Subgroups — C₂×C₃×C₅×D₁₀₉⋊C₃ #34087142

Order 19620.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₁₀₉⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅×D₁₀₉⋊C₃	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×C₅×D₁₀₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×D₁₀₉⋊C₃	C₃	1	3	yes	maximal, direct product factor
5	C₃×C₅×C₁₀₉⋊C₃	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₂×C₃×D₁₀₉⋊C₃	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₅×D₁₀₉⋊C₃	C₂×C₃	1	6	yes	direct product factor
8	C₃×C₅×D₁₀₉	C₂×C₃	1	2	yes	complemented
9	C₂×C₅×C₁₀₉⋊C₃	C₂×C₃	1	3	yes	complemented
10	C₂×C₃×C₅×C₁₀₉	C₂×C₃	1	1	yes	characteristic, complemented
11	C₂×C₅×D₁₀₉	C₃²	1	1	yes	characteristic, complemented, Hall
12	C₂×C₃×C₁₀₉⋊C₃	C₂×C₅	1	1	yes	characteristic, complemented
13	C₃×D₁₀₉⋊C₃	C₂×C₅	1	2	yes	direct product factor
14	C₃×C₅×C₁₀₉	C₂²×C₃	1	1	yes	characteristic, complemented
15	C₅×C₁₀₉⋊C₃	C₂²×C₃	1	3	yes	complemented
16	C₂×C₃×D₁₀₉	C₃×C₅	1	1	yes	characteristic, complemented
17	C₂×D₁₀₉⋊C₃	C₃×C₅	1	3	yes	direct product factor
18	C₂×C₅×C₁₀₉	C₂×C₃²	1	1	yes	characteristic, complemented
19	C₅×D₁₀₉	C₂×C₃²	1	2	yes	complemented
20	C₃×C₁₀₉⋊C₃	C₂²×C₅	1	1	yes	characteristic, complemented, Hall
21	C₂×C₃×C₁₀₉	C₂×C₃×C₅	1	1	yes	characteristic, complemented
22	C₃×D₁₀₉	C₂×C₃×C₅	1	2	yes	complemented
23	C₂×C₁₀₉⋊C₃	C₂×C₃×C₅	1	3	yes	complemented
24	D₁₀₉⋊C₃	C₂×C₃×C₅	1	6	yes	direct product factor
25	C₅×C₁₀₉	C₂²×C₃²	1	1	yes	characteristic, complemented, Hall
26	C₂×D₁₀₉	C₃²×C₅	1	1	yes	characteristic, complemented, Hall
27	C₃×C₁₀₉	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
28	C₁₀₉⋊C₃	C₂²×C₃×C₅	1	3	yes	complemented
29	C₂×C₁₀₉	C₂×C₃²×C₅	1	1	yes	characteristic, complemented
30	D₁₀₉	C₂×C₃²×C₅	1	2	yes	complemented
31	C₂²×C₃²×C₅	—	109	1	no	maximal, Hall
32	C₁₀₉	C₂²×C₃²×C₅	1	1	yes	characteristic, complemented, Sylow
33	C₂×C₃²×C₅	—	109	2	no	—
34	C₂×C₃²×C₅	—	109	1	no	—
35	C₂²×C₃×C₅	—	327	1	no	—
36	C₂²×C₃×C₅	—	109	3	no	—
37	C₃²×C₅	—	109	1	no	Hall
38	C₂²×C₃²	—	109	1	no	Hall
39	C₂×C₃×C₅	—	327	2	no	—
40	C₂×C₃×C₅	—	109	6	no	—
41	C₂×C₃×C₅	—	109	3	no	—
42	C₂×C₃×C₅	D₁₀₉⋊C₃	1	1	yes	characteristic, direct product factor
43	C₂²×C₅	—	327	1	no	Hall
44	C₂×C₃²	—	109	2	no	—
45	C₂×C₃²	—	109	1	no	—
46	C₃×C₅	—	109	3	no	—
47	C₃×C₅	C₂×D₁₀₉⋊C₃	1	1	yes	characteristic, direct product factor
48	C₂²×C₃	—	327	1	no	—
49	C₂²×C₃	—	109	3	no	—
50	C₂×C₅	—	327	2	no	—
51	C₂×C₅	C₃×D₁₀₉⋊C₃	1	1	yes	characteristic, direct product factor
52	C₃²	—	109	1	no	Sylow
53	C₂×C₃	—	327	2	no	—
54	C₂×C₃	—	109	6	no	—
55	C₂×C₃	—	109	3	no	—
56	C₂×C₃	C₅×D₁₀₉⋊C₃	1	1	yes	characteristic, direct product factor
57	C₅	C₂×C₃×D₁₀₉⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
58	C₂²	—	327	1	no	Sylow
59	C₃	—	109	3	no	—
60	C₃	C₂×C₅×D₁₀₉⋊C₃	1	1	yes	characteristic, direct product factor
61	C₂	—	327	2	no	—
62	C₂	C₃×C₅×D₁₀₉⋊C₃	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃×C₅	D₁₀₉⋊C₃
commutator, cosocle, hypocenter	C₁₀₉	C₂²×C₃²×C₅
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₅×C₁₀₉	C₂×C₃
LCS	G ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₃×C₅

