Subgroups — C₃×C₅₃×C₃⋊(C₅⋊D₅) #37558732

Order 23850.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅²×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₅₃×C₃⋊(C₅⋊D₅)	C₃	1	1	yes	maximal, characteristic, direct product factor
3	C₃×C₅⋊D₅×C₅₃	—	3	1	no	maximal
4	C₃×D₁₅×C₅₃	—	5	6	no	maximal
5	C₃×C₅²×C₅₃	—	2	1	no	—
6	C₃×C₅²×C₅₃	D₃	1	1	yes	characteristic, complemented
7	C₃×C₅²×C₅₃	C₂×C₃	1	1	yes	characteristic, complemented
8	C₅⋊D₅×C₅₃	—	3	1	no	Hall
9	C₃²×C₅×C₅₃	D₅	1	6	yes	complemented
10	D₁₅×C₅₃	—	5	6	no	—
11	C₃×D₅×C₅₃	—	15	6	no	—
12	C₅²×C₅₃	C₃×D₃	1	1	yes	characteristic, complemented, Hall
13	C₃×D₃×C₅₃	—	25	1	no	Hall
14	C₃×C₅×C₅₃	—	2	6	no	—
15	C₃×C₅×C₅₃	C₃×D₅	1	6	yes	complemented
16	C₃×C₅×C₅₃	D₁₅	1	6	yes	complemented
17	D₅×C₅₃	—	15	6	no	—
18	C₃²×C₅₃	C₅⋊D₅	1	1	yes	characteristic, complemented, Hall
19	C₃×C₃⋊(C₅⋊D₅)	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₂×C₃×C₅₃	—	75	1	no	—
21	D₃×C₅₃	—	25	1	no	—
22	C₅×C₅₃	C₃×D₁₅	1	6	yes	complemented
23	C₃²×C₅²	C₂×C₅₃	1	1	yes	characteristic, complemented, Hall
24	C₃×C₅₃	—	2	1	no	—
25	C₃×C₅₃	C₃×C₅⋊D₅	1	1	yes	characteristic, complemented
26	C₃×C₅₃	C₃⋊(C₅⋊D₅)	1	1	yes	characteristic, direct product factor
27	C₃⋊(C₅⋊D₅)	C₃×C₅₃	1	1	yes	characteristic, direct product factor
28	C₃×C₅⋊D₅	—	3	1	no	—
29	C₂×C₅₃	—	75	1	no	Hall
30	C₃×D₁₅	—	5	6	no	—
31	C₃×C₅²	—	2	1	no	—
32	C₃×C₅²	D₃×C₅₃	1	1	yes	characteristic, complemented
33	C₃×C₅²	C₂×C₃×C₅₃	1	1	yes	characteristic, complemented
34	C₅₃	C₃×C₃⋊(C₅⋊D₅)	1	1	yes	characteristic, direct product factor, Sylow
35	C₅⋊D₅	—	3	1	no	Hall
36	C₃²×C₅	D₅×C₅₃	1	6	yes	complemented
37	D₁₅	—	5	6	no	—
38	C₃×D₅	—	15	6	no	—
39	C₅²	C₃×D₃×C₅₃	1	1	yes	characteristic, complemented, Sylow
40	C₃×D₃	—	25	1	no	Hall
41	C₃×C₅	—	2	6	no	—
42	C₃×C₅	C₃×D₅×C₅₃	1	6	yes	complemented
43	C₃×C₅	D₁₅×C₅₃	1	6	yes	complemented
44	D₅	—	15	6	no	—
45	C₃²	C₅⋊D₅×C₅₃	1	1	yes	characteristic, complemented, Sylow
46	C₂×C₃	—	75	1	no	—
47	D₃	—	25	1	no	—
48	C₅	C₃×D₁₅×C₅₃	1	6	yes	complemented
49	C₃	—	2	1	no	—
50	C₃	C₃×C₅⋊D₅×C₅₃	1	1	yes	characteristic, complemented
51	C₃	C₅₃×C₃⋊(C₅⋊D₅)	1	1	yes	characteristic, direct product factor
52	C₂	—	75	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃×C₅₃	C₃⋊(C₅⋊D₅)
commutator, cosocle, hypocenter	C₃×C₅²	C₂×C₃×C₅₃
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₅₃	C₂
LCS	G ▷ C₃×C₅²
UCS	C₁ ◁ C₃×C₅₃

Complements

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

N	H ≅ G/N	type
1. C₃²×C₅²×C₅₃	C₂	semidirect
2. C₅₃×C₃⋊(C₅⋊D₅)	C₃	direct (N × H)
6. C₃×C₅²×C₅₃	D₃	semidirect
7. C₃×C₅²×C₅₃	C₂×C₃	semidirect
9. C₃²×C₅×C₅₃	D₅	semidirect
12. C₅²×C₅₃	C₃×D₃	semidirect
15. C₃×C₅×C₅₃	C₃×D₅	semidirect
16. C₃×C₅×C₅₃	D₁₅	semidirect
18. C₃²×C₅₃	C₅⋊D₅	semidirect
19. C₃×C₃⋊(C₅⋊D₅)	C₅₃	direct (N × H)
22. C₅×C₅₃	C₃×D₁₅	semidirect
23. C₃²×C₅²	C₂×C₅₃	semidirect
25. C₃×C₅₃	C₃×C₅⋊D₅	semidirect
26. C₃×C₅₃	C₃⋊(C₅⋊D₅)	direct (N × H)
27. C₃⋊(C₅⋊D₅)	C₃×C₅₃	direct (N × H)
32. C₃×C₅²	D₃×C₅₃	semidirect
33. C₃×C₅²	C₂×C₃×C₅₃	semidirect
34. C₅₃	C₃×C₃⋊(C₅⋊D₅)	direct (N × H)
36. C₃²×C₅	D₅×C₅₃	semidirect
39. C₅²	C₃×D₃×C₅₃	semidirect
42. C₃×C₅	C₃×D₅×C₅₃	semidirect
43. C₃×C₅	D₁₅×C₅₃	semidirect
45. C₃²	C₅⋊D₅×C₅₃	semidirect
48. C₅	C₃×D₁₅×C₅₃	semidirect
50. C₃	C₃×C₅⋊D₅×C₅₃	semidirect
51. C₃	C₅₃×C₃⋊(C₅⋊D₅)	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×C₅²×C₅₃	2
C₅₃×C₃⋊(C₅⋊D₅)	3
C₃×C₅⋊D₅×C₅₃	3
C₃×D₁₅×C₅₃	5
C₃×C₃⋊(C₅⋊D₅)	53

Subgroups — C3×C53×C3⋊(C5⋊D5) #37558732

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅₃×C₃⋊(C₅⋊D₅) #37558732