Subgroups — ΓL(1,𝔽₈)⋊#13093637 #44531270

Order 31332.

24 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×D₃₇₃	C₂	1	1	yes	maximal, characteristic
2	C₇⋊#13093637	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	ΓL(1,𝔽₈)×C₃₇₃	C₄	1	1	yes	characteristic, complemented, Hall
4	C₇×D₃₇₃	C₂×C₃	1	1	yes	characteristic
5	C₃×#13093637	—	7	1	no	maximal, Hall
6	C₇×C₃₇₃	C₃×C₄	1	1	yes	characteristic, complemented, Hall
7	C₃×D₃₇₃	—	7	1	no	—
8	#13093637	—	7	1	no	Hall
9	C₃×C₃₇₃	—	7	1	no	Hall
10	D₃₇₃	AGL(1,𝔽₇)	1	1	yes	characteristic
11	C₃₇₃	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic, complemented, Sylow
12	ΓL(1,𝔽₈)⋊C₄	—	373	1	no	maximal, Hall
13	C₂×ΓL(1,𝔽₈)	—	373	1	no	—
14	Dic₇	—	373	1	no	Hall
15	ΓL(1,𝔽₈)	#13093637	1	1	yes	characteristic, complemented, Hall
16	C₂×C₇	—	373	1	no	—
17	C₃×C₄	—	2611	1	no	Hall
18	C₇	C₃×#13093637	1	1	yes	characteristic, complemented, Sylow
19	C₂×C₃	—	2611	1	no	—
20	C₄	—	2611	1	no	Sylow
21	C₃	—	7	1	no	Sylow
22	C₂	—	373	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₇×C₃₇₃	C₃×C₄
prime radical	ΓL(1,𝔽₈)×D₃₇₃	C₂
cosocle	C₇×D₃₇₃	C₂×C₃
LCS	G ▷ C₇×C₃₇₃
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.