Subgroups — C₂²×C₅₃×C₆₄ #27221247

Order 13568.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×C₆₄	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
4	C₂×C₃₂×C₅₃	C₄	1	3	yes	—
5	C₂×C₃₂×C₅₃	C₂²	1	3	yes	—
6	C₅₃×C₆₄	C₂²	1	4	yes	direct product factor
7	C₂²×C₈×C₅₃	C₈	1	1	yes	characteristic
8	C₂×C₁₆×C₅₃	C₈	1	3	yes	—
9	C₂×C₁₆×C₅₃	C₂×C₄	1	3	yes	—
10	C₃₂×C₅₃	C₂×C₄	1	3	yes	—
11	C₃₂×C₅₃	C₂³	1	1	yes	characteristic
12	C₂²×C₄×C₅₃	C₁₆	1	1	yes	characteristic
13	C₂×C₈×C₅₃	C₁₆	1	3	yes	—
14	C₂×C₈×C₅₃	C₂×C₈	1	3	yes	—
15	C₁₆×C₅₃	C₂×C₈	1	3	yes	—
16	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
17	C₂³×C₅₃	C₃₂	1	1	yes	characteristic
18	C₂×C₄×C₅₃	C₃₂	1	3	yes	—
19	C₂×C₄×C₅₃	C₂×C₁₆	1	3	yes	—
20	C₈×C₅₃	C₂×C₁₆	1	3	yes	—
21	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
22	C₂²×C₆₄	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂²×C₅₃	C₆₄	1	4	yes	direct product factor
24	C₂²×C₅₃	C₂×C₃₂	1	3	yes	—
25	C₄×C₅₃	C₂×C₃₂	1	3	yes	—
26	C₄×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
27	C₂²×C₃₂	C₂×C₅₃	1	1	yes	characteristic
28	C₂×C₆₄	C₂×C₅₃	1	6	yes	direct product factor
29	C₂×C₅₃	C₂×C₆₄	1	6	yes	direct product factor
30	C₂×C₅₃	C₂²×C₃₂	1	1	yes	characteristic
31	C₂²×C₁₆	C₄×C₅₃	1	1	yes	characteristic
32	C₂×C₃₂	C₄×C₅₃	1	3	yes	—
33	C₂×C₃₂	C₂²×C₅₃	1	3	yes	—
34	C₆₄	C₂²×C₅₃	1	4	yes	direct product factor
35	C₅₃	C₂²×C₆₄	1	1	yes	characteristic, direct product factor, Sylow
36	C₂²×C₈	C₈×C₅₃	1	1	yes	characteristic
37	C₂×C₁₆	C₈×C₅₃	1	3	yes	—
38	C₂×C₁₆	C₂×C₄×C₅₃	1	3	yes	—
39	C₃₂	C₂×C₄×C₅₃	1	3	yes	—
40	C₃₂	C₂³×C₅₃	1	1	yes	characteristic
41	C₂²×C₄	C₁₆×C₅₃	1	1	yes	characteristic
42	C₂×C₈	C₁₆×C₅₃	1	3	yes	—
43	C₂×C₈	C₂×C₈×C₅₃	1	3	yes	—
44	C₁₆	C₂×C₈×C₅₃	1	3	yes	—
45	C₁₆	C₂²×C₄×C₅₃	1	1	yes	characteristic
46	C₂³	C₃₂×C₅₃	1	1	yes	characteristic
47	C₂×C₄	C₃₂×C₅₃	1	3	yes	—
48	C₂×C₄	C₂×C₁₆×C₅₃	1	3	yes	—
49	C₈	C₂×C₁₆×C₅₃	1	3	yes	—
50	C₈	C₂²×C₈×C₅₃	1	1	yes	characteristic
51	C₂²	C₅₃×C₆₄	1	4	yes	direct product factor
52	C₂²	C₂×C₃₂×C₅₃	1	3	yes	—
53	C₄	C₂×C₃₂×C₅₃	1	3	yes	—
54	C₄	C₂²×C₁₆×C₅₃	1	1	yes	characteristic
55	C₂	C₂×C₅₃×C₆₄	1	6	yes	direct product factor
56	C₂	C₂²×C₃₂×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃₂	C₂³×C₅₃
socle, prime radical	C₂³×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. C₂×C₅₃×C₆₄	C₂	direct (N × H)
6. C₅₃×C₆₄	C₂²	direct (N × H)
22. C₂²×C₆₄	C₅₃	direct (N × H)
23. C₂²×C₅₃	C₆₄	direct (N × H)
28. C₂×C₆₄	C₂×C₅₃	direct (N × H)
29. C₂×C₅₃	C₂×C₆₄	direct (N × H)
34. C₆₄	C₂²×C₅₃	direct (N × H)
35. C₅₃	C₂²×C₆₄	direct (N × H)
51. C₂²	C₅₃×C₆₄	direct (N × H)
55. C₂	C₂×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
C₂²×C₃₂×C₅₃	2
C₂×C₅₃×C₆₄	2
C₂²×C₆₄	53

Subgroups — C22×C53×C64 #27221247

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅₃×C₆₄ #27221247