Subgroups — #28356610

Order 13851.

1062 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉⋊H(5,𝔽₃)	C₃	1	2	yes	maximal, complemented
2	C₃×#13095331	C₃	1	1	yes	maximal, characteristic, complemented
3	#17987514	C₃	1	24	yes	maximal, complemented
4	C₃×#13095319	C₃	1	12	yes	maximal, complemented
5	C₁₉×H(5,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
6	#13095331	—	3	9	no	—
7	C₃×#10587025	C₃²	1	8	yes	complemented
8	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	24	yes	complemented
9	H(3,𝔽₃)×C₁₉⋊C₃	—	3	18	no	—
10	C₃²×#4294	C₃²	1	4	yes	complemented
11	#13095319	C₃²	1	9	yes	complemented
12	#13095319	C₃²	1	72	yes	complemented
13	C₃×C₁₉×H(3,𝔽₃)	—	3	9	no	—
14	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
15	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	12	yes	complemented
16	#10624932	—	19	1	no	maximal, Sylow
17	C₃³×C₁₉	—	3	12	no	—
18	C₃³×C₁₉	C₃³	1	4	yes	—
19	C₃²×C₁₉⋊C₃	—	3	24	no	—
20	C₁₉×H(3,𝔽₃)	—	3	24	no	—
21	C₁₉×H(3,𝔽₃)	H(3,𝔽₃)	1	9	yes	—
22	C₁₉×H(3,𝔽₃)	C₃³	1	9	yes	—
23	C₃²⋊(C₁₉⋊C₃)	—	3	48	no	—
24	C₃²⋊(C₁₉⋊C₃)	C₃³	1	18	yes	—
25	#10587025	—	3	24	no	—
26	C₃×#4294	—	3	12	no	—
27	C₃×#4294	C₃³	1	1	yes	characteristic
28	C₃×#4294	C₃³	1	8	yes	—
29	H(5,𝔽₃)	—	19	2	no	—
30	H(5,𝔽₃)	C₁₉⋊C₃	1	1	yes	characteristic, complemented
31	C₃×#737	—	19	1	no	—
32	#6911	—	19	24	no	—
33	C₃×#732	—	19	12	no	—
34	C₃²×C₁₉	—	3	9	no	—
35	C₃²×C₁₉	—	9	4	no	—
36	C₃²×C₁₉	—	27	12	no	—
37	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	12	yes	—
38	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
39	C₃×C₁₉⋊C₃	—	3	18	no	—
40	C₃×C₁₉⋊C₃	—	27	24	no	—
41	#4294	—	3	1	no	—
42	#4294	—	3	8	no	—
43	#737	—	57	9	no	—
44	C₃×3₋¹⁺²	—	19	8	no	—
45	C₃×H(3,𝔽₃)	—	3	9	no	—
46	C₃×H(3,𝔽₃)	—	19	24	no	—
47	C₃×H(3,𝔽₃)	—	57	18	no	—
48	C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	1	yes	characteristic
49	C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	12	yes	complemented
50	C₃²×C₉	—	19	4	no	—
51	#732	—	19	9	no	—
52	#732	—	19	72	no	—
53	C₃×C₁₉	—	3	12	no	—
54	C₃×C₁₉	—	3	1	no	—
55	C₃×C₁₉	—	9	9	no	—
56	C₃×C₁₉	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
57	C₁₉⋊C₃	—	9	18	no	—
58	C₃³	—	3	12	no	—
59	C₃³	—	57	24	no	—
60	C₃³	C₃²×C₁₉⋊C₃	1	4	yes	—
61	3₋¹⁺²	—	57	24	no	—
62	H(3,𝔽₃)	—	3	24	no	—
63	H(3,𝔽₃)	—	19	18	no	—
64	H(3,𝔽₃)	—	57	48	no	—
65	H(3,𝔽₃)	C₃²⋊(C₁₉⋊C₃)	1	9	yes	—
66	H(3,𝔽₃)	C₃²×C₁₉⋊C₃	1	9	yes	—
67	C₃×C₉	—	19	1	no	—
68	C₃×C₉	—	19	8	no	—
69	C₃×C₉	—	57	12	no	—
70	C₁₉	#10624932	1	1	yes	characteristic, complemented, Sylow
71	C₃²	—	513	24	no	—
72	C₃²	—	3	9	no	—
73	C₃²	—	9	4	no	—
74	C₃²	—	27	12	no	—
75	C₃²	—	57	18	no	—
76	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	12	yes	—
77	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
78	C₉	—	57	1	no	—
79	C₉	—	57	8	no	—
80	C₃	—	3	12	no	—
81	C₃	—	3	1	no	—
82	C₃	—	9	9	no	—
83	C₃	—	171	18	no	—
84	C₃	C₃²×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₃	C₃²×C₃²⋊(C₁₉⋊C₃)
commutator, cosocle	C₃²×C₁₉	C₃⁴
Frattini	C₃²	C₃³×C₁₉⋊C₃
Fitting	C₁₉×H(5,𝔽₃)	C₃
socle	C₃×C₁₉	C₃²×H(3,𝔽₃)
hypercenter	H(5,𝔽₃)	C₁₉⋊C₃
hypocenter	C₁₉	#10624932
LCS	G ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃×H(3,𝔽₃) ◁ H(5,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. C₁₉⋊H(5,𝔽₃)	C₃	semidirect
2. C₃×#13095331	C₃	semidirect
3. #17987514	C₃	semidirect
4. C₃×#13095319	C₃	semidirect
5. C₁₉×H(5,𝔽₃)	C₃	semidirect
7. C₃×#10587025	C₃²	semidirect
8. C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	semidirect
10. C₃²×#4294	C₃²	semidirect
11. #13095319	C₃²	semidirect
12. #13095319	C₃²	semidirect
15. C₃×C₁₉×H(3,𝔽₃)	C₃²	semidirect
30. H(5,𝔽₃)	C₁₉⋊C₃	semidirect
49. C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	semidirect
70. C₁₉	#10624932	semidirect

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
C₃×#13095331	3
#17987514	3
C₃×#13095319	3
C₁₉×H(5,𝔽₃)	3
#10624932	19