Subgroups — C₁₀₉×#64422 #44545240

Order 32046.

34 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#3862	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	C₂×ΓL(1,𝔽₈)×C₁₀₉	—	7	1	no	maximal
6	ΓL(1,𝔽₈)×C₁₀₉	—	7	1	no	—
7	ΓL(1,𝔽₈)×C₁₀₉	—	7	1	no	—
8	C₂×C₇×C₁₀₉	—	7	1	no	—
9	D₇×C₁₀₉	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
10	C₇×C₁₀₉	—	6	1	no	—
11	C₇×C₁₀₉	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
12	C₇×C₁₀₉	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
13	C₂×C₃×C₁₀₉	—	49	1	no	Hall
14	C₃×C₁₀₉	—	49	1	no	Hall
15	#64422	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂×C₁₀₉	—	7	1	no	Hall
17	#3862	C₂×C₁₀₉	1	1	yes	characteristic, complemented, Hall
18	C₁₀₉	#64422	1	1	yes	characteristic, direct product factor, Sylow
19	C₇×D₇	C₃×C₁₀₉	1	1	yes	characteristic, complemented, Hall
20	C₇²	C₂×C₃×C₁₀₉	1	1	yes	characteristic, complemented, Sylow
21	C₂×ΓL(1,𝔽₈)	—	7	1	no	—
22	AGL(1,𝔽₇)	—	7	1	no	—
23	ΓL(1,𝔽₈)	—	7	1	no	—
24	ΓL(1,𝔽₈)	—	7	1	no	—
25	C₂×C₇	—	7	1	no	—
26	D₇	ΓL(1,𝔽₈)×C₁₀₉	1	1	yes	characteristic, complemented
27	C₇	—	6	1	no	—
28	C₇	C₂×ΓL(1,𝔽₈)×C₁₀₉	1	1	yes	characteristic, complemented
29	C₇	AGL(1,𝔽₇)×C₁₀₉	1	1	yes	characteristic, complemented
30	C₂×C₃	—	49	1	no	Hall
31	C₃	—	49	1	no	Sylow
32	C₂	—	7	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁₀₉	#64422
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. C₁₀₉×#3862	C₂	semidirect
2. C₇×D₇×C₁₀₉	C₃	semidirect
3. C₇²×C₁₀₉	C₂×C₃	semidirect
9. D₇×C₁₀₉	ΓL(1,𝔽₈)	semidirect
11. C₇×C₁₀₉	AGL(1,𝔽₇)	semidirect
12. C₇×C₁₀₉	C₂×ΓL(1,𝔽₈)	semidirect
15. #64422	C₁₀₉	direct (N × H)
17. #3862	C₂×C₁₀₉	semidirect
18. C₁₀₉	#64422	direct (N × H)
19. C₇×D₇	C₃×C₁₀₉	semidirect
20. C₇²	C₂×C₃×C₁₀₉	semidirect
26. D₇	ΓL(1,𝔽₈)×C₁₀₉	semidirect
28. C₇	C₂×ΓL(1,𝔽₈)×C₁₀₉	semidirect
29. 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
C₁₀₉×#3862	2
C₇×D₇×C₁₀₉	3
AGL(1,𝔽₇)×C₁₀₉	7
C₂×ΓL(1,𝔽₈)×C₁₀₉	7
#64422	109

Subgroups — C109×#64422 #44545240

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#64422 #44545240