Subgroups — #14550628

Order 2187.

1994 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10624880	C₃	1	13	yes	maximal, complemented
2	C₃²×H(3,𝔽₃)	C₃²	1	13	yes	—
3	C₃²⋊H(3,𝔽₃)	—	3	156	no	—
4	C₃⁴	H(3,𝔽₃)	1	39	yes	complemented
5	C₃⁴	C₃³	1	1	yes	characteristic
6	C₃×H(3,𝔽₃)	—	3	156	no	—
7	C₃×H(3,𝔽₃)	—	3	351	no	—
8	C₃³	—	3	156	no	—
9	C₃³	—	27	39	no	—
10	C₃³	—	27	1	no	—
11	C₃³	C₃×H(3,𝔽₃)	1	13	yes	—
12	H(3,𝔽₃)	—	9	117	no	—
13	H(3,𝔽₃)	—	27	351	no	—
14	C₃²	—	9	13	no	—
15	C₃²	—	9	468	no	—
16	C₃²	—	9	39	no	—
17	C₃²	C₃²⋊H(3,𝔽₃)	1	13	yes	—
18	C₃	—	3	13	no	—
19	C₃	—	27	39	no	—
20	C₃	#10624562	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃	#10624562
commutator, Frattini, cosocle	C₃⁴	C₃³
LCS	G ▷ C₃⁴ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃ ◁ C₃⁴ ◁ G
LECS	G ▷ C₃⁴ ▷ C₃ ▷ C₁

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #10624880	C₃	semidirect
4. C₃⁴	H(3,𝔽₃)	semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#10624880	3