Subgroups — C₂×D₃×AGL(1,𝔽₅) #6834

Order 240.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂×ΓL(1,𝔽₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic, complemented
4	D₃×AGL(1,𝔽₅)	C₂	1	4	yes	maximal, direct product factor
5	C₂²×AGL(1,𝔽₅)	—	3	1	no	maximal, Hall
6	C₂×D₁₅	C₄	1	1	yes	characteristic, complemented
7	C₂×C₅×D₃	C₄	1	1	yes	characteristic, complemented
8	C₂×C₃×D₅	C₂²	1	1	yes	characteristic
9	D₃×D₅	C₄	1	2	yes	complemented
10	D₃×D₅	C₂²	1	2	yes	—
11	ΓL(1,𝔽₁₆)	C₂²	1	2	yes	complemented
12	C₃×AGL(1,𝔽₅)	C₂²	1	2	yes	complemented
13	C₂×C₄×D₃	—	5	1	no	maximal, Hall
14	C₂²×D₅	—	3	1	no	—
15	C₂×AGL(1,𝔽₅)	—	3	1	no	—
16	C₂×AGL(1,𝔽₅)	—	3	4	no	—
17	C₂×AGL(1,𝔽₅)	D₃	1	1	yes	characteristic, direct product factor
18	C₂×C₃×C₅	C₂×C₄	1	1	yes	characteristic, complemented
19	D₁₅	C₂×C₄	1	2	yes	complemented
20	C₃×D₅	C₂×C₄	1	1	yes	characteristic, complemented
21	C₃×D₅	C₂³	1	1	yes	characteristic
22	C₅×D₃	C₂×C₄	1	2	yes	complemented
23	C₂²×D₃	—	5	1	no	—
24	C₂×C₃×C₄	—	5	1	no	—
25	C₂×Dic₃	—	5	1	no	—
26	C₄×D₃	—	5	4	no	—
27	C₂²×C₅	—	3	1	no	—
28	C₂×D₅	—	3	1	no	—
29	C₂×D₅	—	3	2	no	—
30	C₂×D₅	—	3	2	no	—
31	C₂×D₅	C₂×D₃	1	1	yes	characteristic
32	AGL(1,𝔽₅)	—	3	2	no	—
33	AGL(1,𝔽₅)	C₂×D₃	1	2	yes	direct product factor
34	C₂²×C₄	—	15	1	no	Sylow
35	C₃×C₅	C₂²×C₄	1	1	yes	characteristic, complemented, Hall
36	C₂²×C₃	—	5	1	no	—
37	C₂×D₃	—	5	1	no	—
38	C₂×D₃	—	5	2	no	—
39	C₂×D₃	—	5	2	no	—
40	C₂×D₃	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
41	C₃×C₄	—	5	2	no	—
42	Dic₃	—	5	2	no	—
43	C₂×C₅	—	3	2	no	—
44	C₂×C₅	C₄×D₃	1	1	yes	characteristic, complemented
45	D₅	—	3	2	no	—
46	D₅	C₄×D₃	1	1	yes	characteristic, complemented
47	D₅	C₂²×D₃	1	1	yes	characteristic
48	C₂³	—	15	1	no	—
49	C₂×C₄	—	5	1	no	—
50	C₂×C₄	—	15	4	no	—
51	C₂×C₄	—	15	1	no	—
52	C₂×C₃	—	5	1	no	—
53	C₂×C₃	—	5	1	no	—
54	C₂×C₃	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
55	D₃	—	5	2	no	—
56	D₃	C₂×AGL(1,𝔽₅)	1	2	yes	direct product factor
57	C₅	C₂×C₄×D₃	1	1	yes	characteristic, complemented, Sylow
58	C₂²	—	3	1	no	—
59	C₂²	—	5	1	no	—
60	C₂²	—	15	2	no	—
61	C₂²	—	15	2	no	—
62	C₂²	—	15	1	no	—
63	C₄	—	5	2	no	—
64	C₄	—	15	2	no	—
65	C₃	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
66	C₂	—	3	2	no	—
67	C₂	—	5	1	no	—
68	C₂	—	5	1	no	—
69	C₂	—	15	2	no	—
70	C₂	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	D₃×AGL(1,𝔽₅)
commutator, hypocenter	C₃×C₅	C₂²×C₄
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₅	C₂×C₄
prime radical	C₂×D₃×D₅	C₂
cosocle	C₃×D₅	C₂³
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
2. C₂×ΓL(1,𝔽₁₆)	C₂	semidirect
3. C₂×C₃×AGL(1,𝔽₅)	C₂	semidirect
4. D₃×AGL(1,𝔽₅)	C₂	direct (N × H)
6. C₂×D₁₅	C₄	semidirect
7. C₂×C₅×D₃	C₄	semidirect
9. D₃×D₅	C₄	semidirect
11. ΓL(1,𝔽₁₆)	C₂²	semidirect
12. C₃×AGL(1,𝔽₅)	C₂²	semidirect
17. C₂×AGL(1,𝔽₅)	D₃	direct (N × H)
18. C₂×C₃×C₅	C₂×C₄	semidirect
19. D₁₅	C₂×C₄	semidirect
20. C₃×D₅	C₂×C₄	semidirect
22. C₅×D₃	C₂×C₄	semidirect
33. AGL(1,𝔽₅)	C₂×D₃	direct (N × H)
35. C₃×C₅	C₂²×C₄	semidirect
40. C₂×D₃	AGL(1,𝔽₅)	direct (N × H)
44. C₂×C₅	C₄×D₃	semidirect
46. D₅	C₄×D₃	semidirect
54. C₂×C₃	C₂×AGL(1,𝔽₅)	semidirect
56. D₃	C₂×AGL(1,𝔽₅)	direct (N × H)
57. C₅	C₂×C₄×D₃	semidirect
65. C₃	C₂²×AGL(1,𝔽₅)	semidirect
70. C₂	D₃×AGL(1,𝔽₅)	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×D₃×D₅	2
C₂×ΓL(1,𝔽₁₆)	2
C₂×C₃×AGL(1,𝔽₅)	2
D₃×AGL(1,𝔽₅)	2
C₂²×AGL(1,𝔽₅)	3
C₂×C₄×D₃	5

Subgroups — C2×D3×AGL(1,𝔽5) #6834

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₃×AGL(1,𝔽₅) #6834