Subgroups — C₃²×H(3,𝔽₃)×C₁₉⋊C₃ #28356649

Order 13851.

11744 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⁴×C₁₉⋊C₃	C₃	1	4	yes	maximal, complemented
2	C₃²×C₃²⋊(C₁₉⋊C₃)	C₃	1	8	yes	maximal, complemented
3	C₃×H(3,𝔽₃)×C₁₉⋊C₃	C₃	1	108	yes	maximal, direct product factor
4	C₃²×C₁₉×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃³×C₁₉⋊C₃	—	3	108	no	—
6	C₃³×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
7	C₃³×C₁₉⋊C₃	C₃²	1	144	yes	complemented
8	C₃³×C₁₉⋊C₃	C₃²	1	8	yes	complemented
9	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	288	yes	complemented
10	H(3,𝔽₃)×C₁₉⋊C₃	C₃²	1	729	yes	direct product factor
11	C₃⁴×C₁₉	C₃²	1	4	yes	complemented
12	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	36	yes	complemented
13	C₃³×H(3,𝔽₃)	—	19	1	no	maximal, Sylow
14	C₃³×C₁₉	—	3	36	no	—
15	C₃³×C₁₉	C₃³	1	1	yes	characteristic
16	C₃³×C₁₉	C₃³	1	48	yes	complemented
17	C₃²×C₁₉⋊C₃	—	3	72	no	—
18	C₃²×C₁₉⋊C₃	—	3	1296	no	—
19	C₃²×C₁₉⋊C₃	H(3,𝔽₃)	1	27	yes	direct product factor
20	C₃²×C₁₉⋊C₃	C₃³	1	12	yes	—
21	C₃²×C₁₉⋊C₃	C₃³	1	324	yes	complemented
22	C₃²×C₁₉⋊C₃	C₃³	1	96	yes	complemented
23	C₁₉×H(3,𝔽₃)	C₃³	1	81	yes	complemented
24	C₃²⋊(C₁₉⋊C₃)	C₃³	1	648	yes	complemented
25	C₃⁵	—	19	4	no	—
26	C₃²×H(3,𝔽₃)	—	19	8	no	—
27	C₃²×H(3,𝔽₃)	—	19	108	no	—
28	C₃²×H(3,𝔽₃)	C₁₉⋊C₃	1	1	yes	characteristic, direct product factor
29	C₃²×C₁₉	—	3	144	no	—
30	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	9	yes	complemented
31	C₃²×C₁₉	C₃⁴	1	4	yes	—
32	C₃²×C₁₉	C₃⁴	1	36	yes	complemented
33	C₃×C₁₉⋊C₃	—	3	288	no	—
34	C₃×C₁₉⋊C₃	—	3	972	no	—
35	C₃×C₁₉⋊C₃	C₃×H(3,𝔽₃)	1	108	yes	direct product factor
36	C₃×C₁₉⋊C₃	C₃⁴	1	9	yes	—
37	C₃×C₁₉⋊C₃	C₃⁴	1	72	yes	complemented
38	C₃⁴	—	19	1	no	—
39	C₃⁴	—	19	8	no	—
40	C₃⁴	—	19	144	no	—
41	C₃⁴	—	57	108	no	—
42	C₃⁴	C₃×C₁₉⋊C₃	1	4	yes	complemented
43	C₃×H(3,𝔽₃)	—	19	288	no	—
44	C₃×H(3,𝔽₃)	—	19	729	no	—
45	C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	36	yes	direct product factor
46	C₃×C₁₉	—	3	36	no	—
47	C₃×C₁₉	C₃²×H(3,𝔽₃)	1	12	yes	complemented
48	C₃×C₁₉	C₃⁵	1	1	yes	characteristic
49	C₁₉⋊C₃	—	3	72	no	—
50	C₁₉⋊C₃	C₃²×H(3,𝔽₃)	1	27	yes	direct product factor
51	C₃³	—	3	36	no	—
