Subgroups — C₂³×C₃×D₇ #66756

Order 336.

268 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×D₇	C₂	1	14	yes	maximal, direct product factor
3	C₂³×D₇	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂²×C₃×C₇	C₂²	1	7	yes	complemented
5	C₂×C₃×D₇	C₂²	1	28	yes	direct product factor
6	C₂³×C₇	C₂×C₃	1	1	yes	characteristic, complemented
7	C₂²×D₇	C₂×C₃	1	14	yes	direct product factor
8	C₂⁴×C₃	—	7	1	no	maximal, Hall
9	C₂×C₃×C₇	C₂³	1	7	yes	complemented
10	C₃×D₇	C₂³	1	8	yes	direct product factor
11	C₂²×C₇	C₂²×C₃	1	7	yes	complemented
12	C₂×D₇	C₂²×C₃	1	28	yes	direct product factor
13	C₂³×C₃	—	7	14	no	—
14	C₂³×C₃	D₇	1	1	yes	characteristic, direct product factor
15	C₃×C₇	C₂⁴	1	1	yes	characteristic, complemented, Hall
16	C₂⁴	—	7	1	no	Sylow
17	C₂×C₇	C₂³×C₃	1	7	yes	complemented
18	D₇	C₂³×C₃	1	8	yes	direct product factor
19	C₂²×C₃	—	7	28	no	—
20	C₂²×C₃	C₂×D₇	1	7	yes	direct product factor
21	C₂³	—	7	14	no	—
22	C₂³	C₃×D₇	1	1	yes	characteristic, direct product factor
23	C₇	C₂⁴×C₃	1	1	yes	characteristic, complemented, Sylow
24	C₂×C₃	—	7	8	no	—
25	C₂×C₃	C₂²×D₇	1	7	yes	direct product factor
26	C₂²	—	7	28	no	—
27	C₂²	C₂×C₃×D₇	1	7	yes	direct product factor
28	C₃	C₂³×D₇	1	1	yes	characteristic, direct product factor, Sylow
29	C₂	—	7	8	no	—
30	C₂	C₂²×C₃×D₇	1	7	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂³×C₃	D₇
commutator, cosocle, hypocenter	C₇	C₂⁴×C₃
Frattini	C₁	G
Fitting, socle	C₂³×C₃×C₇	C₂
LCS	G ▷ C₇
UCS	C₁ ◁ C₂³×C₃

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.