Subgroups — Dic₃×D₇ #4242

Order 168.

32 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃×D₇
commutator, hypocenter	C₃×C₇	C₂×C₄
Fitting, socle, cosocle	C₂×C₃×C₇	C₂²
prime radical	C₂×C₃×D₇	C₂
LCS	G ▷ 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.