Subgroups — C₅×D₅×D₃₀₉ #44525518

Order 30900.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×D₅×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅²×D₃₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×D₁₅₄₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×D₅×D₁₀₃	—	3	1	no	maximal, Hall
5	C₃×C₅²×C₁₀₃	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₂×C₅×D₃₀₉	—	5	1	no	maximal
7	D₅×D₃₀₉	C₅	1	1	yes	maximal, characteristic, direct product factor
8	C₅×D₅₁₅	—	3	1	no	—
9	C₅×D₅×C₁₀₃	D₃	1	1	yes	characteristic, complemented
10	C₅²×D₁₀₃	—	3	1	no	—
11	D₁₅₄₅	C₂×C₅	1	1	yes	characteristic, complemented
12	C₅×D₃₀₉	—	2	2	no	—
13	C₅×D₃₀₉	—	5	1	no	—
14	C₅×D₃₀₉	D₅	1	1	yes	characteristic, direct product factor
15	C₅×D₃₀₉	C₂×C₅	1	1	yes	characteristic, complemented
16	C₃×D₅×C₁₀₃	C₂×C₅	1	1	yes	characteristic, complemented
17	C₂×C₃×C₅×C₁₀₃	—	5	1	no	—
18	C₅²×C₁₀₃	C₂×D₃	1	1	yes	characteristic, complemented, Hall
19	C₂×C₅×D₁₀₃	—	15	1	no	—
20	D₅×D₁₀₃	—	3	1	no	—
21	C₃×C₅×C₁₀₃	—	2	2	no	—
22	C₃×C₅×C₁₀₃	C₂×D₅	1	1	yes	characteristic, complemented
23	C₃×C₅×C₁₀₃	C₂²×C₅	1	1	yes	characteristic, complemented
24	C₂×D₃₀₉	—	5	1	no	Hall
25	C₂×C₅×C₁₀₃	—	5	1	no	—
26	D₅₁₅	—	3	1	no	—
27	C₅×D₁₀₃	—	3	1	no	—
28	C₅×D₁₀₃	—	3	1	no	—
29	C₅×D₁₀₃	—	6	2	no	—
30	C₅×D₁₀₃	—	15	1	no	—
31	D₅×C₁₀₃	C₅×D₃	1	1	yes	characteristic, complemented
32	C₂×C₃×C₁₀₃	—	5	1	no	—
33	D₃₀₉	—	5	1	no	—
34	D₃₀₉	C₅×D₅	1	1	yes	characteristic, direct product factor
35	C₅×C₁₀₃	—	2	2	no	—
36	C₅×C₁₀₃	D₃×D₅	1	1	yes	characteristic, complemented
37	C₅×C₁₀₃	C₂×C₅×D₃	1	1	yes	characteristic, complemented
38	C₂×D₁₀₃	—	15	1	no	Hall
39	C₃×C₁₀₃	C₂×C₅×D₅	1	1	yes	characteristic, complemented, Hall
40	C₅×D₃×D₅	—	103	1	no	maximal, Hall
41	C₂×C₁₀₃	—	5	1	no	—
42	D₁₀₃	—	3	1	no	—
43	D₁₀₃	—	15	1	no	—
44	C₅×D₁₅	—	103	1	no	—
45	C₅²×D₃	—	103	1	no	—
46	C₃×C₅×D₅	D₁₀₃	1	1	yes	characteristic, complemented
47	C₁₀₃	C₅×D₃×D₅	1	1	yes	characteristic, complemented, Sylow
48	C₂×C₅×D₅	—	309	1	no	Hall
49	C₃×C₅²	C₂×D₁₀₃	1	1	yes	characteristic, complemented, Hall
50	C₂×C₅×D₃	—	515	1	no	—
51	D₃×D₅	—	103	1	no	—
52	C₂×C₅²	—	309	1	no	—
53	C₅×D₅	—	309	1	no	—
54	C₅×D₅	D₃₀₉	1	1	yes	characteristic, direct product factor
55	C₂×C₃×C₅	—	5	1	no	—
56	D₁₅	—	103	1	no	—
57	C₃×D₅	C₅×D₁₀₃	1	1	yes	characteristic, complemented
58	C₅×D₃	—	515	1	no	—
59	C₅×D₃	—	103	1	no	—
60	C₅×D₃	—	103	1	no	—
61	C₅×D₃	—	206	2	no	—
62	C₅²	C₂×D₃₀₉	1	1	yes	characteristic, complemented, Sylow
63	C₂²×C₅	—	1545	1	no	—
64	C₂×D₅	—	309	1	no	—
65	C₃×C₅	—	2	2	no	—
66	C₃×C₅	D₅×D₁₀₃	1	1	yes	characteristic, complemented
67	C₃×C₅	C₂×C₅×D₁₀₃	1	1	yes	characteristic, complemented
68	C₂×D₃	—	515	1	no	Hall
69	C₂×C₅	—	5	1	no	—
70	C₂×C₅	—	1545	1	no	—
71	C₂×C₅	—	309	1	no	—
72	C₂×C₅	—	309	1	no	—
73	C₂×C₅	—	618	2	no	—
74	D₅	—	309	1	no	—
75	D₅	C₅×D₃₀₉	1	1	yes	characteristic, direct product factor
76	C₂×C₃	—	5	1	no	—
77	D₃	—	515	1	no	—
78	D₃	—	103	1	no	—
79	C₅	—	2	2	no	—
80	C₅	D₅×D₃₀₉	1	1	yes	characteristic, direct product factor
81	C₅	C₂×C₅×D₃₀₉	1	1	yes	characteristic, complemented
82	C₂²	—	1545	1	no	Sylow
83	C₃	C₅×D₅×D₁₀₃	1	1	yes	characteristic, complemented, Sylow
84	C₂	—	5	1	no	—
85	C₂	—	1545	1	no	—
86	C₂	—	309	1	no	—

