Subgroups — C₂²⋊AΓL(1,𝔽₈) #10621675

Order 672.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×AGL(1,𝔽₈)	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	AΓL(1,𝔽₈)	—	4	1	no	maximal
3	C₂×AGL(1,𝔽₈)	—	3	1	no	—
4	C₂×C₂²⋊A₄	—	7	1	no	maximal, Hall
5	C₂²⋊ΓL(1,𝔽₈)	—	8	1	no	maximal
6	AGL(1,𝔽₈)	A₄	1	1	yes	characteristic, complemented
7	C₂²⋊A₄	—	7	1	no	—
8	C₂⁵	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
9	C₂²×C₇	—	8	1	no	—
10	C₂×A₄	—	28	3	no	—
11	C₂×A₄	—	28	1	no	—
12	C₂×A₄	—	28	1	no	—
13	ΓL(1,𝔽₈)	—	32	1	no	Hall
14	C₂⁴	—	3	1	no	—
15	C₂⁴	—	7	1	no	—
16	C₂⁴	—	21	1	no	—
17	C₂×C₇	—	24	1	no	—
18	A₄	—	28	1	no	—
19	A₄	—	28	3	no	—
20	A₄	—	28	1	no	—
21	C₂³	—	7	3	no	—
22	C₂³	—	7	1	no	—
23	C₂³	—	21	1	no	—
24	C₂³	—	21	1	no	—
25	C₂³	—	21	1	no	—
26	C₂³	—	21	3	no	—
27	C₂³	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
28	C₇	—	8	1	no	Sylow
29	C₂×C₃	—	112	1	no	—
30	C₂²	—	7	1	no	—
31	C₂²	—	7	3	no	—
32	C₂²	—	21	1	no	—
33	C₂²	—	21	1	no	—
34	C₂²	—	21	3	no	—
35	C₂²	—	21	1	no	—
36	C₂²	AΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
37	C₃	—	112	1	no	Sylow
38	C₂	—	3	1	no	—
39	C₂	—	7	1	no	—
40	C₂	—	21	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₂²×AGL(1,𝔽₈)	C₃
Fitting, socle	C₂⁵	ΓL(1,𝔽₈)
LCS	G ▷ C₂²×AGL(1,𝔽₈)
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.