Subgroups — #17987509

Order 4617.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#10587025	C₃	1	2	yes	maximal, complemented
2	C₃×#10587025	C₃	1	1	yes	maximal, characteristic, complemented
3	#13095316	C₃	1	9	yes	maximal, complemented
4	C₃×C₁₉×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
5	C₃³×C₁₉	—	3	1	no	—
6	C₃³×C₁₉	C₃²	1	1	yes	characteristic
7	C₁₉×H(3,𝔽₃)	—	3	2	no	—
8	C₁₉×H(3,𝔽₃)	C₃²	1	3	yes	—
9	#10587025	—	3	2	no	—
10	#10587025	—	3	2	no	—
11	#10587025	—	3	2	no	—
12	#10587025	—	3	1	no	—
13	#10587025	C₃²	1	6	yes	complemented
14	C₃×#4294	—	3	2	no	—
15	C₃×#4294	C₃²	1	3	yes	complemented
16	#6910	—	19	1	no	maximal, Sylow
17	C₃²×C₁₉	—	3	3	no	—
18	C₃²×C₁₉	—	9	1	no	—
19	C₃²×C₁₉	—	9	3	no	—
20	C₃²×C₁₉	H(3,𝔽₃)	1	2	yes	—
21	C₃²×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
22	C₃²×C₁₉	C₃³	1	1	yes	characteristic
23	#4294	—	3	6	no	—
24	#4294	—	3	3	no	—
25	C₃×3₋¹⁺²	—	19	1	no	—
26	C₃×3₋¹⁺²	—	19	2	no	—
27	C₃×H(3,𝔽₃)	C₁₉⋊C₃	1	1	yes	characteristic
28	#731	—	19	9	no	—
29	C₃×C₁₉	—	3	1	no	—
30	C₃×C₁₉	—	3	2	no	—
31	C₃×C₁₉	—	3	1	no	—
32	C₃×C₁₉	—	9	3	no	—
33	C₃×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
34	C₃³	—	3	1	no	—
35	C₃³	C₃×C₁₉⋊C₃	1	1	yes	characteristic
36	3₋¹⁺²	—	19	6	no	—
37	3₋¹⁺²	—	57	2	no	—
38	3₋¹⁺²	—	57	1	no	—
39	3₋¹⁺²	—	57	2	no	—
40	3₋¹⁺²	—	57	2	no	—
41	H(3,𝔽₃)	—	3	2	no	—
42	H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	3	yes	—
43	C₃×C₉	—	19	3	no	—
44	C₃×C₉	—	57	2	no	—
45	C₁₉	#6910	1	1	yes	characteristic, complemented, Sylow
46	C₃²	—	3	3	no	—
47	C₃²	—	9	1	no	—
48	C₃²	—	9	3	no	—
49	C₃²	C₃²⋊(C₁₉⋊C₃)	1	2	yes	—
50	C₃²	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
51	C₃²	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
52	C₉	—	57	3	no	—
53	C₉	—	57	6	no	—
54	C₃	—	3	1	no	—
55	C₃	—	3	2	no	—
56	C₃	—	3	1	no	—
57	C₃	—	9	3	no	—
58	C₃	C₃×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₃	C₃×C₃²⋊(C₁₉⋊C₃)
commutator, cosocle	C₃²×C₁₉	C₃³
Frattini	C₃²	C₃²×C₁₉⋊C₃
Fitting, prime radical	C₃×C₁₉×H(3,𝔽₃)	C₃
socle	C₃×C₁₉	C₃×H(3,𝔽₃)
hypercenter	C₃×H(3,𝔽₃)	C₁₉⋊C₃
hypocenter	C₁₉	#6910
LCS	G ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. C₃×#10587025	C₃	semidirect
2. C₃×#10587025	C₃	semidirect
3. #13095316	C₃	semidirect
13. #10587025	C₃²	semidirect
15. C₃×#4294	C₃²	semidirect
45. C₁₉	#6910	semidirect

Maximal subgroups

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

subgroup	index
C₃×#10587025	3
#13095316	3
C₃×C₁₉×H(3,𝔽₃)	3
#6910	19