Subgroups — C₃×C₄⋊AGL(1,𝔽₅) #6753

Order 240.

52 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)	—	2	2	no	—
6	C₃×C₄×C₅	C₄	1	1	yes	characteristic, complemented
7	C₃×Dic₅	C₄	1	1	yes	characteristic, complemented
8	C₃×C₄⋊C₄	—	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	D₄	1	1	yes	characteristic
13	C₃×D₅	Q₈	1	1	yes	characteristic
14	C₂×C₃×C₄	—	5	1	no	—
15	C₂×C₃×C₄	—	5	2	no	—
16	C₂×D₅	C₂²×C₃	1	1	yes	characteristic
17	AGL(1,𝔽₅)	—	2	2	no	—
18	C₄×C₅	C₃×C₄	1	1	yes	characteristic, complemented
19	Dic₅	C₃×C₄	1	1	yes	characteristic, complemented
20	C₄⋊C₄	—	5	1	no	Sylow
21	C₃×C₅	C₄⋊C₄	1	1	yes	characteristic, complemented, Hall
22	C₂²×C₃	—	5	1	no	—
23	C₃×C₄	—	5	1	no	—
24	C₃×C₄	—	10	2	no	—
25	C₃×C₄	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
26	C₂×C₅	C₂×C₃×C₄	1	1	yes	characteristic
27	D₅	C₃×D₄	1	1	yes	characteristic
28	D₅	C₃×Q₈	1	1	yes	characteristic
29	C₂×C₄	—	5	1	no	—
30	C₂×C₄	—	5	2	no	—
31	C₂×C₃	—	5	1	no	—
32	C₂×C₃	—	5	1	no	—
33	C₂×C₃	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
34	C₅	C₃×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
35	C₂²	—	5	1	no	—
36	C₄	—	5	1	no	—
37	C₄	—	10	2	no	—
38	C₄	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
39	C₃	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor, Sylow
40	C₂	—	5	1	no	—
41	C₂	—	5	1	no	—
42	C₂	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂×C₃×C₄
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
hypercenter	C₃×C₄	AGL(1,𝔽₅)
hypocenter	C₅	C₃×C₄⋊C₄
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃ ◁ 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)
6. C₃×C₄×C₅	C₄	semidirect
7. C₃×Dic₅	C₄	semidirect
18. C₄×C₅	C₃×C₄	semidirect
19. Dic₅	C₃×C₄	semidirect
21. C₃×C₅	C₄⋊C₄	semidirect
25. C₃×C₄	AGL(1,𝔽₅)	semidirect
34. C₅	C₃×C₄⋊C₄	semidirect
38. C₄	C₃×AGL(1,𝔽₅)	semidirect
39. 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×C₄⋊C₄	5

Subgroups — C3×C4⋊AGL(1,𝔽5) #6753

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄⋊AGL(1,𝔽₅) #6753