Subgroups — ΓL(1,𝔽₆₄) #67360

Order 378.

26 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4428	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	D₃×ΓL(1,𝔽₈)	—	3	1	no	maximal
3	C₇×D₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×ΓL(1,𝔽₈)	D₃	1	1	yes	characteristic
5	C₇×C₉	C₂×C₃	1	1	yes	characteristic, complemented
6	C₇⋊C₉	—	2	1	no	—
7	AGL(1,ℤ/9)	—	7	1	no	maximal, Hall
8	D₃×C₇	—	3	1	no	—
9	C₂×ΓL(1,𝔽₈)	—	9	1	no	—
10	3₋¹⁺²	—	7	1	no	Sylow
11	C₃×C₇	C₃×D₃	1	1	yes	characteristic
12	ΓL(1,𝔽₈)	—	3	1	no	—
13	C₃×D₃	—	21	1	no	—
14	D₉	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
15	C₂×C₇	—	9	1	no	Hall
16	C₃²	—	7	1	no	—
17	C₉	—	14	1	no	—
18	C₉	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
19	C₇	AGL(1,ℤ/9)	1	1	yes	characteristic, complemented, Sylow
20	C₂×C₃	—	63	1	no	—
21	D₃	—	3	1	no	—
22	C₃	—	21	1	no	—
23	C₃	D₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
24	C₂	—	9	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₇×C₉	C₂×C₃
Frattini	C₃	D₃×ΓL(1,𝔽₈)
socle	C₃×C₇	C₃×D₃
LCS	G ▷ C₇×C₉
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.