Subgroups — C₃×C₁₉⋊H(5,𝔽₃) #28356660

Order 13851.

6776 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉⋊H(5,𝔽₃)	C₃	1	81	yes	maximal, direct product factor
2	C₃²×C₃²⋊(C₁₉⋊C₃)	C₃	1	12	yes	maximal, complemented
3	C₃×H(3,𝔽₃)×C₁₉⋊C₃	C₃	1	27	yes	maximal, complemented
4	C₃²×C₁₉×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃³×C₁₉⋊C₃	C₃²	1	36	yes	complemented
6	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	72	yes	complemented
7	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	9	yes	complemented
8	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	324	yes	complemented
9	H(3,𝔽₃)×C₁₉⋊C₃	C₃²	1	729	yes	complemented
10	C₃⁴×C₁₉	C₃²	1	4	yes	complemented
11	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	9	yes	complemented
12	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	27	yes	complemented
13	C₃×H(5,𝔽₃)	—	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	complemented
18	C₃²×C₁₉⋊C₃	—	9	108	no	—
19	C₃²×C₁₉⋊C₃	C₃³	1	27	yes	—
20	C₃²×C₁₉⋊C₃	C₃³	1	324	yes	complemented
21	C₁₉×H(3,𝔽₃)	C₃³	1	81	yes	complemented
22	C₃²⋊(C₁₉⋊C₃)	C₃³	1	648	yes	complemented
23	C₃²⋊(C₁₉⋊C₃)	C₃³	1	81	yes	complemented
24	H(5,𝔽₃)	—	19	81	no	—
25	C₃²×H(3,𝔽₃)	—	19	12	no	—
26	C₃²×H(3,𝔽₃)	—	19	27	no	—
27	C₃²×H(3,𝔽₃)	C₁₉⋊C₃	1	1	yes	characteristic, complemented
28	C₃²×C₁₉	—	3	36	no	—
29	C₃²×C₁₉	—	3	3	no	—
30	C₃²×C₁₉	—	9	36	no	—
31	C₃²×C₁₉	C₃⁴	1	3	yes	—
32	C₃²×C₁₉	C₃⁴	1	36	yes	—
33	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
34	C₃×C₁₉⋊C₃	—	3	81	no	—
35	C₃×C₁₉⋊C₃	—	9	324	no	—
36	C₃×C₁₉⋊C₃	C₃⁴	1	81	yes	—
37	C₃⁴	—	19	36	no	—
38	C₃⁴	C₃×C₁₉⋊C₃	1	4	yes	complemented
39	C₃×H(3,𝔽₃)	—	19	9	no	—
40	C₃×H(3,𝔽₃)	—	19	72	no	—
41	C₃×H(3,𝔽₃)	—	19	324	no	—
42	C₃×H(3,𝔽₃)	—	19	729	no	—
43	C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	9	yes	complemented
44	C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	27	yes	complemented
45	C₃×C₁₉	—	3	36	no	—
46	C₃×C₁₉	—	3	3	no	—
47	C₃×C₁₉	H(5,𝔽₃)	1	3	yes	complemented
48	C₃×C₁₉	C₃⁵	1	1	yes	characteristic
49	C₁₉⋊C₃	—	3	81	no	—
50	C₃³	—	9	12	no	—
51	C₃³	—	19	27	no	—
52	C₃³	—	19	324	no	—
53	C₃³	—	171	108	no	—
54	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
55	C₃³	C₃²×C₁₉⋊C₃	1	12	yes	—
56	C₃³	C₃²×C₁₉⋊C₃	1	36	yes	complemented
57	H(3,𝔽₃)	—	19	81	no	—
58	H(3,𝔽₃)	—	19	648	no	—
59	H(3,𝔽₃)	C₃²×C₁₉⋊C₃	1	81	yes	complemented
60	C₁₉	C₃×H(5,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
61	C₃²	—	3	3	no	—
62	C₃²	—	3	36	no	—
63	C₃²	—	9	36	no	—
64	C₃²	—	19	81	no	—
65	C₃²	—	57	81	no	—
66	C₃²	—	171	324	no	—
67	C₃²	C₃³×C₁₉⋊C₃	1	3	yes	—
68	C₃²	C₃³×C₁₉⋊C₃	1	36	yes	—
69	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
70	C₃	—	3	3	no	—
71	C₃	—	3	36	no	—
72	C₃	—	57	81	no	—
73	C₃	C₁₉⋊H(5,𝔽₃)	1	3	yes	direct product factor
74	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	C₃²×C₁₉×H(3,𝔽₃)	C₃
socle	C₃²×C₁₉	C₃⁴
hypercenter	C₃²×H(3,𝔽₃)	C₁₉⋊C₃
hypocenter	C₁₉	C₃×H(5,𝔽₃)
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. C₁₉⋊H(5,𝔽₃)	C₃	direct (N × H)
2. C₃²×C₃²⋊(C₁₉⋊C₃)	C₃	semidirect
3. C₃×H(3,𝔽₃)×C₁₉⋊C₃	C₃	semidirect
4. C₃²×C₁₉×H(3,𝔽₃)	C₃	semidirect
5. C₃³×C₁₉⋊C₃	C₃²	semidirect
6. C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	semidirect
7. C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	semidirect
8. C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	semidirect
9. H(3,𝔽₃)×C₁₉⋊C₃	C₃²	semidirect
10. C₃⁴×C₁₉	C₃²	semidirect
11. C₃×C₁₉×H(3,𝔽₃)	C₃²	semidirect
12. C₃×C₁₉×H(3,𝔽₃)	C₃²	semidirect
17. C₃³×C₁₉	C₃³	semidirect
20. C₃²×C₁₉⋊C₃	C₃³	semidirect
21. C₁₉×H(3,𝔽₃)	C₃³	semidirect
22. C₃²⋊(C₁₉⋊C₃)	C₃³	semidirect
23. C₃²⋊(C₁₉⋊C₃)	C₃³	semidirect
27. C₃²×H(3,𝔽₃)	C₁₉⋊C₃	semidirect
38. C₃⁴	C₃×C₁₉⋊C₃	semidirect
43. C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	semidirect
44. C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	semidirect
47. C₃×C₁₉	H(5,𝔽₃)	semidirect
56. C₃³	C₃²×C₁₉⋊C₃	semidirect
59. H(3,𝔽₃)	C₃²×C₁₉⋊C₃	semidirect
60. C₁₉	C₃×H(5,𝔽₃)	semidirect
73. C₃	C₁₉⋊H(5,𝔽₃)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₉⋊H(5,𝔽₃)	3
C₃²×C₃²⋊(C₁₉⋊C₃)	3
C₃×H(3,𝔽₃)×C₁₉⋊C₃	3
C₃²×C₁₉×H(3,𝔽₃)	3
C₃×H(5,𝔽₃)	19

Subgroups — C3×C19⋊H(5,𝔽3) #28356660

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₁₉⋊H(5,𝔽₃) #28356660