Subgroups — #14554486

Order 2187.

722 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10624886	C₃	1	2	yes	maximal, complemented
2	C₃×#6893	C₃	1	1	yes	maximal, characteristic
3	#10624863	C₃	1	1	yes	maximal, characteristic, complemented
4	#10624810	C₃	1	27	yes	maximal, complemented
5	#10624691	C₃	1	3	yes	maximal, complemented
6	#10624689	C₃	1	3	yes	maximal
7	C₃×#6867	C₃	1	3	yes	maximal, complemented
8	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
9	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
10	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
11	C₃³×C₉	C₃²	1	1	yes	characteristic
12	#6895	C₃²	1	18	yes	—
13	#6894	—	3	18	no	—
14	#6893	—	3	9	no	—
15	C₉×3₋¹⁺²	—	3	9	no	—
16	#6887	C₃²	1	9	yes	—
17	C₃×C₃²⋊C₉	C₃²	1	9	yes	—
18	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
19	C₃×C₃²⋊C₉	C₃²	1	6	yes	—
20	#6868	C₃²	1	27	yes	—
21	#6867	C₃²	1	27	yes	—
22	#6866	C₃²	1	27	yes	—
23	C₃⁴	C₃³	1	3	yes	—
24	C₃⁴	C₃³	1	1	yes	characteristic
25	C₃×3₋¹⁺²	—	3	9	no	—
26	C₃×3₋¹⁺²	—	3	3	no	—
27	C₃×3₋¹⁺²	—	3	9	no	—
28	C₃×3₋¹⁺²	H(3,𝔽₃)	1	9	yes	—
29	C₃×H(3,𝔽₃)	—	3	3	no	—
30	C₃×H(3,𝔽₃)	—	3	9	no	—
31	C₃²×C₉	—	3	3	no	—
32	C₃²×C₉	—	3	9	no	—
33	C₃²×C₉	—	3	3	no	—
34	C₃²×C₉	—	3	6	no	—
35	C₃²×C₉	C₃³	1	3	yes	—
36	C₃²×C₉	C₃³	1	3	yes	—
37	C₃²×C₉	C₃³	1	3	yes	—
38	C₉⋊C₉	—	3	18	no	—
39	C₉⋊C₉	—	3	54	no	—
40	C₃²⋊C₉	—	3	6	no	—
41	C₃²⋊C₉	—	3	12	no	—
42	C₃²⋊C₉	—	3	27	no	—
43	C₃²⋊C₉	C₃³	1	9	yes	—
44	C₃²⋊C₉	C₃³	1	18	yes	—
45	C₉²	—	3	9	no	—
46	C₃³	—	3	3	no	—
47	C₃³	—	3	9	no	—
48	C₃³	—	9	3	no	—
49	C₃³	—	9	9	no	—
50	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
51	C₃³	#737	1	9	yes	—
52	C₃³	C₃⁴	1	1	yes	characteristic
53	3₋¹⁺²	—	3	27	no	—
54	3₋¹⁺²	—	9	9	no	—
55	H(3,𝔽₃)	—	9	9	no	—
56	C₃×C₉	—	3	9	no	—
57	C₃×C₉	—	3	9	no	—
58	C₃×C₉	—	3	3	no	—
59	C₃×C₉	—	3	3	no	—
60	C₃×C₉	—	3	18	no	—
61	C₃×C₉	—	3	9	no	—
62	C₃×C₉	—	3	9	no	—
63	C₃×C₉	—	9	9	no	—
64	C₃×C₉	—	27	3	no	—
65	C₃×C₉	—	27	6	no	—
66	C₃×C₉	C₃×H(3,𝔽₃)	1	9	yes	—
67	C₃²	—	3	3	no	—
68	C₃²	—	3	6	no	—
69	C₃²	—	3	3	no	—
70	C₃²	—	9	9	no	—
71	C₃²	—	9	3	no	—
72	C₃²	—	9	9	no	—
73	C₃²	—	27	9	no	—
74	C₃²	#6893	1	9	yes	—
75	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
76	C₃²	C₃×#737	1	3	yes	—
77	C₉	—	3	9	no	—
78	C₉	—	9	3	no	—
79	C₉	—	9	3	no	—
80	C₉	—	9	9	no	—
81	C₉	—	9	18	no	—
82	C₃	—	3	3	no	—
83	C₃	—	3	9	no	—
84	C₃	—	27	3	no	—
85	C₃	C₃×#6893	1	1	yes	characteristic
86	C₃	#10624935	1	3	yes	—

	subgroup	G/H
center, socle	C₃²	C₃²×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃⁴
prime radical	C₃²×H(3,𝔽₃)	C₃²
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃²×3₋¹⁺² ◁ G
LECS	G ▷ C₃³ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #10624886	C₃	semidirect
3. #10624863	C₃	semidirect
4. #10624810	C₃	semidirect
5. #10624691	C₃	semidirect
7. 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
#10624886	3
C₃×#6893	3
#10624863	3
#10624810	3
#10624691	3
#10624689	3
C₃×#6867	3