Subgroups — C₅×C₁₉×C₃⋊ΓL(1,𝔽₁₆) #30685895

Order 17100.

156 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅×D₅×C₁₉	C₂	1	1	yes	maximal, characteristic
2	C₅×C₁₉×ΓL(1,𝔽₁₆)	—	3	4	no	maximal
3	C₃²×C₅²×C₁₉	C₄	1	1	yes	characteristic, complemented, Hall
4	C₁₉×C₃⋊ΓL(1,𝔽₁₆)	C₅	1	1	yes	maximal, characteristic, direct product factor
5	C₅×C₁₉×C₃⋊Dic₃	—	5	1	no	maximal
6	C₃×C₅×D₅×C₁₉	D₃	1	4	yes	—
7	C₅×C₁₉×AGL(1,𝔽₅)	—	9	1	no	Hall
8	C₂×C₃²×C₅×C₁₉	—	5	1	no	—
9	C₃²×D₅×C₁₉	C₂×C₅	1	1	yes	characteristic
10	C₃×C₅²×C₁₉	Dic₃	1	4	yes	complemented
11	C₁₉×ΓL(1,𝔽₁₆)	—	3	4	no	—
12	C₅×Dic₃×C₁₉	—	15	4	no	—
13	C₅×D₅×C₁₉	C₃⋊D₃	1	1	yes	characteristic
14	C₅×C₃⋊ΓL(1,𝔽₁₆)	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₃²×C₅×C₁₉	—	4	1	no	—
16	C₃²×C₅×C₁₉	C₄×C₅	1	1	yes	characteristic, complemented
17	C₃²×C₅×C₁₉	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
18	C₁₉×C₃⋊Dic₃	—	5	1	no	Hall
19	C₂×C₃×C₅×C₁₉	—	5	4	no	—
20	C₃×D₅×C₁₉	C₅×D₃	1	4	yes	—
21	C₅²×C₁₉	C₃⋊Dic₃	1	1	yes	characteristic, complemented, Hall
22	C₃²×C₅×D₅	C₂×C₁₉	1	1	yes	characteristic
23	C₁₉×AGL(1,𝔽₅)	—	9	1	no	—
24	C₄×C₅×C₁₉	—	45	1	no	—
25	C₂×C₃²×C₁₉	—	5	1	no	—
26	C₅×ΓL(1,𝔽₁₆)	—	3	4	no	—
27	C₃×C₅×C₁₉	—	4	4	no	—
28	C₃×C₅×C₁₉	C₅×Dic₃	1	4	yes	complemented
29	C₃×C₅×C₁₉	ΓL(1,𝔽₁₆)	1	4	yes	complemented
30	Dic₃×C₁₉	—	15	4	no	—
31	C₃²×C₅²	C₄×C₁₉	1	1	yes	characteristic, complemented, Hall
32	C₂×C₅×C₁₉	—	5	1	no	—
33	D₅×C₁₉	C₅×C₃⋊D₃	1	1	yes	characteristic
34	C₃⋊ΓL(1,𝔽₁₆)	C₅×C₁₉	1	1	yes	characteristic, direct product factor
35	C₅×C₃⋊Dic₃	—	5	1	no	—
36	C₃²×C₁₉	C₅×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
37	C₃×C₅×D₅	D₃×C₁₉	1	4	yes	—
38	C₂×C₃×C₁₉	—	5	4	no	—
39	C₅×AGL(1,𝔽₅)	—	9	1	no	Hall
40	C₅×C₁₉	—	4	1	no	—
41	C₅×C₁₉	C₅×C₃⋊Dic₃	1	1	yes	characteristic, complemented
42	C₅×C₁₉	C₃⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic, direct product factor
43	C₂×C₃²×C₅	—	5	1	no	—
44	C₃²×D₅	C₂×C₅×C₁₉	1	1	yes	characteristic
45	C₄×C₁₉	—	45	1	no	Hall
46	C₃×C₅²	Dic₃×C₁₉	1	4	yes	complemented
47	ΓL(1,𝔽₁₆)	—	3	4	no	—
48	C₅×Dic₃	—	15	4	no	—
49	C₃×C₁₉	C₅×ΓL(1,𝔽₁₆)	1	4	yes	complemented
50	C₅×D₅	C₃⋊D₃×C₁₉	1	1	yes	characteristic
51	C₃²×C₅	—	4	1	no	—
52	C₃²×C₅	C₄×C₅×C₁₉	1	1	yes	characteristic, complemented
53	C₃²×C₅	C₁₉×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
54	C₂×C₁₉	—	5	1	no	—
55	C₃⋊Dic₃	—	5	1	no	Hall
56	C₂×C₃×C₅	—	5	4	no	—
57	C₃×D₅	C₅×D₃×C₁₉	1	4	yes	—
58	C₅²	C₁₉×C₃⋊Dic₃	1	1	yes	characteristic, complemented, Sylow
59	AGL(1,𝔽₅)	—	9	1	no	—
60	C₄×C₅	—	45	1	no	—
61	C₁₉	C₅×C₃⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic, direct product factor, Sylow
62	C₂×C₃²	—	5	1	no	—
63	C₃×C₅	—	4	4	no	—
64	C₃×C₅	C₅×Dic₃×C₁₉	1	4	yes	complemented
65	C₃×C₅	C₁₉×ΓL(1,𝔽₁₆)	1	4	yes	complemented
66	Dic₃	—	15	4	no	—
67	C₂×C₅	—	5	1	no	—
68	D₅	C₅×C₃⋊D₃×C₁₉	1	1	yes	characteristic
69	C₃²	C₅×C₁₉×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₃	—	5	4	no	—
71	C₅	—	4	1	no	—
72	C₅	C₅×C₁₉×C₃⋊Dic₃	1	1	yes	characteristic, complemented
73	C₅	C₁₉×C₃⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic, direct product factor
74	C₄	—	45	1	no	Sylow
75	C₃	C₅×C₁₉×ΓL(1,𝔽₁₆)	1	4	yes	complemented
76	C₂	—	5	1	no	—