52	C₃³	—	19	27	no	—
53	C₃³	—	19	12	no	—
54	C₃³	—	19	96	no	—
55	C₃³	—	19	324	no	—
56	C₃³	—	57	1296	no	—
57	C₃³	—	57	72	no	—
58	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
59	C₃³	C₃²×C₁₉⋊C₃	1	48	yes	complemented
60	H(3,𝔽₃)	—	19	648	no	—
61	H(3,𝔽₃)	C₃²×C₁₉⋊C₃	1	81	yes	direct product factor
62	C₁₉	C₃³×H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
63	C₃²	—	3	144	no	—
64	C₃²	—	19	108	no	—
65	C₃²	—	19	9	no	—
66	C₃²	—	19	72	no	—
67	C₃²	—	57	972	no	—
68	C₃²	—	57	288	no	—
69	C₃²	H(3,𝔽₃)×C₁₉⋊C₃	1	9	yes	direct product factor
70	C₃²	C₃³×C₁₉⋊C₃	1	4	yes	—
71	C₃²	C₃³×C₁₉⋊C₃	1	36	yes	complemented
72	C₃	—	3	36	no	—
73	C₃	—	19	27	no	—
74	C₃	—	57	72	no	—
75	C₃	C₃×H(3,𝔽₃)×C₁₉⋊C₃	1	12	yes	direct product factor
76	C₃	C₃⁴×C₁₉⋊C₃	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₃⁴×C₁₉⋊C₃	C₃	semidirect
2. C₃²×C₃²⋊(C₁₉⋊C₃)	C₃	semidirect
3. C₃×H(3,𝔽₃)×C₁₉⋊C₃	C₃	direct (N × H)
4. C₃²×C₁₉×H(3,𝔽₃)	C₃	semidirect
7. C₃³×C₁₉⋊C₃	C₃²	semidirect
8. C₃³×C₁₉⋊C₃	C₃²	semidirect
9. C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	semidirect
10. H(3,𝔽₃)×C₁₉⋊C₃	C₃²	direct (N × H)
11. C₃⁴×C₁₉	C₃²	semidirect
12. C₃×C₁₉×H(3,𝔽₃)	C₃²	semidirect
16. C₃³×C₁₉	C₃³	semidirect
19. C₃²×C₁₉⋊C₃	H(3,𝔽₃)	direct (N × H)
21. C₃²×C₁₉⋊C₃	C₃³	semidirect
22. C₃²×C₁₉⋊C₃	C₃³	semidirect
23. C₁₉×H(3,𝔽₃)	C₃³	semidirect
24. C₃²⋊(C₁₉⋊C₃)	C₃³	semidirect
28. C₃²×H(3,𝔽₃)	C₁₉⋊C₃	direct (N × H)
30. C₃²×C₁₉	C₃×H(3,𝔽₃)	semidirect
32. C₃²×C₁₉	C₃⁴	semidirect
35. C₃×C₁₉⋊C₃	C₃×H(3,𝔽₃)	direct (N × H)
37. C₃×C₁₉⋊C₃	C₃⁴	semidirect
42. C₃⁴	C₃×C₁₉⋊C₃	semidirect
45. C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	direct (N × H)
47. C₃×C₁₉	C₃²×H(3,𝔽₃)	semidirect
50. C₁₉⋊C₃	C₃²×H(3,𝔽₃)	direct (N × H)
59. C₃³	C₃²×C₁₉⋊C₃	semidirect
61. H(3,𝔽₃)	C₃²×C₁₉⋊C₃	direct (N × H)
62. C₁₉	C₃³×H(3,𝔽₃)	semidirect
69. C₃²	H(3,𝔽₃)×C₁₉⋊C₃	direct (N × H)
71. C₃²	C₃³×C₁₉⋊C₃	semidirect
75. C₃	C₃×H(3,𝔽₃)×C₁₉⋊C₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃⁴×C₁₉⋊C₃	3
C₃²×C₃²⋊(C₁₉⋊C₃)	3
C₃×H(3,𝔽₃)×C₁₉⋊C₃	3
C₃²×C₁₉×H(3,𝔽₃)	3
C₃³×H(3,𝔽₃)	19

Subgroups — C32×H(3,𝔽3)×C19⋊C3 #28356649

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×H(3,𝔽₃)×C₁₉⋊C₃ #28356649