Subgroups — #14546232

Order 2187.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#6903	C₃	1	1	yes	maximal, characteristic
2	#10624502	C₃	1	3	yes	maximal, complemented
3	C₃×#737	C₉	1	1	yes	characteristic
4	#6903	—	3	9	no	—
5	C₃×C₂₇⋊C₃	—	3	1	no	—
6	C₃×C₂₇⋊C₃	—	3	1	no	—
7	C₃×C₂₇⋊C₃	C₉	1	2	yes	—
8	C₃²×C₂₇	—	3	1	no	—
9	C₃²×C₂₇	C₃²	1	1	yes	characteristic
10	C₈₁⋊C₃	—	3	3	no	—
11	#737	—	3	6	no	—
12	#737	—	3	3	no	—
13	C₃×3₋¹⁺²	—	3	2	no	—
14	C₃×3₋¹⁺²	C₂₇	1	2	yes	—
15	C₃×H(3,𝔽₃)	C₂₇	1	1	yes	characteristic
16	C₃²×C₉	—	3	1	no	—
17	C₃²×C₉	C₃×C₉	1	1	yes	characteristic
18	C₂₇⋊C₃	—	3	9	no	—
19	C₂₇⋊C₃	—	3	9	no	—
20	C₂₇⋊C₃	—	3	6	no	—
21	C₃×C₂₇	—	3	9	no	—
22	C₃×C₂₇	—	3	2	no	—
23	C₃×C₂₇	—	3	3	no	—
24	C₃×C₂₇	H(3,𝔽₃)	1	1	yes	characteristic
25	C₃×C₂₇	3₋¹⁺²	1	1	yes	characteristic
26	C₃×C₂₇	3₋¹⁺²	1	1	yes	characteristic
27	C₈₁	—	9	3	no	—
28	C₃³	—	3	1	no	—
29	C₃³	C₃×C₂₇	1	1	yes	characteristic
30	3₋¹⁺²	—	3	18	no	—
31	3₋¹⁺²	—	3	6	no	—
32	H(3,𝔽₃)	—	3	3	no	—
33	C₃×C₉	—	3	2	no	—
34	C₃×C₉	—	3	3	no	—
35	C₃×C₉	—	3	1	no	—
36	C₃×C₉	—	3	6	no	—
37	C₃×C₉	—	3	2	no	—
38	C₃×C₉	C₃²⋊C₉	1	1	yes	characteristic
39	C₃×C₉	C₂₇⋊C₃	1	2	yes	—
40	C₂₇	—	3	1	no	—
41	C₂₇	—	3	6	no	—
42	C₂₇	—	3	1	no	—
43	C₂₇	AΣL(1,𝔽₂₇)	1	3	yes	—
44	C₃²	—	3	1	no	—
45	C₃²	—	3	3	no	—
46	C₃²	—	9	1	no	—
47	C₃²	—	9	3	no	—
48	C₃²	C₃²⋊C₂₇	1	1	yes	characteristic
49	C₉	—	3	6	no	—
50	C₉	—	3	2	no	—
51	C₉	#6866	1	1	yes	characteristic
52	C₉	#6869	1	2	yes	—
53	C₃	—	3	1	no	—
54	C₃	—	9	1	no	—
55	C₃	—	9	3	no	—
56	C₃	#10624444	1	1	yes	characteristic

	subgroup	G/H
center	C₉	#6866
commutator	C₃³	C₃×C₂₇
Frattini, cosocle	C₃²×C₂₇	C₃²
socle	C₃	#10624444
prime radical	C₃×H(3,𝔽₃)	C₂₇
LCS	G ▷ C₃³ ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₉ ◁ C₃×C₂₇ ◁ C₃²×C₂₇ ◁ G
LECS	G ▷ C₃²×C₂₇ ▷ C₃×C₉ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #10624502	C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃×#6903	3
#10624502	3