Subgroups — C₃×#10624738 #14554261

Order 2187.

2276 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
2	C₃²×#733	C₃	1	9	yes	maximal, complemented
3	C₃×#6891	C₃	1	3	yes	maximal
4	#10624738	C₃	1	27	yes	maximal, direct product factor
5	C₃⁵	C₃²	1	1	yes	characteristic
6	C₃²×3₋¹⁺²	—	3	6	no	—
7	C₃²×3₋¹⁺²	—	3	3	no	—
8	C₃²×3₋¹⁺²	—	3	27	no	—
9	C₃²×3₋¹⁺²	C₃²	1	9	yes	—
10	C₃²×3₋¹⁺²	C₃²	1	9	yes	—
11	C₃³×C₉	C₃²	1	3	yes	—
12	C₃×#733	—	3	81	no	—
13	C₃×#733	C₃²	1	81	yes	complemented
14	#6891	—	3	18	no	—
15	#6891	C₃²	1	27	yes	—
16	C₃×C₃²⋊C₉	—	3	9	no	—
17	C₃⁴	—	3	9	no	—
18	C₃⁴	—	9	9	no	—
19	C₃⁴	H(3,𝔽₃)	1	3	yes	—
20	C₃⁴	H(3,𝔽₃)	1	6	yes	—
21	C₃⁴	C₃³	1	1	yes	characteristic
22	C₃⁴	C₃³	1	3	yes	—
23	C₃×3₋¹⁺²	—	3	162	no	—
24	C₃×3₋¹⁺²	—	3	81	no	—
25	C₃×3₋¹⁺²	—	3	27	no	—
26	C₃×3₋¹⁺²	—	3	18	no	—
27	C₃×3₋¹⁺²	—	3	81	no	—
28	C₃×3₋¹⁺²	—	3	81	no	—
29	C₃×3₋¹⁺²	C₃³	1	27	yes	—
30	C₃²×C₉	—	3	6	no	—
31	C₃²×C₉	—	3	18	no	—
32	C₃²×C₉	—	3	3	no	—
33	C₃²×C₉	—	3	9	no	—
34	C₃²×C₉	H(3,𝔽₃)	1	27	yes	—
35	C₃²×C₉	C₃³	1	9	yes	—
36	#733	—	3	243	no	—
37	C₃²⋊C₉	—	3	81	no	—
38	C₃³	—	3	9	no	—
39	C₃³	—	3	27	no	—
40	C₃³	—	3	9	no	—
41	C₃³	—	3	18	no	—
42	C₃³	—	3	9	no	—
43	C₃³	—	9	18	no	—
44	C₃³	—	9	9	no	—
45	C₃³	—	9	81	no	—
46	C₃³	C₃×H(3,𝔽₃)	1	9	yes	—
47	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
48	C₃³	C₃×H(3,𝔽₃)	1	9	yes	—
49	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
50	C₃³	C₃⁴	1	1	yes	characteristic
51	3₋¹⁺²	—	3	243	no	—
52	3₋¹⁺²	—	3	81	no	—
53	3₋¹⁺²	—	9	54	no	—
54	C₃×C₉	—	3	18	no	—
55	C₃×C₉	—	3	9	no	—
56	C₃×C₉	—	3	54	no	—
57	C₃×C₉	—	3	27	no	—
58	C₃×C₉	—	3	27	no	—
59	C₃×C₉	—	9	27	no	—
60	C₃×C₉	C₃×H(3,𝔽₃)	1	27	yes	—
61	C₃²	—	3	6	no	—
62	C₃²	—	3	3	no	—
63	C₃²	—	3	27	no	—
64	C₃²	—	3	9	no	—
65	C₃²	—	3	27	no	—
66	C₃²	—	9	27	no	—
67	C₃²	—	9	54	no	—
68	C₃²	—	9	27	no	—
69	C₃²	C₃²⋊H(3,𝔽₃)	1	6	yes	—
70	C₃²	C₃²⋊H(3,𝔽₃)	1	3	yes	—
71	C₃²	#6912	1	9	yes	—
72	C₃²	C₃²×H(3,𝔽₃)	1	3	yes	—
73	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
74	C₉	—	3	27	no	—
75	C₉	—	9	27	no	—
76	C₃	—	3	3	no	—
77	C₃	—	3	6	no	—
78	C₃	—	9	9	no	—
79	C₃	#10624738	1	9	yes	direct product factor
80	C₃	C₃×C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
81	C₃	C₃×#6912	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₃²×#733	C₃	semidirect
4. #10624738	C₃	direct (N × H)
13. C₃×#733	C₃²	semidirect
79. C₃	#10624738	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×3₋¹⁺²	3
C₃²×#733	3
C₃×#6891	3
#10624738	3

Subgroups — C3×#10624738 #14554261

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10624738 #14554261