Subgroups — #17987545

Order 4617.

460 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13095333	C₃	1	16	yes	maximal, complemented
2	#13095332	C₃	1	8	yes	maximal, complemented
3	#13095331	C₃	1	2	yes	maximal, complemented
4	C₃×#10587022	C₃	1	4	yes	maximal, complemented
5	3₋¹⁺²×C₁₉⋊C₃	C₃	1	8	yes	maximal, complemented
6	H(3,𝔽₃)×C₁₉⋊C₃	C₃	1	1	yes	maximal, characteristic
7	C₁₉×#737	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃²×C₁₉⋊C₃	C₃²	1	4	yes	—
9	C₁₉×3₋¹⁺²	C₃²	1	8	yes	complemented
10	C₁₉×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
11	C₃×C₉×C₁₉	C₃²	1	4	yes	complemented
12	C₃²⋊(C₁₉⋊C₃)	C₃²	1	8	yes	—
13	#10587025	C₃²	1	16	yes	complemented
14	C₃×#4294	C₃²	1	8	yes	complemented
15	#10587023	C₃²	1	24	yes	complemented
16	#10587022	C₃²	1	8	yes	—
17	#10587022	C₃²	1	1	yes	characteristic
18	#10587022	C₃²	1	8	yes	complemented
19	#10587022	C₃²	1	16	yes	complemented
20	C₉×C₁₉⋊C₃	C₃²	1	24	yes	complemented
21	3₋¹⁺⁴	—	19	1	no	maximal, Sylow
22	C₃²×C₁₉	C₃³	1	4	yes	—
23	C₃×C₁₉⋊C₃	—	9	4	no	—
24	C₃×C₁₉⋊C₃	C₃³	1	8	yes	—
25	C₃×C₁₉⋊C₃	C₃³	1	1	yes	characteristic
26	C₉×C₁₉	C₃³	1	8	yes	—
27	C₉×C₁₉	C₃³	1	1	yes	characteristic
28	#4294	C₃³	1	16	yes	—
29	#4294	C₃³	1	2	yes	—
30	#737	—	19	16	no	—
31	#737	—	19	2	no	—
32	#737	—	19	8	no	—
33	#737	C₁₉⋊C₃	1	1	yes	characteristic, complemented
34	C₃×3₋¹⁺²	—	19	8	no	—
35	C₃×3₋¹⁺²	—	19	4	no	—
36	C₃×H(3,𝔽₃)	—	19	1	no	—
37	C₃×C₁₉	—	3	4	no	—
38	C₃×C₁₉	C₃⁴	1	1	yes	characteristic
39	C₁₉⋊C₃	—	3	1	no	—
40	C₁₉⋊C₃	—	3	8	no	—
41	C₃³	—	19	4	no	—
42	3₋¹⁺²	—	19	16	no	—
43	3₋¹⁺²	—	19	8	no	—
44	3₋¹⁺²	—	19	1	no	—
45	3₋¹⁺²	—	19	16	no	—
46	3₋¹⁺²	—	19	8	no	—
47	3₋¹⁺²	—	19	24	no	—
48	3₋¹⁺²	C₃×C₁₉⋊C₃	1	8	yes	complemented
49	H(3,𝔽₃)	—	19	8	no	—
50	H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	1	yes	characteristic
51	C₃×C₉	—	19	8	no	—
52	C₃×C₉	—	19	24	no	—
53	C₃×C₉	C₃×C₁₉⋊C₃	1	4	yes	complemented
54	C₁₉	3₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
55	C₃²	—	19	8	no	—
56	C₃²	—	19	1	no	—
57	C₃²	—	171	4	no	—
58	C₃²	C₃²×C₁₉⋊C₃	1	4	yes	—
59	C₉	—	19	2	no	—
60	C₉	—	19	16	no	—
61	C₉	C₃²×C₁₉⋊C₃	1	8	yes	—
62	C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
63	C₃	—	3	4	no	—
64	C₃	—	57	1	no	—
65	C₃	—	57	8	no	—
66	C₃	C₃³×C₁₉⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃	C₃³×C₁₉⋊C₃
commutator, socle, cosocle	C₃×C₁₉	C₃⁴
Fitting	C₁₉×#737	C₃
prime radical	H(3,𝔽₃)×C₁₉⋊C₃	C₃
hypercenter	#737	C₁₉⋊C₃
hypocenter	C₁₉	3₋¹⁺⁴
LCS	G ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ #737

Complements

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

N	H ≅ G/N	type
1. #13095333	C₃	semidirect
2. #13095332	C₃	semidirect
3. #13095331	C₃	semidirect
4. C₃×#10587022	C₃	semidirect
5. 3₋¹⁺²×C₁₉⋊C₃	C₃	semidirect
7. C₁₉×#737	C₃	semidirect
9. C₁₉×3₋¹⁺²	C₃²	semidirect
11. C₃×C₉×C₁₉	C₃²	semidirect
13. #10587025	C₃²	semidirect
14. C₃×#4294	C₃²	semidirect
15. #10587023	C₃²	semidirect
18. #10587022	C₃²	semidirect
19. #10587022	C₃²	semidirect
20. C₉×C₁₉⋊C₃	C₃²	semidirect
33. #737	C₁₉⋊C₃	semidirect
48. 3₋¹⁺²	C₃×C₁₉⋊C₃	semidirect
53. C₃×C₉	C₃×C₁₉⋊C₃	semidirect
54. C₁₉	3₋¹⁺⁴	semidirect

Maximal subgroups

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

subgroup	index
#13095333	3
#13095332	3
#13095331	3
C₃×#10587022	3
3₋¹⁺²×C₁₉⋊C₃	3
H(3,𝔽₃)×C₁₉⋊C₃	3
C₁₉×#737	3
3₋¹⁺⁴	19