Subgroups — C₂×C₃×C₅₃×C₆₄ #72197913

Order 20352.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅₃×C₆₄	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×C₁₆×C₅₃	C₄	1	1	yes	characteristic
4	C₃×C₃₂×C₅₃	C₄	1	1	yes	characteristic
5	C₃×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
6	C₂×C₃×C₈×C₅₃	C₈	1	1	yes	characteristic
7	C₃×C₁₆×C₅₃	C₈	1	1	yes	characteristic
8	C₃×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
9	C₂×C₅₃×C₆₄	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂×C₃₂×C₅₃	C₂×C₃	1	1	yes	characteristic
11	C₅₃×C₆₄	C₂×C₃	1	2	yes	direct product factor
12	C₂×C₁₆×C₅₃	C₃×C₄	1	1	yes	characteristic
13	C₃₂×C₅₃	C₃×C₄	1	1	yes	characteristic
14	C₃₂×C₅₃	C₂²×C₃	1	1	yes	characteristic
15	C₂×C₃×C₄×C₅₃	C₁₆	1	1	yes	characteristic
16	C₃×C₈×C₅₃	C₁₆	1	1	yes	characteristic
17	C₃×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
18	C₂×C₈×C₅₃	C₃×C₈	1	1	yes	characteristic
19	C₁₆×C₅₃	C₃×C₈	1	1	yes	characteristic
20	C₁₆×C₅₃	C₂×C₃×C₄	1	1	yes	characteristic
21	C₂²×C₃×C₅₃	C₃₂	1	1	yes	characteristic
22	C₃×C₄×C₅₃	C₃₂	1	1	yes	characteristic
23	C₃×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
24	C₂×C₄×C₅₃	C₃×C₁₆	1	1	yes	characteristic
25	C₈×C₅₃	C₃×C₁₆	1	1	yes	characteristic
26	C₈×C₅₃	C₂×C₃×C₈	1	1	yes	characteristic
27	C₂×C₃×C₆₄	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Hall
28	C₂×C₃×C₅₃	C₆₄	1	2	yes	direct product factor
29	C₂×C₃×C₅₃	C₂×C₃₂	1	1	yes	characteristic
30	C₂²×C₅₃	C₃×C₃₂	1	1	yes	characteristic
31	C₄×C₅₃	C₃×C₃₂	1	1	yes	characteristic
32	C₄×C₅₃	C₂×C₃×C₁₆	1	1	yes	characteristic
33	C₂×C₃×C₃₂	C₂×C₅₃	1	1	yes	characteristic
34	C₃×C₆₄	C₂×C₅₃	1	2	yes	direct product factor
35	C₃×C₅₃	C₂×C₆₄	1	1	yes	characteristic, direct product factor, Hall
36	C₂×C₆₄	C₃×C₅₃	1	1	yes	characteristic, direct product factor, Sylow
37	C₂×C₅₃	C₃×C₆₄	1	2	yes	direct product factor
38	C₂×C₅₃	C₂×C₃×C₃₂	1	1	yes	characteristic
39	C₂×C₃×C₁₆	C₄×C₅₃	1	1	yes	characteristic
40	C₃×C₃₂	C₄×C₅₃	1	1	yes	characteristic
41	C₃×C₃₂	C₂²×C₅₃	1	1	yes	characteristic
42	C₂×C₃₂	C₂×C₃×C₅₃	1	1	yes	characteristic
43	C₆₄	C₂×C₃×C₅₃	1	2	yes	direct product factor
44	C₅₃	C₂×C₃×C₆₄	1	1	yes	characteristic, direct product factor, Sylow
45	C₂×C₃×C₈	C₈×C₅₃	1	1	yes	characteristic
46	C₃×C₁₆	C₈×C₅₃	1	1	yes	characteristic
47	C₃×C₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
48	C₂×C₁₆	C₃×C₄×C₅₃	1	1	yes	characteristic
49	C₃₂	C₃×C₄×C₅₃	1	1	yes	characteristic
50	C₃₂	C₂²×C₃×C₅₃	1	1	yes	characteristic
51	C₂×C₃×C₄	C₁₆×C₅₃	1	1	yes	characteristic
52	C₃×C₈	C₁₆×C₅₃	1	1	yes	characteristic
53	C₃×C₈	C₂×C₈×C₅₃	1	1	yes	characteristic
54	C₂×C₈	C₃×C₈×C₅₃	1	1	yes	characteristic
55	C₁₆	C₃×C₈×C₅₃	1	1	yes	characteristic
56	C₁₆	C₂×C₃×C₄×C₅₃	1	1	yes	characteristic
57	C₂²×C₃	C₃₂×C₅₃	1	1	yes	characteristic
58	C₃×C₄	C₃₂×C₅₃	1	1	yes	characteristic
59	C₃×C₄	C₂×C₁₆×C₅₃	1	1	yes	characteristic
60	C₂×C₄	C₃×C₁₆×C₅₃	1	1	yes	characteristic
61	C₈	C₃×C₁₆×C₅₃	1	1	yes	characteristic
62	C₈	C₂×C₃×C₈×C₅₃	1	1	yes	characteristic
63	C₂×C₃	C₅₃×C₆₄	1	2	yes	direct product factor
64	C₂×C₃	C₂×C₃₂×C₅₃	1	1	yes	characteristic
65	C₂²	C₃×C₃₂×C₅₃	1	1	yes	characteristic
66	C₄	C₃×C₃₂×C₅₃	1	1	yes	characteristic
67	C₄	C₂×C₃×C₁₆×C₅₃	1	1	yes	characteristic
68	C₃	C₂×C₅₃×C₆₄	1	1	yes	characteristic, direct product factor, Sylow
69	C₂	C₃×C₅₃×C₆₄	1	2	yes	direct product factor
70	C₂	C₂×C₃×C₃₂×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃₂	C₂²×C₃×C₅₃
socle, prime radical	C₂²×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. C₃×C₅₃×C₆₄	C₂	direct (N × H)
9. C₂×C₅₃×C₆₄	C₃	direct (N × H)
11. C₅₃×C₆₄	C₂×C₃	direct (N × H)
27. C₂×C₃×C₆₄	C₅₃	direct (N × H)
28. C₂×C₃×C₅₃	C₆₄	direct (N × H)
34. C₃×C₆₄	C₂×C₅₃	direct (N × H)
35. C₃×C₅₃	C₂×C₆₄	direct (N × H)
36. C₂×C₆₄	C₃×C₅₃	direct (N × H)
37. C₂×C₅₃	C₃×C₆₄	direct (N × H)
43. C₆₄	C₂×C₃×C₅₃	direct (N × H)
44. C₅₃	C₂×C₃×C₆₄	direct (N × H)
63. C₂×C₃	C₅₃×C₆₄	direct (N × H)
68. C₃	C₂×C₅₃×C₆₄	direct (N × H)
69. C₂	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
C₂×C₃×C₃₂×C₅₃	2
C₃×C₅₃×C₆₄	2
C₂×C₅₃×C₆₄	3
C₂×C₃×C₆₄	53

Subgroups — C2×C3×C53×C64 #72197913

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅₃×C₆₄ #72197913