Subgroups — C₅×D₃×D₅×C₁₀₉ #45658331

Order 32700.

96 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	C₅×D₁₅×C₁₀₉	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	C₂×C₅×D₃×C₁₀₉	—	5	1	no	maximal
7	D₃×D₅×C₁₀₉	C₅	1	1	yes	maximal, characteristic, direct product factor
8	C₅×D₅×C₁₀₉	—	3	1	no	—
9	C₅×D₅×C₁₀₉	D₃	1	1	yes	characteristic, direct product factor
10	C₂×C₅²×C₁₀₉	—	3	1	no	—
11	D₁₅×C₁₀₉	C₂×C₅	1	1	yes	characteristic, complemented
12	C₅×D₃×C₁₀₉	—	2	2	no	—
13	C₅×D₃×C₁₀₉	—	5	1	no	—
14	C₅×D₃×C₁₀₉	D₅	1	1	yes	characteristic, direct product factor
15	C₅×D₃×C₁₀₉	C₂×C₅	1	1	yes	characteristic, complemented
16	C₃×D₅×C₁₀₉	C₂×C₅	1	1	yes	characteristic, complemented
17	C₂×C₃×C₅×C₁₀₉	—	5	1	no	—
18	C₅²×C₁₀₉	C₂×D₃	1	1	yes	characteristic, complemented, Hall
19	C₂×D₅×C₁₀₉	—	3	1	no	—
20	C₂²×C₅×C₁₀₉	—	15	1	no	—
21	C₃×C₅×C₁₀₉	—	2	2	no	—
22	C₃×C₅×C₁₀₉	C₂×D₅	1	1	yes	characteristic, complemented
23	C₃×C₅×C₁₀₉	C₂²×C₅	1	1	yes	characteristic, complemented
24	C₂×D₃×C₁₀₉	—	5	1	no	Hall
25	C₂×C₅×C₁₀₉	—	3	1	no	—
26	C₂×C₅×C₁₀₉	—	3	1	no	—
27	C₂×C₅×C₁₀₉	—	5	1	no	—
28	C₂×C₅×C₁₀₉	—	6	2	no	—
29	C₂×C₅×C₁₀₉	—	15	1	no	—
30	D₅×C₁₀₉	—	3	1	no	—
31	D₅×C₁₀₉	C₅×D₃	1	1	yes	characteristic, direct product factor
32	C₂×C₃×C₁₀₉	—	5	1	no	—
33	D₃×C₁₀₉	—	5	1	no	—
34	D₃×C₁₀₉	C₅×D₅	1	1	yes	characteristic, direct product factor
35	C₅×C₁₀₉	—	2	2	no	—
36	C₅×C₁₀₉	D₃×D₅	1	1	yes	characteristic, direct product factor
37	C₅×C₁₀₉	C₂×C₅×D₃	1	1	yes	characteristic, complemented
38	C₂²×C₁₀₉	—	15	1	no	Hall
39	C₃×C₁₀₉	C₂×C₅×D₅	1	1	yes	characteristic, complemented, Hall
40	C₅×D₃×D₅	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Hall
41	C₂×C₁₀₉	—	3	1	no	—
42	C₂×C₁₀₉	—	5	1	no	—
43	C₂×C₁₀₉	—	15	1	no	—
44	C₅×D₁₅	C₂×C₁₀₉	1	1	yes	characteristic, complemented
45	C₅²×D₃	C₂×C₁₀₉	1	1	yes	characteristic, complemented
46	C₃×C₅×D₅	C₂×C₁₀₉	1	1	yes	characteristic, complemented
47	C₁₀₉	C₅×D₃×D₅	1	1	yes	characteristic, direct product factor, Sylow
48	C₂×C₅×D₅	—	3	1	no	Hall
49	C₃×C₅²	C₂²×C₁₀₉	1	1	yes	characteristic, complemented, Hall
50	C₂×C₅×D₃	—	5	1	no	—
51	D₃×D₅	C₅×C₁₀₉	1	1	yes	characteristic, direct product factor
52	C₂×C₅²	—	3	1	no	—
53	C₅×D₅	—	3	1	no	—
54	C₅×D₅	D₃×C₁₀₉	1	1	yes	characteristic, direct product factor
55	C₂×C₃×C₅	—	5	1	no	—
56	D₁₅	C₂×C₅×C₁₀₉	1	1	yes	characteristic, complemented
57	C₃×D₅	C₂×C₅×C₁₀₉	1	1	yes	characteristic, complemented
58	C₅×D₃	—	2	2	no	—
59	C₅×D₃	—	5	1	no	—
60	C₅×D₃	D₅×C₁₀₉	1	1	yes	characteristic, direct product factor
61	C₅×D₃	C₂×C₅×C₁₀₉	1	1	yes	characteristic, complemented
62	C₅²	C₂×D₃×C₁₀₉	1	1	yes	characteristic, complemented, Sylow
63	C₂²×C₅	—	15	1	no	—
64	C₂×D₅	—	3	1	no	—
65	C₃×C₅	—	2	2	no	—
66	C₃×C₅	C₂²×C₅×C₁₀₉	1	1	yes	characteristic, complemented
67	C₃×C₅	C₂×D₅×C₁₀₉	1	1	yes	characteristic, complemented
68	C₂×D₃	—	5	1	no	Hall
69	C₂×C₅	—	3	1	no	—
70	C₂×C₅	—	3	1	no	—
71	C₂×C₅	—	5	1	no	—
72	C₂×C₅	—	6	2	no	—
73	C₂×C₅	—	15	1	no	—
74	D₅	—	3	1	no	—
75	D₅	C₅×D₃×C₁₀₉	1	1	yes	characteristic, direct product factor
76	C₂×C₃	—	5	1	no	—
77	D₃	—	5	1	no	—
78	D₃	C₅×D₅×C₁₀₉	1	1	yes	characteristic, direct product factor
79	C₅	—	2	2	no	—
80	C₅	D₃×D₅×C₁₀₉	1	1	yes	characteristic, direct product factor
81	C₅	C₂×C₅×D₃×C₁₀₉	1	1	yes	characteristic, complemented
82	C₂²	—	15	1	no	Sylow
83	C₃	C₂×C₅×D₅×C₁₀₉	1	1	yes	characteristic, complemented, Sylow
84	C₂	—	3	1	no	—
85	C₂	—	5	1	no	—
86	C₂	—	15	1	no	—

