Subgroups — #17987528

Order 4617.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13095332	C₃	1	3	yes	maximal, complemented
2	#13095322	C₃	1	2	yes	maximal, complemented
3	#13095320	C₃	1	1	yes	maximal, characteristic, complemented
4	#13095317	C₃	1	3	yes	maximal, complemented
5	#13095314	C₃	1	3	yes	maximal, complemented
6	C₃×C₁₉×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃³×C₁₉	C₃²	1	1	yes	characteristic
8	C₁₉×3₋¹⁺²	—	3	3	no	—
9	C₃×C₉×C₁₉	C₃²	1	1	yes	characteristic
10	C₃×C₉×C₁₉	C₃²	1	1	yes	characteristic
11	C₃×C₉×C₁₉	C₃²	1	1	yes	characteristic
12	C₃²⋊(C₁₉⋊C₃)	C₃²	1	3	yes	—
13	#10587025	C₃²	1	6	yes	—
14	#10587023	—	3	3	no	—
15	#10587022	—	3	3	no	—
16	C₉×C₁₉⋊C₃	—	3	3	no	—
17	#6913	—	19	1	no	maximal, Sylow
18	C₃²×C₁₉	—	3	1	no	—
19	C₃²×C₁₉	—	9	1	no	—
20	C₃²×C₁₉	C₃³	1	1	yes	characteristic
21	C₃×C₁₉⋊C₃	—	3	3	no	—
22	C₉×C₁₉	—	3	1	no	—
23	C₉×C₁₉	—	3	1	no	—
24	C₉×C₁₉	H(3,𝔽₃)	1	3	yes	—
25	#4294	—	3	6	no	—
26	#737	—	19	3	no	—
27	C₃×3₋¹⁺²	C₁₉⋊C₃	1	1	yes	characteristic, complemented
28	#733	—	19	2	no	—
29	#732	—	19	1	no	—
30	#731	—	19	3	no	—
31	AΣL(1,𝔽₂₇)	—	19	3	no	—
32	C₃×C₁₉	—	3	1	no	—
33	C₃×C₁₉	—	9	1	no	—
34	C₃×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
35	C₁₉⋊C₃	—	9	3	no	—
36	C₃³	C₃×C₁₉⋊C₃	1	1	yes	characteristic
37	3₋¹⁺²	—	3	3	no	—
38	3₋¹⁺²	—	19	6	no	—
39	3₋¹⁺²	—	57	3	no	—
40	3₋¹⁺²	—	57	3	no	—
41	H(3,𝔽₃)	—	19	3	no	—
42	C₃×C₉	—	57	3	no	—
43	C₃×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
44	C₃×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
45	C₃×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
46	C₁₉	#6913	1	1	yes	characteristic, complemented, Sylow
47	C₃²	—	3	1	no	—
48	C₃²	—	9	1	no	—
49	C₃²	—	57	3	no	—
50	C₃²	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
51	C₉	—	3	1	no	—
52	C₉	—	3	1	no	—
53	C₉	—	57	6	no	—
54	C₉	C₃²⋊(C₁₉⋊C₃)	1	3	yes	—
55	C₃	—	3	1	no	—
56	C₃	—	9	1	no	—
57	C₃	—	171	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	C₃×C₁₉×3₋¹⁺²	C₃
socle	C₃×C₁₉	C₃×H(3,𝔽₃)
hypercenter	C₃×3₋¹⁺²	C₁₉⋊C₃
hypocenter	C₁₉	#6913
LCS	G ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃×C₉ ◁ C₃×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
1. #13095332	C₃	semidirect
2. #13095322	C₃	semidirect
3. #13095320	C₃	semidirect
4. #13095317	C₃	semidirect
5. #13095314	C₃	semidirect
6. C₃×C₁₉×3₋¹⁺²	C₃	semidirect
27. C₃×3₋¹⁺²	C₁₉⋊C₃	semidirect
46. C₁₉	#6913	semidirect

Maximal subgroups

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

subgroup	index
#13095332	3
#13095322	3
#13095320	3
#13095317	3
#13095314	3
C₃×C₁₉×3₋¹⁺²	3
#6913	19