Subgroups — C₃×C₃⋊AGL(1,𝔽₅₃) #37613664

Order 24804.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×D₅₃⋊C₁₃	C₂	1	1	yes	maximal, characteristic
2	C₃⋊AGL(1,𝔽₅₃)	C₃	1	1	yes	maximal, characteristic, direct product factor
3	C₃×AGL(1,𝔽₅₃)	—	3	1	no	maximal
4	C₃²×C₅₃⋊C₁₃	C₄	1	1	yes	characteristic, complemented, Hall
5	C₃×D₅₃⋊C₁₃	—	2	1	no	—
6	C₃×D₅₃⋊C₁₃	D₃	1	1	yes	characteristic
7	C₃×D₅₃⋊C₁₃	C₂×C₃	1	1	yes	characteristic
8	AGL(1,𝔽₅₃)	—	3	1	no	Hall
9	C₃×C₅₃⋊C₁₃	—	2	1	no	—
10	C₃×C₅₃⋊C₁₃	Dic₃	1	1	yes	characteristic, complemented
11	C₃×C₅₃⋊C₁₃	C₃×C₄	1	1	yes	characteristic, complemented
12	C₃×C₃⋊#6157	C₁₃	1	1	yes	maximal, characteristic, complemented, Hall
13	D₅₃⋊C₁₃	C₃×D₃	1	1	yes	characteristic
14	C₃²×D₅₃	C₂×C₁₃	1	1	yes	characteristic
15	C₅₃⋊C₁₃	C₃×Dic₃	1	1	yes	characteristic, complemented, Hall
16	C₃⋊#6157	C₃×C₁₃	1	1	yes	characteristic, complemented
17	C₃×#6157	—	3	1	no	—
18	C₃²×C₅₃	C₄×C₁₃	1	1	yes	characteristic, complemented, Hall
19	C₃×Dic₃×C₁₃	—	53	1	no	maximal, Hall
20	C₃×D₅₃	—	2	1	no	—
21	C₃×D₅₃	D₃×C₁₃	1	1	yes	characteristic
22	C₃×D₅₃	C₂×C₃×C₁₃	1	1	yes	characteristic
23	C₂×C₃²×C₁₃	—	53	1	no	—
24	#6157	—	3	1	no	Hall
25	C₃×C₅₃	—	2	1	no	—
26	C₃×C₅₃	Dic₃×C₁₃	1	1	yes	characteristic, complemented
27	C₃×C₅₃	C₃×C₄×C₁₃	1	1	yes	characteristic, complemented
28	C₃×C₄×C₁₃	—	159	1	no	—
29	Dic₃×C₁₃	—	53	1	no	—
30	C₃²×C₁₃	—	53	1	no	Hall
31	D₅₃	C₃×D₃×C₁₃	1	1	yes	characteristic
32	C₂×C₃×C₁₃	—	53	1	no	—
33	C₂×C₃×C₁₃	—	53	1	no	—
34	C₂×C₃×C₁₃	—	106	1	no	—
35	C₅₃	C₃×Dic₃×C₁₃	1	1	yes	characteristic, complemented, Sylow
36	C₄×C₁₃	—	159	1	no	Hall
37	C₃×C₁₃	—	53	1	no	—
38	C₃×C₁₃	—	53	1	no	—
39	C₃×C₁₃	—	106	1	no	—
40	C₃×Dic₃	—	689	1	no	Hall
41	C₂×C₁₃	—	53	1	no	—
42	C₂×C₃²	—	689	1	no	—
43	C₁₃	—	53	1	no	Sylow
44	C₃×C₄	—	2067	1	no	—
45	Dic₃	—	689	1	no	—
46	C₃²	AGL(1,𝔽₅₃)	1	1	yes	characteristic, complemented, Sylow
47	C₂×C₃	—	1378	1	no	—
48	C₂×C₃	—	689	1	no	—
49	C₂×C₃	—	689	1	no	—
50	C₄	—	2067	1	no	Sylow
51	C₃	—	2	1	no	—
52	C₃	C₃×AGL(1,𝔽₅₃)	1	1	yes	characteristic, complemented
53	C₃	C₃⋊AGL(1,𝔽₅₃)	1	1	yes	characteristic, direct product factor
54	C₂	—	689	1	no	—

	subgroup	G/H
center, hypercenter	C₃	C₃⋊AGL(1,𝔽₅₃)
commutator, hypocenter	C₃×C₅₃	C₃×C₄×C₁₃
Frattini	C₁	G
Fitting, socle	C₃²×C₅₃	C₄×C₁₃
prime radical	C₃²×D₅₃⋊C₁₃	C₂
cosocle	C₃×D₅₃	C₂×C₃×C₁₃
LCS	G ▷ C₃×C₅₃
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
2. C₃⋊AGL(1,𝔽₅₃)	C₃	direct (N × H)
4. C₃²×C₅₃⋊C₁₃	C₄	semidirect
10. C₃×C₅₃⋊C₁₃	Dic₃	semidirect
11. C₃×C₅₃⋊C₁₃	C₃×C₄	semidirect
12. C₃×C₃⋊#6157	C₁₃	semidirect
15. C₅₃⋊C₁₃	C₃×Dic₃	semidirect
16. C₃⋊#6157	C₃×C₁₃	semidirect
18. C₃²×C₅₃	C₄×C₁₃	semidirect
26. C₃×C₅₃	Dic₃×C₁₃	semidirect
27. C₃×C₅₃	C₃×C₄×C₁₃	semidirect
35. C₅₃	C₃×Dic₃×C₁₃	semidirect
46. C₃²	AGL(1,𝔽₅₃)	semidirect
52. C₃	C₃×AGL(1,𝔽₅₃)	semidirect
53. C₃	C₃⋊AGL(1,𝔽₅₃)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×D₅₃⋊C₁₃	2
C₃⋊AGL(1,𝔽₅₃)	3
C₃×AGL(1,𝔽₅₃)	3
C₃×C₃⋊#6157	13
C₃×Dic₃×C₁₃	53

Subgroups — C3×C3⋊AGL(1,𝔽53) #37613664

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₃⋊AGL(1,𝔽₅₃) #37613664