Subgroups — C₂×C₅²×D₃₀₉ #44525554

Order 30900.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×D₃₀₉	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₃×C₅²×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅²×D₁₀₃	—	3	1	no	maximal, Hall
4	C₃×C₅²×C₁₀₃	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₅×D₃₀₉	C₅	1	6	yes	maximal, direct product factor
6	C₅²×D₁₀₃	—	3	2	no	—
7	C₂×C₅²×C₁₀₃	D₃	1	1	yes	characteristic, complemented
8	C₅×D₃₀₉	C₂×C₅	1	12	yes	direct product factor
9	C₂×C₃×C₅×C₁₀₃	C₂×C₅	1	6	yes	complemented
10	C₅²×C₁₀₃	C₂×D₃	1	1	yes	characteristic, complemented, Hall
11	C₂×C₅×D₁₀₃	—	3	6	no	—
12	C₃×C₅×C₁₀₃	C₂²×C₅	1	6	yes	complemented
13	C₂×D₃₀₉	C₅²	1	1	yes	characteristic, direct product factor, Hall
14	C₂×C₅×C₁₀₃	C₅×D₃	1	6	yes	complemented
15	C₅×D₁₀₃	—	3	12	no	—
16	C₂×C₃×C₁₀₃	C₂×C₅²	1	1	yes	characteristic, complemented
17	D₃₀₉	C₂×C₅²	1	2	yes	direct product factor
18	C₅×C₁₀₃	C₂×C₅×D₃	1	6	yes	complemented
19	C₂×D₁₀₃	—	3	1	no	Hall
20	C₃×C₁₀₃	C₂²×C₅²	1	1	yes	characteristic, complemented, Hall
21	C₂×C₅²×D₃	—	103	1	no	maximal, Hall
22	C₂×C₁₀₃	C₅²×D₃	1	1	yes	characteristic, complemented
23	D₁₀₃	—	3	2	no	—
24	C₂×C₃×C₅²	D₁₀₃	1	1	yes	characteristic, complemented
25	C₅²×D₃	—	103	2	no	—
26	C₁₀₃	C₂×C₅²×D₃	1	1	yes	characteristic, complemented, Sylow
27	C₂²×C₅²	—	309	1	no	Hall
28	C₃×C₅²	C₂×D₁₀₃	1	1	yes	characteristic, complemented, Hall
29	C₂×C₅×D₃	—	103	6	no	—
30	C₂×C₅²	—	309	2	no	—
31	C₂×C₅²	D₃₀₉	1	1	yes	characteristic, direct product factor
32	C₂×C₃×C₅	C₅×D₁₀₃	1	6	yes	complemented
33	C₅×D₃	—	103	12	no	—
34	C₅²	C₂×D₃₀₉	1	1	yes	characteristic, direct product factor, Sylow
35	C₂²×C₅	—	309	6	no	—
36	C₃×C₅	C₂×C₅×D₁₀₃	1	6	yes	complemented
37	C₂×D₃	—	103	1	no	Hall
38	C₂×C₅	—	309	12	no	—
39	C₂×C₅	C₅×D₃₀₉	1	6	yes	direct product factor
40	C₂×C₃	C₅²×D₁₀₃	1	1	yes	characteristic, complemented
41	D₃	—	103	2	no	—
42	C₅	C₂×C₅×D₃₀₉	1	6	yes	direct product factor
43	C₂²	—	309	1	no	Sylow
44	C₃	C₂×C₅²×D₁₀₃	1	1	yes	characteristic, complemented, Sylow
45	C₂	—	309	2	no	—
46	C₂	C₅²×D₃₀₉	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₅²	D₃₀₉
commutator, cosocle, hypocenter	C₃×C₁₀₃	C₂²×C₅²
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₅²×C₁₀₃	C₂
LCS	G ▷ C₃×C₁₀₃
UCS	C₁ ◁ C₂×C₅²

Complements

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

N	H ≅ G/N	type
1. C₅²×D₃₀₉	C₂	direct (N × H)
2. C₂×C₃×C₅²×C₁₀₃	C₂	semidirect
4. C₃×C₅²×C₁₀₃	C₂²	semidirect
5. C₂×C₅×D₃₀₉	C₅	direct (N × H)
7. C₂×C₅²×C₁₀₃	D₃	semidirect
8. C₅×D₃₀₉	C₂×C₅	direct (N × H)
9. C₂×C₃×C₅×C₁₀₃	C₂×C₅	semidirect
10. C₅²×C₁₀₃	C₂×D₃	semidirect
12. C₃×C₅×C₁₀₃	C₂²×C₅	semidirect
13. C₂×D₃₀₉	C₅²	direct (N × H)
14. C₂×C₅×C₁₀₃	C₅×D₃	semidirect
16. C₂×C₃×C₁₀₃	C₂×C₅²	semidirect
17. D₃₀₉	C₂×C₅²	direct (N × H)
18. C₅×C₁₀₃	C₂×C₅×D₃	semidirect
20. C₃×C₁₀₃	C₂²×C₅²	semidirect
22. C₂×C₁₀₃	C₅²×D₃	semidirect
24. C₂×C₃×C₅²	D₁₀₃	semidirect
26. C₁₀₃	C₂×C₅²×D₃	semidirect
28. C₃×C₅²	C₂×D₁₀₃	semidirect
31. C₂×C₅²	D₃₀₉	direct (N × H)
32. C₂×C₃×C₅	C₅×D₁₀₃	semidirect
34. C₅²	C₂×D₃₀₉	direct (N × H)
36. C₃×C₅	C₂×C₅×D₁₀₃	semidirect
39. C₂×C₅	C₅×D₃₀₉	direct (N × H)
40. C₂×C₃	C₅²×D₁₀₃	semidirect
42. C₅	C₂×C₅×D₃₀₉	direct (N × H)
44. C₃	C₂×C₅²×D₁₀₃	semidirect
46. C₂	C₅²×D₃₀₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅²×D₃₀₉	2
C₂×C₃×C₅²×C₁₀₃	2
C₂×C₅²×D₁₀₃	3
C₂×C₅×D₃₀₉	5
C₂×C₅²×D₃	103

Subgroups — C2×C52×D309 #44525554

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅²×D₃₀₉ #44525554