Subgroups — #14552192

Order 2187.

2678 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×3₋¹⁺²	C₃	1	2	yes	maximal
2	C₃×#6891	C₃	1	2	yes	maximal
3	#10624571	C₃	1	9	yes	maximal, complemented
4	#10624567	C₃	1	9	yes	maximal, complemented
5	#10624558	C₃	1	18	yes	maximal, complemented
6	C₃⁵	C₃²	1	1	yes	characteristic
7	C₃²×3₋¹⁺²	—	3	18	no	—
8	C₃²×3₋¹⁺²	—	3	54	no	—
9	C₃²×3₋¹⁺²	C₃²	1	6	yes	—
10	C₃²×3₋¹⁺²	C₃²	1	6	yes	—
11	C₃²×3₋¹⁺²	C₃²	1	6	yes	—
12	C₃²×3₋¹⁺²	C₃²	1	6	yes	—
13	C₃³×C₉	C₃²	1	6	yes	—
14	#6891	—	3	18	no	—
15	C₃×C₃²⋊C₉	C₃²	1	12	yes	—
16	C₃×C₃²⋊C₉	C₃²	1	6	yes	—
17	#6855	C₃²	1	81	yes	complemented
18	C₃⁴	—	3	18	no	—
19	C₃⁴	—	9	6	no	—
20	C₃⁴	H(3,𝔽₃)	1	9	yes	—
21	C₃⁴	C₃³	1	1	yes	characteristic
22	C₃⁴	C₃³	1	1	yes	characteristic
23	C₃⁴	C₃³	1	2	yes	—
24	C₃×3₋¹⁺²	—	3	486	no	—
25	C₃×3₋¹⁺²	—	3	18	no	—
26	C₃×3₋¹⁺²	—	3	18	no	—
27	C₃×3₋¹⁺²	—	3	18	no	—
28	C₃×3₋¹⁺²	—	3	36	no	—
29	C₃×3₋¹⁺²	—	3	18	no	—
30	C₃×3₋¹⁺²	—	3	54	no	—
31	C₃×3₋¹⁺²	—	3	54	no	—
32	C₃×3₋¹⁺²	—	9	18	no	—
33	C₃×3₋¹⁺²	3₋¹⁺²	1	54	yes	complemented
34	C₃²×C₉	—	3	18	no	—
35	C₃²×C₉	—	3	54	no	—
36	C₃²×C₉	C₃³	1	18	yes	—
37	C₃²×C₉	C₃³	1	18	yes	—
38	C₃²⋊C₉	—	3	18	no	—
39	C₃²⋊C₉	—	3	36	no	—
40	C₃³	—	3	6	no	—
41	C₃³	—	3	12	no	—
42	C₃³	—	3	6	no	—
43	C₃³	—	3	6	no	—
44	C₃³	—	3	6	no	—
45	C₃³	—	3	6	no	—
46	C₃³	—	3	54	no	—
47	C₃³	—	9	18	no	—
48	C₃³	—	9	81	no	—
49	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
50	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
51	C₃³	C₃×H(3,𝔽₃)	1	6	yes	—
52	C₃³	C₃×3₋¹⁺²	1	18	yes	—
53	C₃³	C₃⁴	1	1	yes	characteristic
54	3₋¹⁺²	—	3	162	no	—
55	3₋¹⁺²	—	3	162	no	—
56	3₋¹⁺²	—	3	162	no	—
57	3₋¹⁺²	—	9	54	no	—
58	C₃×C₉	—	3	162	no	—
59	C₃×C₉	—	3	18	no	—
60	C₃×C₉	—	3	36	no	—
61	C₃×C₉	—	3	18	no	—
62	C₃×C₉	—	3	54	no	—
63	C₃×C₉	C₃×3₋¹⁺²	1	54	yes	complemented
64	C₃²	—	3	6	no	—
65	C₃²	—	3	12	no	—
66	C₃²	—	3	54	no	—
67	C₃²	—	3	6	no	—
68	C₃²	—	3	6	no	—
69	C₃²	—	3	6	no	—
70	C₃²	—	3	6	no	—
71	C₃²	—	9	81	no	—
72	C₃²	—	9	9	no	—
73	C₃²	—	9	9	no	—
74	C₃²	#6891	1	18	yes	—
75	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
76	C₃²	C₃²×3₋¹⁺²	1	6	yes	—
77	C₃²	3₋¹⁺⁴	1	6	yes	—
78	C₉	—	3	54	no	—
79	C₉	—	9	18	no	—
80	C₃	—	3	18	no	—
81	C₃	—	9	3	no	—
82	C₃	—	9	3	no	—
83	C₃	C₃×#6891	1	2	yes	—
84	C₃	#10624881	1	2	yes	—
85	C₃	3₋¹⁺²²	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. #10624571	C₃	semidirect
4. #10624567	C₃	semidirect
5. #10624558	C₃	semidirect
17. #6855	C₃²	semidirect
33. C₃×3₋¹⁺²	3₋¹⁺²	semidirect
63. C₃×C₉	C₃×3₋¹⁺²	semidirect

Maximal subgroups

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

subgroup	index
C₃³×3₋¹⁺²	3
C₃×#6891	3
#10624571	3
#10624567	3
#10624558	3