Subgroups — #67385

Order 378.

32 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃⋊D₂₁	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×D₂₁	—	3	1	no	maximal
4	C₃²×C₇	—	2	1	no	—
5	C₃²×C₇	D₃	1	1	yes	characteristic, complemented
6	C₃²×C₇	C₂×C₃	1	1	yes	characteristic, complemented
7	(C₃⋊D₃)⋊C₃	—	7	1	no	maximal, Hall
8	D₂₁	—	3	1	no	—
9	D₂₁	—	9	1	no	—
10	C₃×D₇	—	9	1	no	—
11	H(3,𝔽₃)	D₇	1	1	yes	characteristic, complemented, Sylow
12	C₃×C₇	—	3	1	no	—
13	C₃×C₇	—	3	1	no	—
14	C₃×C₇	—	6	1	no	—
15	C₃×C₇	C₃×D₃	1	1	yes	characteristic
16	C₃⋊D₃	—	7	1	no	—
17	C₃×D₃	—	21	1	no	—
18	D₇	—	9	1	no	Hall
19	C₃²	—	2	1	no	—
20	C₃²	C₃×D₇	1	1	yes	characteristic, complemented
21	C₃²	D₂₁	1	1	yes	characteristic, complemented
22	C₇	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic, complemented, Sylow
23	C₂×C₃	—	63	1	no	—
24	D₃	—	21	1	no	—
25	D₃	—	63	1	no	—
26	C₃	—	3	1	no	—
27	C₃	—	3	1	no	—
28	C₃	—	6	1	no	—
29	C₃	C₃×D₂₁	1	1	yes	characteristic
30	C₂	—	63	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃²×C₇	C₂×C₃
Frattini	C₃	C₃×D₂₁
Fitting	C₇×H(3,𝔽₃)	C₂
socle	C₃×C₇	C₃×D₃
LCS	G ▷ C₃²×C₇
UCS	C₁

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.