Subgroups — #14550782

Order 2187.

344 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10624549	C₃	1	1	yes	maximal, characteristic
2	#10624549	C₃	1	2	yes	maximal
3	C₃×#6855	C₃	1	1	yes	maximal, characteristic
4	#10624466	C₃	1	6	yes	maximal
5	#10624466	C₃	1	3	yes	maximal
6	C₃³×C₉	C₃²	1	3	yes	—
7	C₃³×C₉	C₃²	1	1	yes	characteristic
8	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
9	C₃×C₃²⋊C₉	C₃²	1	6	yes	—
10	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
11	#6855	—	3	3	no	—
12	#6855	—	3	3	no	—
13	#6855	—	3	6	no	—
14	C₃⁴	C₃³	1	1	yes	characteristic
15	C₃²×C₉	—	3	3	no	—
16	C₃²×C₉	—	3	3	no	—
17	C₃²×C₉	—	3	6	no	—
18	C₃²×C₉	—	3	3	no	—
19	C₃²×C₉	—	3	9	no	—
20	C₃²×C₉	—	3	2	no	—
21	C₃²×C₉	—	3	6	no	—
22	C₃²×C₉	—	3	1	no	—
23	C₃²×C₉	—	3	9	no	—
24	C₃²×C₉	—	9	6	no	—
25	C₃²×C₉	H(3,𝔽₃)	1	3	yes	—
26	C₃²⋊C₉	—	3	3	no	—
27	C₃²⋊C₉	—	3	6	no	—
28	C₃²⋊C₉	—	3	18	no	—
29	C₃³	—	3	6	no	—
30	C₃³	—	3	3	no	—
31	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
32	C₃³	#737	1	3	yes	—
33	C₃³	#737	1	6	yes	—
34	C₃³	#737	1	3	yes	—
35	C₃×C₉	—	3	3	no	—
36	C₃×C₉	—	3	3	no	—
37	C₃×C₉	—	3	6	no	—
38	C₃×C₉	—	3	9	no	—
39	C₃×C₉	—	9	6	no	—
40	C₃×C₉	—	9	3	no	—
41	C₃×C₉	—	9	9	no	—
42	C₃×C₉	—	9	27	no	—
43	C₃×C₉	—	9	6	no	—
44	C₃×C₉	—	9	12	no	—
45	C₃×C₉	—	9	1	no	—
46	C₃×C₉	—	9	2	no	—
47	C₃×C₉	—	9	6	no	—
48	C₃×C₉	—	9	3	no	—
49	C₃×C₉	—	9	6	no	—
50	C₃²	—	3	9	no	—
51	C₃²	—	3	3	no	—
52	C₃²	—	3	6	no	—
53	C₃²	—	3	18	no	—
54	C₃²	#6893	1	1	yes	characteristic
55	C₃²	#6893	1	1	yes	characteristic
56	C₃²	#6893	1	2	yes	—
57	C₃²	#6898	1	9	yes	—
58	C₃²	#6908	1	3	yes	—
59	C₃²	#6913	1	6	yes	—
60	C₉	—	9	9	no	—
61	C₉	—	9	3	no	—
62	C₉	—	27	1	no	—
63	C₉	—	27	2	no	—
64	C₉	—	27	6	no	—
65	C₃	—	3	9	no	—
66	C₃	#10624569	1	1	yes	characteristic
67	C₃	#10624810	1	3	yes	—
68	C₃	#10624813	1	1	yes	characteristic
69	C₃	#10624817	1	2	yes	—
70	C₃	#10624827	1	6	yes	—

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

Maximal subgroups

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

subgroup	index
#10624549	3
C₃×#6855	3
#10624466	3