Subgroups — C₅₃×C₄⋊C₆₄ #27215055

Order 13568.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×C₆₄	C₂	1	2	yes	maximal
2	C₄×C₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₂×C₅₃	C₄	1	1	yes	characteristic
4	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
5	C₄×C₁₆×C₅₃	C₄	1	1	yes	characteristic
6	C₅₃×C₆₄	—	2	2	no	—
7	C₂×C₁₆×C₅₃	C₈	1	1	yes	characteristic
8	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
9	C₄×C₈×C₅₃	C₈	1	1	yes	characteristic
10	C₃₂×C₅₃	—	2	1	no	—
11	C₃₂×C₅₃	D₄	1	1	yes	characteristic
12	C₃₂×C₅₃	Q₈	1	1	yes	characteristic
13	C₂×C₈×C₅₃	C₁₆	1	1	yes	characteristic
14	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
15	C₄²×C₅₃	C₁₆	1	1	yes	characteristic
16	C₁₆×C₅₃	—	2	1	no	—
17	C₁₆×C₅₃	C₄⋊C₄	1	1	yes	characteristic
18	C₁₆×C₅₃	M₁₆	1	1	yes	characteristic
19	C₂×C₄×C₅₃	C₃₂	1	1	yes	characteristic
20	C₂×C₄×C₅₃	C₃₂	1	1	yes	characteristic
21	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
22	C₈×C₅₃	—	2	1	no	—
23	C₈×C₅₃	C₄⋊C₈	1	1	yes	characteristic
24	C₈×C₅₃	M₃₂	1	1	yes	characteristic
25	C₄⋊C₆₄	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂²×C₅₃	C₂×C₃₂	1	1	yes	characteristic
27	C₄×C₅₃	—	2	1	no	—
28	C₄×C₅₃	C₆₄	1	2	yes	complemented
29	C₄×C₅₃	C₄⋊C₁₆	1	1	yes	characteristic
30	C₄×C₅₃	M₆₄	1	1	yes	characteristic
31	C₂×C₆₄	C₂×C₅₃	1	2	yes	—
32	C₄×C₃₂	C₂×C₅₃	1	1	yes	characteristic
33	C₂×C₅₃	C₄⋊C₃₂	1	1	yes	characteristic
34	C₂×C₅₃	C₂×C₆₄	1	1	yes	characteristic
35	C₂×C₅₃	M₁₂₈	1	1	yes	characteristic
36	C₂×C₃₂	C₄×C₅₃	1	1	yes	characteristic
37	C₂×C₃₂	C₂²×C₅₃	1	1	yes	characteristic
38	C₄×C₁₆	C₄×C₅₃	1	1	yes	characteristic
39	C₆₄	—	2	2	no	—
40	C₅₃	C₄⋊C₆₄	1	1	yes	characteristic, direct product factor, Sylow
41	C₂×C₁₆	C₈×C₅₃	1	1	yes	characteristic
42	C₂×C₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
43	C₄×C₈	C₈×C₅₃	1	1	yes	characteristic
44	C₃₂	—	2	1	no	—
45	C₃₂	D₄×C₅₃	1	1	yes	characteristic
46	C₃₂	Q₈×C₅₃	1	1	yes	characteristic
47	C₂×C₈	C₁₆×C₅₃	1	1	yes	characteristic
48	C₂×C₈	C₂×C₈×C₅₃	1	1	yes	characteristic
49	C₄²	C₁₆×C₅₃	1	1	yes	characteristic
50	C₁₆	—	2	1	no	—
51	C₁₆	C₄⋊C₄×C₅₃	1	1	yes	characteristic
52	C₁₆	M₁₆×C₅₃	1	1	yes	characteristic
53	C₂×C₄	C₃₂×C₅₃	1	1	yes	characteristic
54	C₂×C₄	C₃₂×C₅₃	1	1	yes	characteristic
55	C₂×C₄	C₂×C₁₆×C₅₃	1	1	yes	characteristic
56	C₈	—	2	1	no	—
57	C₈	C₄⋊C₈×C₅₃	1	1	yes	characteristic
58	C₈	M₃₂×C₅₃	1	1	yes	characteristic
59	C₂²	C₂×C₃₂×C₅₃	1	1	yes	characteristic
60	C₄	—	2	1	no	—
61	C₄	C₅₃×C₆₄	1	2	yes	complemented
62	C₄	C₅₃×C₄⋊C₁₆	1	1	yes	characteristic
63	C₄	C₅₃×M₆₄	1	1	yes	characteristic
64	C₂	C₅₃×C₄⋊C₃₂	1	1	yes	characteristic
65	C₂	C₂×C₅₃×C₆₄	1	1	yes	characteristic
66	C₂	C₅₃×M₁₂₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃₂×C₅₃	C₂²
commutator	C₂	C₂×C₅₃×C₆₄
Frattini, cosocle	C₂×C₃₂	C₂²×C₅₃
socle, prime radical	C₂²×C₅₃	C₂×C₃₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₂×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
25. C₄⋊C₆₄	C₅₃	direct (N × H)
28. C₄×C₅₃	C₆₄	semidirect
40. C₅₃	C₄⋊C₆₄	direct (N × H)
61. C₄	C₅₃×C₆₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×C₆₄	2
C₄×C₃₂×C₅₃	2
C₄⋊C₆₄	53

Subgroups — C53×C4⋊C64 #27215055

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×C₄⋊C₆₄ #27215055