Subgroups — #14550792

Order 2187.

398 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#6895	C₃	1	1	yes	maximal, characteristic
2	#10624549	C₃	1	1	yes	maximal, characteristic
3	#10624549	C₃	1	1	yes	maximal, characteristic
4	C₃×#6855	C₃	1	1	yes	maximal, characteristic
5	#10624466	C₃	1	6	yes	maximal
6	#10624466	C₃	1	3	yes	maximal
7	C₃³×C₉	C₃²	1	3	yes	—
8	C₃³×C₉	C₃²	1	1	yes	characteristic
9	#6895	—	3	9	no	—
10	C₃×C₉⋊C₉	—	3	3	no	—
11	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
12	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
13	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
14	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
15	#6855	—	3	3	no	—
16	#6855	—	3	3	no	—
17	#6855	—	3	3	no	—
18	C₃⁴	C₃³	1	1	yes	characteristic
19	C₃²×C₉	—	3	3	no	—
20	C₃²×C₉	—	3	3	no	—
21	C₃²×C₉	—	3	3	no	—
22	C₃²×C₉	—	3	3	no	—
23	C₃²×C₉	—	3	3	no	—
24	C₃²×C₉	—	3	3	no	—
25	C₃²×C₉	—	3	9	no	—
26	C₃²×C₉	—	3	2	no	—
27	C₃²×C₉	—	3	3	no	—
28	C₃²×C₉	—	3	1	no	—
29	C₃²×C₉	—	3	9	no	—
30	C₃²×C₉	—	9	6	no	—
31	C₃²×C₉	H(3,𝔽₃)	1	3	yes	—
32	C₉⋊C₉	—	3	27	no	—
33	C₃²⋊C₉	—	3	6	no	—
34	C₃²⋊C₉	—	3	3	no	—
35	C₃²⋊C₉	—	3	6	no	—
36	C₃²⋊C₉	—	3	9	no	—
37	C₃²⋊C₉	3₋¹⁺²	1	9	yes	—
38	C₃³	—	3	6	no	—
39	C₃³	—	3	3	no	—
40	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
41	C₃³	C₃×3₋¹⁺²	1	3	yes	—
42	C₃³	#737	1	3	yes	—
43	C₃³	#737	1	3	yes	—
44	C₃³	#737	1	3	yes	—
45	C₃×C₉	—	3	9	no	—
46	C₃×C₉	—	3	3	no	—
47	C₃×C₉	—	3	3	no	—
48	C₃×C₉	—	3	3	no	—
49	C₃×C₉	—	3	3	no	—
50	C₃×C₉	—	3	9	no	—
51	C₃×C₉	—	9	6	no	—
52	C₃×C₉	—	9	6	no	—
53	C₃×C₉	—	9	3	no	—
54	C₃×C₉	—	9	9	no	—
55	C₃×C₉	—	9	3	no	—
56	C₃×C₉	—	9	27	no	—
57	C₃×C₉	—	9	6	no	—
58	C₃×C₉	—	9	6	no	—
59	C₃×C₉	—	9	1	no	—
60	C₃×C₉	—	9	2	no	—
61	C₃×C₉	—	9	3	no	—
62	C₃×C₉	—	9	3	no	—
63	C₃×C₉	—	9	3	no	—
64	C₃²	—	3	9	no	—
65	C₃²	—	3	9	no	—
66	C₃²	—	3	3	no	—
67	C₃²	—	3	6	no	—
68	C₃²	—	3	9	no	—
69	C₃²	#6891	1	1	yes	characteristic
70	C₃²	#6893	1	1	yes	characteristic
71	C₃²	#6893	1	1	yes	characteristic
72	C₃²	#6893	1	1	yes	characteristic
73	C₃²	#6899	1	9	yes	—
74	C₃²	#6908	1	3	yes	—
75	C₃²	#6913	1	6	yes	—
76	C₉	—	9	9	no	—
77	C₉	—	9	9	no	—
78	C₉	—	9	3	no	—
79	C₉	—	27	1	no	—
80	C₉	—	27	2	no	—
81	C₉	—	27	3	no	—
82	C₃	—	3	9	no	—
83	C₃	#10624570	1	1	yes	characteristic
84	C₃	#10624780	1	3	yes	—
85	C₃	#10624810	1	3	yes	—
86	C₃	#10624813	1	1	yes	characteristic
87	C₃	#10624817	1	2	yes	—
88	C₃	#10624827	1	3	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
C₃×#6895	3
#10624549	3
C₃×#6855	3
#10624466	3