	subgroup	G/H
center, hypercenter	C₅	D₅×D₃₀₉
commutator, cosocle, hypocenter	C₃×C₅×C₁₀₃	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₁₀₃	C₂²
LCS	G ▷ C₃×C₅×C₁₀₃
UCS	C₁ ◁ C₅

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×D₅×C₁₀₃	C₂	semidirect
2. C₅²×D₃₀₉	C₂	semidirect
3. C₅×D₁₅₄₅	C₂	semidirect
5. C₃×C₅²×C₁₀₃	C₂²	semidirect
7. D₅×D₃₀₉	C₅	direct (N × H)
9. C₅×D₅×C₁₀₃	D₃	semidirect
11. D₁₅₄₅	C₂×C₅	semidirect
14. C₅×D₃₀₉	D₅	direct (N × H)
15. C₅×D₃₀₉	C₂×C₅	semidirect
16. C₃×D₅×C₁₀₃	C₂×C₅	semidirect
18. C₅²×C₁₀₃	C₂×D₃	semidirect
22. C₃×C₅×C₁₀₃	C₂×D₅	semidirect
23. C₃×C₅×C₁₀₃	C₂²×C₅	semidirect
31. D₅×C₁₀₃	C₅×D₃	semidirect
34. D₃₀₉	C₅×D₅	direct (N × H)
36. C₅×C₁₀₃	D₃×D₅	semidirect
37. C₅×C₁₀₃	C₂×C₅×D₃	semidirect
39. C₃×C₁₀₃	C₂×C₅×D₅	semidirect
46. C₃×C₅×D₅	D₁₀₃	semidirect
47. C₁₀₃	C₅×D₃×D₅	semidirect
49. C₃×C₅²	C₂×D₁₀₃	semidirect
54. C₅×D₅	D₃₀₉	direct (N × H)
57. C₃×D₅	C₅×D₁₀₃	semidirect
62. C₅²	C₂×D₃₀₉	semidirect
66. C₃×C₅	D₅×D₁₀₃	semidirect
67. C₃×C₅	C₂×C₅×D₁₀₃	semidirect
75. D₅	C₅×D₃₀₉	direct (N × H)
80. C₅	D₅×D₃₀₉	direct (N × H)
81. C₅	C₂×C₅×D₃₀₉	semidirect
83. C₃	C₅×D₅×D₁₀₃	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×C₁₀₃	2
C₅²×D₃₀₉	2
C₅×D₁₅₄₅	2
C₅×D₅×D₁₀₃	3
C₂×C₅×D₃₀₉	5
D₅×D₃₀₉	5
C₅×D₃×D₅	103

Subgroups — C5×D5×D309 #44525518

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×D₅×D₃₀₉ #44525518