Subgroups — ΓL(1,𝔽₈)⋊D₂₇ #11765613

Order 1134.

38 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×C₂₇	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	ΓL(1,𝔽₈)⋊D₉	—	3	1	no	maximal
3	D₁₈₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₉×ΓL(1,𝔽₈)	D₃	1	1	yes	characteristic
5	C₇×C₂₇	C₂×C₃	1	1	yes	characteristic, complemented
6	C₇⋊C₂₇	—	2	1	no	—
7	C₃×D₂₇	—	7	1	no	maximal, Hall
8	C₃⋊AGL(1,𝔽₇)	—	9	1	no	—
9	D₆₃	—	3	1	no	—
10	C₃×C₂₇	—	7	1	no	Sylow
11	C₃×ΓL(1,𝔽₈)	D₉	1	1	yes	characteristic
12	C₇×C₉	C₃×D₃	1	1	yes	characteristic
13	C₇⋊C₉	—	2	1	no	—
14	C₃×D₉	—	21	1	no	—
15	D₂₇	—	7	1	no	—
16	D₂₁	—	9	1	no	—
17	AGL(1,𝔽₇)	—	27	1	no	—
18	C₃×C₉	—	7	1	no	—
19	C₂₇	—	14	1	no	—
20	C₂₇	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
21	C₃×C₇	C₃×D₉	1	1	yes	characteristic
22	ΓL(1,𝔽₈)	—	2	1	no	—
23	ΓL(1,𝔽₈)	D₂₇	1	1	yes	characteristic, complemented
24	C₃×D₃	—	63	1	no	—
25	D₉	—	21	1	no	—
26	D₇	—	27	1	no	Hall
27	C₃²	—	7	1	no	—
28	C₉	—	14	1	no	—
29	C₉	C₃⋊AGL(1,𝔽₇)	1	1	yes	characteristic
30	C₇	C₃×D₂₇	1	1	yes	characteristic, complemented, Sylow
31	C₂×C₃	—	189	1	no	—
32	D₃	—	63	1	no	—
33	C₃	—	7	1	no	—
34	C₃	—	14	1	no	—
35	C₃	ΓL(1,𝔽₈)⋊D₉	1	1	yes	characteristic
36	C₂	—	189	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₇×C₂₇	C₂×C₃
Frattini	C₉	C₃⋊AGL(1,𝔽₇)
socle	C₃×C₇	C₃×D₉
LCS	G ▷ C₇×C₂₇
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.