Subgroups — C₂×C₁₉²×ΓL(1,𝔽₈) #29503151

Order 15162.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉²×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, direct product factor, Hall
2	C₂×C₇×C₁₉²	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₇×C₁₉²	C₂×C₃	1	1	yes	characteristic, complemented, Hall
4	C₂×C₃×C₁₉²	—	7	1	no	maximal, Hall
5	C₃×C₁₉²	—	7	1	no	Hall
6	C₂×C₁₉×ΓL(1,𝔽₈)	C₁₉	1	20	yes	maximal, direct product factor
7	C₂×C₁₉²	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
8	C₁₉×ΓL(1,𝔽₈)	C₂×C₁₉	1	20	yes	direct product factor
9	C₁₉²	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
10	C₂×C₇×C₁₉	C₃×C₁₉	1	20	yes	complemented
11	C₇×C₁₉	C₂×C₃×C₁₉	1	20	yes	complemented
12	C₂×C₃×C₁₉	—	7	20	no	—
13	C₃×C₁₉	—	7	20	no	—
14	C₂×ΓL(1,𝔽₈)	C₁₉²	1	1	yes	characteristic, direct product factor, Hall
15	C₂×C₁₉	C₁₉×ΓL(1,𝔽₈)	1	20	yes	direct product factor
16	ΓL(1,𝔽₈)	C₂×C₁₉²	1	1	yes	characteristic, direct product factor, Hall
17	C₁₉	C₂×C₁₉×ΓL(1,𝔽₈)	1	20	yes	direct product factor
18	C₂×C₇	C₃×C₁₉²	1	1	yes	characteristic, complemented, Hall
19	C₇	C₂×C₃×C₁₉²	1	1	yes	characteristic, complemented, Sylow
20	C₂×C₃	—	7	1	no	Hall
21	C₃	—	7	1	no	Sylow
22	C₂	C₁₉²×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center, hypercenter	C₂×C₁₉²	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₂×C₃×C₁₉²
Frattini	C₁	G
Fitting, socle	C₂×C₇×C₁₉²	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₁₉²

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.