Subgroups — C₃²×C₅²×C₆₄ #72192541

Order 14400.

336 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃₂	C₂×C₃²×C₅²
socle, prime radical	C₂×C₃²×C₅²	C₃₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₅²×C₆₄	C₃	direct (N × H)
4. C₃²×C₅×C₆₄	C₅	direct (N × H)
7. C₅²×C₆₄	C₃²	direct (N × H)
10. C₃×C₅×C₆₄	C₃×C₅	direct (N × H)
15. C₃²×C₆₄	C₅²	direct (N × H)
20. C₅×C₆₄	C₃²×C₅	direct (N × H)
24. C₃²×C₅²	C₆₄	direct (N × H)
26. C₃×C₆₄	C₃×C₅²	direct (N × H)
36. C₃×C₅²	C₃×C₆₄	direct (N × H)
38. C₆₄	C₃²×C₅²	direct (N × H)
42. C₃²×C₅	C₅×C₆₄	direct (N × H)
47. C₅²	C₃²×C₆₄	direct (N × H)
52. C₃×C₅	C₃×C₅×C₆₄	direct (N × H)
55. C₃²	C₅²×C₆₄	direct (N × H)
58. C₅	C₃²×C₅×C₆₄	direct (N × H)
60. C₃	C₃×C₅²×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
C₃²×C₅²×C₃₂	2
C₃×C₅²×C₆₄	3
C₃²×C₅×C₆₄	5

Subgroups — C32×C52×C64 #72192541

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₅²×C₆₄ #72192541