Subgroups — C₅₃×#7511 #27215023

Order 13568.

74 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	C₅₃×#1397	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	C₄	1	4	yes	complemented
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	C₈	1	2	yes	—
11	C₃₂×C₅₃	C₂×C₄	1	2	yes	—
12	C₂×C₈×C₅₃	C₁₆	1	1	yes	characteristic
13	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
14	C₄²×C₅₃	C₁₆	1	1	yes	characteristic
15	C₁₆×C₅₃	C₁₆	1	2	yes	—
16	C₁₆×C₅₃	C₄²	1	1	yes	characteristic
17	C₁₆×C₅₃	C₂×C₈	1	1	yes	characteristic
18	C₂×C₄×C₅₃	—	2	1	no	—
19	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
20	C₈×C₅₃	—	2	1	no	—
21	C₈×C₅₃	C₄×C₈	1	1	yes	characteristic
22	C₈×C₅₃	C₂×C₁₆	1	1	yes	characteristic
23	#7511	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂²×C₅₃	M₆₄	1	1	yes	characteristic
25	C₄×C₅₃	—	4	1	no	—
26	C₄×C₅₃	C₄×C₁₆	1	1	yes	characteristic
27	C₄×C₅₃	M₆₄	1	1	yes	characteristic
28	M₁₂₈	C₂×C₅₃	1	2	yes	—
29	#1397	C₂×C₅₃	1	1	yes	characteristic
30	C₂×C₅₃	—	2	1	no	—
31	C₂×C₅₃	#1397	1	1	yes	characteristic
32	C₂×C₃₂	C₄×C₅₃	1	1	yes	characteristic
33	C₂×C₃₂	C₂²×C₅₃	1	1	yes	characteristic
34	C₄×C₁₆	C₄×C₅₃	1	1	yes	characteristic
35	C₆₄	C₄×C₅₃	1	4	yes	complemented
36	C₅₃	#7511	1	1	yes	characteristic, direct product factor, Sylow
37	C₂×C₁₆	C₈×C₅₃	1	1	yes	characteristic
38	C₂×C₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
39	C₄×C₈	C₈×C₅₃	1	1	yes	characteristic
40	C₃₂	C₈×C₅₃	1	2	yes	—
41	C₃₂	C₂×C₄×C₅₃	1	2	yes	—
42	C₂×C₈	C₁₆×C₅₃	1	1	yes	characteristic
43	C₂×C₈	C₂×C₈×C₅₃	1	1	yes	characteristic
44	C₄²	C₁₆×C₅₃	1	1	yes	characteristic
45	C₁₆	C₁₆×C₅₃	1	2	yes	—
46	C₁₆	C₄²×C₅₃	1	1	yes	characteristic
47	C₁₆	C₂×C₈×C₅₃	1	1	yes	characteristic
48	C₂×C₄	—	2	1	no	—
49	C₂×C₄	C₂×C₁₆×C₅₃	1	1	yes	characteristic
50	C₈	—	2	1	no	—
51	C₈	C₄×C₈×C₅₃	1	1	yes	characteristic
52	C₈	C₂×C₁₆×C₅₃	1	1	yes	characteristic
53	C₂²	C₅₃×M₆₄	1	1	yes	characteristic
54	C₄	—	4	1	no	—
55	C₄	C₄×C₁₆×C₅₃	1	1	yes	characteristic
56	C₄	C₅₃×M₆₄	1	1	yes	characteristic
57	C₂	—	2	1	no	—
58	C₂	C₅₃×#1397	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆×C₅₃	C₄²
commutator	C₄	C₄×C₁₆×C₅₃
Frattini, cosocle	C₂×C₃₂	C₂²×C₅₃
socle	C₂×C₅₃	#1397
prime radical	C₂²×C₅₃	M₆₄
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₁₆×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₅₃×C₆₄	C₄	semidirect
23. #7511	C₅₃	direct (N × H)
35. C₆₄	C₄×C₅₃	semidirect
36. C₅₃	#7511	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
C₅₃×#1397	2
#7511	53

Subgroups — C53×#7511 #27215023

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7511 #27215023