Subgroups — C₂×C₃²×AGL(1,𝔽₅) #67215

Order 360.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×D₅	C₂	1	1	yes	maximal, characteristic
2	C₃²×AGL(1,𝔽₅)	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×AGL(1,𝔽₅)	C₃	1	4	yes	maximal, direct product factor
4	C₂×C₃²×C₅	C₄	1	1	yes	characteristic, complemented
5	C₃²×D₅	C₄	1	1	yes	characteristic, complemented
6	C₃²×D₅	C₂²	1	1	yes	characteristic
7	C₂×C₃²×C₄	—	5	1	no	maximal, Hall
8	C₂×C₃×D₅	C₂×C₃	1	4	yes	—
9	C₃×AGL(1,𝔽₅)	C₂×C₃	1	8	yes	direct product factor
10	C₃²×C₅	C₂×C₄	1	1	yes	characteristic, complemented, Hall
11	C₂×AGL(1,𝔽₅)	C₃²	1	1	yes	characteristic, direct product factor, Hall
12	C₂²×C₃²	—	5	1	no	—
13	C₃²×C₄	—	5	2	no	—
14	C₂×C₃×C₅	C₃×C₄	1	4	yes	complemented
15	C₃×D₅	C₃×C₄	1	4	yes	complemented
16	C₃×D₅	C₂²×C₃	1	4	yes	—
17	C₂×C₃×C₄	—	5	4	no	—
18	C₂×D₅	C₂×C₃²	1	1	yes	characteristic
19	AGL(1,𝔽₅)	C₂×C₃²	1	2	yes	direct product factor
20	C₂×C₃²	—	5	1	no	—
21	C₂×C₃²	—	5	1	no	—
22	C₂×C₃²	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
23	C₃×C₅	C₂×C₃×C₄	1	4	yes	complemented
24	C₂²×C₃	—	5	4	no	—
25	C₃×C₄	—	5	8	no	—
26	C₂×C₅	C₃²×C₄	1	1	yes	characteristic, complemented
27	D₅	C₃²×C₄	1	1	yes	characteristic, complemented
28	D₅	C₂²×C₃²	1	1	yes	characteristic
29	C₃²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor, Sylow
30	C₂×C₄	—	5	1	no	Sylow
31	C₂×C₃	—	5	4	no	—
32	C₂×C₃	—	5	4	no	—
33	C₂×C₃	C₃×AGL(1,𝔽₅)	1	4	yes	direct product factor
34	C₅	C₂×C₃²×C₄	1	1	yes	characteristic, complemented, Sylow
35	C₂²	—	5	1	no	—
36	C₄	—	5	2	no	—
37	C₃	C₂×C₃×AGL(1,𝔽₅)	1	4	yes	direct product factor
38	C₂	—	5	1	no	—
39	C₂	—	5	1	no	—
40	C₂	C₃²×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃²	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₂×C₃²×C₄
Frattini	C₁	G
Fitting, socle	C₂×C₃²×C₅	C₄
prime radical	C₂×C₃²×D₅	C₂
cosocle	D₅	C₂²×C₃²
LCS	G ▷ C₅
UCS	C₁ ◁ C₂×C₃²

Complements

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

N	H ≅ G/N	type
2. C₃²×AGL(1,𝔽₅)	C₂	direct (N × H)
3. C₂×C₃×AGL(1,𝔽₅)	C₃	direct (N × H)
4. C₂×C₃²×C₅	C₄	semidirect
5. C₃²×D₅	C₄	semidirect
9. C₃×AGL(1,𝔽₅)	C₂×C₃	direct (N × H)
10. C₃²×C₅	C₂×C₄	semidirect
11. C₂×AGL(1,𝔽₅)	C₃²	direct (N × H)
14. C₂×C₃×C₅	C₃×C₄	semidirect
15. C₃×D₅	C₃×C₄	semidirect
19. AGL(1,𝔽₅)	C₂×C₃²	direct (N × H)
22. C₂×C₃²	AGL(1,𝔽₅)	direct (N × H)
23. C₃×C₅	C₂×C₃×C₄	semidirect
26. C₂×C₅	C₃²×C₄	semidirect
27. D₅	C₃²×C₄	semidirect
29. C₃²	C₂×AGL(1,𝔽₅)	direct (N × H)
33. C₂×C₃	C₃×AGL(1,𝔽₅)	direct (N × H)
34. C₅	C₂×C₃²×C₄	semidirect
37. C₃	C₂×C₃×AGL(1,𝔽₅)	direct (N × H)
40. C₂	C₃²×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×C₃²×D₅	2
C₃²×AGL(1,𝔽₅)	2
C₂×C₃×AGL(1,𝔽₅)	3
C₂×C₃²×C₄	5

Subgroups — C2×C32×AGL(1,𝔽5) #67215

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×AGL(1,𝔽₅) #67215