Subgroups — ΓL(1,𝔽₈)×#11715871 #30633872

Order 16212.

24 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×D₁₉₃	C₂	1	1	yes	maximal, characteristic
2	C₇×#11715871	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	ΓL(1,𝔽₈)×C₁₉₃	C₄	1	1	yes	characteristic, complemented, Hall
4	C₇×D₁₉₃	C₂×C₃	1	1	yes	characteristic
5	C₃×#11715871	—	7	1	no	maximal, Hall
6	C₇×C₁₉₃	C₃×C₄	1	1	yes	characteristic, complemented, Hall
7	C₃×D₁₉₃	—	7	1	no	—
8	#11715871	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
9	C₃×C₁₉₃	—	7	1	no	Hall
10	D₁₉₃	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
11	C₁₉₃	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
12	C₄×ΓL(1,𝔽₈)	—	193	1	no	maximal, Hall
13	C₂×ΓL(1,𝔽₈)	—	193	1	no	—
14	C₄×C₇	—	193	1	no	Hall
15	ΓL(1,𝔽₈)	#11715871	1	1	yes	characteristic, direct product factor, Hall
16	C₂×C₇	—	193	1	no	—
17	C₃×C₄	—	1351	1	no	Hall
18	C₇	C₃×#11715871	1	1	yes	characteristic, complemented, Sylow
19	C₂×C₃	—	1351	1	no	—
20	C₄	—	193	1	no	Sylow
21	C₃	—	7	1	no	Sylow
22	C₂	—	193	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₇×C₁₉₃	C₃×C₄
prime radical	ΓL(1,𝔽₈)×D₁₉₃	C₂
cosocle	C₇×D₁₉₃	C₂×C₃
LCS	G ▷ C₇×C₁₉₃
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.