Subgroups — #14550787

Order 2187.

249 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10624885	C₃	1	1	yes	maximal, characteristic
2	#10624692	C₃	1	3	yes	maximal
3	#10624470	C₃	1	6	yes	maximal, complemented
4	H(3,ℤ/9)	C₃	1	3	yes	maximal, complemented
5	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
6	#6896	—	3	3	no	—
7	#6896	—	3	6	no	—
8	#6895	—	3	1	no	—
9	#6895	—	3	1	no	—
10	#6891	—	3	1	no	—
11	#6887	C₃²	1	9	yes	—
12	#6887	C₃²	1	1	yes	characteristic
13	#6887	C₃²	1	2	yes	—
14	#6868	—	3	3	no	—
15	#6867	—	3	3	no	—
16	#6866	—	3	3	no	—
17	C₃⁴	H(3,𝔽₃)	1	1	yes	characteristic
18	C₃×3₋¹⁺²	—	3	3	no	—
19	C₃×3₋¹⁺²	—	3	1	no	—
20	C₃×3₋¹⁺²	—	3	2	no	—
21	C₃×3₋¹⁺²	—	3	9	no	—
22	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
23	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
24	C₃²×C₉	C₃³	1	1	yes	characteristic
25	C₉⋊C₉	—	3	9	no	—
26	C₉⋊C₉	—	3	9	no	—
27	C₉⋊C₉	—	3	2	no	—
28	C₉⋊C₉	—	3	2	no	—
29	C₉⋊C₉	—	3	2	no	—
30	C₉⋊C₉	—	3	18	no	—
31	C₃²⋊C₉	—	3	6	no	—
32	C₃²⋊C₉	—	3	1	no	—
33	C₃²⋊C₉	—	3	2	no	—
34	C₃²⋊C₉	—	3	9	no	—
35	C₉²	—	3	9	no	—
36	C₉²	3₋¹⁺²	1	3	yes	—
37	C₉²	3₋¹⁺²	1	6	yes	—
38	C₃³	—	3	1	no	—
39	C₃³	—	3	3	no	—
40	C₃³	—	27	1	no	—
41	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
42	3₋¹⁺²	—	9	3	no	—
43	3₋¹⁺²	—	9	6	no	—
44	3₋¹⁺²	—	27	3	no	—
45	C₃×C₉	—	3	3	no	—
46	C₃×C₉	—	3	6	no	—
47	C₃×C₉	—	3	1	no	—
48	C₃×C₉	—	3	1	no	—
49	C₃×C₉	—	3	3	no	—
50	C₃×C₉	—	3	3	no	—
51	C₃×C₉	—	9	9	no	—
52	C₃×C₉	C₃×3₋¹⁺²	1	1	yes	characteristic
53	C₃×C₉	C₃×3₋¹⁺²	1	2	yes	—
54	C₃×C₉	#737	1	9	yes	—
55	C₃²	—	9	1	no	—
56	C₃²	—	9	1	no	—
57	C₃²	—	9	2	no	—
58	C₃²	—	9	1	no	—
59	C₃²	—	27	3	no	—
60	C₃²	#6891	1	1	yes	characteristic
61	C₃²	#6893	1	3	yes	—
62	C₉	—	9	3	no	—
63	C₉	—	9	6	no	—
64	C₉	—	9	1	no	—
65	C₉	—	9	1	no	—
66	C₉	—	27	9	no	—
67	C₉	#6899	1	3	yes	—
68	C₉	#6899	1	6	yes	—
69	C₃	—	3	1	no	—
70	C₃	—	3	3	no	—
71	C₃	—	27	1	no	—
72	C₃	#10624570	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃	#10624570
commutator, Frattini, cosocle	C₃²×C₉	C₃³
prime radical	C₃⁴	H(3,𝔽₃)
LCS	G ▷ C₃²×C₉ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃ ◁ C₃²×C₉ ◁ G
LECS	G ▷ C₃²×C₉ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #10624470	C₃	semidirect
4. H(3,ℤ/9)	C₃	semidirect

Maximal subgroups

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

subgroup	index
#10624885	3
#10624692	3
#10624470	3
H(3,ℤ/9)	3