Subgroups — #14550370

Order 2187.

713 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#6893	C₃	1	1	yes	maximal, characteristic
2	C₃²×C₉²	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×3₋¹⁺²⋊C₉	C₃	1	6	yes	maximal, complemented
4	C₃×H(3,𝔽₃)⋊C₉	C₃	1	3	yes	maximal
5	#10624549	C₃	1	2	yes	maximal, complemented
6	C₃²×3₋¹⁺²	C₃²	1	2	yes	—
7	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
8	C₃³×C₉	C₃²	1	3	yes	—
9	C₃³×C₉	C₃²	1	1	yes	characteristic
10	#6893	—	3	6	no	—
11	#6893	C₉	1	9	yes	complemented
12	C₃×C₃²⋊C₉	—	3	3	no	—
13	C₃×C₃²⋊C₉	C₃²	1	6	yes	—
14	C₃×C₉²	—	3	9	no	—
15	C₃×C₉²	—	3	3	no	—
16	3₋¹⁺²⋊C₉	—	3	18	no	—
17	H(3,𝔽₃)⋊C₉	—	3	9	no	—
18	#6855	—	3	6	no	—
19	C₃⁴	—	3	1	no	—
20	C₃⁴	C₃³	1	1	yes	characteristic
21	C₃×3₋¹⁺²	—	3	4	no	—
22	C₃×3₋¹⁺²	—	3	6	no	—
23	C₃×3₋¹⁺²	—	3	12	no	—
24	C₃×3₋¹⁺²	C₃×C₉	1	6	yes	—
25	C₃×H(3,𝔽₃)	—	3	3	no	—
26	C₃×H(3,𝔽₃)	—	3	6	no	—
27	C₃×H(3,𝔽₃)	—	3	2	no	—
28	C₃×H(3,𝔽₃)	C₃×C₉	1	3	yes	—
29	C₃²×C₉	—	3	6	no	—
30	C₃²×C₉	—	3	3	no	—
31	C₃²×C₉	—	3	3	no	—
32	C₃²×C₉	—	3	6	no	—
33	C₃²×C₉	—	3	27	no	—
34	C₃²×C₉	—	3	2	no	—
35	C₃²×C₉	—	3	6	no	—
36	C₃²×C₉	C₃×C₉	1	3	yes	—
37	C₃²×C₉	H(3,𝔽₃)	1	9	yes	—
38	C₃²×C₉	H(3,𝔽₃)	1	3	yes	—
39	C₃²×C₉	3₋¹⁺²	1	6	yes	—
40	C₃²⋊C₉	—	3	27	no	—
41	C₃²⋊C₉	—	3	6	no	—
42	C₃²⋊C₉	—	3	12	no	—
43	C₉²	—	3	27	no	—
44	C₃³	—	3	9	no	—
45	C₃³	—	3	3	no	—
46	C₃³	—	9	1	no	—
47	C₃³	—	9	2	no	—
48	C₃³	—	9	9	no	—
49	C₃³	C₃²×C₉	1	1	yes	characteristic
50	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
51	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
52	C₃³	C₃×3₋¹⁺²	1	2	yes	—
53	C₃³	#737	1	6	yes	—
54	3₋¹⁺²	—	3	18	no	—
55	3₋¹⁺²	—	9	12	no	—
56	H(3,𝔽₃)	—	3	9	no	—
57	H(3,𝔽₃)	—	9	6	no	—
58	C₃×C₉	—	3	54	no	—
59	C₃×C₉	—	3	9	no	—
60	C₃×C₉	—	3	3	no	—
61	C₃×C₉	—	3	9	no	—
62	C₃×C₉	—	3	18	no	—
63	C₃×C₉	—	3	6	no	—
64	C₃×C₉	—	3	27	no	—
65	C₃×C₉	—	3	27	no	—
66	C₃×C₉	—	3	6	no	—
67	C₃×C₉	—	9	2	no	—
68	C₃×C₉	—	9	4	no	—
69	C₃×C₉	—	9	6	no	—
70	C₃×C₉	—	9	18	no	—
71	C₃×C₉	C₃²⋊C₉	1	9	yes	—
72	C₃²	—	3	18	no	—
73	C₃²	—	3	3	no	—
74	C₃²	—	3	6	no	—
75	C₃²	—	3	6	no	—
76	C₃²	—	3	3	no	—
77	C₃²	—	9	3	no	—
78	C₃²	—	9	9	no	—
79	C₃²	—	27	6	no	—
80	C₃²	—	27	3	no	—
81	C₃²	C₃×C₃²⋊C₉	1	1	yes	characteristic
82	C₃²	C₉×H(3,𝔽₃)	1	3	yes	—
83	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
84	C₃²	C₉⋊H(3,𝔽₃)	1	6	yes	—
85	C₃²	#6893	1	2	yes	—
86	C₃²	#6908	1	9	yes	—
87	C₉	—	3	9	no	—
88	C₉	—	3	27	no	—
89	C₉	—	9	6	no	—
90	C₉	—	27	6	no	—
91	C₃	—	3	3	no	—
92	C₃	—	3	6	no	—
93	C₃	—	27	3	no	—
94	C₃	#10624555	1	1	yes	characteristic
95	C₃	#10624697	1	3	yes	—
96	C₃	#10624734	1	3	yes	—
97	C₃	#10624813	1	6	yes	—

	subgroup	G/H
center, socle	C₃³	C₃×H(3,𝔽₃)
commutator	C₃³	C₃²×C₉
Frattini, cosocle	C₃⁴	C₃³
prime radical	C₃²×H(3,𝔽₃)	C₃²
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃³ ◁ C₃³×C₉ ◁ G
LECS	G ▷ C₃⁴ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₃²×C₉²	C₃	semidirect
3. C₃×3₋¹⁺²⋊C₉	C₃	semidirect
5. #10624549	C₃	semidirect
11. #6893	C₉	semidirect

Maximal subgroups

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

subgroup	index
C₃×#6893	3
C₃²×C₉²	3
C₃×3₋¹⁺²⋊C₉	3
C₃×H(3,𝔽₃)⋊C₉	3
#10624549	3