Subgroups — ΓL(1,𝔽₈)×Dic₃₃₇ #43246722

Order 28308.

24 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×ΓL(1,𝔽₈)×C₃₃₇	C₂	1	1	yes	maximal, characteristic
2	C₇×Dic₃₃₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	ΓL(1,𝔽₈)×C₃₃₇	C₄	1	1	yes	characteristic, complemented, Hall
4	C₂×C₇×C₃₃₇	C₂×C₃	1	1	yes	characteristic
5	C₃×Dic₃₃₇	—	7	1	no	maximal, Hall
6	C₇×C₃₃₇	C₃×C₄	1	1	yes	characteristic, complemented, Hall
7	C₂×C₃×C₃₃₇	—	7	1	no	—
8	Dic₃₃₇	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
9	C₃×C₃₃₇	—	7	1	no	Hall
10	C₂×C₃₃₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
11	C₃₃₇	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
12	C₄×ΓL(1,𝔽₈)	—	337	1	no	maximal, Hall
13	C₂×ΓL(1,𝔽₈)	D₃₃₇	1	1	yes	characteristic
14	C₄×C₇	—	337	1	no	Hall
15	ΓL(1,𝔽₈)	Dic₃₃₇	1	1	yes	characteristic, direct product factor, Hall
16	C₂×C₇	C₃×D₃₃₇	1	1	yes	characteristic
17	C₃×C₄	—	2359	1	no	Hall
18	C₇	C₃×Dic₃₃₇	1	1	yes	characteristic, complemented, Sylow
19	C₂×C₃	—	7	1	no	—
20	C₄	—	337	1	no	Sylow
21	C₃	—	7	1	no	Sylow
22	C₂	ΓL(1,𝔽₈)×D₃₃₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	ΓL(1,𝔽₈)×D₃₃₇
commutator, hypocenter	C₇×C₃₃₇	C₃×C₄
Fitting, socle, cosocle	C₂×C₇×C₃₃₇	C₂×C₃
prime radical	C₂×ΓL(1,𝔽₈)×C₃₃₇	C₂
LCS	G ▷ C₇×C₃₃₇
UCS	C₁ ◁ C₂

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.