	subgroup	G/H
center, hypercenter	C₅×C₁₀₉	D₃×D₅
commutator, cosocle, hypocenter	C₃×C₅	C₂²×C₅×C₁₀₉
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₁₀₉	C₂²
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₅×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. C₅×D₁₅×C₁₀₉	C₂	semidirect
5. C₃×C₅²×C₁₀₉	C₂²	semidirect
7. D₃×D₅×C₁₀₉	C₅	direct (N × H)
9. C₅×D₅×C₁₀₉	D₃	direct (N × H)
11. D₁₅×C₁₀₉	C₂×C₅	semidirect
14. C₅×D₃×C₁₀₉	D₅	direct (N × H)
15. C₅×D₃×C₁₀₉	C₂×C₅	semidirect
16. C₃×D₅×C₁₀₉	C₂×C₅	semidirect
18. C₅²×C₁₀₉	C₂×D₃	semidirect
22. C₃×C₅×C₁₀₉	C₂×D₅	semidirect
23. C₃×C₅×C₁₀₉	C₂²×C₅	semidirect
31. D₅×C₁₀₉	C₅×D₃	direct (N × H)
34. D₃×C₁₀₉	C₅×D₅	direct (N × H)
36. C₅×C₁₀₉	D₃×D₅	direct (N × H)
37. C₅×C₁₀₉	C₂×C₅×D₃	semidirect
39. C₃×C₁₀₉	C₂×C₅×D₅	semidirect
40. C₅×D₃×D₅	C₁₀₉	direct (N × H)
44. C₅×D₁₅	C₂×C₁₀₉	semidirect
45. C₅²×D₃	C₂×C₁₀₉	semidirect
46. C₃×C₅×D₅	C₂×C₁₀₉	semidirect
47. C₁₀₉	C₅×D₃×D₅	direct (N × H)
49. C₃×C₅²	C₂²×C₁₀₉	semidirect
51. D₃×D₅	C₅×C₁₀₉	direct (N × H)
54. C₅×D₅	D₃×C₁₀₉	direct (N × H)
56. D₁₅	C₂×C₅×C₁₀₉	semidirect
57. C₃×D₅	C₂×C₅×C₁₀₉	semidirect
60. C₅×D₃	D₅×C₁₀₉	direct (N × H)
61. C₅×D₃	C₂×C₅×C₁₀₉	semidirect
62. C₅²	C₂×D₃×C₁₀₉	semidirect
66. C₃×C₅	C₂²×C₅×C₁₀₉	semidirect
67. C₃×C₅	C₂×D₅×C₁₀₉	semidirect
75. D₅	C₅×D₃×C₁₀₉	direct (N × H)
78. D₃	C₅×D₅×C₁₀₉	direct (N × H)
80. C₅	D₃×D₅×C₁₀₉	direct (N × H)
81. C₅	C₂×C₅×D₃×C₁₀₉	semidirect
83. 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
C₅×D₁₅×C₁₀₉	2
C₂×C₅×D₅×C₁₀₉	3
C₂×C₅×D₃×C₁₀₉	5
D₃×D₅×C₁₀₉	5
C₅×D₃×D₅	109

Subgroups — C5×D3×D5×C109 #45658331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×D₃×D₅×C₁₀₉ #45658331