Subgroups — A₄×ΓL(1,𝔽₈) #6988

Order 252.

28 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊ΓL(1,𝔽₈)	C₃	1	2	yes	maximal, complemented
2	C₇×A₄	C₃	1	1	yes	maximal, characteristic, complemented
3	C₂²×ΓL(1,𝔽₈)	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×ΓL(1,𝔽₈)	—	4	1	no	maximal, Hall
5	C₂×ΓL(1,𝔽₈)	—	3	1	no	—
6	C₃×A₄	—	7	1	no	maximal, Hall
7	C₂²×C₇	C₃²	1	1	yes	characteristic, complemented, Hall
8	C₃×C₇	—	4	1	no	—
9	ΓL(1,𝔽₈)	—	4	2	no	—
10	ΓL(1,𝔽₈)	A₄	1	1	yes	characteristic, direct product factor
11	C₂×C₇	—	3	1	no	—
12	C₂²×C₃	—	7	1	no	—
13	A₄	—	7	2	no	—
14	A₄	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
15	C₃²	—	28	1	no	Sylow
16	C₇	C₃×A₄	1	1	yes	characteristic, complemented, Sylow
17	C₂×C₃	—	21	1	no	—
18	C₂²	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
19	C₃	—	4	1	no	—
20	C₃	—	7	1	no	—
21	C₃	—	28	2	no	—
22	C₂	—	3	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₂²×C₇	C₃²
LCS	G ▷ C₂²×C₇
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.