Subgroups — Dic₃×AGL(1,𝔽₅) #6734

Order 240.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×ΓL(1,𝔽₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic
3	D₅×Dic₃	C₂	1	1	yes	maximal, characteristic
4	C₄×AGL(1,𝔽₅)	—	3	1	no	maximal, Hall
5	C₂×C₃×D₅	C₂²	1	1	yes	characteristic
6	ΓL(1,𝔽₁₆)	C₄	1	2	yes	complemented
7	C₃×AGL(1,𝔽₅)	C₄	1	2	yes	complemented
8	Dic₁₅	C₄	1	1	yes	characteristic, complemented
9	C₅×Dic₃	C₄	1	1	yes	characteristic, complemented
10	C₄×Dic₃	—	5	1	no	maximal, Hall
11	C₂×AGL(1,𝔽₅)	—	3	1	no	—
12	C₂×AGL(1,𝔽₅)	D₃	1	1	yes	characteristic
13	C₄×D₅	—	3	1	no	—
14	C₂×C₃×C₅	C₂×C₄	1	1	yes	characteristic
15	C₃×D₅	C₂×C₄	1	1	yes	characteristic
16	C₃×D₅	C₂×C₄	1	1	yes	characteristic
17	C₂×C₃×C₄	—	5	1	no	—
18	C₂×Dic₃	—	5	1	no	—
19	C₂×Dic₃	—	5	1	no	—
20	C₂×D₅	C₂×D₃	1	1	yes	characteristic
21	AGL(1,𝔽₅)	—	3	2	no	—
22	AGL(1,𝔽₅)	Dic₃	1	2	yes	direct product factor
23	C₄×C₅	—	3	1	no	—
24	Dic₅	—	3	1	no	—
25	C₄²	—	15	1	no	Sylow
26	C₃×C₅	C₄²	1	1	yes	characteristic, complemented, Hall
27	C₂²×C₃	—	5	1	no	—
28	C₃×C₄	—	5	2	no	—
29	Dic₃	—	5	2	no	—
30	Dic₃	—	5	1	no	—
31	Dic₃	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
32	C₂×C₅	C₄×D₃	1	1	yes	characteristic
33	D₅	C₄×D₃	1	1	yes	characteristic
34	D₅	C₂×Dic₃	1	1	yes	characteristic
35	C₂×C₄	—	5	1	no	—
36	C₂×C₄	—	15	1	no	—
37	C₂×C₄	—	15	1	no	—
38	C₂×C₃	—	5	1	no	—
39	C₂×C₃	—	5	1	no	—
40	C₂×C₃	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
41	C₅	C₄×Dic₃	1	1	yes	characteristic, complemented, Sylow
42	C₂²	—	5	1	no	—
43	C₄	—	3	1	no	—
44	C₄	—	5	2	no	—
45	C₄	—	15	2	no	—
46	C₄	—	15	1	no	—
47	C₃	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
48	C₂	—	5	1	no	—
49	C₂	—	5	1	no	—
50	C₂	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃×AGL(1,𝔽₅)
commutator, hypocenter	C₃×C₅	C₄²
Fitting, socle	C₂×C₃×C₅	C₂×C₄
prime radical, cosocle	C₂×C₃×D₅	C₂²
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
6. ΓL(1,𝔽₁₆)	C₄	semidirect
7. C₃×AGL(1,𝔽₅)	C₄	semidirect
8. Dic₁₅	C₄	semidirect
9. C₅×Dic₃	C₄	semidirect
22. AGL(1,𝔽₅)	Dic₃	direct (N × H)
26. C₃×C₅	C₄²	semidirect
31. Dic₃	AGL(1,𝔽₅)	direct (N × H)
41. C₅	C₄×Dic₃	semidirect
47. C₃	C₄×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
C₂×ΓL(1,𝔽₁₆)	2
C₂×C₃×AGL(1,𝔽₅)	2
D₅×Dic₃	2
C₄×AGL(1,𝔽₅)	3
C₄×Dic₃	5

Subgroups — Dic3×AGL(1,𝔽5) #6734

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₃×AGL(1,𝔽₅) #6734