Subgroups — C₂×C₅²×C₃⋊D₅₁ #29506075

Order 15300.

480 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×C₃⋊D₅₁	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₃²×C₅²×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅²×D₅₁	—	3	4	no	maximal
4	C₃²×C₅²×C₁₇	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₅×C₃⋊D₅₁	C₅	1	6	yes	maximal, direct product factor
6	C₅²×D₅₁	—	3	8	no	—
7	C₂×C₃×C₅²×C₁₇	D₃	1	4	yes	complemented
8	C₂×C₅²×D₁₇	—	9	1	no	Hall
9	C₂×C₃²×C₅×C₁₇	C₂×C₅	1	6	yes	complemented
10	C₅×C₃⋊D₅₁	C₂×C₅	1	12	yes	direct product factor
11	C₃×C₅²×C₁₇	C₂×D₃	1	4	yes	complemented
12	C₂×C₅×D₅₁	—	3	24	no	—
13	C₂×C₅²×C₃⋊D₃	—	17	1	no	maximal, Hall
14	C₂×C₅²×C₁₇	C₃⋊D₃	1	1	yes	characteristic, complemented
15	C₅²×D₁₇	—	9	2	no	—
16	C₃²×C₅×C₁₇	C₂²×C₅	1	6	yes	complemented
17	C₂×C₃⋊D₅₁	C₅²	1	1	yes	characteristic, direct product factor, Hall
18	C₂×C₃×C₅×C₁₇	C₅×D₃	1	24	yes	complemented
19	C₅×D₅₁	—	3	48	no	—
20	C₂×C₃²×C₅²	D₁₇	1	1	yes	characteristic, complemented
21	C₅²×C₃⋊D₃	—	17	2	no	—
22	C₅²×C₁₇	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented, Hall
23	C₂×C₅×D₁₇	—	9	6	no	—
24	C₂×C₃²×C₁₇	C₂×C₅²	1	1	yes	characteristic, complemented
25	C₃⋊D₅₁	C₂×C₅²	1	2	yes	direct product factor
26	C₂×C₅²×D₃	—	51	4	no	—
27	C₃×C₅×C₁₇	C₂×C₅×D₃	1	24	yes	complemented
28	C₃²×C₅²	C₂×D₁₇	1	1	yes	characteristic, complemented, Hall
29	C₂×D₅₁	—	3	4	no	—
30	C₂×C₅×C₃⋊D₃	—	17	6	no	—
31	C₂×C₅×C₁₇	C₅×C₃⋊D₃	1	6	yes	complemented
32	C₅×D₁₇	—	9	12	no	—
33	C₃²×C₁₇	C₂²×C₅²	1	1	yes	characteristic, complemented, Hall
34	C₂×C₃×C₅²	D₅₁	1	4	yes	complemented
35	C₅²×D₃	—	51	8	no	—
36	C₂×C₃×C₁₇	C₅²×D₃	1	4	yes	complemented
37	D₅₁	—	3	8	no	—
38	C₂²×C₅²	—	153	1	no	Hall
39	C₂×C₃²×C₅	C₅×D₁₇	1	6	yes	complemented
40	C₅×C₃⋊D₃	—	17	12	no	—
41	C₅×C₁₇	C₂×C₅×C₃⋊D₃	1	6	yes	complemented
42	C₃×C₅²	C₂×D₅₁	1	4	yes	complemented
43	C₂×D₁₇	—	9	1	no	Hall
44	C₂×C₅×D₃	—	51	24	no	—
45	C₃×C₁₇	C₂×C₅²×D₃	1	4	yes	complemented
46	C₂×C₅²	—	153	2	no	—
47	C₂×C₅²	C₃⋊D₅₁	1	1	yes	characteristic, direct product factor
48	C₃²×C₅	C₂×C₅×D₁₇	1	6	yes	complemented
49	C₂×C₃⋊D₃	—	17	1	no	Hall
50	C₂×C₁₇	C₅²×C₃⋊D₃	1	1	yes	characteristic, complemented
51	D₁₇	—	9	2	no	—
52	C₂×C₃×C₅	C₅×D₅₁	1	24	yes	complemented
53	C₅×D₃	—	51	48	no	—
54	C₅²	C₂×C₃⋊D₅₁	1	1	yes	characteristic, direct product factor, Sylow
55	C₂²×C₅	—	153	6	no	—
56	C₂×C₃²	C₅²×D₁₇	1	1	yes	characteristic, complemented
57	C₃⋊D₃	—	17	2	no	—
58	C₁₇	C₂×C₅²×C₃⋊D₃	1	1	yes	characteristic, complemented, Sylow
59	C₃×C₅	C₂×C₅×D₅₁	1	24	yes	complemented
60	C₂×D₃	—	51	4	no	—
61	C₂×C₅	—	153	12	no	—
62	C₂×C₅	C₅×C₃⋊D₅₁	1	6	yes	direct product factor
63	C₃²	C₂×C₅²×D₁₇	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₃	C₅²×D₅₁	1	4	yes	complemented
65	D₃	—	51	8	no	—
66	C₅	C₂×C₅×C₃⋊D₅₁	1	6	yes	direct product factor
67	C₂²	—	153	1	no	Sylow
68	C₃	C₂×C₅²×D₅₁	1	4	yes	complemented
69	C₂	—	153	2	no	—
70	C₂	C₅²×C₃⋊D₅₁	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₅²	C₃⋊D₅₁
commutator, cosocle, hypocenter	C₃²×C₁₇	C₂²×C₅²
Frattini	C₁	G
Fitting, socle	C₂×C₃²×C₅²×C₁₇	C₂
LCS	G ▷ C₃²×C₁₇
UCS	C₁ ◁ C₂×C₅²

