Subgroups — C₁₃×C₃⋊D₃×D₅₃ #37613745

Order 24804.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₁₃×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×C₃⋊D₃×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×C₃⋊D₁₅₉	C₂	1	1	yes	maximal, characteristic, complemented
4	D₃×C₁₃×D₅₃	—	3	4	no	maximal
5	C₃²×C₁₃×C₅₃	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₁₃×D₁₅₉	—	3	4	no	—
7	D₃×C₁₃×C₅₃	—	3	4	no	—
8	C₃×C₁₃×D₅₃	D₃	1	4	yes	complemented
9	C₂×C₁₃×D₅₃	—	9	1	no	Hall
10	C₃×C₁₃×C₅₃	C₂×D₃	1	4	yes	complemented
11	C₃⋊D₃×D₅₃	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₂×C₁₃×C₅₃	—	9	1	no	—
13	C₁₃×D₅₃	—	9	1	no	—
14	C₁₃×D₅₃	C₃⋊D₃	1	1	yes	characteristic, direct product factor
15	C₃⋊D₁₅₉	C₂×C₁₃	1	1	yes	characteristic, complemented
16	C₃⋊D₃×C₅₃	C₂×C₁₃	1	1	yes	characteristic, complemented
17	C₃²×D₅₃	C₂×C₁₃	1	1	yes	characteristic, complemented
18	C₁₃×C₅₃	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented, Hall
19	D₃×D₅₃	—	3	4	no	—
20	C₃²×C₅₃	C₂²×C₁₃	1	1	yes	characteristic, complemented, Hall
21	C₂×C₁₃×C₃⋊D₃	—	53	1	no	maximal, Hall
22	D₁₅₉	—	3	4	no	—
23	C₃×D₅₃	D₃×C₁₃	1	4	yes	complemented
24	D₃×C₅₃	—	3	4	no	—
25	C₂×C₃²×C₁₃	—	53	1	no	—
26	C₁₃×C₃⋊D₃	—	53	1	no	—
27	C₁₃×C₃⋊D₃	D₅₃	1	1	yes	characteristic, direct product factor
28	C₂×D₅₃	—	9	1	no	Hall
29	C₃×C₅₃	C₂×D₃×C₁₃	1	4	yes	complemented
30	C₂×D₃×C₁₃	—	159	4	no	—
31	C₃²×C₁₃	C₂×D₅₃	1	1	yes	characteristic, complemented, Hall
32	C₂×C₅₃	—	9	1	no	—
33	D₅₃	—	9	1	no	—
34	D₅₃	C₁₃×C₃⋊D₃	1	1	yes	characteristic, direct product factor
35	C₂×C₃×C₁₃	—	53	4	no	—
36	D₃×C₁₃	—	3	4	no	—
37	D₃×C₁₃	—	159	4	no	—
38	C₅₃	C₂×C₁₃×C₃⋊D₃	1	1	yes	characteristic, complemented, Sylow
39	C₂²×C₁₃	—	477	1	no	Hall
40	C₃×C₁₃	D₃×D₅₃	1	4	yes	complemented
41	C₂×C₃⋊D₃	—	53	1	no	Hall
42	C₂×C₁₃	—	9	1	no	—
43	C₂×C₁₃	—	53	1	no	—
44	C₂×C₁₃	—	477	1	no	—
45	C₂×C₃²	—	53	1	no	—
46	C₃⋊D₃	—	53	1	no	—
47	C₃⋊D₃	C₁₃×D₅₃	1	1	yes	characteristic, direct product factor
48	C₁₃	C₃⋊D₃×D₅₃	1	1	yes	characteristic, direct product factor, Sylow
49	C₂×D₃	—	159	4	no	—
50	C₃²	C₂×C₁₃×D₅₃	1	1	yes	characteristic, complemented, Sylow
51	C₂×C₃	—	53	4	no	—
52	D₃	—	3	4	no	—
53	D₃	—	159	4	no	—
54	C₂²	—	477	1	no	Sylow
55	C₃	D₃×C₁₃×D₅₃	1	4	yes	complemented
56	C₂	—	9	1	no	—
57	C₂	—	53	1	no	—
58	C₂	—	477	1	no	—

	subgroup	G/H
center, hypercenter	C₁₃	C₃⋊D₃×D₅₃
commutator, cosocle, hypocenter	C₃²×C₅₃	C₂²×C₁₃
Frattini	C₁	G
Fitting, socle	C₃²×C₁₃×C₅₃	C₂²
LCS	G ▷ C₃²×C₅₃
UCS	C₁ ◁ C₁₃

Complements

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

N	H ≅ G/N	type
1. C₃²×C₁₃×D₅₃	C₂	semidirect
2. C₁₃×C₃⋊D₃×C₅₃	C₂	semidirect
3. C₁₃×C₃⋊D₁₅₉	C₂	semidirect
5. C₃²×C₁₃×C₅₃	C₂²	semidirect
8. C₃×C₁₃×D₅₃	D₃	semidirect
10. C₃×C₁₃×C₅₃	C₂×D₃	semidirect
11. C₃⋊D₃×D₅₃	C₁₃	direct (N × H)
14. C₁₃×D₅₃	C₃⋊D₃	direct (N × H)
15. C₃⋊D₁₅₉	C₂×C₁₃	semidirect
16. C₃⋊D₃×C₅₃	C₂×C₁₃	semidirect
17. C₃²×D₅₃	C₂×C₁₃	semidirect
18. C₁₃×C₅₃	C₂×C₃⋊D₃	semidirect
20. C₃²×C₅₃	C₂²×C₁₃	semidirect
23. C₃×D₅₃	D₃×C₁₃	semidirect
27. C₁₃×C₃⋊D₃	D₅₃	direct (N × H)
29. C₃×C₅₃	C₂×D₃×C₁₃	semidirect
31. C₃²×C₁₃	C₂×D₅₃	semidirect
34. D₅₃	C₁₃×C₃⋊D₃	direct (N × H)
38. C₅₃	C₂×C₁₃×C₃⋊D₃	semidirect
40. C₃×C₁₃	D₃×D₅₃	semidirect
47. C₃⋊D₃	C₁₃×D₅₃	direct (N × H)
48. C₁₃	C₃⋊D₃×D₅₃	direct (N × H)
50. C₃²	C₂×C₁₃×D₅₃	semidirect
55. C₃	D₃×C₁₃×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₁₃×D₅₃	2
C₁₃×C₃⋊D₃×C₅₃	2
C₁₃×C₃⋊D₁₅₉	2
D₃×C₁₃×D₅₃	3
C₃⋊D₃×D₅₃	13
C₂×C₁₃×C₃⋊D₃	53

Subgroups — C13×C3⋊D3×D53 #37613745

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×C₃⋊D₃×D₅₃ #37613745