Subgroups — C₂×C₁₇²×AGL(1,𝔽₇) #37566246

Order 24276.

400 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₇²×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₇²×AGL(1,𝔽₇)	C₂	1	2	yes	maximal, direct product factor
3	C₂×D₇×C₁₇²	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₁₇²×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented, Hall
5	D₇×C₁₇²	C₂×C₃	1	2	yes	complemented
6	C₂×C₇×C₁₇²	C₂×C₃	1	1	yes	characteristic, complemented
7	C₂²×C₃×C₁₇²	—	7	1	no	maximal, Hall
8	C₇×C₁₇²	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
9	C₂×C₃×C₁₇²	—	7	2	no	—
10	C₂×C₃×C₁₇²	—	7	1	no	—
11	C₂×C₁₇×AGL(1,𝔽₇)	C₁₇	1	18	yes	maximal, direct product factor
12	C₂²×C₁₇²	—	7	1	no	Hall
13	C₃×C₁₇²	—	7	1	no	Hall
14	C₂×C₁₇×ΓL(1,𝔽₈)	C₂×C₁₇	1	18	yes	complemented
15	C₁₇×AGL(1,𝔽₇)	C₂×C₁₇	1	36	yes	direct product factor
16	C₂×C₁₇²	—	7	2	no	—
17	C₂×C₁₇²	AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
18	C₂×D₇×C₁₇	C₃×C₁₇	1	18	yes	complemented
19	C₁₇×ΓL(1,𝔽₈)	C₂²×C₁₇	1	18	yes	complemented
20	C₁₇²	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor, Sylow
21	C₂×C₇×C₁₇	C₂×C₃×C₁₇	1	18	yes	complemented
22	D₇×C₁₇	C₂×C₃×C₁₇	1	36	yes	complemented
23	C₂²×C₃×C₁₇	—	7	18	no	—
24	C₇×C₁₇	C₂²×C₃×C₁₇	1	18	yes	complemented
25	C₂×C₃×C₁₇	—	7	36	no	—
26	C₂×C₃×C₁₇	—	7	18	no	—
27	C₂×AGL(1,𝔽₇)	C₁₇²	1	1	yes	characteristic, direct product factor, Hall
28	C₂²×C₁₇	—	7	18	no	—
29	C₃×C₁₇	—	7	18	no	—
30	C₂×ΓL(1,𝔽₈)	C₂×C₁₇²	1	1	yes	characteristic, complemented
31	AGL(1,𝔽₇)	C₂×C₁₇²	1	2	yes	direct product factor
32	C₂×C₁₇	—	7	36	no	—
33	C₂×C₁₇	C₁₇×AGL(1,𝔽₇)	1	18	yes	direct product factor
34	C₂×D₇	C₃×C₁₇²	1	1	yes	characteristic, complemented, Hall
35	ΓL(1,𝔽₈)	C₂²×C₁₇²	1	1	yes	characteristic, complemented, Hall
36	C₁₇	C₂×C₁₇×AGL(1,𝔽₇)	1	18	yes	direct product factor
37	C₂×C₇	C₂×C₃×C₁₇²	1	1	yes	characteristic, complemented
38	D₇	C₂×C₃×C₁₇²	1	2	yes	complemented
39	C₂²×C₃	—	7	1	no	Hall
40	C₇	C₂²×C₃×C₁₇²	1	1	yes	characteristic, complemented, Sylow
41	C₂×C₃	—	7	2	no	—
42	C₂×C₃	—	7	1	no	—
43	C₂²	—	7	1	no	Sylow
44	C₃	—	7	1	no	Sylow
45	C₂	—	7	2	no	—
46	C₂	C₁₇²×AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₁₇²	AGL(1,𝔽₇)
commutator, cosocle, hypocenter	C₇	C₂²×C₃×C₁₇²
Frattini	C₁	G
Fitting, socle	C₂×C₇×C₁₇²	C₂×C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₁₇²

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₇²×ΓL(1,𝔽₈)	C₂	semidirect
2. C₁₇²×AGL(1,𝔽₇)	C₂	direct (N × H)
3. C₂×D₇×C₁₇²	C₃	semidirect
4. C₁₇²×ΓL(1,𝔽₈)	C₂²	semidirect
5. D₇×C₁₇²	C₂×C₃	semidirect
6. C₂×C₇×C₁₇²	C₂×C₃	semidirect
8. C₇×C₁₇²	C₂²×C₃	semidirect
11. C₂×C₁₇×AGL(1,𝔽₇)	C₁₇	direct (N × H)
14. C₂×C₁₇×ΓL(1,𝔽₈)	C₂×C₁₇	semidirect
15. C₁₇×AGL(1,𝔽₇)	C₂×C₁₇	direct (N × H)
17. C₂×C₁₇²	AGL(1,𝔽₇)	direct (N × H)
18. C₂×D₇×C₁₇	C₃×C₁₇	semidirect
19. C₁₇×ΓL(1,𝔽₈)	C₂²×C₁₇	semidirect
20. C₁₇²	C₂×AGL(1,𝔽₇)	direct (N × H)
21. C₂×C₇×C₁₇	C₂×C₃×C₁₇	semidirect
22. D₇×C₁₇	C₂×C₃×C₁₇	semidirect
24. C₇×C₁₇	C₂²×C₃×C₁₇	semidirect
27. C₂×AGL(1,𝔽₇)	C₁₇²	direct (N × H)
30. C₂×ΓL(1,𝔽₈)	C₂×C₁₇²	semidirect
31. AGL(1,𝔽₇)	C₂×C₁₇²	direct (N × H)
33. C₂×C₁₇	C₁₇×AGL(1,𝔽₇)	direct (N × H)
34. C₂×D₇	C₃×C₁₇²	semidirect
35. ΓL(1,𝔽₈)	C₂²×C₁₇²	semidirect
36. C₁₇	C₂×C₁₇×AGL(1,𝔽₇)	direct (N × H)
37. C₂×C₇	C₂×C₃×C₁₇²	semidirect
38. D₇	C₂×C₃×C₁₇²	semidirect
40. C₇	C₂²×C₃×C₁₇²	semidirect
46. C₂	C₁₇²×AGL(1,𝔽₇)	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×C₁₇²×ΓL(1,𝔽₈)	2
C₁₇²×AGL(1,𝔽₇)	2
C₂×D₇×C₁₇²	3
C₂²×C₃×C₁₇²	7
C₂×C₁₇×AGL(1,𝔽₇)	17

Subgroups — C2×C172×AGL(1,𝔽7) #37566246

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₇²×AGL(1,𝔽₇) #37566246