Subgroups — C₃×C₅²×C₁₃×C₁₆ #72193627

Order 15600.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅²×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
2	C₅²×C₁₃×C₁₆	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₃×C₅×C₁₃×C₁₆	C₅	1	6	yes	maximal, direct product factor
4	C₅²×C₈×C₁₃	C₂×C₃	1	1	yes	characteristic
5	C₃×C₄×C₅²×C₁₃	C₄	1	1	yes	characteristic
6	C₂×C₃×C₅²×C₁₃	C₈	1	1	yes	characteristic
7	C₃×C₅×C₈×C₁₃	C₂×C₅	1	6	yes	—
8	C₄×C₅²×C₁₃	C₃×C₄	1	1	yes	characteristic
9	C₃×C₅²×C₁₆	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₅×C₁₃×C₁₆	C₃×C₅	1	6	yes	direct product factor
11	C₃×C₅²×C₁₃	C₁₆	1	1	yes	characteristic, direct product factor, Hall
12	C₃×C₄×C₅×C₁₃	C₄×C₅	1	6	yes	—
13	C₂×C₅²×C₁₃	C₃×C₈	1	1	yes	characteristic
14	C₃×C₁₃×C₁₆	C₅²	1	1	yes	characteristic, direct product factor, Hall
15	C₃×C₅²×C₈	C₂×C₁₃	1	1	yes	characteristic
16	C₅×C₈×C₁₃	C₂×C₃×C₅	1	6	yes	—
17	C₅²×C₁₆	C₃×C₁₃	1	1	yes	characteristic, direct product factor, Hall
18	C₂×C₃×C₅×C₁₃	C₅×C₈	1	6	yes	—
19	C₅²×C₁₃	C₃×C₁₆	1	1	yes	characteristic, direct product factor, Hall
20	C₃×C₈×C₁₃	C₂×C₅²	1	1	yes	characteristic
21	C₃×C₄×C₅²	C₄×C₁₃	1	1	yes	characteristic
22	C₄×C₅×C₁₃	C₃×C₄×C₅	1	6	yes	—
23	C₃×C₅×C₁₆	C₅×C₁₃	1	6	yes	direct product factor
24	C₁₃×C₁₆	C₃×C₅²	1	1	yes	characteristic, direct product factor, Hall
25	C₅²×C₈	C₂×C₃×C₁₃	1	1	yes	characteristic
26	C₃×C₅×C₁₃	C₅×C₁₆	1	6	yes	direct product factor
27	C₃×C₄×C₁₃	C₄×C₅²	1	1	yes	characteristic
28	C₂×C₃×C₅²	C₈×C₁₃	1	1	yes	characteristic
29	C₂×C₅×C₁₃	C₃×C₅×C₈	1	6	yes	—
30	C₃×C₅×C₈	C₂×C₅×C₁₃	1	6	yes	—
31	C₈×C₁₃	C₂×C₃×C₅²	1	1	yes	characteristic
32	C₄×C₅²	C₃×C₄×C₁₃	1	1	yes	characteristic
33	C₅×C₁₆	C₃×C₅×C₁₃	1	6	yes	direct product factor
34	C₂×C₃×C₁₃	C₅²×C₈	1	1	yes	characteristic
35	C₃×C₅²	C₁₃×C₁₆	1	1	yes	characteristic, direct product factor, Hall
36	C₅×C₁₃	C₃×C₅×C₁₆	1	6	yes	direct product factor
37	C₃×C₄×C₅	C₄×C₅×C₁₃	1	6	yes	—
38	C₄×C₁₃	C₃×C₄×C₅²	1	1	yes	characteristic
39	C₂×C₅²	C₃×C₈×C₁₃	1	1	yes	characteristic
40	C₃×C₁₆	C₅²×C₁₃	1	1	yes	characteristic, direct product factor, Hall
41	C₅×C₈	C₂×C₃×C₅×C₁₃	1	6	yes	—
42	C₃×C₁₃	C₅²×C₁₆	1	1	yes	characteristic, direct product factor, Hall
43	C₂×C₃×C₅	C₅×C₈×C₁₃	1	6	yes	—
44	C₂×C₁₃	C₃×C₅²×C₈	1	1	yes	characteristic
45	C₅²	C₃×C₁₃×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
46	C₃×C₈	C₂×C₅²×C₁₃	1	1	yes	characteristic
47	C₄×C₅	C₃×C₄×C₅×C₁₃	1	6	yes	—
48	C₁₆	C₃×C₅²×C₁₃	1	1	yes	characteristic, direct product factor, Sylow
49	C₃×C₅	C₅×C₁₃×C₁₆	1	6	yes	direct product factor
50	C₁₃	C₃×C₅²×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
51	C₃×C₄	C₄×C₅²×C₁₃	1	1	yes	characteristic
52	C₂×C₅	C₃×C₅×C₈×C₁₃	1	6	yes	—
53	C₈	C₂×C₃×C₅²×C₁₃	1	1	yes	characteristic
54	C₂×C₃	C₅²×C₈×C₁₃	1	1	yes	characteristic
55	C₅	C₃×C₅×C₁₃×C₁₆	1	6	yes	direct product factor
56	C₄	C₃×C₄×C₅²×C₁₃	1	1	yes	characteristic
57	C₃	C₅²×C₁₃×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
58	C₂	C₃×C₅²×C₈×C₁₃	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₈	C₂×C₃×C₅²×C₁₃
socle, prime radical	C₂×C₃×C₅²×C₁₃	C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅²×C₁₃×C₁₆	C₃	direct (N × H)
3. C₃×C₅×C₁₃×C₁₆	C₅	direct (N × H)
9. C₃×C₅²×C₁₆	C₁₃	direct (N × H)
10. C₅×C₁₃×C₁₆	C₃×C₅	direct (N × H)
11. C₃×C₅²×C₁₃	C₁₆	direct (N × H)
14. C₃×C₁₃×C₁₆	C₅²	direct (N × H)
17. C₅²×C₁₆	C₃×C₁₃	direct (N × H)
19. C₅²×C₁₃	C₃×C₁₆	direct (N × H)
23. C₃×C₅×C₁₆	C₅×C₁₃	direct (N × H)
24. C₁₃×C₁₆	C₃×C₅²	direct (N × H)
26. C₃×C₅×C₁₃	C₅×C₁₆	direct (N × H)
33. C₅×C₁₆	C₃×C₅×C₁₃	direct (N × H)
35. C₃×C₅²	C₁₃×C₁₆	direct (N × H)
36. C₅×C₁₃	C₃×C₅×C₁₆	direct (N × H)
40. C₃×C₁₆	C₅²×C₁₃	direct (N × H)
42. C₃×C₁₃	C₅²×C₁₆	direct (N × H)
45. C₅²	C₃×C₁₃×C₁₆	direct (N × H)
48. C₁₆	C₃×C₅²×C₁₃	direct (N × H)
49. C₃×C₅	C₅×C₁₃×C₁₆	direct (N × H)
50. C₁₃	C₃×C₅²×C₁₆	direct (N × H)
55. C₅	C₃×C₅×C₁₃×C₁₆	direct (N × H)
57. C₃	C₅²×C₁₃×C₁₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅²×C₈×C₁₃	2
C₅²×C₁₃×C₁₆	3
C₃×C₅×C₁₃×C₁₆	5
C₃×C₅²×C₁₆	13

Subgroups — C3×C52×C13×C16 #72193627

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅²×C₁₃×C₁₆ #72193627