Subgroups — C₂²×C₃×AGL(1,𝔽₅) #6839

Order 240.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×AGL(1,𝔽₅)	C₂	1	6	yes	maximal, direct product factor
3	C₂²×AGL(1,𝔽₅)	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂²×C₃×C₅	C₄	1	1	yes	characteristic, complemented
5	C₂×C₃×D₅	C₄	1	3	yes	complemented
6	C₂×C₃×D₅	C₂²	1	3	yes	—
7	C₃×AGL(1,𝔽₅)	C₂²	1	4	yes	direct product factor
8	C₂²×C₃×C₄	—	5	1	no	maximal, Hall
9	C₂²×D₅	C₂×C₃	1	1	yes	characteristic
10	C₂×AGL(1,𝔽₅)	C₂×C₃	1	6	yes	direct product factor
11	C₂×C₃×C₅	C₂×C₄	1	3	yes	complemented
12	C₃×D₅	C₂×C₄	1	3	yes	complemented
13	C₃×D₅	C₂³	1	1	yes	characteristic
14	C₂³×C₃	—	5	1	no	—
15	C₂×C₃×C₄	—	5	6	no	—
16	C₂²×C₅	C₃×C₄	1	1	yes	characteristic, complemented
17	C₂×D₅	C₃×C₄	1	3	yes	complemented
18	C₂×D₅	C₂²×C₃	1	3	yes	—
19	AGL(1,𝔽₅)	C₂²×C₃	1	4	yes	direct product factor
20	C₂²×C₄	—	5	1	no	Sylow
21	C₃×C₅	C₂²×C₄	1	1	yes	characteristic, complemented, Hall
22	C₂²×C₃	—	5	3	no	—
23	C₂²×C₃	—	5	3	no	—
24	C₂²×C₃	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
25	C₃×C₄	—	5	4	no	—
26	C₂×C₅	C₂×C₃×C₄	1	3	yes	complemented
27	D₅	C₂×C₃×C₄	1	3	yes	complemented
28	D₅	C₂³×C₃	1	1	yes	characteristic
29	C₂³	—	5	1	no	—
30	C₂×C₄	—	5	6	no	—
31	C₂×C₃	—	5	1	no	—
32	C₂×C₃	—	5	3	no	—
33	C₂×C₃	C₂×AGL(1,𝔽₅)	1	3	yes	direct product factor
34	C₅	C₂²×C₃×C₄	1	1	yes	characteristic, complemented, Sylow
35	C₂²	—	5	3	no	—
36	C₂²	—	5	3	no	—
37	C₂²	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
38	C₄	—	5	4	no	—
39	C₃	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor, Sylow
40	C₂	—	5	1	no	—
41	C₂	—	5	3	no	—
42	C₂	C₂×C₃×AGL(1,𝔽₅)	1	3	yes	direct product factor

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×AGL(1,𝔽₅)	C₂	direct (N × H)
3. C₂²×AGL(1,𝔽₅)	C₃	direct (N × H)
4. C₂²×C₃×C₅	C₄	semidirect
5. C₂×C₃×D₅	C₄	semidirect
7. C₃×AGL(1,𝔽₅)	C₂²	direct (N × H)
10. C₂×AGL(1,𝔽₅)	C₂×C₃	direct (N × H)
11. C₂×C₃×C₅	C₂×C₄	semidirect
12. C₃×D₅	C₂×C₄	semidirect
16. C₂²×C₅	C₃×C₄	semidirect
17. C₂×D₅	C₃×C₄	semidirect
19. AGL(1,𝔽₅)	C₂²×C₃	direct (N × H)
21. C₃×C₅	C₂²×C₄	semidirect
24. C₂²×C₃	AGL(1,𝔽₅)	direct (N × H)
26. C₂×C₅	C₂×C₃×C₄	semidirect
27. D₅	C₂×C₃×C₄	semidirect
33. C₂×C₃	C₂×AGL(1,𝔽₅)	direct (N × H)
34. C₅	C₂²×C₃×C₄	semidirect
37. C₂²	C₃×AGL(1,𝔽₅)	direct (N × H)
39. C₃	C₂²×AGL(1,𝔽₅)	direct (N × H)
42. C₂	C₂×C₃×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
C₂²×C₃×D₅	2
C₂×C₃×AGL(1,𝔽₅)	2
C₂²×AGL(1,𝔽₅)	3
C₂²×C₃×C₄	5

Subgroups — C22×C3×AGL(1,𝔽5) #6839

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×AGL(1,𝔽₅) #6839