Subgroups — #28356612

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	3	yes	maximal, complemented
2	#17987515	C₃	1	24	yes	maximal, complemented
3	C₃×#13095320	C₃	1	12	yes	maximal, complemented
4	C₃×C₁₉×#737	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	36	yes	complemented
6	H(3,𝔽₃)×C₁₉⋊C₃	—	3	27	no	—
7	#13095320	C₃²	1	9	yes	complemented
8	#13095320	C₃²	1	72	yes	complemented
9	C₁₉×#737	—	3	9	no	—
10	C₃×C₁₉×3₋¹⁺²	C₃²	1	8	yes	complemented
11	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
12	C₃²×C₉×C₁₉	C₃²	1	4	yes	complemented
13	#10624932	—	19	1	no	maximal, Sylow
14	C₃³×C₁₉	C₃³	1	4	yes	—
15	C₃²×C₁₉⋊C₃	—	3	36	no	—
16	C₁₉×3₋¹⁺²	—	3	24	no	—
17	C₁₉×H(3,𝔽₃)	H(3,𝔽₃)	1	9	yes	—
18	C₃×C₉×C₁₉	—	3	12	no	—
19	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
20	C₃×C₉×C₁₉	C₃³	1	8	yes	—
21	C₃²⋊(C₁₉⋊C₃)	—	3	72	no	—
22	C₃²⋊(C₁₉⋊C₃)	C₃³	1	27	yes	—
23	H(5,𝔽₃)	—	19	3	no	—
24	C₃×#737	C₁₉⋊C₃	1	1	yes	characteristic, complemented
25	#6911	—	19	24	no	—
26	C₃×#732	—	19	12	no	—
27	C₃²×C₁₉	—	9	4	no	—
28	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	12	yes	—
29	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
30	C₃×C₁₉⋊C₃	—	3	27	no	—
31	C₃×C₁₉⋊C₃	—	27	36	no	—
32	C₉×C₁₉	—	3	1	no	—
33	C₉×C₁₉	—	3	8	no	—
34	#737	—	3	9	no	—
35	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	8	yes	complemented
36	C₃×H(3,𝔽₃)	—	19	36	no	—
37	C₃×H(3,𝔽₃)	—	57	27	no	—
38	C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	1	yes	characteristic
39	C₃²×C₉	C₃×C₁₉⋊C₃	1	4	yes	complemented
40	#732	—	19	9	no	—
41	#732	—	19	72	no	—
42	C₃×C₁₉	—	3	12	no	—
43	C₃×C₁₉	—	3	1	no	—
44	C₃×C₁₉	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
45	C₁₉⋊C₃	—	9	27	no	—
46	C₃³	—	57	36	no	—
47	C₃³	C₃²×C₁₉⋊C₃	1	4	yes	—
48	3₋¹⁺²	—	3	24	no	—
49	H(3,𝔽₃)	—	19	27	no	—
50	H(3,𝔽₃)	—	57	72	no	—
51	H(3,𝔽₃)	C₃²⋊(C₁₉⋊C₃)	1	9	yes	—
52	C₃×C₉	—	3	12	no	—
53	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
54	C₃×C₉	C₃²×C₁₉⋊C₃	1	8	yes	—
55	C₁₉	#10624932	1	1	yes	characteristic, complemented, Sylow
56	C₃²	—	513	36	no	—
57	C₃²	—	9	4	no	—
58	C₃²	—	57	27	no	—
59	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	12	yes	—
60	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
61	C₉	—	3	1	no	—
62	C₉	—	3	8	no	—
63	C₃	—	3	12	no	—
64	C₃	—	3	1	no	—
65	C₃	—	171	27	no	—
66	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)
hypercenter	C₃×#737	C₁₉⋊C₃
hypocenter	C₁₉	#10624932
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. C₁₉⋊H(5,𝔽₃)	C₃	semidirect
2. #17987515	C₃	semidirect
3. C₃×#13095320	C₃	semidirect
4. C₃×C₁₉×#737	C₃	semidirect
5. C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	semidirect
7. #13095320	C₃²	semidirect
8. #13095320	C₃²	semidirect
10. C₃×C₁₉×3₋¹⁺²	C₃²	semidirect
12. C₃²×C₉×C₁₉	C₃²	semidirect
24. C₃×#737	C₁₉⋊C₃	semidirect
35. C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	semidirect
39. C₃²×C₉	C₃×C₁₉⋊C₃	semidirect
55. 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
#17987515	3
C₃×#13095320	3
C₃×C₁₉×#737	3
#10624932	19