Subgroups — #14549766

Order 2187.

851 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₃²⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
2	#10624547	C₃	1	3	yes	maximal, complemented
3	#10624449	C₃	1	9	yes	maximal, complemented
4	C₃⁵	C₃²	1	1	yes	characteristic
5	C₃²×H(3,𝔽₃)	—	3	3	no	—
6	C₃²×H(3,𝔽₃)	C₃²	1	3	yes	—
7	C₃²⋊H(3,𝔽₃)	—	3	9	no	—
8	C₃×C₃²⋊C₉	—	3	3	no	—
9	C₃×C₃²⋊C₉	—	3	6	no	—
10	C₃×C₃²⋊C₉	C₃²	1	9	yes	—
11	C₃⁴	—	3	9	no	—
12	C₃⁴	—	3	3	no	—
13	C₃⁴	—	3	3	no	—
14	C₃⁴	—	9	3	no	—
15	C₃⁴	—	9	3	no	—
16	C₃⁴	—	9	3	no	—
17	C₃⁴	H(3,𝔽₃)	1	3	yes	—
18	C₃⁴	C₃³	1	1	yes	characteristic
19	C₃×H(3,𝔽₃)	—	3	27	no	—
20	C₃×H(3,𝔽₃)	—	3	6	no	—
21	C₃×H(3,𝔽₃)	—	9	9	no	—
22	C₃×H(3,𝔽₃)	—	9	9	no	—
23	C₃×H(3,𝔽₃)	—	9	9	no	—
24	C₃×H(3,𝔽₃)	—	9	9	no	—
25	C₃×H(3,𝔽₃)	3₋¹⁺²	1	9	yes	—
26	C₃²×C₉	—	3	9	no	—
27	C₃²⋊C₉	—	9	9	no	—
28	C₃²⋊C₉	—	9	18	no	—
29	C₃²⋊C₉	—	9	9	no	—
30	C₃³	—	3	3	no	—
31	C₃³	—	3	6	no	—
32	C₃³	—	3	3	no	—
33	C₃³	—	3	3	no	—
34	C₃³	—	3	3	no	—
35	C₃³	—	3	3	no	—
36	C₃³	—	3	3	no	—
37	C₃³	—	3	6	no	—
38	C₃³	—	3	6	no	—
39	C₃³	—	3	6	no	—
40	C₃³	—	3	3	no	—
41	C₃³	—	3	3	no	—
42	C₃³	—	3	9	no	—
43	C₃³	—	9	27	no	—
44	C₃³	—	9	27	no	—
45	C₃³	—	9	9	no	—
46	C₃³	—	9	27	no	—
47	C₃³	—	9	27	no	—
48	C₃³	—	9	9	no	—
49	C₃³	—	9	9	no	—
50	C₃³	—	9	9	no	—
51	C₃³	—	9	9	no	—
52	C₃³	#731	1	9	yes	—
53	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
54	C₃³	C₃×3₋¹⁺²	1	3	yes	—
55	H(3,𝔽₃)	—	9	9	no	—
56	H(3,𝔽₃)	—	9	18	no	—
57	H(3,𝔽₃)	—	9	81	no	—
58	C₃×C₉	—	9	9	no	—
59	C₃×C₉	—	9	9	no	—
60	C₃×C₉	—	9	18	no	—
61	C₃²	—	3	3	no	—
62	C₃²	—	3	9	no	—
63	C₃²	—	3	9	no	—
64	C₃²	—	3	18	no	—
65	C₃²	—	3	9	no	—
66	C₃²	—	9	81	no	—
67	C₃²	—	9	3	no	—
68	C₃²	—	9	6	no	—
69	C₃²	—	9	3	no	—
70	C₃²	—	9	3	no	—
71	C₃²	—	9	6	no	—
72	C₃²	—	9	3	no	—
73	C₃²	—	9	3	no	—
74	C₃²	—	9	3	no	—
75	C₃²	—	9	6	no	—
76	C₃²	—	9	9	no	—
77	C₃²	—	9	27	no	—
78	C₃²	—	27	9	no	—
79	C₃²	—	27	9	no	—
80	C₃²	—	27	9	no	—
81	C₃²	#6891	1	1	yes	characteristic
82	C₃²	C₃×#731	1	3	yes	—
83	C₃²	#6909	1	3	yes	—
84	C₃²	#6909	1	3	yes	—
85	C₃²	#6911	1	3	yes	—
86	C₉	—	27	9	no	—
87	C₃	—	9	3	no	—
88	C₃	—	9	9	no	—
89	C₃	—	27	9	no	—
90	C₃	#10624620	1	3	yes	—
91	C₃	#10624639	1	3	yes	—
92	C₃	#10624665	1	3	yes	—
93	C₃	#10624769	1	1	yes	characteristic
94	C₃	#10624778	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
2. #10624547	C₃	semidirect
3. #10624449	C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₃²⋊H(3,𝔽₃)	3
#10624547	3
#10624449	3