Subgroups — C₅₃×#7544 #27215056

Order 13568.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×M₁₂₈	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	—	2	1	no	—
20	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
21	C₈×C₅₃	C₃₂	1	2	yes	—
22	C₈×C₅₃	C₄⋊C₈	1	1	yes	characteristic
23	C₈×C₅₃	M₃₂	1	1	yes	characteristic
24	#7544	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂²×C₅₃	M₆₄	1	1	yes	characteristic
26	C₄×C₅₃	—	2	2	no	—
27	C₄×C₅₃	C₄⋊C₁₆	1	1	yes	characteristic
28	C₄×C₅₃	C₂×C₃₂	1	1	yes	characteristic
29	M₁₂₈	C₂×C₅₃	1	2	yes	—
30	C₄×C₃₂	C₂×C₅₃	1	1	yes	characteristic
31	C₂×C₅₃	—	2	1	no	—
32	C₂×C₅₃	C₄⋊C₃₂	1	1	yes	characteristic
33	C₂×C₃₂	C₄×C₅₃	1	1	yes	characteristic
34	C₂×C₃₂	C₂²×C₅₃	1	1	yes	characteristic
35	C₄×C₁₆	C₄×C₅₃	1	1	yes	characteristic
36	C₆₄	—	2	2	no	—
37	C₅₃	#7544	1	1	yes	characteristic, direct product factor, Sylow
38	C₂×C₁₆	C₈×C₅₃	1	1	yes	characteristic
39	C₂×C₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
40	C₄×C₈	C₈×C₅₃	1	1	yes	characteristic
41	C₃₂	—	2	1	no	—
42	C₃₂	D₄×C₅₃	1	1	yes	characteristic
43	C₃₂	Q₈×C₅₃	1	1	yes	characteristic
44	C₂×C₈	C₁₆×C₅₃	1	1	yes	characteristic
45	C₂×C₈	C₂×C₈×C₅₃	1	1	yes	characteristic
46	C₄²	C₁₆×C₅₃	1	1	yes	characteristic
47	C₁₆	—	2	1	no	—
48	C₁₆	C₄⋊C₄×C₅₃	1	1	yes	characteristic
49	C₁₆	M₁₆×C₅₃	1	1	yes	characteristic
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	C₂×C₁₆×C₅₃	1	1	yes	characteristic
52	C₈	C₃₂×C₅₃	1	2	yes	—
53	C₈	C₄⋊C₈×C₅₃	1	1	yes	characteristic
54	C₈	M₃₂×C₅₃	1	1	yes	characteristic
55	C₂²	C₅₃×M₆₄	1	1	yes	characteristic
56	C₄	—	2	2	no	—
57	C₄	C₅₃×C₄⋊C₁₆	1	1	yes	characteristic
58	C₄	C₂×C₃₂×C₅₃	1	1	yes	characteristic
59	C₂	—	2	1	no	—
60	C₂	C₅₃×C₄⋊C₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₃₂×C₅₃	D₄
commutator	C₄	C₂×C₃₂×C₅₃
Frattini, cosocle	C₂×C₃₂	C₂²×C₅₃
socle	C₂×C₅₃	C₄⋊C₃₂
prime radical	C₂²×C₅₃	M₆₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃₂×C₅₃ ◁ C₂×C₃₂×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
24. #7544	C₅₃	direct (N × H)
37. C₅₃	#7544	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×M₁₂₈	2
C₄×C₃₂×C₅₃	2
#7544	53

Subgroups — C53×#7544 #27215056

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7544 #27215056