Subgroups — C₃×C₅²×A₄×C₁₇ #29506069

Order 15300.

224 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×A₄×C₁₇	C₃	1	3	yes	maximal, direct product factor
2	C₂²×C₃×C₅²×C₁₇	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃²×C₅²×C₁₇	—	4	1	no	maximal, Hall
4	C₃×C₅×A₄×C₁₇	C₅	1	6	yes	maximal, direct product factor
5	C₂×C₃×C₅²×C₁₇	—	3	1	no	—
6	C₂²×C₅²×C₁₇	C₃²	1	1	yes	characteristic, complemented, Hall
7	C₃×C₅²×C₁₇	—	4	3	no	—
8	C₃×C₅²×C₁₇	A₄	1	1	yes	characteristic, direct product factor
9	C₂²×C₃×C₅×C₁₇	C₃×C₅	1	6	yes	complemented
10	C₅×A₄×C₁₇	C₃×C₅	1	18	yes	direct product factor
11	C₃×C₅²×A₄	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₂×C₅²×C₁₇	—	3	1	no	—
13	C₃²×C₅×C₁₇	—	4	6	no	—
14	C₃×A₄×C₁₇	C₅²	1	1	yes	characteristic, direct product factor, Hall
15	C₂×C₃×C₅×C₁₇	—	3	6	no	—
16	C₅²×C₁₇	C₃×A₄	1	1	yes	characteristic, direct product factor, Hall
17	C₂²×C₅×C₁₇	C₃²×C₅	1	6	yes	complemented
18	C₂²×C₃×C₅²	C₃×C₁₇	1	1	yes	characteristic, complemented
19	C₅²×A₄	C₃×C₁₇	1	3	yes	direct product factor
20	C₃×C₅×C₁₇	—	4	18	no	—
21	C₃×C₅×C₁₇	C₅×A₄	1	6	yes	direct product factor
22	C₃²×C₅²	—	4	1	no	Hall
23	C₂²×C₃×C₁₇	C₃×C₅²	1	1	yes	characteristic, complemented
24	A₄×C₁₇	C₃×C₅²	1	3	yes	direct product factor
25	C₃×C₅×A₄	C₅×C₁₇	1	6	yes	direct product factor
26	C₂×C₅×C₁₇	—	3	6	no	—
27	C₃²×C₁₇	—	4	1	no	Hall
28	C₂×C₃×C₅²	—	3	1	no	—
29	C₂×C₃×C₁₇	—	3	1	no	—
30	C₂²×C₅²	C₃²×C₁₇	1	1	yes	characteristic, complemented, Hall
31	C₅×C₁₇	C₃×C₅×A₄	1	6	yes	direct product factor
32	C₃×C₅²	—	4	3	no	—
33	C₃×C₅²	A₄×C₁₇	1	1	yes	characteristic, direct product factor
34	C₂²×C₁₇	C₃²×C₅²	1	1	yes	characteristic, complemented, Hall
35	C₂²×C₃×C₅	C₃×C₅×C₁₇	1	6	yes	complemented
36	C₅×A₄	C₃×C₅×C₁₇	1	18	yes	direct product factor
37	C₃×C₁₇	—	4	3	no	—
38	C₃×C₁₇	C₅²×A₄	1	1	yes	characteristic, direct product factor
39	C₂×C₅²	—	3	1	no	—
40	C₃²×C₅	—	4	6	no	—
41	C₃×A₄	C₅²×C₁₇	1	1	yes	characteristic, direct product factor, Hall
42	C₂×C₁₇	—	3	1	no	—
43	C₂×C₃×C₅	—	3	6	no	—
44	C₅²	C₃×A₄×C₁₇	1	1	yes	characteristic, direct product factor, Sylow
45	C₂²×C₅	C₃²×C₅×C₁₇	1	6	yes	complemented
46	C₁₇	C₃×C₅²×A₄	1	1	yes	characteristic, direct product factor, Sylow
47	C₃×C₅	—	4	18	no	—
48	C₃×C₅	C₅×A₄×C₁₇	1	6	yes	direct product factor
49	C₂²×C₃	C₃×C₅²×C₁₇	1	1	yes	characteristic, complemented
50	A₄	C₃×C₅²×C₁₇	1	3	yes	direct product factor
51	C₂×C₅	—	3	6	no	—
52	C₃²	—	4	1	no	Sylow
53	C₂×C₃	—	3	1	no	—
54	C₅	C₃×C₅×A₄×C₁₇	1	6	yes	direct product factor
55	C₂²	C₃²×C₅²×C₁₇	1	1	yes	characteristic, complemented, Sylow
56	C₃	—	4	3	no	—
57	C₃	C₅²×A₄×C₁₇	1	1	yes	characteristic, direct product factor
58	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₃×C₅²×C₁₇	A₄
commutator, cosocle, hypocenter	C₂²	C₃²×C₅²×C₁₇
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₅²×C₁₇	C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₃×C₅²×C₁₇

