Subgroups — D₃²×C₇ #6996

Order 252.

44 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×D₃×C₇	C₂	1	2	yes	maximal, complemented
3	C₂×D₃×C₇	—	3	2	no	maximal
4	C₃²×C₇	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₃×C₇	—	3	2	no	—
6	D₃×C₇	—	3	2	no	—
7	D₃×C₇	—	6	1	no	—
8	D₃×C₇	D₃	1	2	yes	direct product factor
9	D₃²	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂²×C₇	—	9	1	no	Hall
11	C₃×C₇	—	2	1	no	—
12	C₃×C₇	C₂×D₃	1	2	yes	complemented
13	C₃⋊D₃	C₂×C₇	1	1	yes	characteristic, complemented
14	C₃×D₃	C₂×C₇	1	2	yes	complemented
15	C₂×C₇	—	3	2	no	—
16	C₂×C₇	—	9	1	no	—
17	C₂×D₃	—	3	2	no	—
18	C₃²	C₂²×C₇	1	1	yes	characteristic, complemented, Sylow
19	C₇	D₃²	1	1	yes	characteristic, direct product factor, Sylow
20	C₂×C₃	—	3	2	no	—
21	D₃	—	3	2	no	—
22	D₃	—	6	1	no	—
23	D₃	D₃×C₇	1	2	yes	direct product factor
24	C₂²	—	9	1	no	Sylow
25	C₃	—	2	1	no	—
26	C₃	C₂×D₃×C₇	1	2	yes	complemented
27	C₂	—	3	2	no	—
28	C₂	—	9	1	no	—

	subgroup	G/H
center, hypercenter	C₇	D₃²
commutator, cosocle, hypocenter	C₃²	C₂²×C₇
Frattini	C₁	G
Fitting, socle	C₃²×C₇	C₂²
LCS	G ▷ C₃²
UCS	C₁ ◁ C₇

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.