Subgroups — #28356607

Order 13851.

978 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17987547	C₃	1	2	yes	maximal, complemented
2	C₁₉⋊H(5,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	#17987510	C₃	1	24	yes	maximal, complemented
4	C₃×#13095317	C₃	1	12	yes	maximal, complemented
5	C₃×C₁₉×#737	C₃	1	1	yes	maximal, characteristic, complemented
6	#13095331	—	3	18	no	—
7	C₃×#10587025	C₃²	1	24	yes	complemented
8	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	12	yes	—
9	H(3,𝔽₃)×C₁₉⋊C₃	—	3	9	no	—
10	#13095317	C₃²	1	9	yes	complemented
11	#13095317	C₃²	1	72	yes	complemented
12	C₁₉×#737	—	3	9	no	—
13	C₃×C₁₉×3₋¹⁺²	C₃²	1	8	yes	complemented
14	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
15	C₃²×C₉×C₁₉	C₃²	1	4	yes	complemented
16	#10624931	—	19	1	no	maximal, Sylow
17	C₃³×C₁₉	C₃³	1	4	yes	—
18	C₃²×C₁₉⋊C₃	—	3	12	no	—
19	C₁₉×3₋¹⁺²	—	3	24	no	—
20	C₁₉×H(3,𝔽₃)	H(3,𝔽₃)	1	9	yes	—
21	C₃×C₉×C₁₉	—	3	12	no	—
22	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
23	C₃×C₉×C₁₉	C₃³	1	8	yes	—
24	C₃²⋊(C₁₉⋊C₃)	—	3	24	no	—
25	C₃²⋊(C₁₉⋊C₃)	C₃³	1	9	yes	—
26	#10587025	—	3	48	no	—
27	#10587025	C₃³	1	18	yes	—
28	C₃×#4294	—	3	24	no	—
29	3₋¹⁺⁴	—	19	2	no	—
30	H(5,𝔽₃)	—	19	1	no	—
31	C₃×#737	C₁₉⋊C₃	1	1	yes	characteristic, complemented
32	#6910	—	19	24	no	—
33	C₃×#731	—	19	12	no	—
34	C₃²×C₁₉	—	9	4	no	—
35	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	12	yes	—
36	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
37	C₃×C₁₉⋊C₃	—	3	9	no	—
38	C₃×C₁₉⋊C₃	—	27	12	no	—
39	C₉×C₁₉	—	3	1	no	—
40	C₉×C₁₉	—	3	8	no	—
41	#4294	—	3	18	no	—
42	#737	—	3	9	no	—
43	#737	—	57	18	no	—
44	C₃×3₋¹⁺²	—	19	24	no	—
45	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	8	yes	complemented
46	C₃×H(3,𝔽₃)	—	19	12	no	—
47	C₃×H(3,𝔽₃)	—	57	9	no	—
48	C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	1	yes	characteristic
49	C₃²×C₉	C₃×C₁₉⋊C₃	1	4	yes	complemented
50	#731	—	19	9	no	—
51	#731	—	19	72	no	—
52	C₃×C₁₉	—	3	12	no	—
53	C₃×C₁₉	—	3	1	no	—
54	C₃×C₁₉	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
55	C₁₉⋊C₃	—	9	9	no	—
56	C₃³	—	57	12	no	—
57	C₃³	C₃²×C₁₉⋊C₃	1	4	yes	—
58	3₋¹⁺²	—	3	24	no	—
59	3₋¹⁺²	—	19	18	no	—
60	3₋¹⁺²	—	57	48	no	—
61	H(3,𝔽₃)	—	19	9	no	—
62	H(3,𝔽₃)	—	57	24	no	—
63	H(3,𝔽₃)	C₃²⋊(C₁₉⋊C₃)	1	9	yes	—
64	C₃×C₉	—	3	12	no	—
65	C₃×C₉	—	57	24	no	—
66	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
67	C₃×C₉	C₃²×C₁₉⋊C₃	1	8	yes	—
68	C₁₉	#10624931	1	1	yes	characteristic, complemented, Sylow
69	C₃²	—	513	12	no	—
70	C₃²	—	9	4	no	—
71	C₃²	—	57	9	no	—
72	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	12	yes	—
73	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
74	C₉	—	3	1	no	—
75	C₉	—	3	8	no	—
76	C₉	—	57	18	no	—
77	C₃	—	3	12	no	—
78	C₃	—	3	1	no	—
79	C₃	—	171	9	no	—
80	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	C₃×C₁₉×#737	C₃
socle	C₃×C₁₉	C₃²×H(3,𝔽₃)
prime radical	C₁₉⋊H(5,𝔽₃)	C₃
hypercenter	C₃×#737	C₁₉⋊C₃
hypocenter	C₁₉	#10624931
LCS	G ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃×H(3,𝔽₃) ◁ C₃×#737

Complements

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

N	H ≅ G/N	type
1. #17987547	C₃	semidirect
3. #17987510	C₃	semidirect
4. C₃×#13095317	C₃	semidirect
5. C₃×C₁₉×#737	C₃	semidirect
7. C₃×#10587025	C₃²	semidirect
10. #13095317	C₃²	semidirect
11. #13095317	C₃²	semidirect
13. C₃×C₁₉×3₋¹⁺²	C₃²	semidirect
15. C₃²×C₉×C₁₉	C₃²	semidirect
31. C₃×#737	C₁₉⋊C₃	semidirect
45. C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	semidirect
49. C₃²×C₉	C₃×C₁₉⋊C₃	semidirect
68. 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
#17987510	3
C₃×#13095317	3
C₃×C₁₉×#737	3
#10624931	19