Subgroups — C₃×C₅×C₁₃×C₆₄ #72190818

Order 12480.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×C₁₃×C₃₂	C₂	1	1	yes	maximal, characteristic
2	C₅×C₁₃×C₆₄	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₃×C₅×C₁₃×C₁₆	C₄	1	1	yes	characteristic
4	C₃×C₁₃×C₆₄	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₅×C₁₃×C₃₂	C₂×C₃	1	1	yes	characteristic
6	C₃×C₅×C₈×C₁₃	C₈	1	1	yes	characteristic
7	C₃×C₁₃×C₃₂	C₂×C₅	1	1	yes	characteristic
8	C₅×C₁₃×C₁₆	C₃×C₄	1	1	yes	characteristic
9	C₃×C₅×C₆₄	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₁₃×C₆₄	C₃×C₅	1	1	yes	characteristic, direct product factor, Hall
11	C₃×C₄×C₅×C₁₃	C₁₆	1	1	yes	characteristic
12	C₃×C₁₃×C₁₆	C₄×C₅	1	1	yes	characteristic
13	C₅×C₈×C₁₃	C₃×C₈	1	1	yes	characteristic
14	C₃×C₅×C₃₂	C₂×C₁₃	1	1	yes	characteristic
15	C₁₃×C₃₂	C₂×C₃×C₅	1	1	yes	characteristic
16	C₂×C₃×C₅×C₁₃	C₃₂	1	1	yes	characteristic
17	C₅×C₆₄	C₃×C₁₃	1	1	yes	characteristic, direct product factor, Hall
18	C₃×C₈×C₁₃	C₅×C₈	1	1	yes	characteristic
19	C₄×C₅×C₁₃	C₃×C₁₆	1	1	yes	characteristic
20	C₃×C₅×C₁₆	C₄×C₁₃	1	1	yes	characteristic
21	C₁₃×C₁₆	C₃×C₄×C₅	1	1	yes	characteristic
22	C₃×C₅×C₁₃	C₆₄	1	1	yes	characteristic, direct product factor, Hall
23	C₃×C₆₄	C₅×C₁₃	1	1	yes	characteristic, direct product factor, Hall
24	C₅×C₃₂	C₂×C₃×C₁₃	1	1	yes	characteristic
25	C₃×C₄×C₁₃	C₅×C₁₆	1	1	yes	characteristic
26	C₂×C₅×C₁₃	C₃×C₃₂	1	1	yes	characteristic
27	C₃×C₅×C₈	C₈×C₁₃	1	1	yes	characteristic
28	C₈×C₁₃	C₃×C₅×C₈	1	1	yes	characteristic
29	C₃×C₃₂	C₂×C₅×C₁₃	1	1	yes	characteristic
30	C₅×C₁₆	C₃×C₄×C₁₃	1	1	yes	characteristic
31	C₂×C₃×C₁₃	C₅×C₃₂	1	1	yes	characteristic
32	C₅×C₁₃	C₃×C₆₄	1	1	yes	characteristic, direct product factor, Hall
33	C₆₄	C₃×C₅×C₁₃	1	1	yes	characteristic, direct product factor, Sylow
34	C₃×C₄×C₅	C₁₃×C₁₆	1	1	yes	characteristic
35	C₄×C₁₃	C₃×C₅×C₁₆	1	1	yes	characteristic
36	C₃×C₁₆	C₄×C₅×C₁₃	1	1	yes	characteristic
37	C₅×C₈	C₃×C₈×C₁₃	1	1	yes	characteristic
38	C₃×C₁₃	C₅×C₆₄	1	1	yes	characteristic, direct product factor, Hall
39	C₃₂	C₂×C₃×C₅×C₁₃	1	1	yes	characteristic
40	C₂×C₃×C₅	C₁₃×C₃₂	1	1	yes	characteristic
41	C₂×C₁₃	C₃×C₅×C₃₂	1	1	yes	characteristic
42	C₃×C₈	C₅×C₈×C₁₃	1	1	yes	characteristic
43	C₄×C₅	C₃×C₁₃×C₁₆	1	1	yes	characteristic
44	C₁₆	C₃×C₄×C₅×C₁₃	1	1	yes	characteristic
45	C₃×C₅	C₁₃×C₆₄	1	1	yes	characteristic, direct product factor, Hall
46	C₁₃	C₃×C₅×C₆₄	1	1	yes	characteristic, direct product factor, Sylow
47	C₃×C₄	C₅×C₁₃×C₁₆	1	1	yes	characteristic
48	C₂×C₅	C₃×C₁₃×C₃₂	1	1	yes	characteristic
49	C₈	C₃×C₅×C₈×C₁₃	1	1	yes	characteristic
50	C₂×C₃	C₅×C₁₃×C₃₂	1	1	yes	characteristic
51	C₅	C₃×C₁₃×C₆₄	1	1	yes	characteristic, direct product factor, Sylow
52	C₄	C₃×C₅×C₁₃×C₁₆	1	1	yes	characteristic
53	C₃	C₅×C₁₃×C₆₄	1	1	yes	characteristic, direct product factor, Sylow
54	C₂	C₃×C₅×C₁₃×C₃₂	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃₂	C₂×C₃×C₅×C₁₃
socle, prime radical	C₂×C₃×C₅×C₁₃	C₃₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×C₁₃×C₆₄	C₃	direct (N × H)
4. C₃×C₁₃×C₆₄	C₅	direct (N × H)
9. C₃×C₅×C₆₄	C₁₃	direct (N × H)
10. C₁₃×C₆₄	C₃×C₅	direct (N × H)
17. C₅×C₆₄	C₃×C₁₃	direct (N × H)
22. C₃×C₅×C₁₃	C₆₄	direct (N × H)
23. C₃×C₆₄	C₅×C₁₃	direct (N × H)
32. C₅×C₁₃	C₃×C₆₄	direct (N × H)
33. C₆₄	C₃×C₅×C₁₃	direct (N × H)
38. C₃×C₁₃	C₅×C₆₄	direct (N × H)
45. C₃×C₅	C₁₃×C₆₄	direct (N × H)
46. C₁₃	C₃×C₅×C₆₄	direct (N × H)
51. C₅	C₃×C₁₃×C₆₄	direct (N × H)
53. C₃	C₅×C₁₃×C₆₄	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×C₅×C₁₃×C₃₂	2
C₅×C₁₃×C₆₄	3
C₃×C₁₃×C₆₄	5
C₃×C₅×C₆₄	13

Subgroups — C3×C5×C13×C64 #72190818

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×C₁₃×C₆₄ #72190818