Subgroups — C₃×C₂²⋊AGL(1,𝔽₅) #6756

Order 240.

68 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	2	yes	maximal, complemented
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	—	2	2	no	—
6	C₂×C₃×D₅	C₄	1	1	yes	characteristic, complemented
7	C₂×C₃×D₅	C₂²	1	1	yes	characteristic
8	C₃×AGL(1,𝔽₅)	—	2	2	no	—
9	C₃×C₂²⋊C₄	—	5	1	no	maximal, Hall
10	C₂²×D₅	C₂×C₃	1	1	yes	characteristic
11	C₂×AGL(1,𝔽₅)	C₂×C₃	1	2	yes	complemented
12	C₂×C₃×C₅	—	2	1	no	—
13	C₂×C₃×C₅	C₂×C₄	1	1	yes	characteristic
14	C₃×D₅	—	2	1	no	—
15	C₃×D₅	D₄	1	2	yes	—
16	C₂³×C₃	—	5	1	no	—
17	C₂×C₃×C₄	—	5	2	no	—
18	C₂²×C₅	C₃×C₄	1	1	yes	characteristic, complemented
19	C₂×D₅	—	2	2	no	—
20	C₂×D₅	C₃×C₄	1	1	yes	characteristic, complemented
21	C₂×D₅	C₂²×C₃	1	1	yes	characteristic
22	AGL(1,𝔽₅)	—	2	2	no	—
23	C₂²⋊C₄	—	5	1	no	Sylow
24	C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic, complemented, Hall
25	C₂²×C₃	—	5	1	no	—
26	C₂²×C₃	—	5	1	no	—
27	C₂²×C₃	—	10	2	no	—
28	C₂²×C₃	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
29	C₃×C₄	—	10	2	no	—
30	C₂×C₅	—	2	1	no	—
31	C₂×C₅	C₂×C₃×C₄	1	1	yes	characteristic
32	D₅	—	2	1	no	—
33	D₅	C₃×D₄	1	2	yes	—
34	C₂³	—	5	1	no	—
35	C₂×C₄	—	5	2	no	—
36	C₂×C₃	—	2	1	no	—
37	C₂×C₃	—	5	2	no	—
38	C₂×C₃	—	10	1	no	—
39	C₂×C₃	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
40	C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
41	C₂²	—	5	1	no	—
42	C₂²	—	5	1	no	—
43	C₂²	—	10	2	no	—
44	C₂²	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
45	C₄	—	10	2	no	—
46	C₃	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor, Sylow
47	C₂	—	2	1	no	—
48	C₂	—	5	2	no	—
49	C₂	—	10	1	no	—
50	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	C₂²×C₃×C₅	C₄
socle	C₂×C₃×C₅	C₂×C₄
prime radical	C₂²×C₃×D₅	C₂
cosocle	C₂×D₅	C₂²×C₃
hypercenter	C₂²×C₃	AGL(1,𝔽₅)
hypocenter	C₅	C₃×C₂²⋊C₄
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃ ◁ 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	semidirect
3. C₂²⋊AGL(1,𝔽₅)	C₃	direct (N × H)
4. C₂²×C₃×C₅	C₄	semidirect
6. C₂×C₃×D₅	C₄	semidirect
11. C₂×AGL(1,𝔽₅)	C₂×C₃	semidirect
18. C₂²×C₅	C₃×C₄	semidirect
20. C₂×D₅	C₃×C₄	semidirect
24. C₃×C₅	C₂²⋊C₄	semidirect
28. C₂²×C₃	AGL(1,𝔽₅)	semidirect
40. C₅	C₃×C₂²⋊C₄	semidirect
44. C₂²	C₃×AGL(1,𝔽₅)	semidirect
46. C₃	C₂²⋊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
C₃×C₂²⋊C₄	5

Subgroups — C3×C22⋊AGL(1,𝔽5) #6756

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₂²⋊AGL(1,𝔽₅) #6756