Subgroups — C₂×D₃×ΓL(1,𝔽₈) #6990

Order 252.

44 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
2	D₃×ΓL(1,𝔽₈)	C₂	1	2	yes	maximal, direct product factor
3	C₂×D₃×C₇	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂²×ΓL(1,𝔽₈)	—	3	1	no	maximal
5	C₃×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₂×C₃×C₇	C₂×C₃	1	1	yes	characteristic, complemented
7	D₃×C₇	C₂×C₃	1	2	yes	complemented
8	C₂×ΓL(1,𝔽₈)	—	2	1	no	—
9	C₂×ΓL(1,𝔽₈)	—	3	2	no	—
10	C₂×ΓL(1,𝔽₈)	D₃	1	1	yes	characteristic, direct product factor
11	C₂×C₃×D₃	—	7	1	no	maximal, Hall
12	C₂²×C₇	—	3	1	no	Hall
13	C₃×C₇	C₂²×C₃	1	1	yes	characteristic, complemented
14	ΓL(1,𝔽₈)	—	2	1	no	—
15	ΓL(1,𝔽₈)	C₂×D₃	1	1	yes	characteristic, direct product factor
16	C₂×C₃²	—	7	1	no	—
17	C₃×D₃	—	7	2	no	—
18	C₂×C₇	—	3	2	no	—
19	C₂×C₇	C₃×D₃	1	1	yes	characteristic, complemented
20	C₂²×C₃	—	21	1	no	—
21	C₂×D₃	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
22	C₃²	—	7	1	no	Sylow
23	C₇	C₂×C₃×D₃	1	1	yes	characteristic, complemented, Sylow
24	C₂×C₃	—	7	1	no	—
25	C₂×C₃	—	14	1	no	—
26	C₂×C₃	—	21	2	no	—
27	C₂×C₃	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
28	D₃	C₂×ΓL(1,𝔽₈)	1	2	yes	direct product factor
29	C₂²	—	3	1	no	Sylow
30	C₃	—	7	1	no	—
31	C₃	—	14	1	no	—
32	C₃	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
33	C₂	—	3	2	no	—
34	C₂	D₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	D₃×ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₃×C₇	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₇	C₂×C₃
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₂

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.