Subgroups — #1044×#4294 #30685540

Order 17100.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅⋊D₅×#4294	C₂	1	1	yes	maximal, characteristic
2	C₃×C₁₉×#1044	C₃	1	1	yes	maximal, characteristic
3	C₅²×#4294	C₄	1	1	yes	characteristic, complemented, Hall
4	AGL(1,𝔽₅)×#4294	—	5	2	no	maximal
5	C₃×C₁₉×C₅⋊D₅	C₂×C₃	1	1	yes	characteristic
6	C₁₉×#1044	C₉	1	1	yes	characteristic, complemented, Hall
7	D₅×#4294	—	5	2	no	—
8	D₅×#4294	—	10	2	no	—
9	C₃×C₅²×C₁₉	C₃×C₄	1	1	yes	characteristic
10	C₃×C₁₉×AGL(1,𝔽₅)	—	5	2	no	—
11	C₁₉×C₅⋊D₅	C₂×C₉	1	1	yes	characteristic
12	C₉×#1044	—	19	1	no	maximal, Hall
13	C₅×#4294	—	2	2	no	—
14	C₅×#4294	AGL(1,𝔽₅)	1	2	yes	complemented
15	C₄×#4294	—	25	1	no	Hall
16	C₃×D₅×C₁₉	—	5	2	no	—
17	C₃×D₅×C₁₉	—	10	2	no	—
18	C₅²×C₁₉	C₄×C₉	1	1	yes	characteristic, complemented, Hall
19	C₉×C₅⋊D₅	—	19	1	no	—
20	C₁₉×AGL(1,𝔽₅)	—	5	2	no	—
21	C₂×#4294	—	25	1	no	—
22	C₃×#1044	C₁₉⋊C₃	1	1	yes	characteristic
23	C₃×C₅×C₁₉	—	2	2	no	—
24	C₃×C₅×C₁₉	C₃×AGL(1,𝔽₅)	1	2	yes	—
25	C₃×C₄×C₁₉	—	25	1	no	—
26	C₅²×C₉	—	19	1	no	Hall
27	D₅×C₁₉	—	5	2	no	—
28	D₅×C₁₉	—	10	2	no	—
29	C₉×AGL(1,𝔽₅)	—	95	2	no	—
30	#4294	#1044	1	1	yes	characteristic, direct product factor, Hall
31	C₃×C₅⋊D₅	C₂×C₁₉⋊C₃	1	1	yes	characteristic
32	C₂×C₃×C₁₉	—	25	1	no	—
33	#1044	#4294	1	1	yes	characteristic, direct product factor, Hall
34	C₅×C₁₉	—	2	2	no	—
35	C₅×C₁₉	C₉×AGL(1,𝔽₅)	1	2	yes	complemented
36	C₉×D₅	—	95	2	no	—
37	C₉×D₅	—	190	2	no	—
38	C₄×C₁₉	—	25	1	no	Hall
39	C₃×C₅²	C₄×C₁₉⋊C₃	1	1	yes	characteristic
40	C₃×AGL(1,𝔽₅)	—	5	2	no	—
41	C₃×C₁₉	C₃×#1044	1	1	yes	characteristic
42	C₅⋊D₅	C₂×#4294	1	1	yes	characteristic
43	C₅×C₉	—	19	2	no	—
44	C₅×C₉	—	38	2	no	—
45	C₂×C₁₉	—	25	1	no	—
46	C₄×C₉	—	475	1	no	Hall
47	C₃×D₅	—	5	2	no	—
48	C₃×D₅	—	10	2	no	—
49	C₅²	C₄×#4294	1	1	yes	characteristic, complemented, Sylow
50	AGL(1,𝔽₅)	—	5	2	no	—
51	C₁₉	C₉×#1044	1	1	yes	characteristic, complemented, Sylow
52	C₂×C₉	—	475	1	no	—
53	C₃×C₅	—	2	2	no	—
54	C₃×C₅	AGL(1,𝔽₅)×C₁₉⋊C₃	1	2	yes	—
55	C₃×C₄	—	25	1	no	—
56	D₅	—	5	2	no	—
57	D₅	—	10	2	no	—
58	C₉	—	19	1	no	Sylow
59	C₂×C₃	—	25	1	no	—
60	C₅	—	2	2	no	—
61	C₅	AGL(1,𝔽₅)×#4294	1	2	yes	complemented
62	C₄	—	25	1	no	Sylow
63	C₃	C₁₉⋊C₃×#1044	1	1	yes	characteristic
64	C₂	—	25	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₃	C₁₉⋊C₃×#1044
commutator, hypocenter	C₅²×C₁₉	C₄×C₉
Fitting, socle	C₃×C₅²×C₁₉	C₃×C₄
prime radical, cosocle	C₃×C₁₉×C₅⋊D₅	C₂×C₃
LCS	G ▷ C₅²×C₁₉
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
3. C₅²×#4294	C₄	semidirect
6. C₁₉×#1044	C₉	semidirect
14. C₅×#4294	AGL(1,𝔽₅)	semidirect
18. C₅²×C₁₉	C₄×C₉	semidirect
30. #4294	#1044	direct (N × H)
33. #1044	#4294	direct (N × H)
35. C₅×C₁₉	C₉×AGL(1,𝔽₅)	semidirect
49. C₅²	C₄×#4294	semidirect
51. C₁₉	C₉×#1044	semidirect
61. C₅	AGL(1,𝔽₅)×#4294	semidirect

Maximal subgroups

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

subgroup	index
C₅⋊D₅×#4294	2
C₃×C₁₉×#1044	3
AGL(1,𝔽₅)×#4294	5
C₉×#1044	19