Subgroups — C₅₃×#13737 #27221249

Order 13568.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2258	C₂	1	1	yes	maximal, characteristic
2	C₅₃×M₁₂₈	C₂	1	3	yes	maximal, complemented
3	C₂×C₅₃×C₆₄	C₂	1	3	yes	maximal, complemented
4	C₅₃×#504	C₄	1	1	yes	characteristic
5	C₅₃×M₆₄	C₄	1	3	yes	—
6	C₂×C₃₂×C₅₃	C₂²	1	3	yes	—
7	C₅₃×C₆₄	C₂²	1	3	yes	—
8	C₅₃×C₆₄	C₂²	1	1	yes	characteristic
9	#131×C₅₃	C₈	1	1	yes	characteristic
10	M₃₂×C₅₃	C₈	1	3	yes	—
11	C₂×C₁₆×C₅₃	C₂×C₄	1	3	yes	—
12	C₃₂×C₅₃	C₂×C₄	1	3	yes	—
13	C₃₂×C₅₃	C₂³	1	1	yes	characteristic
14	#41×C₅₃	C₁₆	1	1	yes	characteristic
15	M₁₆×C₅₃	C₁₆	1	3	yes	—
16	C₂×C₈×C₅₃	C₂×C₈	1	3	yes	—
17	C₁₆×C₅₃	C₂×C₈	1	3	yes	—
18	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
19	Q₈×C₅₃	C₃₂	1	1	yes	characteristic
20	D₄×C₅₃	C₃₂	1	3	yes	—
21	C₂×C₄×C₅₃	C₂×C₁₆	1	3	yes	—
22	C₈×C₅₃	C₂×C₁₆	1	3	yes	—
23	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
24	#13737	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂²×C₅₃	C₂×C₃₂	1	3	yes	—
26	C₄×C₅₃	C₂×C₃₂	1	3	yes	—
27	C₄×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
28	#2258	C₂×C₅₃	1	1	yes	characteristic
29	M₁₂₈	C₂×C₅₃	1	3	yes	complemented
30	C₂×C₆₄	C₂×C₅₃	1	3	yes	complemented
31	C₂×C₅₃	—	2	3	no	—
32	C₂×C₅₃	C₂²×C₃₂	1	1	yes	characteristic
33	#504	C₄×C₅₃	1	1	yes	characteristic
34	M₆₄	C₄×C₅₃	1	3	yes	—
35	C₂×C₃₂	C₂²×C₅₃	1	3	yes	—
36	C₆₄	C₂²×C₅₃	1	3	yes	—
37	C₆₄	C₂²×C₅₃	1	1	yes	characteristic
38	C₅₃	#13737	1	1	yes	characteristic, direct product factor, Sylow
39	#131	C₈×C₅₃	1	1	yes	characteristic
40	M₃₂	C₈×C₅₃	1	3	yes	—
41	C₂×C₁₆	C₂×C₄×C₅₃	1	3	yes	—
42	C₃₂	C₂×C₄×C₅₃	1	3	yes	—
43	C₃₂	C₂³×C₅₃	1	1	yes	characteristic
44	#41	C₁₆×C₅₃	1	1	yes	characteristic
45	M₁₆	C₁₆×C₅₃	1	3	yes	—
46	C₂×C₈	C₂×C₈×C₅₃	1	3	yes	—
47	C₁₆	C₂×C₈×C₅₃	1	3	yes	—
48	C₁₆	C₂²×C₄×C₅₃	1	1	yes	characteristic
49	Q₈	C₃₂×C₅₃	1	1	yes	characteristic
50	D₄	C₃₂×C₅₃	1	3	yes	—
51	C₂×C₄	C₂×C₁₆×C₅₃	1	3	yes	—
52	C₈	C₂×C₁₆×C₅₃	1	3	yes	—
53	C₈	C₂²×C₈×C₅₃	1	1	yes	characteristic
54	C₂²	C₂×C₃₂×C₅₃	1	3	yes	—
55	C₄	C₂×C₃₂×C₅₃	1	3	yes	—
56	C₄	C₂²×C₁₆×C₅₃	1	1	yes	characteristic
57	C₂	—	2	3	no	—
58	C₂	C₂²×C₃₂×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₅₃×C₆₄	C₂²
commutator	C₂	C₂²×C₃₂×C₅₃
Frattini, cosocle	C₃₂	C₂³×C₅₃
socle	C₂×C₅₃	C₂²×C₃₂
prime radical	#41×C₅₃	C₁₆
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₅₃×C₆₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×M₁₂₈	C₂	semidirect
3. C₂×C₅₃×C₆₄	C₂	semidirect
24. #13737	C₅₃	direct (N × H)
29. M₁₂₈	C₂×C₅₃	semidirect
30. C₂×C₆₄	C₂×C₅₃	semidirect
38. C₅₃	#13737	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×#2258	2
C₅₃×M₁₂₈	2
C₂×C₅₃×C₆₄	2
#13737	53

Subgroups — C53×#13737 #27221249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#13737 #27221249