Subgroups — C₃²×C₁₉×#1043 #30685858

Order 17100.

288 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₁₉×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic
2	C₃×C₁₉×#1043	C₃	1	4	yes	maximal, direct product factor
3	C₃²×C₅²×C₁₉	C₄	1	1	yes	characteristic, complemented, Hall
4	C₃²×C₁₉×AGL(1,𝔽₅)	—	5	6	no	maximal
5	C₃×C₁₉×C₅⋊D₅	C₂×C₃	1	4	yes	—
6	C₁₉×#1043	C₃²	1	1	yes	characteristic, direct product factor, Hall
7	C₃²×D₅×C₁₉	—	5	6	no	—
8	C₃×C₅²×C₁₉	C₃×C₄	1	4	yes	complemented
9	C₃×C₁₉×AGL(1,𝔽₅)	—	5	24	no	—
10	C₁₉×C₅⋊D₅	C₂×C₃²	1	1	yes	characteristic
11	C₃²×#1043	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₃²×C₅×C₁₉	AGL(1,𝔽₅)	1	6	yes	complemented
13	C₃²×C₄×C₁₉	—	25	1	no	Hall
14	C₃×D₅×C₁₉	—	5	24	no	—
15	C₅²×C₁₉	C₃²×C₄	1	1	yes	characteristic, complemented, Hall
16	C₃²×C₅⋊D₅	C₂×C₁₉	1	1	yes	characteristic
17	C₁₉×AGL(1,𝔽₅)	—	5	6	no	—
18	C₂×C₃²×C₁₉	—	25	1	no	—
19	C₃×#1043	C₃×C₁₉	1	4	yes	direct product factor
20	C₃×C₅×C₁₉	C₃×AGL(1,𝔽₅)	1	24	yes	complemented
21	C₃×C₄×C₁₉	—	25	4	no	—
22	C₃²×C₅²	C₄×C₁₉	1	1	yes	characteristic, complemented, Hall
23	D₅×C₁₉	—	5	6	no	—
24	C₃²×AGL(1,𝔽₅)	—	5	6	no	—
25	C₃²×C₁₉	#1043	1	1	yes	characteristic, direct product factor, Hall
26	C₃×C₅⋊D₅	C₂×C₃×C₁₉	1	4	yes	—
27	C₂×C₃×C₁₉	—	25	4	no	—
28	#1043	C₃²×C₁₉	1	1	yes	characteristic, direct product factor, Hall
29	C₅×C₁₉	C₃²×AGL(1,𝔽₅)	1	6	yes	complemented
30	C₃²×D₅	—	5	6	no	—
31	C₄×C₁₉	—	25	1	no	Hall
32	C₃×C₅²	C₃×C₄×C₁₉	1	4	yes	complemented
33	C₃×AGL(1,𝔽₅)	—	5	24	no	—
34	C₃×C₁₉	C₃×#1043	1	4	yes	direct product factor
35	C₅⋊D₅	C₂×C₃²×C₁₉	1	1	yes	characteristic
36	C₃²×C₅	C₁₉×AGL(1,𝔽₅)	1	6	yes	complemented
37	C₂×C₁₉	—	25	1	no	—
38	C₃²×C₄	—	25	1	no	Hall
39	C₃×D₅	—	5	24	no	—
40	C₅²	C₃²×C₄×C₁₉	1	1	yes	characteristic, complemented, Sylow
41	AGL(1,𝔽₅)	—	5	6	no	—
42	C₁₉	C₃²×#1043	1	1	yes	characteristic, direct product factor, Sylow
43	C₂×C₃²	—	25	1	no	—
44	C₃×C₅	C₃×C₁₉×AGL(1,𝔽₅)	1	24	yes	complemented
45	C₃×C₄	—	25	4	no	—
46	D₅	—	5	6	no	—
47	C₃²	C₁₉×#1043	1	1	yes	characteristic, direct product factor, Sylow
48	C₂×C₃	—	25	4	no	—
49	C₅	C₃²×C₁₉×AGL(1,𝔽₅)	1	6	yes	complemented
50	C₄	—	25	1	no	Sylow
51	C₃	C₃×C₁₉×#1043	1	4	yes	direct product factor
52	C₂	—	25	1	no	—

	subgroup	G/H
center, hypercenter	C₃²×C₁₉	#1043
commutator, hypocenter	C₅²	C₃²×C₄×C₁₉
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₁₉	C₄
prime radical	C₃²×C₁₉×C₅⋊D₅	C₂
cosocle	C₅⋊D₅	C₂×C₃²×C₁₉
LCS	G ▷ C₅²
UCS	C₁ ◁ C₃²×C₁₉

Complements

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

N	H ≅ G/N	type
2. C₃×C₁₉×#1043	C₃	direct (N × H)
3. C₃²×C₅²×C₁₉	C₄	semidirect
6. C₁₉×#1043	C₃²	direct (N × H)
8. C₃×C₅²×C₁₉	C₃×C₄	semidirect
11. C₃²×#1043	C₁₉	direct (N × H)
12. C₃²×C₅×C₁₉	AGL(1,𝔽₅)	semidirect
15. C₅²×C₁₉	C₃²×C₄	semidirect
19. C₃×#1043	C₃×C₁₉	direct (N × H)
20. C₃×C₅×C₁₉	C₃×AGL(1,𝔽₅)	semidirect
22. C₃²×C₅²	C₄×C₁₉	semidirect
25. C₃²×C₁₉	#1043	direct (N × H)
28. #1043	C₃²×C₁₉	direct (N × H)
29. C₅×C₁₉	C₃²×AGL(1,𝔽₅)	semidirect
32. C₃×C₅²	C₃×C₄×C₁₉	semidirect
34. C₃×C₁₉	C₃×#1043	direct (N × H)
36. C₃²×C₅	C₁₉×AGL(1,𝔽₅)	semidirect
40. C₅²	C₃²×C₄×C₁₉	semidirect
42. C₁₉	C₃²×#1043	direct (N × H)
44. C₃×C₅	C₃×C₁₉×AGL(1,𝔽₅)	semidirect
47. C₃²	C₁₉×#1043	direct (N × H)
49. C₅	C₃²×C₁₉×AGL(1,𝔽₅)	semidirect
51. C₃	C₃×C₁₉×#1043	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×C₁₉×C₅⋊D₅	2
C₃×C₁₉×#1043	3
C₃²×C₁₉×AGL(1,𝔽₅)	5
C₃²×#1043	19

Subgroups — C32×C19×#1043 #30685858

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₁₉×#1043 #30685858