Subgroups — ΓL(1,𝔽₈)×#11764308 #37551639

Order 23268.

24 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×D₂₇₇	C₂	1	1	yes	maximal, characteristic
2	C₇×#11764308	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	ΓL(1,𝔽₈)×C₂₇₇	C₄	1	1	yes	characteristic, complemented, Hall
4	C₇×D₂₇₇	C₂×C₃	1	1	yes	characteristic
5	C₃×#11764308	—	7	1	no	maximal, Hall
6	C₇×C₂₇₇	C₃×C₄	1	1	yes	characteristic, complemented, Hall
7	C₃×D₂₇₇	—	7	1	no	—
8	#11764308	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
9	C₃×C₂₇₇	—	7	1	no	Hall
10	D₂₇₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
11	C₂₇₇	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
12	C₄×ΓL(1,𝔽₈)	—	277	1	no	maximal, Hall
13	C₂×ΓL(1,𝔽₈)	—	277	1	no	—
14	C₄×C₇	—	277	1	no	Hall
15	ΓL(1,𝔽₈)	#11764308	1	1	yes	characteristic, direct product factor, Hall
16	C₂×C₇	—	277	1	no	—
17	C₃×C₄	—	1939	1	no	Hall
18	C₇	C₃×#11764308	1	1	yes	characteristic, complemented, Sylow
19	C₂×C₃	—	1939	1	no	—
20	C₄	—	277	1	no	Sylow
21	C₃	—	7	1	no	Sylow
22	C₂	—	277	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₇×C₂₇₇	C₃×C₄
prime radical	ΓL(1,𝔽₈)×D₂₇₇	C₂
cosocle	C₇×D₂₇₇	C₂×C₃
LCS	G ▷ C₇×C₂₇₇
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.