Subgroups — C₃×C₅×C₈²×C₁₃ #72190822

Order 12480.

296 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₅×C₈×C₁₃	C₂	1	3	yes	maximal
2	C₅×C₈²×C₁₃	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₂×C₃×C₅×C₈×C₁₃	C₄	1	6	yes	—
4	C₃×C₄²×C₅×C₁₃	C₂²	1	1	yes	characteristic
5	C₃×C₈²×C₁₃	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₄×C₅×C₈×C₁₃	C₂×C₃	1	3	yes	—
7	C₂×C₃×C₄×C₅×C₁₃	C₂×C₄	1	3	yes	—
8	C₃×C₅×C₈×C₁₃	C₈	1	12	yes	direct product factor
9	C₃×C₄×C₈×C₁₃	C₂×C₅	1	3	yes	—
10	C₂×C₅×C₈×C₁₃	C₃×C₄	1	6	yes	—
11	C₄²×C₅×C₁₃	C₂²×C₃	1	1	yes	characteristic
12	C₃×C₅×C₈²	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₈²×C₁₃	C₃×C₅	1	1	yes	characteristic, direct product factor, Hall
14	C₂²×C₃×C₅×C₁₃	C₄²	1	1	yes	characteristic
15	C₃×C₄×C₅×C₁₃	C₂×C₈	1	6	yes	—
16	C₂×C₃×C₈×C₁₃	C₄×C₅	1	6	yes	—
17	C₃×C₄²×C₁₃	C₂²×C₅	1	1	yes	characteristic
18	C₂×C₄×C₅×C₁₃	C₂×C₃×C₄	1	3	yes	—
19	C₅×C₈×C₁₃	C₃×C₈	1	12	yes	direct product factor
20	C₃×C₄×C₅×C₈	C₂×C₁₃	1	3	yes	—
21	C₄×C₈×C₁₃	C₂×C₃×C₅	1	3	yes	—
22	C₂×C₃×C₅×C₁₃	C₄×C₈	1	3	yes	—
23	C₅×C₈²	C₃×C₁₃	1	1	yes	characteristic, direct product factor, Hall
24	C₂×C₃×C₄×C₁₃	C₂×C₄×C₅	1	3	yes	—
25	C₃×C₈×C₁₃	C₅×C₈	1	12	yes	direct product factor
26	C₂²×C₅×C₁₃	C₃×C₄²	1	1	yes	characteristic
27	C₄×C₅×C₁₃	C₂×C₃×C₈	1	6	yes	—
28	C₂×C₃×C₅×C₈	C₄×C₁₃	1	6	yes	—
29	C₃×C₄²×C₅	C₂²×C₁₃	1	1	yes	characteristic
30	C₂×C₈×C₁₃	C₃×C₄×C₅	1	6	yes	—
31	C₄²×C₁₃	C₂²×C₃×C₅	1	1	yes	characteristic
32	C₃×C₅×C₁₃	C₈²	1	1	yes	characteristic, direct product factor, Hall
33	C₃×C₈²	C₅×C₁₃	1	1	yes	characteristic, direct product factor, Hall
34	C₄×C₅×C₈	C₂×C₃×C₁₃	1	3	yes	—
35	C₂²×C₃×C₁₃	C₄²×C₅	1	1	yes	characteristic
36	C₃×C₄×C₁₃	C₂×C₅×C₈	1	6	yes	—
37	C₂×C₅×C₁₃	C₃×C₄×C₈	1	3	yes	—
38	C₂×C₃×C₄×C₅	C₂×C₄×C₁₃	1	3	yes	—
39	C₃×C₅×C₈	C₈×C₁₃	1	12	yes	direct product factor
40	C₂×C₄×C₁₃	C₂×C₃×C₄×C₅	1	3	yes	—
41	C₈×C₁₃	C₃×C₅×C₈	1	12	yes	direct product factor
42	C₃×C₄×C₈	C₂×C₅×C₁₃	1	3	yes	—
43	C₂×C₅×C₈	C₃×C₄×C₁₃	1	6	yes	—
44	C₄²×C₅	C₂²×C₃×C₁₃	1	1	yes	characteristic
45	C₂×C₃×C₁₃	C₄×C₅×C₈	1	3	yes	—
46	C₅×C₁₃	C₃×C₈²	1	1	yes	characteristic, direct product factor, Hall
47	C₈²	C₃×C₅×C₁₃	1	1	yes	characteristic, direct product factor, Sylow
48	C₂²×C₃×C₅	C₄²×C₁₃	1	1	yes	characteristic
49	C₃×C₄×C₅	C₂×C₈×C₁₃	1	6	yes	—
50	C₂²×C₁₃	C₃×C₄²×C₅	1	1	yes	characteristic
51	C₄×C₁₃	C₂×C₃×C₅×C₈	1	6	yes	—
