Subgroups — ΓL(1,𝔽₈)×D₈₅₁ #49088367

Order 35742.

32 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×C₂₃×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇×D₈₅₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₇×C₂₃×C₃₇	C₂×C₃	1	1	yes	characteristic, complemented, Hall
4	C₃×D₈₅₁	—	7	1	no	maximal, Hall
5	C₃×C₂₃×C₃₇	—	7	1	no	Hall
6	D₈₅₁	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
7	ΓL(1,𝔽₈)×D₃₇	—	23	1	no	maximal, Hall
8	ΓL(1,𝔽₈)×D₂₃	—	37	1	no	maximal, Hall
9	C₂₃×C₃₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
10	ΓL(1,𝔽₈)×C₃₇	D₂₃	1	1	yes	characteristic, complemented, Hall
11	C₇×D₃₇	—	23	1	no	Hall
12	ΓL(1,𝔽₈)×C₂₃	D₃₇	1	1	yes	characteristic, complemented, Hall
13	C₇×D₂₃	—	37	1	no	Hall
14	C₇×C₃₇	C₃×D₂₃	1	1	yes	characteristic, complemented, Hall
15	C₃×D₃₇	—	161	1	no	Hall
16	C₇×C₂₃	C₃×D₃₇	1	1	yes	characteristic, complemented, Hall
17	C₃×D₂₃	—	259	1	no	Hall
18	C₃×C₃₇	—	7	1	no	Hall
19	D₃₇	—	23	1	no	Hall
20	C₃×C₂₃	—	7	1	no	Hall
21	D₂₃	—	37	1	no	Hall
22	C₂×ΓL(1,𝔽₈)	—	851	1	no	Hall
23	C₃₇	ΓL(1,𝔽₈)×D₂₃	1	1	yes	characteristic, complemented, Sylow
24	C₂₃	ΓL(1,𝔽₈)×D₃₇	1	1	yes	characteristic, complemented, Sylow
25	ΓL(1,𝔽₈)	D₈₅₁	1	1	yes	characteristic, direct product factor, Hall
26	C₂×C₇	—	851	1	no	Hall
27	C₇	C₃×D₈₅₁	1	1	yes	characteristic, complemented, Sylow
28	C₂×C₃	—	5957	1	no	Hall
29	C₃	—	7	1	no	Sylow
30	C₂	—	851	1	no	Sylow

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₂₃×C₃₇	C₂×C₃
LCS	G ▷ C₇×C₂₃×C₃₇
UCS	C₁

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

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