Subgroups — #14550719

Order 2187.

794 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic
2	C₃×#6867	C₃	1	9	yes	maximal, complemented
3	#10624547	C₃	1	2	yes	maximal
4	#10624547	C₃	1	1	yes	maximal, characteristic
5	C₃⁵	C₃²	1	1	yes	characteristic
6	C₃³×C₉	C₃²	1	3	yes	—
7	C₃×C₃²⋊C₉	—	3	3	no	—
8	C₃×C₃²⋊C₉	—	3	9	no	—
9	C₃×C₃²⋊C₉	—	3	6	no	—
10	C₃×C₃²⋊C₉	—	3	3	no	—
11	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
12	C₃×C₃²⋊C₉	C₃²	1	6	yes	—
13	#6867	—	3	27	no	—
14	C₃⁴	—	3	9	no	—
15	C₃⁴	—	3	3	no	—
16	C₃⁴	—	3	3	no	—
17	C₃⁴	—	3	6	no	—
18	C₃⁴	—	9	6	no	—
19	C₃⁴	H(3,𝔽₃)	1	3	yes	—
20	C₃⁴	C₃³	1	1	yes	characteristic
21	C₃²×C₉	—	3	6	no	—
22	C₃²×C₉	—	3	3	no	—
23	C₃²×C₉	—	3	9	no	—
24	C₃²×C₉	—	3	18	no	—
25	C₃²×C₉	—	3	3	no	—
26	C₃²×C₉	—	3	6	no	—
27	C₃²×C₉	H(3,𝔽₃)	1	9	yes	—
28	C₃²⋊C₉	—	3	27	no	—
29	C₃²⋊C₉	—	3	9	no	—
30	C₃²⋊C₉	—	3	18	no	—
31	C₃²⋊C₉	—	9	18	no	—
32	C₃²⋊C₉	—	9	9	no	—
33	C₃³	—	3	3	no	—
34	C₃³	—	3	6	no	—
35	C₃³	—	3	27	no	—
36	C₃³	—	3	3	no	—
37	C₃³	—	3	3	no	—
38	C₃³	—	3	6	no	—
39	C₃³	—	3	3	no	—
40	C₃³	—	3	3	no	—
41	C₃³	—	3	6	no	—
42	C₃³	—	3	6	no	—
43	C₃³	—	3	6	no	—
44	C₃³	—	9	27	no	—
45	C₃³	—	9	54	no	—
46	C₃³	—	9	18	no	—
47	C₃³	—	9	9	no	—
48	C₃³	#731	1	9	yes	—
49	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
50	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
51	C₃³	#737	1	3	yes	—
52	C₃³	#737	1	6	yes	—
53	C₃×C₉	—	3	9	no	—
54	C₃×C₉	—	3	27	no	—
55	C₃×C₉	—	9	18	no	—
56	C₃×C₉	—	9	3	no	—
57	C₃×C₉	—	9	6	no	—
58	C₃×C₉	—	9	3	no	—
59	C₃×C₉	—	9	6	no	—
60	C₃×C₉	—	9	9	no	—
61	C₃×C₉	—	9	18	no	—
62	C₃²	—	3	9	no	—
63	C₃²	—	3	18	no	—
64	C₃²	—	3	3	no	—
65	C₃²	—	3	6	no	—
66	C₃²	—	3	9	no	—
67	C₃²	—	3	18	no	—
68	C₃²	—	3	9	no	—
69	C₃²	—	9	81	no	—
70	C₃²	—	9	3	no	—
71	C₃²	—	9	3	no	—
72	C₃²	—	9	6	no	—
73	C₃²	—	9	6	no	—
74	C₃²	—	9	3	no	—
75	C₃²	—	9	6	no	—
76	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
77	C₃²	#6893	1	9	yes	—
78	C₃²	#6893	1	1	yes	characteristic
79	C₃²	#6893	1	2	yes	—
80	C₃²	C₃×#731	1	3	yes	—
81	C₃²	#6910	1	6	yes	—
82	C₉	—	9	9	no	—
83	C₉	—	27	3	no	—
84	C₉	—	27	6	no	—
85	C₃	—	3	9	no	—
86	C₃	—	9	9	no	—
87	C₃	#10624564	1	1	yes	characteristic
88	C₃	#10624731	1	1	yes	characteristic
89	C₃	#10624736	1	2	yes	—
90	C₃	#10624808	1	3	yes	—
91	C₃	#10624809	1	6	yes	—

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

Maximal subgroups

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

subgroup	index
C₃²×C₃²⋊C₉	3
C₃×#6867	3
#10624547	3