Subgroups — C₁₀₉×ΓL(1,𝔽₈)⋊D₇ #44545211

Order 32046.

40 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×ΓL(1,𝔽₈)×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇⋊D₇×C₁₀₉	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₇²×C₁₀₉	C₂×C₃	1	1	yes	characteristic, complemented, Hall
4	AGL(1,𝔽₇)×C₁₀₉	—	7	1	no	maximal
5	C₃×D₇×C₁₀₉	—	7	1	no	maximal
6	C₃×C₇×C₁₀₉	—	7	1	no	—
7	ΓL(1,𝔽₈)×C₁₀₉	D₇	1	1	yes	characteristic, complemented
8	D₇×C₁₀₉	—	7	1	no	—
9	D₇×C₁₀₉	—	7	1	no	—
10	D₇×C₁₀₉	—	21	2	no	—
11	C₇×C₁₀₉	—	3	2	no	—
12	C₇×C₁₀₉	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
13	C₇×C₁₀₉	C₃×D₇	1	1	yes	characteristic, complemented
14	C₂×C₃×C₁₀₉	—	49	1	no	Hall
15	C₃×C₁₀₉	—	7	1	no	Hall
16	ΓL(1,𝔽₈)⋊D₇	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂×C₁₀₉	—	49	1	no	Hall
18	C₇×ΓL(1,𝔽₈)	C₂×C₁₀₉	1	1	yes	characteristic, complemented, Hall
19	C₁₀₉	ΓL(1,𝔽₈)⋊D₇	1	1	yes	characteristic, direct product factor, Sylow
20	C₇⋊D₇	C₃×C₁₀₉	1	1	yes	characteristic, complemented, Hall
21	C₇²	C₂×C₃×C₁₀₉	1	1	yes	characteristic, complemented, Sylow
22	C₃×D₇	—	7	1	no	—
23	AGL(1,𝔽₇)	—	7	1	no	—
24	C₃×C₇	—	7	1	no	—
25	ΓL(1,𝔽₈)	D₇×C₁₀₉	1	1	yes	characteristic, complemented
26	D₇	—	7	1	no	—
27	D₇	—	7	1	no	—
28	D₇	—	21	2	no	—
29	C₇	—	3	2	no	—
30	C₇	C₃×D₇×C₁₀₉	1	1	yes	characteristic, complemented
31	C₇	AGL(1,𝔽₇)×C₁₀₉	1	1	yes	characteristic, complemented
32	C₂×C₃	—	49	1	no	Hall
33	C₃	—	7	1	no	Sylow
34	C₂	—	49	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁₀₉	ΓL(1,𝔽₈)⋊D₇
commutator, cosocle, hypocenter	C₇²	C₂×C₃×C₁₀₉
Frattini	C₁	G
Fitting, socle	C₇²×C₁₀₉	C₂×C₃
LCS	G ▷ C₇²
UCS	C₁ ◁ C₁₀₉

Complements

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

N	H ≅ G/N	type
1. C₇×ΓL(1,𝔽₈)×C₁₀₉	C₂	semidirect
2. C₇⋊D₇×C₁₀₉	C₃	semidirect
3. C₇²×C₁₀₉	C₂×C₃	semidirect
7. ΓL(1,𝔽₈)×C₁₀₉	D₇	semidirect
12. C₇×C₁₀₉	AGL(1,𝔽₇)	semidirect
13. C₇×C₁₀₉	C₃×D₇	semidirect
16. ΓL(1,𝔽₈)⋊D₇	C₁₀₉	direct (N × H)
18. C₇×ΓL(1,𝔽₈)	C₂×C₁₀₉	semidirect
19. C₁₀₉	ΓL(1,𝔽₈)⋊D₇	direct (N × H)
20. C₇⋊D₇	C₃×C₁₀₉	semidirect
21. C₇²	C₂×C₃×C₁₀₉	semidirect
25. ΓL(1,𝔽₈)	D₇×C₁₀₉	semidirect
30. C₇	C₃×D₇×C₁₀₉	semidirect
31. C₇	AGL(1,𝔽₇)×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₇×ΓL(1,𝔽₈)×C₁₀₉	2
C₇⋊D₇×C₁₀₉	3
AGL(1,𝔽₇)×C₁₀₉	7
C₃×D₇×C₁₀₉	7
ΓL(1,𝔽₈)⋊D₇	109

Subgroups — C109×ΓL(1,𝔽8)⋊D7 #44545211

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×ΓL(1,𝔽₈)⋊D₇ #44545211