52	C₂×C₃×C₈	C₄×C₅×C₁₃	1	6	yes	—
53	C₃×C₄²	C₂²×C₅×C₁₃	1	1	yes	characteristic
54	C₂×C₄×C₅	C₂×C₃×C₄×C₁₃	1	3	yes	—
55	C₅×C₈	C₃×C₈×C₁₃	1	12	yes	direct product factor
56	C₃×C₁₃	C₅×C₈²	1	1	yes	characteristic, direct product factor, Hall
57	C₄×C₈	C₂×C₃×C₅×C₁₃	1	3	yes	—
58	C₂×C₃×C₅	C₄×C₈×C₁₃	1	3	yes	—
59	C₂×C₁₃	C₃×C₄×C₅×C₈	1	3	yes	—
60	C₂×C₃×C₄	C₂×C₄×C₅×C₁₃	1	3	yes	—
61	C₃×C₈	C₅×C₈×C₁₃	1	12	yes	direct product factor
62	C₂²×C₅	C₃×C₄²×C₁₃	1	1	yes	characteristic
63	C₄×C₅	C₂×C₃×C₈×C₁₃	1	6	yes	—
64	C₂×C₈	C₃×C₄×C₅×C₁₃	1	6	yes	—
65	C₄²	C₂²×C₃×C₅×C₁₃	1	1	yes	characteristic
66	C₃×C₅	C₈²×C₁₃	1	1	yes	characteristic, direct product factor, Hall
67	C₁₃	C₃×C₅×C₈²	1	1	yes	characteristic, direct product factor, Sylow
68	C₂²×C₃	C₄²×C₅×C₁₃	1	1	yes	characteristic
69	C₃×C₄	C₂×C₅×C₈×C₁₃	1	6	yes	—
70	C₂×C₅	C₃×C₄×C₈×C₁₃	1	3	yes	—
71	C₂×C₄	C₂×C₃×C₄×C₅×C₁₃	1	3	yes	—
72	C₈	C₃×C₅×C₈×C₁₃	1	12	yes	direct product factor
73	C₂×C₃	C₄×C₅×C₈×C₁₃	1	3	yes	—
74	C₅	C₃×C₈²×C₁₃	1	1	yes	characteristic, direct product factor, Sylow
75	C₂²	C₃×C₄²×C₅×C₁₃	1	1	yes	characteristic
76	C₄	C₂×C₃×C₅×C₈×C₁₃	1	6	yes	—
77	C₃	C₅×C₈²×C₁₃	1	1	yes	characteristic, direct product factor, Sylow
78	C₂	C₃×C₄×C₅×C₈×C₁₃	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄²	C₂²×C₃×C₅×C₁₃
socle, prime radical	C₂²×C₃×C₅×C₁₃	C₄²
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×C₈²×C₁₃	C₃	direct (N × H)
5. C₃×C₈²×C₁₃	C₅	direct (N × H)
8. C₃×C₅×C₈×C₁₃	C₈	direct (N × H)
12. C₃×C₅×C₈²	C₁₃	direct (N × H)
13. C₈²×C₁₃	C₃×C₅	direct (N × H)
19. C₅×C₈×C₁₃	C₃×C₈	direct (N × H)
23. C₅×C₈²	C₃×C₁₃	direct (N × H)
25. C₃×C₈×C₁₃	C₅×C₈	direct (N × H)
32. C₃×C₅×C₁₃	C₈²	direct (N × H)
33. C₃×C₈²	C₅×C₁₃	direct (N × H)
39. C₃×C₅×C₈	C₈×C₁₃	direct (N × H)
41. C₈×C₁₃	C₃×C₅×C₈	direct (N × H)
46. C₅×C₁₃	C₃×C₈²	direct (N × H)
47. C₈²	C₃×C₅×C₁₃	direct (N × H)
55. C₅×C₈	C₃×C₈×C₁₃	direct (N × H)
56. C₃×C₁₃	C₅×C₈²	direct (N × H)
61. C₃×C₈	C₅×C₈×C₁₃	direct (N × H)
66. C₃×C₅	C₈²×C₁₃	direct (N × H)
67. C₁₃	C₃×C₅×C₈²	direct (N × H)
72. C₈	C₃×C₅×C₈×C₁₃	direct (N × H)
74. C₅	C₃×C₈²×C₁₃	direct (N × H)
77. C₃	C₅×C₈²×C₁₃	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×C₄×C₅×C₈×C₁₃	2
C₅×C₈²×C₁₃	3
C₃×C₈²×C₁₃	5
C₃×C₅×C₈²	13

Subgroups — C3×C5×C82×C13 #72190822

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×C₈²×C₁₃ #72190822