Subgroups — C₅₃×C₃⋊(Dic₁₃⋊C₃) #37613643

Order 24804.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₁₃⋊C₃×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×Dic₁₃⋊C₃	—	3	1	no	maximal
3	C₅₃×Dic₃₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₁₃⋊C₃×C₅₃	C₄	1	1	yes	characteristic, complemented, Hall
5	C₂×C₁₃⋊C₃×C₅₃	—	2	1	no	—
6	C₂×C₁₃⋊C₃×C₅₃	D₃	1	1	yes	characteristic
7	C₂×C₃×C₁₃×C₅₃	C₂×C₃	1	1	yes	characteristic
8	Dic₁₃×C₅₃	—	3	1	no	Hall
9	C₃×C₁₃×C₅₃	C₃×C₄	1	1	yes	characteristic, complemented
10	C₁₃⋊C₃×C₅₃	—	2	1	no	—
11	C₁₃⋊C₃×C₅₃	Dic₃	1	1	yes	characteristic, complemented
12	C₃×Dic₃×C₅₃	—	13	1	no	maximal, Hall
13	C₂×C₁₃×C₅₃	C₃×D₃	1	1	yes	characteristic
14	C₂×C₃²×C₅₃	—	13	1	no	—
15	C₁₃×C₅₃	C₃×Dic₃	1	1	yes	characteristic, complemented, Hall
16	C₃×C₄×C₅₃	—	39	1	no	—
17	Dic₃×C₅₃	—	13	1	no	—
18	C₃²×C₅₃	—	13	1	no	Hall
19	C₃⋊(Dic₁₃⋊C₃)	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₂×C₃×C₅₃	—	13	1	no	—
21	C₂×C₃×C₅₃	—	26	1	no	—
22	C₂×C₃×C₅₃	D₁₃⋊C₃	1	1	yes	characteristic
23	C₂×C₃×C₁₃⋊C₃	C₂×C₅₃	1	1	yes	characteristic
24	C₄×C₅₃	—	39	1	no	Hall
25	C₃×C₅₃	—	13	1	no	—
26	C₃×C₅₃	—	26	1	no	—
27	C₃×C₅₃	Dic₁₃⋊C₃	1	1	yes	characteristic, complemented
28	Dic₃₉	C₃×C₅₃	1	1	yes	characteristic, complemented
29	Dic₁₃⋊C₃	—	3	1	no	—
30	C₃×C₁₃⋊C₃	C₄×C₅₃	1	1	yes	characteristic, complemented, Hall
31	C₂×C₅₃	C₃⋊(D₁₃⋊C₃)	1	1	yes	characteristic
32	C₂×C₃×C₁₃	C₂×C₃×C₅₃	1	1	yes	characteristic
33	C₂×C₁₃⋊C₃	—	2	1	no	—
34	C₂×C₁₃⋊C₃	D₃×C₅₃	1	1	yes	characteristic
35	C₅₃	C₃⋊(Dic₁₃⋊C₃)	1	1	yes	characteristic, direct product factor, Sylow
36	Dic₁₃	—	3	1	no	Hall
37	C₃×C₁₃	C₃×C₄×C₅₃	1	1	yes	characteristic, complemented
38	C₁₃⋊C₃	—	2	1	no	—
39	C₁₃⋊C₃	Dic₃×C₅₃	1	1	yes	characteristic, complemented
40	C₃×Dic₃	—	13	1	no	Hall
41	C₂×C₁₃	C₃×D₃×C₅₃	1	1	yes	characteristic
42	C₂×C₃²	—	13	1	no	—
43	C₁₃	C₃×Dic₃×C₅₃	1	1	yes	characteristic, complemented, Sylow
44	C₃×C₄	—	39	1	no	—
45	Dic₃	—	13	1	no	—
46	C₃²	—	13	1	no	Sylow
47	C₂×C₃	—	13	1	no	—
48	C₂×C₃	—	26	1	no	—
49	C₂×C₃	C₅₃×D₁₃⋊C₃	1	1	yes	characteristic
50	C₄	—	39	1	no	Sylow
51	C₃	—	13	1	no	—
52	C₃	—	26	1	no	—
53	C₃	C₅₃×Dic₁₃⋊C₃	1	1	yes	characteristic, complemented
54	C₂	C₅₃×C₃⋊(D₁₃⋊C₃)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₅₃	C₃⋊(D₁₃⋊C₃)
commutator, hypocenter	C₃×C₁₃	C₃×C₄×C₅₃
Frattini	C₂	C₅₃×C₃⋊(D₁₃⋊C₃)
Fitting, socle	C₂×C₃×C₁₃×C₅₃	C₂×C₃
prime radical	C₂×C₃×C₁₃⋊C₃×C₅₃	C₂
cosocle	C₂×C₃×C₁₃	C₂×C₃×C₅₃
LCS	G ▷ C₃×C₁₃
UCS	C₁ ◁ C₂×C₅₃

Complements

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

N	H ≅ G/N	type
3. C₅₃×Dic₃₉	C₃	semidirect
4. C₃×C₁₃⋊C₃×C₅₃	C₄	semidirect
9. C₃×C₁₃×C₅₃	C₃×C₄	semidirect
11. C₁₃⋊C₃×C₅₃	Dic₃	semidirect
15. C₁₃×C₅₃	C₃×Dic₃	semidirect
19. C₃⋊(Dic₁₃⋊C₃)	C₅₃	direct (N × H)
27. C₃×C₅₃	Dic₁₃⋊C₃	semidirect
28. Dic₃₉	C₃×C₅₃	semidirect
30. C₃×C₁₃⋊C₃	C₄×C₅₃	semidirect
35. C₅₃	C₃⋊(Dic₁₃⋊C₃)	direct (N × H)
37. C₃×C₁₃	C₃×C₄×C₅₃	semidirect
39. C₁₃⋊C₃	Dic₃×C₅₃	semidirect
43. C₁₃	C₃×Dic₃×C₅₃	semidirect
53. C₃	C₅₃×Dic₁₃⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₁₃⋊C₃×C₅₃	2
C₅₃×Dic₁₃⋊C₃	3
C₅₃×Dic₃₉	3
C₃×Dic₃×C₅₃	13
C₃⋊(Dic₁₃⋊C₃)	53

Subgroups — C53×C3⋊(Dic13⋊C3) #37613643

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×C₃⋊(Dic₁₃⋊C₃) #37613643