Subgroups — #14550277

Order 2187.

2006 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×#6867	C₃	1	9	yes	maximal, complemented
3	#10624547	C₃	1	3	yes	maximal
4	C₃⁵	C₃²	1	1	yes	characteristic
5	C₃³×C₉	—	3	3	no	—
6	C₃³×C₉	—	3	6	no	—
7	C₃³×C₉	—	3	9	no	—
8	C₃³×C₉	—	3	18	no	—
9	C₃³×C₉	C₉	1	9	yes	complemented
10	C₃³×C₉	C₃²	1	3	yes	—
11	C₃×C₃²⋊C₉	—	3	9	no	—
12	C₃×C₃²⋊C₉	C₃²	1	9	yes	—
13	#6867	—	3	27	no	—
14	C₃⁴	—	3	27	no	—
15	C₃⁴	—	3	3	no	—
16	C₃⁴	—	3	6	no	—
17	C₃⁴	C₃×C₉	1	3	yes	—
18	C₃⁴	H(3,𝔽₃)	1	3	yes	—
19	C₃⁴	3₋¹⁺²	1	6	yes	—
20	C₃⁴	C₃³	1	1	yes	characteristic
21	C₃²×C₉	—	3	27	no	—
22	C₃²×C₉	—	3	162	no	—
23	C₃²×C₉	—	3	54	no	—
24	C₃²×C₉	—	3	27	no	—
25	C₃²×C₉	—	3	6	no	—
26	C₃²×C₉	—	3	3	no	—
27	C₃²×C₉	—	3	6	no	—
28	C₃²×C₉	—	3	12	no	—
29	C₃²×C₉	—	3	81	no	—
30	C₃²×C₉	—	3	9	no	—
31	C₃²×C₉	C₃×C₉	1	9	yes	—
32	C₃²×C₉	H(3,𝔽₃)	1	9	yes	—
33	C₃²×C₉	3₋¹⁺²	1	18	yes	—
34	C₃²⋊C₉	—	3	27	no	—
35	C₃²⋊C₉	—	9	27	no	—
36	C₃³	—	3	81	no	—
37	C₃³	—	3	162	no	—
38	C₃³	—	3	27	no	—
39	C₃³	—	3	54	no	—
40	C₃³	—	3	9	no	—
41	C₃³	—	3	3	no	—
42	C₃³	—	3	6	no	—
43	C₃³	—	3	18	no	—
44	C₃³	—	3	18	no	—
45	C₃³	—	3	9	no	—
46	C₃³	C₃²⋊C₉	1	9	yes	—
47	C₃³	#731	1	27	yes	—
48	C₃³	C₃²×C₉	1	1	yes	characteristic
49	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
50	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
51	C₃³	C₃×3₋¹⁺²	1	2	yes	—
52	C₃³	C₃×3₋¹⁺²	1	6	yes	—
53	C₃×C₉	—	3	243	no	—
54	C₃×C₉	—	3	18	no	—
55	C₃×C₉	—	3	9	no	—
56	C₃×C₉	—	3	9	no	—
57	C₃×C₉	—	3	18	no	—
58	C₃×C₉	—	3	36	no	—
59	C₃×C₉	—	3	18	no	—
60	C₃×C₉	—	9	9	no	—
61	C₃×C₉	—	9	27	no	—
62	C₃×C₉	C₃²⋊C₉	1	27	yes	complemented
63	C₃²	—	3	81	no	—
64	C₃²	—	3	162	no	—
65	C₃²	—	3	9	no	—
66	C₃²	—	3	18	no	—
67	C₃²	—	3	3	no	—
68	C₃²	—	3	6	no	—
69	C₃²	—	3	27	no	—
70	C₃²	—	3	9	no	—
71	C₃²	—	3	54	no	—
72	C₃²	—	3	18	no	—
73	C₃²	#6867	1	27	yes	complemented
74	C₃²	C₃×C₃²⋊C₉	1	3	yes	—
75	C₃²	C₃×C₃²⋊C₉	1	1	yes	characteristic
76	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
77	C₃²	#6891	1	6	yes	—
78	C₃²	#6891	1	2	yes	—
79	C₃²	C₃×#731	1	9	yes	—
80	C₉	—	3	27	no	—
81	C₉	—	27	9	no	—
82	C₃	—	3	27	no	—
83	C₃	—	3	3	no	—
84	C₃	—	3	6	no	—
85	C₃	#10624547	1	1	yes	characteristic
86	C₃	C₃×#6867	1	3	yes	—
87	C₃	#10624731	1	3	yes	—
88	C₃	#10624767	1	6	yes	—

	subgroup	G/H
center, socle	C₃³	C₃×H(3,𝔽₃)
commutator	C₃³	C₃²×C₉
Frattini, cosocle	C₃⁴	C₃³
prime radical	C₃⁵	C₃²
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃³ ◁ C₃⁵ ◁ G
LECS	G ▷ C₃⁴ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₃×#6867	C₃	semidirect
9. C₃³×C₉	C₉	semidirect
62. C₃×C₉	C₃²⋊C₉	semidirect
73. C₃²	#6867	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×#6867	3
#10624547	3