Subgroups — C₃×C₅³×C₆₄ #72201264

Order 24000.

896 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅³×C₃₂	C₂	1	1	yes	maximal, characteristic
2	C₅³×C₆₄	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₃×C₅³×C₁₆	C₄	1	1	yes	characteristic
4	C₃×C₅²×C₆₄	C₅	1	31	yes	maximal, direct product factor
5	C₅³×C₃₂	C₂×C₃	1	1	yes	characteristic
6	C₃×C₅³×C₈	C₈	1	1	yes	characteristic
7	C₃×C₅²×C₃₂	C₂×C₅	1	31	yes	—
8	C₅³×C₁₆	C₃×C₄	1	1	yes	characteristic
9	C₅²×C₆₄	C₃×C₅	1	31	yes	direct product factor
10	C₃×C₄×C₅³	C₁₆	1	1	yes	characteristic
11	C₃×C₅²×C₁₆	C₄×C₅	1	31	yes	—
12	C₅³×C₈	C₃×C₈	1	1	yes	characteristic
13	C₃×C₅×C₆₄	C₅²	1	31	yes	direct product factor
14	C₅²×C₃₂	C₂×C₃×C₅	1	31	yes	—
15	C₂×C₃×C₅³	C₃₂	1	1	yes	characteristic
16	C₃×C₅²×C₈	C₅×C₈	1	31	yes	—
17	C₄×C₅³	C₃×C₁₆	1	1	yes	characteristic
18	C₃×C₅×C₃₂	C₂×C₅²	1	31	yes	—
19	C₅²×C₁₆	C₃×C₄×C₅	1	31	yes	—
20	C₃×C₅³	C₆₄	1	1	yes	characteristic, direct product factor, Hall
21	C₅×C₆₄	C₃×C₅²	1	31	yes	direct product factor
22	C₃×C₄×C₅²	C₅×C₁₆	1	31	yes	—
23	C₂×C₅³	C₃×C₃₂	1	1	yes	characteristic
24	C₃×C₅×C₁₆	C₄×C₅²	1	31	yes	—
25	C₅²×C₈	C₃×C₅×C₈	1	31	yes	—
26	C₃×C₆₄	C₅³	1	1	yes	characteristic, direct product factor, Hall
27	C₅×C₃₂	C₂×C₃×C₅²	1	31	yes	—
28	C₂×C₃×C₅²	C₅×C₃₂	1	31	yes	—
29	C₅³	C₃×C₆₄	1	1	yes	characteristic, direct product factor, Sylow
30	C₃×C₅×C₈	C₅²×C₈	1	31	yes	—
31	C₄×C₅²	C₃×C₅×C₁₆	1	31	yes	—
32	C₃×C₃₂	C₂×C₅³	1	1	yes	characteristic
33	C₅×C₁₆	C₃×C₄×C₅²	1	31	yes	—
34	C₃×C₅²	C₅×C₆₄	1	31	yes	direct product factor
35	C₆₄	C₃×C₅³	1	1	yes	characteristic, direct product factor, Sylow
36	C₃×C₄×C₅	C₅²×C₁₆	1	31	yes	—
37	C₂×C₅²	C₃×C₅×C₃₂	1	31	yes	—
38	C₃×C₁₆	C₄×C₅³	1	1	yes	characteristic
39	C₅×C₈	C₃×C₅²×C₈	1	31	yes	—
40	C₃₂	C₂×C₃×C₅³	1	1	yes	characteristic
41	C₂×C₃×C₅	C₅²×C₃₂	1	31	yes	—
42	C₅²	C₃×C₅×C₆₄	1	31	yes	direct product factor
43	C₃×C₈	C₅³×C₈	1	1	yes	characteristic
44	C₄×C₅	C₃×C₅²×C₁₆	1	31	yes	—
45	C₁₆	C₃×C₄×C₅³	1	1	yes	characteristic
46	C₃×C₅	C₅²×C₆₄	1	31	yes	direct product factor
47	C₃×C₄	C₅³×C₁₆	1	1	yes	characteristic
48	C₂×C₅	C₃×C₅²×C₃₂	1	31	yes	—
49	C₈	C₃×C₅³×C₈	1	1	yes	characteristic
50	C₂×C₃	C₅³×C₃₂	1	1	yes	characteristic
51	C₅	C₃×C₅²×C₆₄	1	31	yes	direct product factor
52	C₄	C₃×C₅³×C₁₆	1	1	yes	characteristic
53	C₃	C₅³×C₆₄	1	1	yes	characteristic, direct product factor, Sylow
54	C₂	C₃×C₅³×C₃₂	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃₂	C₂×C₃×C₅³
socle, prime radical	C₂×C₃×C₅³	C₃₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅³×C₆₄	C₃	direct (N × H)
4. C₃×C₅²×C₆₄	C₅	direct (N × H)
9. C₅²×C₆₄	C₃×C₅	direct (N × H)
13. C₃×C₅×C₆₄	C₅²	direct (N × H)
20. C₃×C₅³	C₆₄	direct (N × H)
21. C₅×C₆₄	C₃×C₅²	direct (N × H)
26. C₃×C₆₄	C₅³	direct (N × H)
29. C₅³	C₃×C₆₄	direct (N × H)
34. C₃×C₅²	C₅×C₆₄	direct (N × H)
35. C₆₄	C₃×C₅³	direct (N × H)
42. C₅²	C₃×C₅×C₆₄	direct (N × H)
46. C₃×C₅	C₅²×C₆₄	direct (N × H)
51. C₅	C₃×C₅²×C₆₄	direct (N × H)
53. C₃	C₅³×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×C₅³×C₃₂	2
C₅³×C₆₄	3
C₃×C₅²×C₆₄	5

Subgroups — C3×C53×C64 #72201264

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅³×C₆₄ #72201264