Subgroups — Dic₁₄ #284

Order 56.

12 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇	C₂	1	1	yes	maximal, characteristic
2	Dic₇	C₂	1	2	yes	maximal
3	C₂×C₇	C₂²	1	1	yes	characteristic
4	Q₈	—	7	1	no	maximal, Sylow
5	C₇	Q₈	1	1	yes	characteristic, complemented, Sylow
6	C₄	—	7	2	no	—
7	C₄	D₇	1	1	yes	characteristic
8	C₂	C₂×D₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×D₇
commutator, socle, prime radical, cosocle	C₂×C₇	C₂²
Fitting	C₄×C₇	C₂
hypercenter	C₄	D₇
hypocenter	C₇	Q₈
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₄

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
5. C₇	Q₈	semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.