Subgroups — C₃²×AGL(1,𝔽₇) #67394

Order 378.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃²×D₇	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×AGL(1,𝔽₇)	C₃	1	12	yes	maximal, direct product factor
4	C₃²×C₇	C₂×C₃	1	1	yes	characteristic, complemented
5	C₃×ΓL(1,𝔽₈)	C₂×C₃	1	12	yes	complemented
6	C₂×C₃³	—	7	1	no	maximal, Hall
7	C₃×D₇	C₃²	1	4	yes	complemented
8	AGL(1,𝔽₇)	C₃²	1	9	yes	direct product factor
9	C₃³	—	7	1	no	Sylow
10	C₃×C₇	C₂×C₃²	1	4	yes	complemented
11	ΓL(1,𝔽₈)	C₂×C₃²	1	9	yes	complemented
12	C₂×C₃²	—	7	12	no	—
13	C₂×C₃²	—	7	1	no	—
14	D₇	C₃³	1	1	yes	characteristic, complemented, Hall
15	C₃²	—	7	12	no	—
16	C₃²	AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
17	C₇	C₂×C₃³	1	1	yes	characteristic, complemented, Sylow
18	C₂×C₃	—	7	9	no	—
19	C₂×C₃	—	7	4	no	—
20	C₃	—	7	9	no	—
21	C₃	C₃×AGL(1,𝔽₇)	1	4	yes	direct product factor
22	C₂	—	7	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃²	AGL(1,𝔽₇)
commutator, cosocle, hypocenter	C₇	C₂×C₃³
Frattini	C₁	G
Fitting, socle	C₃²×C₇	C₂×C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₃²

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.