Subgroups — C₂×C₃×ΓL(1,𝔽₁₆) #67216

Order 360.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, hypercenter	C₂×C₃	ΓL(1,𝔽₁₆)
commutator, hypocenter	C₃×C₅	C₂×C₃×C₄
Frattini	C₁	G
Fitting, socle	C₂×C₃²×C₅	C₄
prime radical	C₂×C₃²×D₅	C₂
cosocle	C₃×D₅	C₂²×C₃
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₃

Complements

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

N	H ≅ G/N	type
2. C₃×ΓL(1,𝔽₁₆)	C₂	direct (N × H)
3. C₂×ΓL(1,𝔽₁₆)	C₃	direct (N × H)
5. C₂×C₃²×C₅	C₄	semidirect
6. C₃²×D₅	C₄	semidirect
12. ΓL(1,𝔽₁₆)	C₂×C₃	direct (N × H)
14. C₃²×C₅	C₂×C₄	semidirect
19. C₂×C₃×C₅	Dic₃	semidirect
20. C₂×C₃×C₅	C₃×C₄	semidirect
23. C₃×D₅	Dic₃	semidirect
24. C₃×D₅	C₃×C₄	semidirect
33. C₂×C₃²	AGL(1,𝔽₅)	semidirect
35. C₃×C₅	C₂×Dic₃	semidirect
36. C₃×C₅	C₂×C₃×C₄	semidirect
42. C₂×C₅	C₃×Dic₃	semidirect
43. D₅	C₃×Dic₃	semidirect
45. C₃²	C₂×AGL(1,𝔽₅)	semidirect
54. C₂×C₃	C₃×AGL(1,𝔽₅)	semidirect
55. C₂×C₃	ΓL(1,𝔽₁₆)	direct (N × H)
56. C₅	C₂×C₃×Dic₃	semidirect
60. C₃	C₂×C₃×AGL(1,𝔽₅)	semidirect
61. C₃	C₂×ΓL(1,𝔽₁₆)	direct (N × H)
64. C₂	C₃×ΓL(1,𝔽₁₆)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃²×D₅	2
C₃×ΓL(1,𝔽₁₆)	2
C₂×ΓL(1,𝔽₁₆)	3
C₂×C₃×AGL(1,𝔽₅)	3
C₂×C₃×Dic₃	5

Subgroups — C2×C3×ΓL(1,𝔽16) #67216

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×ΓL(1,𝔽₁₆) #67216