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×C₁₀₉⋊C₃	C₂	semidirect
2. C₃×C₅×D₁₀₉⋊C₃	C₂	direct (N × H)
3. C₂×C₃×C₅×D₁₀₉	C₃	semidirect
4. C₂×C₅×D₁₀₉⋊C₃	C₃	direct (N × H)
5. C₃×C₅×C₁₀₉⋊C₃	C₂²	semidirect
6. C₂×C₃×D₁₀₉⋊C₃	C₅	direct (N × H)
7. C₅×D₁₀₉⋊C₃	C₂×C₃	direct (N × H)
8. C₃×C₅×D₁₀₉	C₂×C₃	semidirect
9. C₂×C₅×C₁₀₉⋊C₃	C₂×C₃	semidirect
10. C₂×C₃×C₅×C₁₀₉	C₂×C₃	semidirect
11. C₂×C₅×D₁₀₉	C₃²	semidirect
12. C₂×C₃×C₁₀₉⋊C₃	C₂×C₅	semidirect
13. C₃×D₁₀₉⋊C₃	C₂×C₅	direct (N × H)
14. C₃×C₅×C₁₀₉	C₂²×C₃	semidirect
15. C₅×C₁₀₉⋊C₃	C₂²×C₃	semidirect
16. C₂×C₃×D₁₀₉	C₃×C₅	semidirect
17. C₂×D₁₀₉⋊C₃	C₃×C₅	direct (N × H)
18. C₂×C₅×C₁₀₉	C₂×C₃²	semidirect
19. C₅×D₁₀₉	C₂×C₃²	semidirect
20. C₃×C₁₀₉⋊C₃	C₂²×C₅	semidirect
21. C₂×C₃×C₁₀₉	C₂×C₃×C₅	semidirect
22. C₃×D₁₀₉	C₂×C₃×C₅	semidirect
23. C₂×C₁₀₉⋊C₃	C₂×C₃×C₅	semidirect
24. D₁₀₉⋊C₃	C₂×C₃×C₅	direct (N × H)
25. C₅×C₁₀₉	C₂²×C₃²	semidirect
26. C₂×D₁₀₉	C₃²×C₅	semidirect
27. C₃×C₁₀₉	C₂²×C₃×C₅	semidirect
28. C₁₀₉⋊C₃	C₂²×C₃×C₅	semidirect
29. C₂×C₁₀₉	C₂×C₃²×C₅	semidirect
30. D₁₀₉	C₂×C₃²×C₅	semidirect
32. C₁₀₉	C₂²×C₃²×C₅	semidirect
42. C₂×C₃×C₅	D₁₀₉⋊C₃	direct (N × H)
47. C₃×C₅	C₂×D₁₀₉⋊C₃	direct (N × H)
51. C₂×C₅	C₃×D₁₀₉⋊C₃	direct (N × H)
56. C₂×C₃	C₅×D₁₀₉⋊C₃	direct (N × H)
57. C₅	C₂×C₃×D₁₀₉⋊C₃	direct (N × H)
60. C₃	C₂×C₅×D₁₀₉⋊C₃	direct (N × H)
62. C₂	C₃×C₅×D₁₀₉⋊C₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×C₁₀₉⋊C₃	2
C₃×C₅×D₁₀₉⋊C₃	2
C₂×C₃×C₅×D₁₀₉	3
C₂×C₅×D₁₀₉⋊C₃	3
C₂×C₃×D₁₀₉⋊C₃	5
C₂²×C₃²×C₅	109

Subgroups — C2×C3×C5×D109⋊C3 #34087142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×D₁₀₉⋊C₃ #34087142