Subgroups — C₅²×C₉×C₆₄ #72192539

Order 14400.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×C₉×C₃₂	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅²×C₆₄	C₃	1	1	yes	maximal, characteristic
3	C₅²×C₉×C₁₆	C₄	1	1	yes	characteristic
4	C₅×C₉×C₆₄	C₅	1	6	yes	maximal, direct product factor
5	C₃×C₅²×C₃₂	C₂×C₃	1	1	yes	characteristic
6	C₅²×C₈×C₉	C₈	1	1	yes	characteristic
7	C₅²×C₆₄	C₉	1	1	yes	characteristic, direct product factor, Hall
8	C₅×C₉×C₃₂	C₂×C₅	1	6	yes	—
9	C₃×C₅²×C₁₆	C₃×C₄	1	1	yes	characteristic
10	C₃×C₅×C₆₄	C₃×C₅	1	6	yes	—
11	C₄×C₅²×C₉	C₁₆	1	1	yes	characteristic
12	C₅²×C₃₂	C₂×C₉	1	1	yes	characteristic
13	C₅×C₉×C₁₆	C₄×C₅	1	6	yes	—
14	C₃×C₅²×C₈	C₃×C₈	1	1	yes	characteristic
15	C₉×C₆₄	C₅²	1	1	yes	characteristic, direct product factor, Hall
16	C₃×C₅×C₃₂	C₂×C₃×C₅	1	6	yes	—
17	C₂×C₅²×C₉	C₃₂	1	1	yes	characteristic
18	C₅²×C₁₆	C₄×C₉	1	1	yes	characteristic
19	C₅×C₈×C₉	C₅×C₈	1	6	yes	—
20	C₅×C₆₄	C₅×C₉	1	6	yes	direct product factor
21	C₃×C₄×C₅²	C₃×C₁₆	1	1	yes	characteristic
22	C₉×C₃₂	C₂×C₅²	1	1	yes	characteristic
23	C₃×C₅×C₁₆	C₃×C₄×C₅	1	6	yes	—
24	C₅²×C₉	C₆₄	1	1	yes	characteristic, direct product factor, Hall
25	C₅²×C₈	C₈×C₉	1	1	yes	characteristic
26	C₃×C₆₄	C₃×C₅²	1	1	yes	characteristic
27	C₄×C₅×C₉	C₅×C₁₆	1	6	yes	—
28	C₅×C₃₂	C₂×C₅×C₉	1	6	yes	—
29	C₂×C₃×C₅²	C₃×C₃₂	1	1	yes	characteristic
30	C₉×C₁₆	C₄×C₅²	1	1	yes	characteristic
31	C₃×C₅×C₈	C₃×C₅×C₈	1	6	yes	—
32	C₄×C₅²	C₉×C₁₆	1	1	yes	characteristic
33	C₃×C₃₂	C₂×C₃×C₅²	1	1	yes	characteristic
34	C₂×C₅×C₉	C₅×C₃₂	1	6	yes	—
35	C₅×C₁₆	C₄×C₅×C₉	1	6	yes	—
36	C₃×C₅²	C₃×C₆₄	1	1	yes	characteristic
37	C₈×C₉	C₅²×C₈	1	1	yes	characteristic
38	C₆₄	C₅²×C₉	1	1	yes	characteristic, direct product factor, Sylow
39	C₃×C₄×C₅	C₃×C₅×C₁₆	1	6	yes	—
40	C₂×C₅²	C₉×C₃₂	1	1	yes	characteristic
41	C₃×C₁₆	C₃×C₄×C₅²	1	1	yes	characteristic
42	C₅×C₉	C₅×C₆₄	1	6	yes	direct product factor
43	C₅×C₈	C₅×C₈×C₉	1	6	yes	—
44	C₄×C₉	C₅²×C₁₆	1	1	yes	characteristic
45	C₃₂	C₂×C₅²×C₉	1	1	yes	characteristic
46	C₂×C₃×C₅	C₃×C₅×C₃₂	1	6	yes	—
47	C₅²	C₉×C₆₄	1	1	yes	characteristic, direct product factor, Sylow
48	C₃×C₈	C₃×C₅²×C₈	1	1	yes	characteristic
49	C₄×C₅	C₅×C₉×C₁₆	1	6	yes	—
50	C₂×C₉	C₅²×C₃₂	1	1	yes	characteristic
51	C₁₆	C₄×C₅²×C₉	1	1	yes	characteristic
52	C₃×C₅	C₃×C₅×C₆₄	1	6	yes	—
53	C₃×C₄	C₃×C₅²×C₁₆	1	1	yes	characteristic
54	C₂×C₅	C₅×C₉×C₃₂	1	6	yes	—
55	C₉	C₅²×C₆₄	1	1	yes	characteristic, direct product factor, Sylow
56	C₈	C₅²×C₈×C₉	1	1	yes	characteristic
57	C₂×C₃	C₃×C₅²×C₃₂	1	1	yes	characteristic
58	C₅	C₅×C₉×C₆₄	1	6	yes	direct product factor
59	C₄	C₅²×C₉×C₁₆	1	1	yes	characteristic
60	C₃	C₃×C₅²×C₆₄	1	1	yes	characteristic
61	C₂	C₅²×C₉×C₃₂	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃×C₃₂	C₂×C₃×C₅²
socle, prime radical	C₂×C₃×C₅²	C₃×C₃₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅×C₉×C₆₄	C₅	direct (N × H)
7. C₅²×C₆₄	C₉	direct (N × H)
15. C₉×C₆₄	C₅²	direct (N × H)
20. C₅×C₆₄	C₅×C₉	direct (N × H)
24. C₅²×C₉	C₆₄	direct (N × H)
38. C₆₄	C₅²×C₉	direct (N × H)
42. C₅×C₉	C₅×C₆₄	direct (N × H)
47. C₅²	C₉×C₆₄	direct (N × H)
55. C₉	C₅²×C₆₄	direct (N × H)
58. C₅	C₅×C₉×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
C₅²×C₉×C₃₂	2
C₃×C₅²×C₆₄	3
C₅×C₉×C₆₄	5

Subgroups — C52×C9×C64 #72192539

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×C₉×C₆₄ #72192539