Subgroups — D₃×C₅⋊D₅×C₁₀₉ #45658352

Order 32700.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅⋊D₅×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅²×D₃×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₀₉×C₃⋊(C₅⋊D₅)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅⋊D₅×C₁₀₉	—	3	1	no	maximal, Hall
5	C₃×C₅²×C₁₀₉	C₂²	1	1	yes	characteristic, complemented, Hall
6	D₃×D₅×C₁₀₉	—	5	6	no	maximal
7	C₅⋊D₅×C₁₀₉	—	3	1	no	—
8	C₅⋊D₅×C₁₀₉	D₃	1	1	yes	characteristic, direct product factor
9	C₂×C₅²×C₁₀₉	—	3	1	no	—
10	D₁₅×C₁₀₉	—	5	6	no	—
11	C₅×D₃×C₁₀₉	D₅	1	6	yes	complemented
12	C₃×D₅×C₁₀₉	—	5	6	no	—
13	C₅²×C₁₀₉	C₂×D₃	1	1	yes	characteristic, complemented, Hall
14	C₂×D₅×C₁₀₉	—	15	6	no	—
15	C₃×C₅×C₁₀₉	C₂×D₅	1	6	yes	complemented
16	C₂×D₃×C₁₀₉	—	25	1	no	Hall
17	C₂×C₅×C₁₀₉	—	3	6	no	—
18	D₅×C₁₀₉	—	5	6	no	—
19	D₅×C₁₀₉	—	15	6	no	—
20	C₂×C₃×C₁₀₉	—	25	1	no	—
21	D₃×C₁₀₉	—	25	1	no	—
22	D₃×C₁₀₉	C₅⋊D₅	1	1	yes	characteristic, direct product factor
23	C₅×C₁₀₉	D₃×D₅	1	6	yes	complemented
24	C₂²×C₁₀₉	—	75	1	no	Hall
25	C₃×C₁₀₉	C₂×C₅⋊D₅	1	1	yes	characteristic, complemented, Hall
26	D₃×C₅⋊D₅	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Hall
27	C₂×C₁₀₉	—	3	1	no	—
28	C₂×C₁₀₉	—	25	1	no	—
29	C₂×C₁₀₉	—	75	1	no	—
30	C₃⋊(C₅⋊D₅)	C₂×C₁₀₉	1	1	yes	characteristic, complemented
31	C₅²×D₃	C₂×C₁₀₉	1	1	yes	characteristic, complemented
32	C₃×C₅⋊D₅	C₂×C₁₀₉	1	1	yes	characteristic, complemented
33	C₁₀₉	D₃×C₅⋊D₅	1	1	yes	characteristic, direct product factor, Sylow
34	C₂×C₅⋊D₅	—	3	1	no	Hall
35	C₃×C₅²	C₂²×C₁₀₉	1	1	yes	characteristic, complemented, Hall
36	D₃×D₅	—	5	6	no	—
37	C₂×C₅²	—	3	1	no	—
38	C₅⋊D₅	—	3	1	no	—
39	C₅⋊D₅	D₃×C₁₀₉	1	1	yes	characteristic, direct product factor
40	D₁₅	—	5	6	no	—
41	C₃×D₅	—	5	6	no	—
42	C₅×D₃	D₅×C₁₀₉	1	6	yes	complemented
43	C₅²	C₂×D₃×C₁₀₉	1	1	yes	characteristic, complemented, Sylow
44	C₂×D₅	—	15	6	no	—
45	C₃×C₅	C₂×D₅×C₁₀₉	1	6	yes	complemented
46	C₂×D₃	—	25	1	no	Hall
47	C₂×C₅	—	3	6	no	—
48	D₅	—	5	6	no	—
49	D₅	—	15	6	no	—
50	C₂×C₃	—	25	1	no	—
51	D₃	—	25	1	no	—
52	D₃	C₅⋊D₅×C₁₀₉	1	1	yes	characteristic, direct product factor
53	C₅	D₃×D₅×C₁₀₉	1	6	yes	complemented
54	C₂²	—	75	1	no	Sylow
55	C₃	C₂×C₅⋊D₅×C₁₀₉	1	1	yes	characteristic, complemented, Sylow
56	C₂	—	3	1	no	—
57	C₂	—	25	1	no	—
58	C₂	—	75	1	no	—

	subgroup	G/H
center, hypercenter	C₁₀₉	D₃×C₅⋊D₅
commutator, cosocle, hypocenter	C₃×C₅²	C₂²×C₁₀₉
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₁₀₉	C₂²
LCS	G ▷ C₃×C₅²
UCS	C₁ ◁ C₁₀₉

Complements

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

N	H ≅ G/N	type
1. C₃×C₅⋊D₅×C₁₀₉	C₂	semidirect
2. C₅²×D₃×C₁₀₉	C₂	semidirect
3. C₁₀₉×C₃⋊(C₅⋊D₅)	C₂	semidirect
5. C₃×C₅²×C₁₀₉	C₂²	semidirect
8. C₅⋊D₅×C₁₀₉	D₃	direct (N × H)
11. C₅×D₃×C₁₀₉	D₅	semidirect
13. C₅²×C₁₀₉	C₂×D₃	semidirect
15. C₃×C₅×C₁₀₉	C₂×D₅	semidirect
22. D₃×C₁₀₉	C₅⋊D₅	direct (N × H)
23. C₅×C₁₀₉	D₃×D₅	semidirect
25. C₃×C₁₀₉	C₂×C₅⋊D₅	semidirect
26. D₃×C₅⋊D₅	C₁₀₉	direct (N × H)
30. C₃⋊(C₅⋊D₅)	C₂×C₁₀₉	semidirect
31. C₅²×D₃	C₂×C₁₀₉	semidirect
32. C₃×C₅⋊D₅	C₂×C₁₀₉	semidirect
33. C₁₀₉	D₃×C₅⋊D₅	direct (N × H)
35. C₃×C₅²	C₂²×C₁₀₉	semidirect
39. C₅⋊D₅	D₃×C₁₀₉	direct (N × H)
42. C₅×D₃	D₅×C₁₀₉	semidirect
43. C₅²	C₂×D₃×C₁₀₉	semidirect
45. C₃×C₅	C₂×D₅×C₁₀₉	semidirect
52. D₃	C₅⋊D₅×C₁₀₉	direct (N × H)
53. C₅	D₃×D₅×C₁₀₉	semidirect
55. C₃	C₂×C₅⋊D₅×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅⋊D₅×C₁₀₉	2
C₅²×D₃×C₁₀₉	2
C₁₀₉×C₃⋊(C₅⋊D₅)	2
C₂×C₅⋊D₅×C₁₀₉	3
D₃×D₅×C₁₀₉	5
D₃×C₅⋊D₅	109

Subgroups — D3×C5⋊D5×C109 #45658352

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×C₅⋊D₅×C₁₀₉ #45658352