	subgroup	G/H
center, hypercenter	C₅×C₁₉	C₃⋊ΓL(1,𝔽₁₆)
commutator, hypocenter	C₃²×C₅	C₄×C₅×C₁₉
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₁₉	C₄
prime radical	C₃²×C₅×D₅×C₁₉	C₂
cosocle	C₃²×D₅	C₂×C₅×C₁₉
LCS	G ▷ C₃²×C₅
UCS	C₁ ◁ C₅×C₁₉

Complements

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

N	H ≅ G/N	type
3. C₃²×C₅²×C₁₉	C₄	semidirect
4. C₁₉×C₃⋊ΓL(1,𝔽₁₆)	C₅	direct (N × H)
10. C₃×C₅²×C₁₉	Dic₃	semidirect
14. C₅×C₃⋊ΓL(1,𝔽₁₆)	C₁₉	direct (N × H)
16. C₃²×C₅×C₁₉	C₄×C₅	semidirect
17. C₃²×C₅×C₁₉	AGL(1,𝔽₅)	semidirect
21. C₅²×C₁₉	C₃⋊Dic₃	semidirect
28. C₃×C₅×C₁₉	C₅×Dic₃	semidirect
29. C₃×C₅×C₁₉	ΓL(1,𝔽₁₆)	semidirect
31. C₃²×C₅²	C₄×C₁₉	semidirect
34. C₃⋊ΓL(1,𝔽₁₆)	C₅×C₁₉	direct (N × H)
36. C₃²×C₁₉	C₅×AGL(1,𝔽₅)	semidirect
41. C₅×C₁₉	C₅×C₃⋊Dic₃	semidirect
42. C₅×C₁₉	C₃⋊ΓL(1,𝔽₁₆)	direct (N × H)
46. C₃×C₅²	Dic₃×C₁₉	semidirect
49. C₃×C₁₉	C₅×ΓL(1,𝔽₁₆)	semidirect
52. C₃²×C₅	C₄×C₅×C₁₉	semidirect
53. C₃²×C₅	C₁₉×AGL(1,𝔽₅)	semidirect
58. C₅²	C₁₉×C₃⋊Dic₃	semidirect
61. C₁₉	C₅×C₃⋊ΓL(1,𝔽₁₆)	direct (N × H)
64. C₃×C₅	C₅×Dic₃×C₁₉	semidirect
65. C₃×C₅	C₁₉×ΓL(1,𝔽₁₆)	semidirect
69. C₃²	C₅×C₁₉×AGL(1,𝔽₅)	semidirect
72. C₅	C₅×C₁₉×C₃⋊Dic₃	semidirect
73. C₅	C₁₉×C₃⋊ΓL(1,𝔽₁₆)	direct (N × H)
75. C₃	C₅×C₁₉×ΓL(1,𝔽₁₆)	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₅×D₅×C₁₉	2
C₅×C₁₉×ΓL(1,𝔽₁₆)	3
C₁₉×C₃⋊ΓL(1,𝔽₁₆)	5
C₅×C₁₉×C₃⋊Dic₃	5
C₅×C₃⋊ΓL(1,𝔽₁₆)	19

Subgroups — C5×C19×C3⋊ΓL(1,𝔽16) #30685895

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₁₉×C₃⋊ΓL(1,𝔽₁₆) #30685895