Subgroups — A₄⋊AGL(1,𝔽₇) #92734

Order 504.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	A₄×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇⋊S₄	C₃	1	1	yes	maximal, characteristic, complemented
3	(C₂²⋊D₇)⋊C₃	—	3	1	no	maximal
4	C₃⋊AGL(1,𝔽₇)	—	4	1	no	maximal
5	C₂²⋊ΓL(1,𝔽₈)	—	2	1	no	—
6	C₇×A₄	C₂×C₃	1	1	yes	characteristic, complemented
7	C₂²×ΓL(1,𝔽₈)	D₃	1	1	yes	characteristic, complemented
8	C₂×AGL(1,𝔽₇)	—	3	1	no	—
9	ΓL(1,𝔽₈)⋊C₄	—	3	1	no	—
10	C₃×S₄	—	7	1	no	maximal, Hall
11	C₃×ΓL(1,𝔽₈)	—	4	1	no	Hall
12	C₂²⋊D₇	—	3	1	no	Hall
13	D₂₁	—	4	1	no	—
14	C₂×ΓL(1,𝔽₈)	—	3	1	no	—
15	AGL(1,𝔽₇)	—	6	1	no	—
16	C₃×A₄	—	7	1	no	—
17	C₂²×C₇	C₃×D₃	1	1	yes	characteristic, complemented
18	C₂×D₇	—	3	1	no	—
19	Dic₇	—	3	1	no	—
20	S₄	—	7	1	no	—
21	C₃×D₄	—	21	1	no	—
22	C₃×C₇	—	4	1	no	—
23	ΓL(1,𝔽₈)	—	8	1	no	—
24	ΓL(1,𝔽₈)	S₄	1	1	yes	characteristic, complemented
25	C₃×D₃	—	28	1	no	—
26	C₂×C₇	—	3	1	no	—
27	D₇	—	6	1	no	—
28	C₂²×C₃	—	7	1	no	—
29	C₂²×C₃	—	21	1	no	—
30	A₄	—	14	1	no	—
31	A₄	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
32	C₃×C₄	—	21	1	no	—
33	C₃²	—	28	1	no	Sylow
34	D₄	—	21	1	no	Sylow
35	C₇	C₃×S₄	1	1	yes	characteristic, complemented, Sylow
36	C₂×C₃	—	21	1	no	—
37	C₂×C₃	—	42	1	no	—
38	D₃	—	28	1	no	—
39	C₂²	—	21	1	no	—
40	C₂²	C₃⋊AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
41	C₄	—	21	1	no	—
42	C₃	—	4	1	no	—
43	C₃	—	7	1	no	—
44	C₃	—	56	1	no	—
45	C₂	—	3	1	no	—
46	C₂	—	42	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₇×A₄	C₂×C₃
Fitting, socle	C₂²×C₇	C₃×D₃
LCS	G ▷ C₇×A₄
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.