Subgroups — #14554128

Order 2187.

1740 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×3₋¹⁺⁴	C₃	1	1	yes	maximal, characteristic
2	#10624862	C₃	1	1	yes	maximal, characteristic
3	#10624862	C₃	1	1	yes	maximal, characteristic
4	C₃×C₉×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
5	#10624699	C₃	1	24	yes	maximal, complemented
6	C₃×3₋¹⁺²⋊C₉	C₃	1	12	yes	maximal, complemented
7	3₋¹⁺⁴	—	3	18	no	—
8	3₋¹⁺⁴	C₉	1	27	yes	complemented
9	C₃×#737	—	3	9	no	—
10	C₃²×3₋¹⁺²	C₃²	1	12	yes	—
11	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
12	C₃³×C₉	C₃²	1	4	yes	—
13	C₉×H(3,𝔽₃)	—	3	9	no	—
14	C₉×H(3,𝔽₃)	—	3	9	no	—
15	C₉×H(3,𝔽₃)	—	3	9	no	—
16	C₃×C₃²⋊C₉	C₃²	1	8	yes	—
17	C₃×C₃²⋊C₉	C₃²	1	12	yes	—
18	C₃×C₃²⋊C₉	C₃²	1	12	yes	—
19	3₋¹⁺²⋊C₉	C₃²	1	9	yes	—
20	3₋¹⁺²⋊C₉	C₃²	1	72	yes	—
21	C₃⁴	C₃³	1	4	yes	—
22	#737	—	3	243	no	—
23	C₃×3₋¹⁺²	—	3	24	no	—
24	C₃×3₋¹⁺²	—	3	72	no	—
25	C₃×3₋¹⁺²	C₃×C₉	1	108	yes	complemented
26	C₃×3₋¹⁺²	C₃³	1	9	yes	—
27	C₃×H(3,𝔽₃)	C₃×C₉	1	9	yes	—
28	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	9	yes	—
29	C₃×H(3,𝔽₃)	3₋¹⁺²	1	18	yes	—
30	C₃²×C₉	—	3	12	no	—
31	C₃²×C₉	—	3	12	no	—
32	C₃²×C₉	—	3	12	no	—
33	C₃²×C₉	—	3	12	no	—
34	C₃²×C₉	—	9	12	no	—
35	C₃²×C₉	C₃³	1	1	yes	characteristic
36	C₃²×C₉	C₃³	1	8	yes	—
37	C₃²⋊C₉	—	3	24	no	—
38	C₃²⋊C₉	—	3	24	no	—
39	C₃²⋊C₉	—	3	24	no	—
40	C₃²⋊C₉	C₃³	1	9	yes	—
41	C₃²⋊C₉	C₃³	1	9	yes	—
42	C₃³	—	9	4	no	—
43	C₃³	—	9	4	no	—
44	C₃³	—	9	4	no	—
45	C₃³	C₃²×C₉	1	12	yes	—
46	C₃³	C₃×H(3,𝔽₃)	1	12	yes	—
47	C₃³	C₃×3₋¹⁺²	1	24	yes	—
48	C₃³	C₃⁴	1	1	yes	characteristic
49	3₋¹⁺²	—	3	18	no	—
50	3₋¹⁺²	—	3	216	no	—
51	3₋¹⁺²	C₃²×C₉	1	27	yes	—
52	H(3,𝔽₃)	C₃²⋊C₉	1	81	yes	—
53	C₃×C₉	—	3	1	no	—
54	C₃×C₉	—	3	9	no	—
55	C₃×C₉	—	3	108	no	—
56	C₃×C₉	—	3	3	no	—
57	C₃×C₉	—	3	8	no	—
58	C₃×C₉	—	3	9	no	—
59	C₃×C₉	—	3	9	no	—
60	C₃×C₉	—	9	36	no	—
61	C₃×C₉	—	9	8	no	—
62	C₃×C₉	—	27	12	no	—
63	C₃×C₉	—	27	12	no	—
64	C₃²	—	3	1	no	—
65	C₃²	—	3	12	no	—
66	C₃²	—	3	12	no	—
67	C₃²	—	3	1	no	—
68	C₃²	—	3	1	no	—
69	C₃²	—	3	12	no	—
70	C₃²	—	9	12	no	—
71	C₃²	—	9	24	no	—
72	C₃²	C₃×C₃²⋊C₉	1	36	yes	—
73	C₃²	C₃³×C₉	1	1	yes	characteristic
74	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
75	C₃²	C₃²×3₋¹⁺²	1	2	yes	—
76	C₉	—	3	27	no	—
77	C₉	—	3	3	no	—
78	C₉	—	9	8	no	—
79	C₉	—	9	9	no	—
80	C₉	—	9	9	no	—
81	C₃	—	3	1	no	—
82	C₃	—	3	2	no	—
83	C₃	—	3	36	no	—
84	C₃	C₃²×C₃²⋊C₉	1	1	yes	characteristic
85	C₃	#10624930	1	1	yes	characteristic
86	C₃	#10624931	1	1	yes	characteristic
87	C₃	#10624933	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
5. #10624699	C₃	semidirect
6. C₃×3₋¹⁺²⋊C₉	C₃	semidirect
8. 3₋¹⁺⁴	C₉	semidirect
25. C₃×3₋¹⁺²	C₃×C₉	semidirect

Maximal subgroups

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

subgroup	index
C₃×3₋¹⁺⁴	3
#10624862	3
C₃×C₉×H(3,𝔽₃)	3
#10624699	3
C₃×3₋¹⁺²⋊C₉	3