Subgroups — ΓL(1,𝔽₈)⋊C₄×C₂₇₇ #37551622

Order 23268.

24 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×ΓL(1,𝔽₈)×C₂₇₇	C₂	1	1	yes	maximal, characteristic
2	Dic₇×C₂₇₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	ΓL(1,𝔽₈)×C₂₇₇	C₄	1	1	yes	characteristic, complemented, Hall
4	C₂×C₇×C₂₇₇	C₂×C₃	1	1	yes	characteristic
5	C₃×C₄×C₂₇₇	—	7	1	no	maximal, Hall
6	C₇×C₂₇₇	C₃×C₄	1	1	yes	characteristic, complemented, Hall
7	C₂×C₃×C₂₇₇	—	7	1	no	—
8	C₄×C₂₇₇	—	7	1	no	Hall
9	C₃×C₂₇₇	—	7	1	no	Hall
10	C₂×C₂₇₇	AGL(1,𝔽₇)	1	1	yes	characteristic
11	C₂₇₇	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
12	ΓL(1,𝔽₈)⋊C₄	C₂₇₇	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×ΓL(1,𝔽₈)	C₂×C₂₇₇	1	1	yes	characteristic
14	Dic₇	C₃×C₂₇₇	1	1	yes	characteristic, complemented, Hall
15	ΓL(1,𝔽₈)	C₄×C₂₇₇	1	1	yes	characteristic, complemented, Hall
16	C₂×C₇	C₂×C₃×C₂₇₇	1	1	yes	characteristic
17	C₃×C₄	—	7	1	no	Hall
18	C₇	C₃×C₄×C₂₇₇	1	1	yes	characteristic, complemented, Sylow
19	C₂×C₃	—	7	1	no	—
20	C₄	—	7	1	no	Sylow
21	C₃	—	7	1	no	Sylow
22	C₂	AGL(1,𝔽₇)×C₂₇₇	1	1	yes	characteristic

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.