Subgroups — #4248

Order 168.

24 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Dic₂₁	C₂	1	1	yes	maximal, characteristic
2	C₃×Dic₇	C₂	1	1	yes	maximal, characteristic
3	C₇×Dic₃	C₂	1	1	yes	maximal, characteristic
4	Dic₁₄	—	3	1	no	maximal, Hall
5	C₂×C₃×C₇	C₂²	1	1	yes	characteristic
6	C₄×C₇	—	3	1	no	—
7	Dic₇	—	3	1	no	—
8	Dic₇	D₃	1	1	yes	characteristic
9	Dic₆	—	7	1	no	maximal, Hall
10	C₃×C₇	Q₈	1	1	yes	characteristic, complemented, Hall
11	C₂×C₇	C₂×D₃	1	1	yes	characteristic
12	C₃×C₄	—	7	1	no	—
13	Dic₃	—	7	1	no	—
14	Dic₃	D₇	1	1	yes	characteristic
15	Q₈	—	21	1	no	Sylow
16	C₇	Dic₆	1	1	yes	characteristic, complemented, Sylow
17	C₂×C₃	C₂×D₇	1	1	yes	characteristic
18	C₄	—	3	1	no	—
19	C₄	—	7	1	no	—
20	C₄	—	21	1	no	—
21	C₃	Dic₁₄	1	1	yes	characteristic, complemented, Sylow
22	C₂	D₃×D₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃×D₇
commutator, Fitting, socle, prime radical, cosocle	C₂×C₃×C₇	C₂²
hypocenter	C₃×C₇	Q₈
LCS	G ▷ C₂×C₃×C₇ ▷ 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.