Subgroups — D₃×AGL(1,𝔽₇) #6987

Order 252.

44 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⋊AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic, complemented
2	D₃×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic, complemented
4	D₃×D₇	C₃	1	1	yes	maximal, characteristic, complemented
5	C₂×AGL(1,𝔽₇)	—	3	1	no	maximal
6	C₃×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented, Hall
7	D₂₁	C₂×C₃	1	1	yes	characteristic, complemented
8	C₃×D₇	C₂×C₃	1	1	yes	characteristic, complemented
9	D₃×C₇	C₂×C₃	1	1	yes	characteristic, complemented
10	C₂×ΓL(1,𝔽₈)	—	3	1	no	—
11	AGL(1,𝔽₇)	—	2	1	no	—
12	AGL(1,𝔽₇)	—	3	1	no	—
13	AGL(1,𝔽₇)	D₃	1	1	yes	characteristic, direct product factor
14	C₂×C₃×D₃	—	7	1	no	maximal, Hall
15	C₂×D₇	—	3	1	no	Hall
16	C₃×C₇	C₂²×C₃	1	1	yes	characteristic, complemented
17	ΓL(1,𝔽₈)	—	2	1	no	—
18	ΓL(1,𝔽₈)	C₂×D₃	1	1	yes	characteristic, complemented
19	C₂×C₃²	—	7	1	no	—
20	C₃×D₃	—	7	1	no	—
21	C₃×D₃	—	7	1	no	—
22	C₂×C₇	—	3	1	no	—
23	D₇	—	3	1	no	—
24	D₇	C₃×D₃	1	1	yes	characteristic, complemented
25	C₂²×C₃	—	21	1	no	—
26	C₂×D₃	—	7	1	no	—
27	C₃²	—	7	1	no	Sylow
28	C₇	C₂×C₃×D₃	1	1	yes	characteristic, complemented, Sylow
29	C₂×C₃	—	7	1	no	—
30	C₂×C₃	—	7	1	no	—
31	C₂×C₃	—	14	1	no	—
32	C₂×C₃	—	21	1	no	—
33	C₂×C₃	—	21	1	no	—
34	D₃	—	7	1	no	—
35	D₃	AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
36	C₂²	—	21	1	no	Sylow
37	C₃	—	7	1	no	—
38	C₃	—	14	1	no	—
39	C₃	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
40	C₂	—	3	1	no	—
41	C₂	—	7	1	no	—
42	C₂	—	21	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₇	C₂²×C₃
LCS	G ▷ C₃×C₇
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₃⋊AGL(1,𝔽₇)	C₂	semidirect
2. D₃×ΓL(1,𝔽₈)	C₂	semidirect
3. C₃×AGL(1,𝔽₇)	C₂	semidirect
4. D₃×D₇	C₃	semidirect
6. C₃×ΓL(1,𝔽₈)	C₂²	semidirect
7. D₂₁	C₂×C₃	semidirect
8. C₃×D₇	C₂×C₃	semidirect
9. D₃×C₇	C₂×C₃	semidirect
13. AGL(1,𝔽₇)	D₃	direct (N × H)
16. C₃×C₇	C₂²×C₃	semidirect
18. ΓL(1,𝔽₈)	C₂×D₃	semidirect
24. D₇	C₃×D₃	semidirect
28. C₇	C₂×C₃×D₃	semidirect
35. D₃	AGL(1,𝔽₇)	direct (N × H)
39. C₃	C₂×AGL(1,𝔽₇)	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃⋊AGL(1,𝔽₇)	2
D₃×ΓL(1,𝔽₈)	2
C₃×AGL(1,𝔽₇)	2
D₃×D₇	3
C₂×AGL(1,𝔽₇)	3
C₂×C₃×D₃	7

Subgroups — D3×AGL(1,𝔽7) #6987

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×AGL(1,𝔽₇) #6987