Subgroups — Dic₆₇₅₁ #41008849

Order 27004.

12 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₃×C₁₅₇	C₂	1	1	yes	maximal, characteristic
2	C₄₃×C₁₅₇	C₄	1	1	yes	characteristic, complemented, Hall
3	Dic₁₅₇	—	43	1	no	maximal, Hall
4	C₂×C₁₅₇	D₄₃	1	1	yes	characteristic
5	Dic₄₃	—	157	1	no	maximal, Hall
6	C₁₅₇	Dic₄₃	1	1	yes	characteristic, complemented, Sylow
7	C₂×C₄₃	D₁₅₇	1	1	yes	characteristic
8	C₄₃	Dic₁₅₇	1	1	yes	characteristic, complemented, Sylow
9	C₄	—	6751	1	no	Sylow
10	C₂	D₆₇₅₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₆₇₅₁
commutator, hypocenter	C₄₃×C₁₅₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₄₃×C₁₅₇	C₂
LCS	G ▷ C₄₃×C₁₅₇
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.