Subgroups — C₃×D₃×AGL(1,𝔽₅) #67196

Order 360.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×D₃×D₅	C₂	1	1	yes	maximal, characteristic
2	C₃×ΓL(1,𝔽₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×AGL(1,𝔽₅)	—	3	1	no	maximal
5	D₃×AGL(1,𝔽₅)	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₃×D₁₅	C₄	1	1	yes	characteristic, complemented
7	C₃×C₅×D₃	C₄	1	1	yes	characteristic, complemented
8	C₃²×D₅	C₂²	1	1	yes	characteristic
9	C₃×C₄×D₃	—	5	1	no	maximal, Hall
10	C₂×C₃×D₅	—	3	1	no	—
11	D₃×D₅	C₂×C₃	1	1	yes	characteristic
12	ΓL(1,𝔽₁₆)	C₂×C₃	1	1	yes	characteristic, complemented
13	C₃×AGL(1,𝔽₅)	—	2	1	no	—
14	C₃×AGL(1,𝔽₅)	—	3	1	no	—
15	C₃×AGL(1,𝔽₅)	D₃	1	1	yes	characteristic, direct product factor
16	C₃×AGL(1,𝔽₅)	C₂×C₃	1	1	yes	characteristic, complemented
17	C₃²×C₅	C₂×C₄	1	1	yes	characteristic, complemented, Hall
18	C₂×AGL(1,𝔽₅)	—	3	1	no	Hall
19	C₂×C₃×D₃	—	5	1	no	—
20	C₃²×C₄	—	5	1	no	—
21	C₃×Dic₃	—	5	1	no	—
22	C₂×C₃×C₅	—	3	1	no	—
23	D₁₅	C₃×C₄	1	1	yes	characteristic, complemented
24	C₃×D₅	—	2	1	no	—
25	C₃×D₅	—	3	1	no	—
26	C₃×D₅	C₂×D₃	1	1	yes	characteristic
27	C₃×D₅	C₂²×C₃	1	1	yes	characteristic
28	C₅×D₃	C₃×C₄	1	1	yes	characteristic, complemented
29	C₂×C₃×C₄	—	15	1	no	—
30	C₄×D₃	—	5	1	no	—
31	C₂×D₅	—	3	1	no	—
32	AGL(1,𝔽₅)	—	3	1	no	—
33	AGL(1,𝔽₅)	C₃×D₃	1	1	yes	characteristic, direct product factor
34	C₂×C₃²	—	5	1	no	—
35	C₃×D₃	—	5	1	no	—
36	C₃×D₃	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
37	C₃×C₅	—	2	1	no	—
38	C₃×C₅	C₄×D₃	1	1	yes	characteristic, complemented
39	C₃×C₅	C₂×C₃×C₄	1	1	yes	characteristic, complemented
40	C₂²×C₃	—	15	1	no	—
41	C₂×D₃	—	5	1	no	—
42	C₃×C₄	—	5	1	no	—
43	C₃×C₄	—	5	1	no	—
44	C₃×C₄	—	10	1	no	—
45	C₃×C₄	—	15	1	no	—
46	Dic₃	—	5	1	no	—
47	C₂×C₅	—	3	1	no	—
48	D₅	—	3	1	no	—
49	D₅	C₂×C₃×D₃	1	1	yes	characteristic
50	C₃²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
51	C₂×C₄	—	15	1	no	Sylow
52	C₂×C₃	—	3	1	no	—
53	C₂×C₃	—	5	1	no	—
54	C₂×C₃	—	5	1	no	—
55	C₂×C₃	—	10	1	no	—
56	C₂×C₃	—	15	1	no	—
57	D₃	—	5	1	no	—
58	D₃	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
59	C₅	C₃×C₄×D₃	1	1	yes	characteristic, complemented, Sylow
60	C₂²	—	15	1	no	—
61	C₄	—	5	1	no	—
62	C₄	—	15	1	no	—
63	C₃	—	2	1	no	—
64	C₃	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
65	C₃	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
66	C₂	—	3	1	no	—
67	C₂	—	5	1	no	—
68	C₂	—	15	1	no	—

	subgroup	G/H
center, hypercenter	C₃	D₃×AGL(1,𝔽₅)
commutator, hypocenter	C₃×C₅	C₂×C₃×C₄
Frattini	C₁	G
Fitting, socle	C₃²×C₅	C₂×C₄
prime radical	C₃×D₃×D₅	C₂
cosocle	C₃×D₅	C₂²×C₃
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
2. C₃×ΓL(1,𝔽₁₆)	C₂	semidirect
3. C₃²×AGL(1,𝔽₅)	C₂	semidirect
5. D₃×AGL(1,𝔽₅)	C₃	direct (N × H)
6. C₃×D₁₅	C₄	semidirect
7. C₃×C₅×D₃	C₄	semidirect
12. ΓL(1,𝔽₁₆)	C₂×C₃	semidirect
15. C₃×AGL(1,𝔽₅)	D₃	direct (N × H)
16. C₃×AGL(1,𝔽₅)	C₂×C₃	semidirect
17. C₃²×C₅	C₂×C₄	semidirect
23. D₁₅	C₃×C₄	semidirect
28. C₅×D₃	C₃×C₄	semidirect
33. AGL(1,𝔽₅)	C₃×D₃	direct (N × H)
36. C₃×D₃	AGL(1,𝔽₅)	direct (N × H)
38. C₃×C₅	C₄×D₃	semidirect
39. C₃×C₅	C₂×C₃×C₄	semidirect
50. C₃²	C₂×AGL(1,𝔽₅)	semidirect
58. D₃	C₃×AGL(1,𝔽₅)	direct (N × H)
59. C₅	C₃×C₄×D₃	semidirect
64. C₃	D₃×AGL(1,𝔽₅)	direct (N × H)
65. C₃	C₂×C₃×AGL(1,𝔽₅)	semidirect

Maximal subgroups

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

subgroup	index
C₃×D₃×D₅	2
C₃×ΓL(1,𝔽₁₆)	2
C₃²×AGL(1,𝔽₅)	2
C₂×C₃×AGL(1,𝔽₅)	3
D₃×AGL(1,𝔽₅)	3
C₃×C₄×D₃	5

Subgroups — C3×D3×AGL(1,𝔽5) #67196

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₃×AGL(1,𝔽₅) #67196