Subgroups — C₃×C₃⋊AGL(1,𝔽₇) #67396

Order 378.

64 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	C₃×D₂₁	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃⋊AGL(1,𝔽₇)	C₃	1	3	yes	maximal, direct product factor
4	C₃×AGL(1,𝔽₇)	—	3	1	no	maximal
5	C₃²×C₇	C₂×C₃	1	1	yes	characteristic, complemented
6	C₃×ΓL(1,𝔽₈)	—	2	1	no	—
7	C₃×ΓL(1,𝔽₈)	—	2	3	no	—
8	C₃×ΓL(1,𝔽₈)	D₃	1	1	yes	characteristic, complemented
9	C₃×ΓL(1,𝔽₈)	C₂×C₃	1	3	yes	complemented
10	C₃²×D₃	—	7	1	no	maximal, Hall
11	D₂₁	C₃²	1	1	yes	characteristic, complemented
12	C₃×D₇	—	3	1	no	—
13	AGL(1,𝔽₇)	—	3	3	no	—
14	C₃³	—	7	1	no	Sylow
15	C₃×C₇	—	2	1	no	—
16	C₃×C₇	C₃×D₃	1	1	yes	characteristic, complemented
17	C₃×C₇	C₂×C₃²	1	1	yes	characteristic, complemented
18	ΓL(1,𝔽₈)	—	2	3	no	—
19	ΓL(1,𝔽₈)	C₃×D₃	1	3	yes	complemented
20	C₂×C₃²	—	21	1	no	—
21	C₃×D₃	—	7	3	no	—
22	C₃×D₃	—	7	1	no	—
23	D₇	—	3	1	no	Hall
24	C₃²	—	7	1	no	—
25	C₃²	—	7	3	no	—
26	C₃²	—	14	3	no	—
27	C₃²	—	14	1	no	—
28	C₃²	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
29	C₇	C₃²×D₃	1	1	yes	characteristic, complemented, Sylow
30	C₂×C₃	—	21	3	no	—
31	C₂×C₃	—	21	1	no	—
32	D₃	—	7	1	no	—
33	C₃	—	2	1	no	—
34	C₃	—	7	3	no	—
35	C₃	—	14	3	no	—
36	C₃	C₃×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
37	C₃	C₃⋊AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
38	C₂	—	21	1	no	Sylow

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

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.