Subgroups — Dic₃×ΓL(1,𝔽₈) #6978

Order 252.

28 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
2	C₇×Dic₃	C₃	1	1	yes	maximal, characteristic, complemented
3	C₄×ΓL(1,𝔽₈)	—	3	1	no	maximal
4	C₃×ΓL(1,𝔽₈)	C₄	1	1	yes	characteristic, complemented, Hall
5	C₂×C₃×C₇	C₂×C₃	1	1	yes	characteristic
6	C₂×ΓL(1,𝔽₈)	—	2	1	no	—
7	C₂×ΓL(1,𝔽₈)	D₃	1	1	yes	characteristic
8	C₃×Dic₃	—	7	1	no	maximal, Hall
9	C₄×C₇	—	3	1	no	Hall
10	C₃×C₇	C₃×C₄	1	1	yes	characteristic, complemented
11	ΓL(1,𝔽₈)	—	2	1	no	—
12	ΓL(1,𝔽₈)	Dic₃	1	1	yes	characteristic, direct product factor
13	C₂×C₃²	—	7	1	no	—
14	C₂×C₇	C₃×D₃	1	1	yes	characteristic
15	C₃×C₄	—	21	1	no	—
16	Dic₃	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
17	C₃²	—	7	1	no	Sylow
18	C₇	C₃×Dic₃	1	1	yes	characteristic, complemented, Sylow
19	C₂×C₃	—	7	1	no	—
20	C₂×C₃	—	14	1	no	—
21	C₂×C₃	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
22	C₄	—	3	1	no	Sylow
23	C₃	—	7	1	no	—
24	C₃	—	14	1	no	—
25	C₃	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
26	C₂	D₃×ΓL(1,𝔽₈)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃×ΓL(1,𝔽₈)
commutator, hypocenter	C₃×C₇	C₃×C₄
Fitting, socle, cosocle	C₂×C₃×C₇	C₂×C₃
prime radical	C₂×C₃×ΓL(1,𝔽₈)	C₂
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.