Subgroups — C₃×C₄×AGL(1,𝔽₅) #6752

Order 240.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

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

Complements

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

N	H ≅ G/N	type
3. C₄×AGL(1,𝔽₅)	C₃	direct (N × H)
5. C₃×AGL(1,𝔽₅)	C₄	direct (N × H)
6. C₃×C₄×C₅	C₄	semidirect
7. C₃×Dic₅	C₄	semidirect
16. AGL(1,𝔽₅)	C₃×C₄	direct (N × H)
17. C₄×C₅	C₃×C₄	semidirect
18. Dic₅	C₃×C₄	semidirect
20. C₃×C₅	C₄²	semidirect
24. C₃×C₄	AGL(1,𝔽₅)	direct (N × H)
31. C₅	C₃×C₄²	semidirect
35. C₄	C₃×AGL(1,𝔽₅)	direct (N × H)
36. C₃	C₄×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×C₃×AGL(1,𝔽₅)	2
C₃×C₄×D₅	2
C₄×AGL(1,𝔽₅)	3
C₃×C₄²	5

Subgroups — C3×C4×AGL(1,𝔽5) #6752

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×AGL(1,𝔽₅) #6752