Complements

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

N	H ≅ G/N	type
1. C₅²×A₄×C₁₇	C₃	direct (N × H)
2. C₂²×C₃×C₅²×C₁₇	C₃	semidirect
4. C₃×C₅×A₄×C₁₇	C₅	direct (N × H)
6. C₂²×C₅²×C₁₇	C₃²	semidirect
8. C₃×C₅²×C₁₇	A₄	direct (N × H)
9. C₂²×C₃×C₅×C₁₇	C₃×C₅	semidirect
10. C₅×A₄×C₁₇	C₃×C₅	direct (N × H)
11. C₃×C₅²×A₄	C₁₇	direct (N × H)
14. C₃×A₄×C₁₇	C₅²	direct (N × H)
16. C₅²×C₁₇	C₃×A₄	direct (N × H)
17. C₂²×C₅×C₁₇	C₃²×C₅	semidirect
18. C₂²×C₃×C₅²	C₃×C₁₇	semidirect
19. C₅²×A₄	C₃×C₁₇	direct (N × H)
21. C₃×C₅×C₁₇	C₅×A₄	direct (N × H)
23. C₂²×C₃×C₁₇	C₃×C₅²	semidirect
24. A₄×C₁₇	C₃×C₅²	direct (N × H)
25. C₃×C₅×A₄	C₅×C₁₇	direct (N × H)
30. C₂²×C₅²	C₃²×C₁₇	semidirect
31. C₅×C₁₇	C₃×C₅×A₄	direct (N × H)
33. C₃×C₅²	A₄×C₁₇	direct (N × H)
34. C₂²×C₁₇	C₃²×C₅²	semidirect
35. C₂²×C₃×C₅	C₃×C₅×C₁₇	semidirect
36. C₅×A₄	C₃×C₅×C₁₇	direct (N × H)
38. C₃×C₁₇	C₅²×A₄	direct (N × H)
41. C₃×A₄	C₅²×C₁₇	direct (N × H)
44. C₅²	C₃×A₄×C₁₇	direct (N × H)
45. C₂²×C₅	C₃²×C₅×C₁₇	semidirect
46. C₁₇	C₃×C₅²×A₄	direct (N × H)
48. C₃×C₅	C₅×A₄×C₁₇	direct (N × H)
49. C₂²×C₃	C₃×C₅²×C₁₇	semidirect
50. A₄	C₃×C₅²×C₁₇	direct (N × H)
54. C₅	C₃×C₅×A₄×C₁₇	direct (N × H)
55. C₂²	C₃²×C₅²×C₁₇	semidirect
57. C₃	C₅²×A₄×C₁₇	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×A₄×C₁₇	3
C₂²×C₃×C₅²×C₁₇	3
C₃²×C₅²×C₁₇	4
C₃×C₅×A₄×C₁₇	5
C₃×C₅²×A₄	17

Subgroups — C3×C52×A4×C17 #29506069

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅²×A₄×C₁₇ #29506069