Subgroups — #28356608

Order 13851.

978 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17987547	C₃	1	1	yes	maximal, characteristic, complemented
2	C₁₉⋊H(5,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	C₃×#13095331	C₃	1	1	yes	maximal, characteristic, complemented
4	#17987511	C₃	1	24	yes	maximal, complemented
5	C₃×#13095318	C₃	1	12	yes	maximal, complemented
6	C₁₉×3₋¹⁺⁴	C₃	1	1	yes	maximal, characteristic, complemented
7	#13095331	—	3	9	no	—
8	#13095331	—	3	9	no	—
9	C₃×#10587025	C₃²	1	12	yes	complemented
10	C₃×#10587025	C₃²	1	8	yes	complemented
11	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	12	yes	—
12	H(3,𝔽₃)×C₁₉⋊C₃	—	3	9	no	—
13	C₃²×#4294	C₃²	1	4	yes	complemented
14	#13095318	C₃²	1	9	yes	complemented
15	#13095318	C₃²	1	72	yes	complemented
16	C₁₉×#737	—	3	9	no	—
17	C₃×C₁₉×3₋¹⁺²	C₃²	1	12	yes	complemented
18	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
19	#10624931	—	19	1	no	maximal, Sylow
20	C₃³×C₁₉	C₃³	1	4	yes	—
21	C₃²×C₁₉⋊C₃	—	3	12	no	—
22	C₁₉×3₋¹⁺²	—	3	24	no	—
23	C₁₉×3₋¹⁺²	C₃³	1	9	yes	—
24	C₁₉×H(3,𝔽₃)	H(3,𝔽₃)	1	9	yes	—
25	C₃×C₉×C₁₉	—	3	12	no	—
26	C₃²⋊(C₁₉⋊C₃)	—	3	24	no	—
27	C₃²⋊(C₁₉⋊C₃)	C₃³	1	9	yes	—
28	#10587025	—	3	24	no	—
29	#10587025	—	3	24	no	—
30	#10587025	C₃³	1	9	yes	—
31	C₃×#4294	—	3	12	no	—
32	C₃×#4294	—	3	12	no	—
33	C₃×#4294	C₃³	1	1	yes	characteristic
34	C₃×#4294	C₃³	1	8	yes	—
35	3₋¹⁺⁴	—	19	1	no	—
36	3₋¹⁺⁴	C₁₉⋊C₃	1	1	yes	characteristic, complemented
37	H(5,𝔽₃)	—	19	1	no	—
38	C₃×#737	—	19	1	no	—
39	#6910	—	19	24	no	—
40	C₃×#731	—	19	12	no	—
41	C₃²×C₁₉	—	9	4	no	—
42	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	12	yes	—
43	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
44	C₃×C₁₉⋊C₃	—	3	9	no	—
45	C₃×C₁₉⋊C₃	—	27	12	no	—
46	C₉×C₁₉	—	3	9	no	—
47	#4294	—	3	1	no	—
48	#4294	—	3	9	no	—
49	#4294	—	3	8	no	—
50	#737	—	3	9	no	—
51	#737	—	57	9	no	—
52	#737	—	57	9	no	—
53	C₃×3₋¹⁺²	—	19	8	no	—
54	C₃×3₋¹⁺²	—	19	12	no	—
55	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	12	yes	complemented
56	C₃×H(3,𝔽₃)	—	19	12	no	—
57	C₃×H(3,𝔽₃)	—	57	9	no	—
58	C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	1	yes	characteristic
59	C₃²×C₉	—	19	4	no	—
60	#731	—	19	9	no	—
61	#731	—	19	72	no	—
62	C₃×C₁₉	—	3	12	no	—
63	C₃×C₁₉	—	3	1	no	—
64	C₃×C₁₉	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
65	C₁₉⋊C₃	—	9	9	no	—
66	C₃³	—	57	12	no	—
67	C₃³	C₃²×C₁₉⋊C₃	1	4	yes	—
68	3₋¹⁺²	—	3	24	no	—
69	3₋¹⁺²	—	19	9	no	—
70	3₋¹⁺²	—	57	24	no	—
71	3₋¹⁺²	—	57	24	no	—
72	3₋¹⁺²	C₃²×C₁₉⋊C₃	1	9	yes	—
73	H(3,𝔽₃)	—	19	9	no	—
74	H(3,𝔽₃)	—	57	24	no	—
75	H(3,𝔽₃)	C₃²⋊(C₁₉⋊C₃)	1	9	yes	—
76	C₃×C₉	—	3	12	no	—
77	C₃×C₉	—	19	1	no	—
78	C₃×C₉	—	19	8	no	—
79	C₃×C₉	—	57	12	no	—
80	C₃×C₉	—	57	12	no	—
81	C₁₉	#10624931	1	1	yes	characteristic, complemented, Sylow
82	C₃²	—	513	12	no	—
83	C₃²	—	9	4	no	—
84	C₃²	—	57	9	no	—
85	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	12	yes	—
86	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
87	C₉	—	3	9	no	—
88	C₉	—	57	1	no	—
89	C₉	—	57	8	no	—
90	C₉	—	57	9	no	—
91	C₃	—	3	12	no	—
92	C₃	—	3	1	no	—
93	C₃	—	171	9	no	—
94	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	C₁₉×3₋¹⁺⁴	C₃
socle	C₃×C₁₉	C₃²×H(3,𝔽₃)
prime radical	C₁₉⋊H(5,𝔽₃)	C₃
hypercenter	3₋¹⁺⁴	C₁₉⋊C₃
hypocenter	C₁₉	#10624931
LCS	G ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃×H(3,𝔽₃) ◁ 3₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. #17987547	C₃	semidirect
3. C₃×#13095331	C₃	semidirect
4. #17987511	C₃	semidirect
5. C₃×#13095318	C₃	semidirect
6. C₁₉×3₋¹⁺⁴	C₃	semidirect
9. C₃×#10587025	C₃²	semidirect
10. C₃×#10587025	C₃²	semidirect
13. C₃²×#4294	C₃²	semidirect
14. #13095318	C₃²	semidirect
15. #13095318	C₃²	semidirect
17. C₃×C₁₉×3₋¹⁺²	C₃²	semidirect
36. 3₋¹⁺⁴	C₁₉⋊C₃	semidirect
55. C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	semidirect
81. C₁₉	#10624931	semidirect

Maximal subgroups

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

subgroup	index
#17987547	3
C₁₉⋊H(5,𝔽₃)	3
C₃×#13095331	3
#17987511	3
C₃×#13095318	3
C₁₉×3₋¹⁺⁴	3
#10624931	19