Complements

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

N	H ≅ G/N	type
1. C₅²×C₃⋊D₅₁	C₂	direct (N × H)
2. C₂×C₃²×C₅²×C₁₇	C₂	semidirect
4. C₃²×C₅²×C₁₇	C₂²	semidirect
5. C₂×C₅×C₃⋊D₅₁	C₅	direct (N × H)
7. C₂×C₃×C₅²×C₁₇	D₃	semidirect
9. C₂×C₃²×C₅×C₁₇	C₂×C₅	semidirect
10. C₅×C₃⋊D₅₁	C₂×C₅	direct (N × H)
11. C₃×C₅²×C₁₇	C₂×D₃	semidirect
14. C₂×C₅²×C₁₇	C₃⋊D₃	semidirect
16. C₃²×C₅×C₁₇	C₂²×C₅	semidirect
17. C₂×C₃⋊D₅₁	C₅²	direct (N × H)
18. C₂×C₃×C₅×C₁₇	C₅×D₃	semidirect
20. C₂×C₃²×C₅²	D₁₇	semidirect
22. C₅²×C₁₇	C₂×C₃⋊D₃	semidirect
24. C₂×C₃²×C₁₇	C₂×C₅²	semidirect
25. C₃⋊D₅₁	C₂×C₅²	direct (N × H)
27. C₃×C₅×C₁₇	C₂×C₅×D₃	semidirect
28. C₃²×C₅²	C₂×D₁₇	semidirect
31. C₂×C₅×C₁₇	C₅×C₃⋊D₃	semidirect
33. C₃²×C₁₇	C₂²×C₅²	semidirect
34. C₂×C₃×C₅²	D₅₁	semidirect
36. C₂×C₃×C₁₇	C₅²×D₃	semidirect
39. C₂×C₃²×C₅	C₅×D₁₇	semidirect
41. C₅×C₁₇	C₂×C₅×C₃⋊D₃	semidirect
42. C₃×C₅²	C₂×D₅₁	semidirect
45. C₃×C₁₇	C₂×C₅²×D₃	semidirect
47. C₂×C₅²	C₃⋊D₅₁	direct (N × H)
48. C₃²×C₅	C₂×C₅×D₁₇	semidirect
50. C₂×C₁₇	C₅²×C₃⋊D₃	semidirect
52. C₂×C₃×C₅	C₅×D₅₁	semidirect
54. C₅²	C₂×C₃⋊D₅₁	direct (N × H)
56. C₂×C₃²	C₅²×D₁₇	semidirect
58. C₁₇	C₂×C₅²×C₃⋊D₃	semidirect
59. C₃×C₅	C₂×C₅×D₅₁	semidirect
62. C₂×C₅	C₅×C₃⋊D₅₁	direct (N × H)
63. C₃²	C₂×C₅²×D₁₇	semidirect
64. C₂×C₃	C₅²×D₅₁	semidirect
66. C₅	C₂×C₅×C₃⋊D₅₁	direct (N × H)
68. C₃	C₂×C₅²×D₅₁	semidirect
70. C₂	C₅²×C₃⋊D₅₁	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×C₃⋊D₅₁	2
C₂×C₃²×C₅²×C₁₇	2
C₂×C₅²×D₅₁	3
C₂×C₅×C₃⋊D₅₁	5
C₂×C₅²×C₃⋊D₃	17

Subgroups — C2×C52×C3⋊D51 #29506075

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅²×C₃⋊D₅₁ #29506075