Subgroups — ΓL(1,𝔽₈)×D₁₀₁⋊C₅ #35234607

Order 21210.

32 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×C₁₀₁⋊C₅	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇×D₁₀₁⋊C₅	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	ΓL(1,𝔽₈)×D₁₀₁	C₅	1	1	yes	maximal, characteristic, complemented, Hall
4	C₇×C₁₀₁⋊C₅	C₂×C₃	1	1	yes	characteristic, complemented, Hall
5	C₃×D₁₀₁⋊C₅	—	7	1	no	maximal, Hall
6	ΓL(1,𝔽₈)×C₁₀₁	C₂×C₅	1	1	yes	characteristic, complemented, Hall
7	C₃×C₁₀₁⋊C₅	—	7	1	no	Hall
8	C₇×D₁₀₁	C₃×C₅	1	1	yes	characteristic, complemented, Hall
9	D₁₀₁⋊C₅	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
10	C₇×C₁₀₁	C₂×C₃×C₅	1	1	yes	characteristic, complemented, Hall
11	C₃×D₁₀₁	—	7	1	no	Hall
12	C₁₀₁⋊C₅	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
13	C₃×C₁₀₁	—	7	1	no	Hall
14	C₂×C₅×ΓL(1,𝔽₈)	—	101	1	no	maximal, Hall
15	D₁₀₁	C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
16	C₅×ΓL(1,𝔽₈)	—	101	1	no	Hall
17	C₁₀₁	C₂×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
18	C₂×C₅×C₇	—	101	1	no	Hall
19	C₂×ΓL(1,𝔽₈)	—	505	1	no	Hall
20	C₅×C₇	—	101	1	no	Hall
21	C₂×C₃×C₅	—	707	1	no	Hall
22	ΓL(1,𝔽₈)	D₁₀₁⋊C₅	1	1	yes	characteristic, direct product factor, Hall
23	C₃×C₅	—	707	1	no	Hall
24	C₂×C₇	—	505	1	no	Hall
25	C₂×C₅	—	101	1	no	Hall
26	C₇	C₃×D₁₀₁⋊C₅	1	1	yes	characteristic, complemented, Sylow
27	C₂×C₃	—	3535	1	no	Hall
28	C₅	—	101	1	no	Sylow
29	C₃	—	7	1	no	Sylow
30	C₂	—	505	1	no	Sylow

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₁₀₁	C₂×C₃×C₅
LCS	G ▷ C₇×C₁₀₁
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. ΓL(1,𝔽₈)×C₁₀₁⋊C₅	C₂	semidirect
2. C₇×D₁₀₁⋊C₅	C₃	semidirect
3. ΓL(1,𝔽₈)×D₁₀₁	C₅	semidirect
4. C₇×C₁₀₁⋊C₅	C₂×C₃	semidirect
6. ΓL(1,𝔽₈)×C₁₀₁	C₂×C₅	semidirect
8. C₇×D₁₀₁	C₃×C₅	semidirect
9. D₁₀₁⋊C₅	ΓL(1,𝔽₈)	direct (N × H)
10. C₇×C₁₀₁	C₂×C₃×C₅	semidirect
12. C₁₀₁⋊C₅	C₂×ΓL(1,𝔽₈)	semidirect
15. D₁₀₁	C₅×ΓL(1,𝔽₈)	semidirect
17. C₁₀₁	C₂×C₅×ΓL(1,𝔽₈)	semidirect
22. ΓL(1,𝔽₈)	D₁₀₁⋊C₅	direct (N × H)
26. C₇	C₃×D₁₀₁⋊C₅	semidirect

Maximal subgroups

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

subgroup	index
ΓL(1,𝔽₈)×C₁₀₁⋊C₅	2
C₇×D₁₀₁⋊C₅	3
ΓL(1,𝔽₈)×D₁₀₁	5
C₃×D₁₀₁⋊C₅	7
C₂×C₅×ΓL(1,𝔽₈)	101

Subgroups — ΓL(1,𝔽8)×D101⋊C5 #35234607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×D₁₀₁⋊C₅ #35234607