Subgroups — C₃×C₅×#11927412 #32913089

Order 18300.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×D₃₀₅	C₂	1	1	yes	maximal, characteristic
2	C₅×#11927412	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₃×C₅²×C₆₁	C₄	1	1	yes	characteristic, complemented, Hall
4	C₃×#11927412	C₅	1	1	yes	maximal, characteristic, direct product factor
5	C₃×C₅×#6923	—	5	1	no	maximal
6	C₅×D₃₀₅	C₂×C₃	1	1	yes	characteristic
7	C₃×D₃₀₅	C₂×C₅	1	1	yes	characteristic
8	C₃×C₅×D₆₁	—	5	1	no	—
9	C₅²×C₆₁	C₃×C₄	1	1	yes	characteristic, complemented, Hall
10	#11927412	C₃×C₅	1	1	yes	characteristic, direct product factor
11	C₅×#6923	—	5	1	no	—
12	C₃×C₅×C₆₁	—	4	1	no	—
13	C₃×C₅×C₆₁	C₄×C₅	1	1	yes	characteristic, complemented
14	C₃×C₅×C₆₁	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
15	C₃×#6923	—	5	1	no	Hall
16	D₃₀₅	C₂×C₃×C₅	1	1	yes	characteristic
17	C₅×D₆₁	—	5	1	no	—
18	C₃×D₆₁	—	5	1	no	—
19	C₅×C₆₁	—	4	1	no	—
20	C₅×C₆₁	C₃×C₄×C₅	1	1	yes	characteristic, complemented
21	C₅×C₆₁	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
22	C₃×C₅×AGL(1,𝔽₅)	—	61	1	no	maximal, Hall
23	#6923	—	5	1	no	Hall
24	C₃×C₆₁	C₅×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
25	C₃×C₅×D₅	—	61	1	no	—
26	D₆₁	—	5	1	no	—
27	C₅×AGL(1,𝔽₅)	—	61	1	no	Hall
28	C₃×C₅²	#6923	1	1	yes	characteristic, complemented, Hall
29	C₆₁	C₃×C₅×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
30	C₃×AGL(1,𝔽₅)	—	61	1	no	—
31	C₃×C₄×C₅	—	305	1	no	—
32	C₅×D₅	—	61	1	no	—
33	C₂×C₃×C₅	—	305	1	no	—
34	C₃×D₅	—	61	1	no	—
35	C₅²	C₃×#6923	1	1	yes	characteristic, complemented, Sylow
36	AGL(1,𝔽₅)	—	61	1	no	—
37	C₄×C₅	—	305	1	no	—
38	C₃×C₅	—	4	1	no	—
39	C₃×C₅	C₅×#6923	1	1	yes	characteristic, complemented
40	C₃×C₅	#11927412	1	1	yes	characteristic, direct product factor
41	C₃×C₄	—	305	1	no	Hall
42	C₂×C₅	—	305	1	no	—
43	D₅	—	61	1	no	—
44	C₂×C₃	—	305	1	no	—
45	C₅	—	4	1	no	—
46	C₅	C₃×C₅×#6923	1	1	yes	characteristic, complemented
47	C₅	C₃×#11927412	1	1	yes	characteristic, direct product factor
48	C₄	—	305	1	no	Sylow
49	C₃	C₅×#11927412	1	1	yes	characteristic, direct product factor, Sylow
50	C₂	—	305	1	no	—

	subgroup	G/H
center, hypercenter	C₃×C₅	#11927412
commutator, hypocenter	C₅×C₆₁	C₃×C₄×C₅
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₆₁	C₄
prime radical	C₃×C₅×D₃₀₅	C₂
cosocle	D₃₀₅	C₂×C₃×C₅
LCS	G ▷ C₅×C₆₁
UCS	C₁ ◁ C₃×C₅

Complements

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

N	H ≅ G/N	type
2. C₅×#11927412	C₃	direct (N × H)
3. C₃×C₅²×C₆₁	C₄	semidirect
4. C₃×#11927412	C₅	direct (N × H)
9. C₅²×C₆₁	C₃×C₄	semidirect
10. #11927412	C₃×C₅	direct (N × H)
13. C₃×C₅×C₆₁	C₄×C₅	semidirect
14. C₃×C₅×C₆₁	AGL(1,𝔽₅)	semidirect
20. C₅×C₆₁	C₃×C₄×C₅	semidirect
21. C₅×C₆₁	C₃×AGL(1,𝔽₅)	semidirect
24. C₃×C₆₁	C₅×AGL(1,𝔽₅)	semidirect
28. C₃×C₅²	#6923	semidirect
29. C₆₁	C₃×C₅×AGL(1,𝔽₅)	semidirect
35. C₅²	C₃×#6923	semidirect
39. C₃×C₅	C₅×#6923	semidirect
40. C₃×C₅	#11927412	direct (N × H)
46. C₅	C₃×C₅×#6923	semidirect
47. C₅	C₃×#11927412	direct (N × H)
49. C₃	C₅×#11927412	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅×D₃₀₅	2
C₅×#11927412	3
C₃×#11927412	5
C₃×C₅×#6923	5
C₃×C₅×AGL(1,𝔽₅)	61

Subgroups — C3×C5×#11927412 #32913089

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#11927412 #32913089