Subgroups — #4246

Order 168.

32 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×D₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	D₂₈	—	3	1	no	maximal, Hall
5	C₂×C₃×C₇	C₂²	1	1	yes	characteristic
6	D₂₁	—	2	1	no	—
7	C₃×D₇	—	2	1	no	—
8	C₂×D₇	—	3	1	no	—
9	C₂×D₇	D₃	1	1	yes	characteristic, complemented
10	C₄×C₇	—	3	1	no	—
11	#67	—	7	1	no	maximal, Hall
12	C₃×C₇	D₄	1	1	yes	characteristic, complemented, Hall
13	C₂×C₇	C₂×D₃	1	1	yes	characteristic
14	D₇	—	2	1	no	—
15	D₇	—	6	1	no	—
16	C₂²×C₃	—	7	1	no	—
17	C₂×D₃	—	7	1	no	—
18	Dic₃	D₇	1	1	yes	characteristic, complemented
19	D₄	—	21	1	no	Sylow
20	C₇	#67	1	1	yes	characteristic, complemented, Sylow
21	C₂×C₃	—	14	1	no	—
22	C₂×C₃	C₂×D₇	1	1	yes	characteristic
23	D₃	—	14	1	no	—
24	C₂²	—	7	1	no	—
25	C₂²	—	21	1	no	—
26	C₄	—	3	1	no	—
27	C₃	D₂₈	1	1	yes	characteristic, complemented, Sylow
28	C₂	—	14	1	no	—
29	C₂	—	42	1	no	—
30	C₂	D₃×D₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃×D₇
commutator, Fitting, socle, cosocle	C₂×C₃×C₇	C₂²
hypocenter	C₃×C₇	D₄
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.