Subgroups — #17987507

Order 4617.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#10587023	C₃	1	1	yes	maximal, characteristic, complemented
2	3₋¹⁺²×C₁₉⋊C₃	C₃	1	2	yes	maximal, complemented
3	H(3,𝔽₃)×C₁₉⋊C₃	C₃	1	1	yes	maximal, characteristic
4	#13095318	C₃	1	4	yes	maximal, complemented
5	#13095317	C₃	1	2	yes	maximal, complemented
6	#13095316	C₃	1	2	yes	maximal, complemented
7	C₁₉×#731	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃²×C₁₉⋊C₃	—	3	1	no	—
9	C₃²×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
10	C₁₉×3₋¹⁺²	C₃²	1	2	yes	complemented
11	C₁₉×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
12	C₃×C₉×C₁₉	C₃²	1	1	yes	characteristic, complemented
13	C₃²⋊(C₁₉⋊C₃)	—	3	2	no	—
14	C₃²⋊(C₁₉⋊C₃)	C₃²	1	2	yes	—
15	#10587025	C₃²	1	4	yes	complemented
16	C₃×#4294	C₃²	1	2	yes	complemented
17	#10587023	—	3	2	no	—
18	#10587023	—	3	2	no	—
19	#10587023	—	3	1	no	—
20	#10587022	—	3	2	no	—
21	C₉×C₁₉⋊C₃	—	3	2	no	—
22	#6910	—	19	1	no	maximal, Sylow
23	C₃²×C₁₉	—	3	1	no	—
24	C₃²×C₁₉	C₃³	1	1	yes	characteristic
25	C₃×C₁₉⋊C₃	—	3	2	no	—
26	C₃×C₁₉⋊C₃	—	9	1	no	—
27	C₃×C₁₉⋊C₃	—	9	3	no	—
28	C₃×C₁₉⋊C₃	H(3,𝔽₃)	1	2	yes	—
29	C₃×C₁₉⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
30	C₉×C₁₉	—	3	2	no	—
31	C₉×C₁₉	—	3	1	no	—
32	#4294	—	3	4	no	—
33	#4294	—	3	2	no	—
34	C₃×3₋¹⁺²	—	19	1	no	—
35	C₃×3₋¹⁺²	—	19	2	no	—
36	C₃×H(3,𝔽₃)	—	19	1	no	—
37	#731	—	19	4	no	—
38	#731	—	19	2	no	—
39	#731	—	19	2	no	—
40	#731	C₁₉⋊C₃	1	1	yes	characteristic, complemented
41	C₃×C₁₉	—	3	1	no	—
42	C₃×C₁₉	—	9	1	no	—
43	C₃×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
44	C₁₉⋊C₃	—	3	1	no	—
45	C₁₉⋊C₃	—	3	2	no	—
46	C₁₉⋊C₃	—	9	2	no	—
47	C₃³	—	19	1	no	—
48	C₃³	—	57	1	no	—
49	3₋¹⁺²	—	19	4	no	—
50	3₋¹⁺²	—	57	2	no	—
51	3₋¹⁺²	—	57	2	no	—
52	3₋¹⁺²	—	57	2	no	—
53	3₋¹⁺²	—	57	1	no	—
54	3₋¹⁺²	C₃×C₁₉⋊C₃	1	2	yes	complemented
55	H(3,𝔽₃)	—	19	2	no	—
56	H(3,𝔽₃)	—	57	2	no	—
57	H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	1	yes	characteristic
58	C₃×C₉	—	19	2	no	—
59	C₃×C₉	—	57	2	no	—
60	C₃×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic, complemented
61	C₁₉	#6910	1	1	yes	characteristic, complemented, Sylow
62	C₃²	—	3	1	no	—
63	C₃²	—	19	2	no	—
64	C₃²	—	19	1	no	—
65	C₃²	—	57	2	no	—
66	C₃²	—	171	1	no	—
67	C₃²	—	171	3	no	—
68	C₃²	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
69	C₉	—	3	2	no	—
70	C₉	—	3	1	no	—
71	C₉	—	57	2	no	—
72	C₉	—	57	4	no	—
73	C₃	—	3	1	no	—
74	C₃	—	9	1	no	—
75	C₃	—	57	1	no	—
76	C₃	—	57	2	no	—
77	C₃	—	171	2	no	—
78	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×#731	C₃
socle	C₃×C₁₉	C₃×H(3,𝔽₃)
prime radical	H(3,𝔽₃)×C₁₉⋊C₃	C₃
hypercenter	#731	C₁₉⋊C₃
hypocenter	C₁₉	#6910
LCS	G ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃² ◁ #731

Complements

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

N	H ≅ G/N	type
1. C₃×#10587023	C₃	semidirect
2. 3₋¹⁺²×C₁₉⋊C₃	C₃	semidirect
4. #13095318	C₃	semidirect
5. #13095317	C₃	semidirect
6. #13095316	C₃	semidirect
7. C₁₉×#731	C₃	semidirect
10. C₁₉×3₋¹⁺²	C₃²	semidirect
12. C₃×C₉×C₁₉	C₃²	semidirect
15. #10587025	C₃²	semidirect
16. C₃×#4294	C₃²	semidirect
40. #731	C₁₉⋊C₃	semidirect
54. 3₋¹⁺²	C₃×C₁₉⋊C₃	semidirect
60. C₃×C₉	C₃×C₁₉⋊C₃	semidirect
61. 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×#10587023	3
3₋¹⁺²×C₁₉⋊C₃	3
H(3,𝔽₃)×C₁₉⋊C₃	3
#13095318	3
#13095317	3
#13095316	3
C₁₉×#731	3
#6910	19