Subgroups — C₅×C₆₄×C₈₁ #72203061

Order 25920.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₃₂×C₈₁	C₂	1	1	yes	maximal, characteristic
2	C₅×C₂₇×C₆₄	C₃	1	1	yes	maximal, characteristic
3	C₅×C₁₆×C₈₁	C₄	1	1	yes	characteristic
4	C₆₄×C₈₁	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₅×C₂₇×C₃₂	C₂×C₃	1	1	yes	characteristic
6	C₅×C₈×C₈₁	C₈	1	1	yes	characteristic
7	C₅×C₉×C₆₄	C₉	1	1	yes	characteristic
8	C₃₂×C₈₁	C₂×C₅	1	1	yes	characteristic
9	C₅×C₁₆×C₂₇	C₃×C₄	1	1	yes	characteristic
10	C₂₇×C₆₄	C₃×C₅	1	1	yes	characteristic
11	C₄×C₅×C₈₁	C₁₆	1	1	yes	characteristic
12	C₅×C₉×C₃₂	C₂×C₉	1	1	yes	characteristic
13	C₁₆×C₈₁	C₄×C₅	1	1	yes	characteristic
14	C₅×C₈×C₂₇	C₃×C₈	1	1	yes	characteristic
15	C₃×C₅×C₆₄	C₂₇	1	1	yes	characteristic
16	C₂₇×C₃₂	C₂×C₃×C₅	1	1	yes	characteristic
17	C₂×C₅×C₈₁	C₃₂	1	1	yes	characteristic
18	C₅×C₉×C₁₆	C₄×C₉	1	1	yes	characteristic
19	C₈×C₈₁	C₅×C₈	1	1	yes	characteristic
20	C₉×C₆₄	C₅×C₉	1	1	yes	characteristic
21	C₄×C₅×C₂₇	C₃×C₁₆	1	1	yes	characteristic
22	C₃×C₅×C₃₂	C₂×C₂₇	1	1	yes	characteristic
23	C₁₆×C₂₇	C₃×C₄×C₅	1	1	yes	characteristic
24	C₅×C₈₁	C₆₄	1	1	yes	characteristic, direct product factor, Hall
25	C₅×C₈×C₉	C₈×C₉	1	1	yes	characteristic
26	C₄×C₈₁	C₅×C₁₆	1	1	yes	characteristic
27	C₅×C₆₄	C₈₁	1	1	yes	characteristic, direct product factor, Hall
28	C₉×C₃₂	C₂×C₅×C₉	1	1	yes	characteristic
29	C₂×C₅×C₂₇	C₃×C₃₂	1	1	yes	characteristic
30	C₃×C₅×C₁₆	C₄×C₂₇	1	1	yes	characteristic
31	C₈×C₂₇	C₃×C₅×C₈	1	1	yes	characteristic
32	C₃×C₆₄	C₅×C₂₇	1	1	yes	characteristic
33	C₄×C₅×C₉	C₉×C₁₆	1	1	yes	characteristic
34	C₂×C₈₁	C₅×C₃₂	1	1	yes	characteristic
35	C₅×C₃₂	C₂×C₈₁	1	1	yes	characteristic
36	C₉×C₁₆	C₄×C₅×C₉	1	1	yes	characteristic
37	C₅×C₂₇	C₃×C₆₄	1	1	yes	characteristic
38	C₃×C₅×C₈	C₈×C₂₇	1	1	yes	characteristic
39	C₄×C₂₇	C₃×C₅×C₁₆	1	1	yes	characteristic
40	C₃×C₃₂	C₂×C₅×C₂₇	1	1	yes	characteristic
41	C₂×C₅×C₉	C₉×C₃₂	1	1	yes	characteristic
42	C₈₁	C₅×C₆₄	1	1	yes	characteristic, direct product factor, Sylow
43	C₅×C₁₆	C₄×C₈₁	1	1	yes	characteristic
44	C₈×C₉	C₅×C₈×C₉	1	1	yes	characteristic
45	C₆₄	C₅×C₈₁	1	1	yes	characteristic, direct product factor, Sylow
46	C₃×C₄×C₅	C₁₆×C₂₇	1	1	yes	characteristic
47	C₂×C₂₇	C₃×C₅×C₃₂	1	1	yes	characteristic
48	C₃×C₁₆	C₄×C₅×C₂₇	1	1	yes	characteristic
49	C₅×C₉	C₉×C₆₄	1	1	yes	characteristic
50	C₅×C₈	C₈×C₈₁	1	1	yes	characteristic
51	C₄×C₉	C₅×C₉×C₁₆	1	1	yes	characteristic
52	C₃₂	C₂×C₅×C₈₁	1	1	yes	characteristic
53	C₂×C₃×C₅	C₂₇×C₃₂	1	1	yes	characteristic
54	C₂₇	C₃×C₅×C₆₄	1	1	yes	characteristic
55	C₃×C₈	C₅×C₈×C₂₇	1	1	yes	characteristic
56	C₄×C₅	C₁₆×C₈₁	1	1	yes	characteristic
57	C₂×C₉	C₅×C₉×C₃₂	1	1	yes	characteristic
58	C₁₆	C₄×C₅×C₈₁	1	1	yes	characteristic
59	C₃×C₅	C₂₇×C₆₄	1	1	yes	characteristic
60	C₃×C₄	C₅×C₁₆×C₂₇	1	1	yes	characteristic
61	C₂×C₅	C₃₂×C₈₁	1	1	yes	characteristic
62	C₉	C₅×C₉×C₆₄	1	1	yes	characteristic
63	C₈	C₅×C₈×C₈₁	1	1	yes	characteristic
64	C₂×C₃	C₅×C₂₇×C₃₂	1	1	yes	characteristic
65	C₅	C₆₄×C₈₁	1	1	yes	characteristic, direct product factor, Sylow
66	C₄	C₅×C₁₆×C₈₁	1	1	yes	characteristic
67	C₃	C₅×C₂₇×C₆₄	1	1	yes	characteristic
68	C₂	C₅×C₃₂×C₈₁	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂₇×C₃₂	C₂×C₃×C₅
socle, prime radical	C₂×C₃×C₅	C₂₇×C₃₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₆₄×C₈₁	C₅	direct (N × H)
24. C₅×C₈₁	C₆₄	direct (N × H)
27. C₅×C₆₄	C₈₁	direct (N × H)
42. C₈₁	C₅×C₆₄	direct (N × H)
45. C₆₄	C₅×C₈₁	direct (N × H)
65. C₅	C₆₄×C₈₁	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₅×C₃₂×C₈₁	2
C₅×C₂₇×C₆₄	3
C₆₄×C₈₁	5

Subgroups — C5×C64×C81 #72203061

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₆₄×C₈₁ #72203061