Subgroups — C₃×(C₅⋊D₅)⋊D₁₀₉ #45658317

Order 32700.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×D₅₄₅	C₂	1	2	yes	maximal, complemented
2	C₃×C₅⋊D₅×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	(C₅⋊D₅)⋊D₁₀₉	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₃×C₅²×C₁₀₉	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₃×D₅×D₁₀₉	—	5	2	no	maximal
6	C₅⋊D₅×C₁₀₉	C₂×C₃	1	1	yes	characteristic, complemented
7	C₅×D₅₄₅	C₂×C₃	1	2	yes	complemented
8	C₃×D₅₄₅	D₅	1	2	yes	complemented
9	C₃×D₅×C₁₀₉	—	5	2	no	—
10	C₃×D₅×C₁₀₉	—	10	2	no	—
11	C₃×C₅×D₁₀₉	—	5	2	no	—
12	C₅²×C₁₀₉	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
13	D₅×D₁₀₉	—	5	2	no	—
14	C₃×C₅×C₁₀₉	—	2	2	no	—
15	C₃×C₅×C₁₀₉	C₂×D₅	1	2	yes	complemented
16	C₂×C₃×D₁₀₉	—	25	1	no	Hall
17	D₅₄₅	C₃×D₅	1	2	yes	complemented
18	C₅×D₁₀₉	—	5	2	no	—
19	D₅×C₁₀₉	—	5	2	no	—
20	D₅×C₁₀₉	—	10	2	no	—
21	C₂×C₃×C₁₀₉	—	25	1	no	—
22	C₃×D₁₀₉	—	5	2	no	—
23	C₅×C₁₀₉	—	2	2	no	—
24	C₅×C₁₀₉	C₂×C₃×D₅	1	2	yes	complemented
25	C₂×D₁₀₉	—	25	1	no	Hall
26	C₃×C₁₀₉	D₅²	1	1	yes	characteristic, complemented, Hall
27	C₃×D₅²	—	109	1	no	maximal, Hall
28	C₂×C₁₀₉	—	25	1	no	—
29	D₁₀₉	—	5	2	no	—
30	C₃×C₅⋊D₅	D₁₀₉	1	1	yes	characteristic, complemented
31	C₃×C₅×D₅	—	109	2	no	—
32	C₁₀₉	C₃×D₅²	1	1	yes	characteristic, complemented, Sylow
33	D₅²	—	109	1	no	Hall
34	C₃×C₅²	C₂×D₁₀₉	1	1	yes	characteristic, complemented, Hall
35	C₂×C₃×D₅	—	545	2	no	—
36	C₅⋊D₅	C₃×D₁₀₉	1	1	yes	characteristic, complemented
37	C₅×D₅	—	109	2	no	—
38	C₂×C₃×C₅	—	545	2	no	—
39	C₃×D₅	—	5	2	no	—
40	C₃×D₅	—	10	2	no	—
41	C₃×D₅	—	109	2	no	—
42	C₅²	C₂×C₃×D₁₀₉	1	1	yes	characteristic, complemented, Sylow
43	C₂×D₅	—	545	2	no	—
44	C₃×C₅	—	2	2	no	—
45	C₃×C₅	D₅×D₁₀₉	1	2	yes	complemented
46	C₂²×C₃	—	2725	1	no	Hall
47	C₂×C₅	—	545	2	no	—
48	D₅	—	5	2	no	—
49	D₅	—	10	2	no	—
50	D₅	—	109	2	no	—
51	C₂×C₃	—	25	1	no	—
52	C₂×C₃	—	545	2	no	—
53	C₅	—	2	2	no	—
54	C₅	C₃×D₅×D₁₀₉	1	2	yes	complemented
55	C₂²	—	2725	1	no	Sylow
56	C₃	(C₅⋊D₅)⋊D₁₀₉	1	1	yes	characteristic, direct product factor, Sylow
57	C₂	—	25	1	no	—
58	C₂	—	545	2	no	—

	subgroup	G/H
center, hypercenter	C₃	(C₅⋊D₅)⋊D₁₀₉
commutator, cosocle, hypocenter	C₅²×C₁₀₉	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₁₀₉	C₂²
LCS	G ▷ C₅²×C₁₀₉
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×D₅₄₅	C₂	semidirect
2. C₃×C₅⋊D₅×C₁₀₉	C₂	semidirect
3. (C₅⋊D₅)⋊D₁₀₉	C₃	direct (N × H)
4. C₃×C₅²×C₁₀₉	C₂²	semidirect
6. C₅⋊D₅×C₁₀₉	C₂×C₃	semidirect
7. C₅×D₅₄₅	C₂×C₃	semidirect
8. C₃×D₅₄₅	D₅	semidirect
12. C₅²×C₁₀₉	C₂²×C₃	semidirect
15. C₃×C₅×C₁₀₉	C₂×D₅	semidirect
17. D₅₄₅	C₃×D₅	semidirect
24. C₅×C₁₀₉	C₂×C₃×D₅	semidirect
26. C₃×C₁₀₉	D₅²	semidirect
30. C₃×C₅⋊D₅	D₁₀₉	semidirect
32. C₁₀₉	C₃×D₅²	semidirect
34. C₃×C₅²	C₂×D₁₀₉	semidirect
36. C₅⋊D₅	C₃×D₁₀₉	semidirect
42. C₅²	C₂×C₃×D₁₀₉	semidirect
45. C₃×C₅	D₅×D₁₀₉	semidirect
54. C₅	C₃×D₅×D₁₀₉	semidirect
56. C₃	(C₅⋊D₅)⋊D₁₀₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅×D₅₄₅	2
C₃×C₅⋊D₅×C₁₀₉	2
(C₅⋊D₅)⋊D₁₀₉	3
C₃×D₅×D₁₀₉	5
C₃×D₅²	109

Subgroups — C3×(C5⋊D5)⋊D109 #45658317

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×(C₅⋊D₅)⋊D₁₀₉ #45658317