Subgroups — #14552617

Order 2187.

2456 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⁴×C₉	C₃	1	1	yes	maximal, characteristic
2	C₃×#6891	C₃	1	3	yes	maximal
3	#10624558	C₃	1	27	yes	maximal, complemented
4	#10624549	C₃	1	9	yes	maximal, complemented
5	C₃⁵	C₃²	1	1	yes	characteristic
6	C₃²×3₋¹⁺²	C₃²	1	9	yes	—
7	C₃³×C₉	—	3	9	no	—
8	C₃³×C₉	—	3	27	no	—
9	C₃³×C₉	C₃²	1	9	yes	—
10	C₃³×C₉	C₃²	1	3	yes	—
11	#6891	—	3	27	no	—
12	C₃×C₃²⋊C₉	C₃²	1	9	yes	—
13	C₃×C₃²⋊C₉	C₃²	1	18	yes	—
14	#6855	C₃²	1	81	yes	complemented
15	C₃⁴	—	3	9	no	—
16	C₃⁴	—	3	27	no	—
17	C₃⁴	H(3,𝔽₃)	1	9	yes	—
18	C₃⁴	C₃³	1	1	yes	characteristic
19	C₃⁴	C₃³	1	3	yes	—
20	C₃×3₋¹⁺²	—	3	27	no	—
21	C₃×3₋¹⁺²	—	3	81	no	—
22	C₃²×C₉	—	3	243	no	—
23	C₃²×C₉	—	3	27	no	—
24	C₃²×C₉	—	3	9	no	—
25	C₃²×C₉	—	3	18	no	—
26	C₃²×C₉	—	3	81	no	—
27	C₃²×C₉	3₋¹⁺²	1	27	yes	complemented
28	C₃²×C₉	C₃³	1	27	yes	—
29	C₃²×C₉	C₃³	1	9	yes	—
30	C₃²⋊C₉	—	3	27	no	—
31	C₃²⋊C₉	—	3	54	no	—
32	C₃³	—	3	243	no	—
33	C₃³	—	3	9	no	—
34	C₃³	—	3	9	no	—
35	C₃³	—	3	18	no	—
36	C₃³	—	3	27	no	—
37	C₃³	—	3	81	no	—
38	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
39	C₃³	C₃×H(3,𝔽₃)	1	9	yes	—
40	C₃³	C₃×3₋¹⁺²	1	9	yes	—
41	C₃³	#737	1	27	yes	—
42	C₃³	C₃⁴	1	1	yes	characteristic
43	3₋¹⁺²	—	9	81	no	—
44	C₃×C₉	—	3	243	no	—
45	C₃×C₉	—	3	9	no	—
46	C₃×C₉	—	3	18	no	—
47	C₃×C₉	—	3	81	no	—
48	C₃×C₉	—	3	27	no	—
49	C₃×C₉	—	3	81	no	—
50	C₃×C₉	C₃×3₋¹⁺²	1	27	yes	—
51	C₃²	—	3	243	no	—
52	C₃²	—	3	9	no	—
53	C₃²	—	3	9	no	—
54	C₃²	—	3	18	no	—
55	C₃²	—	3	27	no	—
56	C₃²	—	3	81	no	—
57	C₃²	#6891	1	9	yes	—
58	C₃²	#6893	1	27	yes	—
59	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
60	C₃²	C₃²×3₋¹⁺²	1	3	yes	—
61	C₃²	C₃×#737	1	9	yes	—
62	C₉	—	3	27	no	—
63	C₉	—	9	27	no	—
64	C₃	—	3	9	no	—
65	C₃	—	3	27	no	—
66	C₃	C₃×#6891	1	1	yes	characteristic
67	C₃	C₃×#6893	1	3	yes	—
68	C₃	#10624872	1	9	yes	—

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₃³	C₃⁴
prime radical	C₃⁵	C₃²
LCS	G ▷ C₃³ ▷ C₁
UCS	C₁ ◁ C₃³ ◁ G
LECS	G ▷ C₃³ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #10624558	C₃	semidirect
4. #10624549	C₃	semidirect
14. #6855	C₃²	semidirect
27. C₃²×C₉	3₋¹⁺²	semidirect

Maximal subgroups

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

subgroup	index
C₃⁴×C₉	3
C₃×#6891	3
#10624558	3
#10624549	3