	characteristic
24	C₃³×C₉	C₃×C₁₉	1	4	yes	complemented
25	C₃²×C₁₉	—	3	144	no	—
26	C₃²×C₁₉	#737	1	9	yes	direct product factor
27	C₃²×C₁₉	C₃⁴	1	36	yes	—
28	C₃²×C₁₉	C₃⁴	1	4	yes	—
29	C₉×C₁₉	C₃⁴	1	9	yes	—
30	C₉×C₁₉	C₃⁴	1	72	yes	—
31	C₃⁴	C₃²×C₁₉	1	4	yes	—
32	#737	C₃²×C₁₉	1	729	yes	direct product factor
33	C₃×3₋¹⁺²	C₃²×C₁₉	1	288	yes	complemented
34	C₃×H(3,𝔽₃)	C₃²×C₁₉	1	36	yes	—
35	C₃²×C₉	C₃²×C₁₉	1	1	yes	characteristic
36	C₃²×C₉	C₃²×C₁₉	1	8	yes	—
37	C₃²×C₉	C₃²×C₁₉	1	144	yes	complemented
38	C₃×C₁₉	—	3	36	no	—
39	C₃×C₁₉	C₃×#737	1	12	yes	direct product factor
40	C₃×C₁₉	C₃⁵	1	1	yes	characteristic
41	C₃³	—	3	36	no	—
42	C₃³	C₃³×C₁₉	1	48	yes	—
43	C₃³	C₃³×C₁₉	1	1	yes	characteristic
44	3₋¹⁺²	C₃³×C₁₉	1	648	yes	complemented
45	H(3,𝔽₃)	C₃³×C₁₉	1	81	yes	—
46	C₃×C₉	C₃³×C₁₉	1	12	yes	—
47	C₃×C₉	C₃³×C₁₉	1	96	yes	—
48	C₃×C₉	C₃³×C₁₉	1	324	yes	complemented
49	C₁₉	C₃²×#737	1	1	yes	characteristic, direct product factor, Sylow
50	C₃²	—	3	144	no	—
51	C₃²	C₁₉×#737	1	9	yes	direct product factor
52	C₃²	C₃⁴×C₁₉	1	36	yes	—
53	C₃²	C₃⁴×C₁₉	1	4	yes	—
54	C₉	C₃⁴×C₁₉	1	9	yes	—
55	C₉	C₃⁴×C₁₉	1	72	yes	—
56	C₃	—	3	36	no	—
57	C₃	C₃×C₁₉×#737	1	12	yes	direct product factor
58	C₃	C₃⁵×C₁₉	1	1	yes	characteristic

	subgroup	G/H
center	C₃²×C₉×C₁₉	C₃²
commutator, Frattini, cosocle	C₃	C₃⁵×C₁₉
socle	C₃³×C₁₉	C₃³
prime radical	C₃²×C₁₉×H(3,𝔽₃)	C₃
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₉×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₁₉×#737	C₃	direct (N × H)
2. C₃²×C₁₉×3₋¹⁺²	C₃	semidirect
4. C₃³×C₉×C₁₉	C₃	semidirect
6. C₁₉×#737	C₃²	direct (N × H)
7. C₃×C₁₉×3₋¹⁺²	C₃²	semidirect
11. C₃²×C₉×C₁₉	C₃²	semidirect
12. C₃²×#737	C₁₉	direct (N × H)
16. C₁₉×3₋¹⁺²	C₃³	semidirect
20. C₃×C₉×C₁₉	C₃³	semidirect
21. C₃×#737	C₃×C₁₉	direct (N × H)
22. C₃²×3₋¹⁺²	C₃×C₁₉	semidirect
24. C₃³×C₉	C₃×C₁₉	semidirect
26. C₃²×C₁₉	#737	direct (N × H)
32. #737	C₃²×C₁₉	direct (N × H)
33. C₃×3₋¹⁺²	C₃²×C₁₉	semidirect
37. C₃²×C₉	C₃²×C₁₉	semidirect
39. C₃×C₁₉	C₃×#737	direct (N × H)
44. 3₋¹⁺²	C₃³×C₁₉	semidirect
48. C₃×C₉	C₃³×C₁₉	semidirect
49. C₁₉	C₃²×#737	direct (N × H)
51. C₃²	C₁₉×#737	direct (N × H)
57. C₃	C₃×C₁₉×#737	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₁₉×#737	3
C₃²×C₁₉×3₋¹⁺²	3
C₃²×C₁₉×H(3,𝔽₃)	3
C₃³×C₉×C₁₉	3
C₃²×#737	19

Subgroups — C32×C19×#737 #28353136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₁₉×#737 #28353136