Subgroups — C₃×C₅×A₄×C₁₀₉ #34087201

Order 19620.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×A₄×C₁₀₉	C₃	1	3	yes	maximal, direct product factor
2	C₂²×C₃×C₅×C₁₀₉	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃²×C₅×C₁₀₉	—	4	1	no	maximal, Hall
4	C₃×A₄×C₁₀₉	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂×C₃×C₅×C₁₀₉	—	3	1	no	—
6	C₂²×C₅×C₁₀₉	C₃²	1	1	yes	characteristic, complemented, Hall
7	C₃×C₅×C₁₀₉	—	4	3	no	—
8	C₃×C₅×C₁₀₉	A₄	1	1	yes	characteristic, direct product factor
9	C₂²×C₃×C₁₀₉	C₃×C₅	1	1	yes	characteristic, complemented
10	A₄×C₁₀₉	C₃×C₅	1	3	yes	direct product factor
11	C₂×C₅×C₁₀₉	—	3	1	no	—
12	C₃²×C₁₀₉	—	4	1	no	Hall
13	C₂×C₃×C₁₀₉	—	3	1	no	—
14	C₅×C₁₀₉	C₃×A₄	1	1	yes	characteristic, direct product factor, Hall
15	C₂²×C₁₀₉	C₃²×C₅	1	1	yes	characteristic, complemented, Hall
16	C₃×C₁₀₉	—	4	3	no	—
17	C₃×C₁₀₉	C₅×A₄	1	1	yes	characteristic, direct product factor
18	C₂×C₁₀₉	—	3	1	no	—
19	C₃×C₅×A₄	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₁₀₉	C₃×C₅×A₄	1	1	yes	characteristic, direct product factor, Sylow
21	C₂²×C₃×C₅	C₃×C₁₀₉	1	1	yes	characteristic, complemented
22	C₅×A₄	C₃×C₁₀₉	1	3	yes	direct product factor
23	C₃²×C₅	—	4	1	no	Hall
24	C₃×A₄	C₅×C₁₀₉	1	1	yes	characteristic, direct product factor, Hall
25	C₂×C₃×C₅	—	3	1	no	—
26	C₂²×C₅	C₃²×C₁₀₉	1	1	yes	characteristic, complemented, Hall
27	C₃×C₅	—	4	3	no	—
28	C₃×C₅	A₄×C₁₀₉	1	1	yes	characteristic, direct product factor
29	C₂²×C₃	C₃×C₅×C₁₀₉	1	1	yes	characteristic, complemented
30	A₄	C₃×C₅×C₁₀₉	1	3	yes	direct product factor
31	C₂×C₅	—	3	1	no	—
32	C₃²	—	4	1	no	Sylow
33	C₂×C₃	—	3	1	no	—
34	C₅	C₃×A₄×C₁₀₉	1	1	yes	characteristic, direct product factor, Sylow
35	C₂²	C₃²×C₅×C₁₀₉	1	1	yes	characteristic, complemented, Sylow
36	C₃	—	4	3	no	—
37	C₃	C₅×A₄×C₁₀₉	1	1	yes	characteristic, direct product factor
38	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₃×C₅×C₁₀₉	A₄
commutator, cosocle, hypocenter	C₂²	C₃²×C₅×C₁₀₉
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₅×C₁₀₉	C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₃×C₅×C₁₀₉

Complements

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

N	H ≅ G/N	type
1. C₅×A₄×C₁₀₉	C₃	direct (N × H)
2. C₂²×C₃×C₅×C₁₀₉	C₃	semidirect
4. C₃×A₄×C₁₀₉	C₅	direct (N × H)
6. C₂²×C₅×C₁₀₉	C₃²	semidirect
8. C₃×C₅×C₁₀₉	A₄	direct (N × H)
9. C₂²×C₃×C₁₀₉	C₃×C₅	semidirect
10. A₄×C₁₀₉	C₃×C₅	direct (N × H)
14. C₅×C₁₀₉	C₃×A₄	direct (N × H)
15. C₂²×C₁₀₉	C₃²×C₅	semidirect
17. C₃×C₁₀₉	C₅×A₄	direct (N × H)
19. C₃×C₅×A₄	C₁₀₉	direct (N × H)
20. C₁₀₉	C₃×C₅×A₄	direct (N × H)
21. C₂²×C₃×C₅	C₃×C₁₀₉	semidirect
22. C₅×A₄	C₃×C₁₀₉	direct (N × H)
24. C₃×A₄	C₅×C₁₀₉	direct (N × H)
26. C₂²×C₅	C₃²×C₁₀₉	semidirect
28. C₃×C₅	A₄×C₁₀₉	direct (N × H)
29. C₂²×C₃	C₃×C₅×C₁₀₉	semidirect
30. A₄	C₃×C₅×C₁₀₉	direct (N × H)
34. C₅	C₃×A₄×C₁₀₉	direct (N × H)
35. C₂²	C₃²×C₅×C₁₀₉	semidirect
37. C₃	C₅×A₄×C₁₀₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×A₄×C₁₀₉	3
C₂²×C₃×C₅×C₁₀₉	3
C₃²×C₅×C₁₀₉	4
C₃×A₄×C₁₀₉	5
C₃×C₅×A₄	109

Subgroups — C3×C5×A4×C109 #34087201

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×A₄×C₁₀₉ #34087201