Subgroups — #17987516

Order 4617.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#10587022	C₃	1	1	yes	maximal, characteristic
2	3₋¹⁺²×C₁₉⋊C₃	C₃	1	3	yes	maximal
3	#13095322	C₃	1	2	yes	maximal, complemented
4	#13095321	C₃	1	6	yes	maximal, complemented
5	C₁₉×#733	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃²×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
7	C₁₉×3₋¹⁺²	C₃²	1	3	yes	—
8	C₃×C₉×C₁₉	C₃²	1	1	yes	characteristic
9	#10587025	C₃²	1	6	yes	—
10	C₃×#4294	C₃²	1	2	yes	—
11	#10587023	—	3	3	no	—
12	#10587022	—	3	3	no	—
13	#10587022	—	3	3	no	—
14	C₉×C₁₉⋊C₃	—	3	3	no	—
15	#6912	—	19	1	no	maximal, Sylow
16	C₃²×C₁₉	C₃³	1	1	yes	characteristic
17	C₃×C₁₉⋊C₃	—	9	1	no	—
18	C₃×C₁₉⋊C₃	H(3,𝔽₃)	1	3	yes	—
19	C₉×C₁₉	—	3	3	no	—
20	C₉×C₁₉	—	3	1	no	—
21	#4294	—	3	6	no	—
22	#4294	—	3	2	no	—
23	C₃×3₋¹⁺²	—	19	1	no	—
24	C₃×3₋¹⁺²	—	19	3	no	—
25	#733	—	19	6	no	—
26	#733	—	19	2	no	—
27	#733	C₁₉⋊C₃	1	1	yes	characteristic, complemented
28	C₃×C₁₉	—	3	1	no	—
29	C₃×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
30	C₁₉⋊C₃	—	3	3	no	—
31	C₃³	—	19	1	no	—
32	3₋¹⁺²	—	19	6	no	—
33	3₋¹⁺²	—	57	3	no	—
34	3₋¹⁺²	—	57	3	no	—
35	3₋¹⁺²	—	57	3	no	—
36	3₋¹⁺²	C₃×C₁₉⋊C₃	1	3	yes	—
37	C₃×C₉	—	19	2	no	—
38	C₃×C₉	—	57	3	no	—
39	C₃×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
40	C₁₉	#6912	1	1	yes	characteristic, complemented, Sylow
41	C₃²	—	19	3	no	—
42	C₃²	—	171	1	no	—
43	C₃²	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
44	C₉	—	3	3	no	—
45	C₉	—	3	1	no	—
46	C₉	—	57	2	no	—
47	C₉	—	57	6	no	—
48	C₃	—	3	1	no	—
49	C₃	—	57	3	no	—
50	C₃	H(3,𝔽₃)×C₁₉⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃	H(3,𝔽₃)×C₁₉⋊C₃
commutator, cosocle	C₃²×C₁₉	C₃³
Frattini	C₃²	C₃²×C₁₉⋊C₃
Fitting	C₁₉×#733	C₃
socle	C₃×C₁₉	C₃×H(3,𝔽₃)
prime radical	C₃²×C₁₉⋊C₃	C₃²
hypercenter	#733	C₁₉⋊C₃
hypocenter	C₁₉	#6912
LCS	G ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃² ◁ #733

Complements

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

N	H ≅ G/N	type
3. #13095322	C₃	semidirect
4. #13095321	C₃	semidirect
5. C₁₉×#733	C₃	semidirect
27. #733	C₁₉⋊C₃	semidirect
40. C₁₉	#6912	semidirect

Maximal subgroups

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

subgroup	index
C₃×#10587022	3
3₋¹⁺²×C₁₉⋊C₃	3
#13095322	3
#13095321	3
C₁₉×#733	3
#6912	19