Subgroups — #14554139

Order 2187.

1691 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#737	C₃	1	1	yes	maximal, characteristic
2	#10624862	C₃	1	3	yes	maximal
3	#10624690	C₃	1	24	yes	maximal, complemented
4	C₃×#6867	C₃	1	12	yes	maximal, complemented
5	C₃×#737	—	3	9	no	—
6	C₃×#737	—	3	18	no	—
7	C₃×#737	C₉	1	27	yes	complemented
8	C₃²×3₋¹⁺²	C₃²	1	8	yes	—
9	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
10	C₃³×C₉	C₃²	1	4	yes	—
11	C₉×H(3,𝔽₃)	—	3	27	no	—
12	C₃×C₃²⋊C₉	C₃²	1	36	yes	—
13	#6867	C₃²	1	9	yes	—
14	#6867	C₃²	1	72	yes	—
15	C₃⁴	C₃³	1	4	yes	—
16	#737	—	3	243	no	—
17	C₃×3₋¹⁺²	—	3	24	no	—
18	C₃×3₋¹⁺²	—	3	48	no	—
19	C₃×3₋¹⁺²	C₃×C₉	1	72	yes	complemented
20	C₃×H(3,𝔽₃)	C₃×C₉	1	9	yes	—
21	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	9	yes	—
22	C₃×H(3,𝔽₃)	3₋¹⁺²	1	18	yes	—
23	C₃²×C₉	—	3	12	no	—
24	C₃²×C₉	—	3	24	no	—
25	C₃²×C₉	—	3	36	no	—
26	C₃²×C₉	C₃×C₉	1	36	yes	complemented
27	C₃²×C₉	C₃³	1	1	yes	characteristic
28	C₃²×C₉	C₃³	1	8	yes	—
29	C₃²⋊C₉	—	3	72	no	—
30	C₃²⋊C₉	C₃³	1	27	yes	—
31	C₃³	—	9	12	no	—
32	C₃³	C₃²×C₉	1	12	yes	—
33	C₃³	C₃×H(3,𝔽₃)	1	12	yes	—
34	C₃³	C₃×3₋¹⁺²	1	24	yes	—
35	C₃³	C₃⁴	1	1	yes	characteristic
36	3₋¹⁺²	—	3	216	no	—
37	H(3,𝔽₃)	C₃²⋊C₉	1	81	yes	—
38	C₃×C₉	—	3	1	no	—
39	C₃×C₉	—	3	8	no	—
40	C₃×C₉	—	3	2	no	—
41	C₃×C₉	—	3	16	no	—
42	C₃×C₉	—	3	108	no	—
43	C₃×C₉	—	3	27	no	—
44	C₃×C₉	—	27	36	no	—
45	C₃×C₉	C₃²×C₉	1	3	yes	—
46	C₃×C₉	C₃²×C₉	1	24	yes	—
47	C₃²	—	3	36	no	—
48	C₃²	—	3	3	no	—
49	C₃²	—	9	12	no	—
50	C₃²	—	9	24	no	—
51	C₃²	C₃×C₃²⋊C₉	1	36	yes	—
52	C₃²	C₃³×C₉	1	1	yes	characteristic
53	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
54	C₃²	C₃²×3₋¹⁺²	1	2	yes	—
55	C₉	—	3	3	no	—
56	C₉	—	3	24	no	—
57	C₉	—	9	27	no	—
58	C₃	—	3	1	no	—
59	C₃	—	3	2	no	—
60	C₃	—	3	36	no	—
61	C₃	C₃²×C₃²⋊C₉	1	1	yes	characteristic
62	C₃	#10624931	1	3	yes	—

	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
3. #10624690	C₃	semidirect
4. C₃×#6867	C₃	semidirect
7. C₃×#737	C₉	semidirect
19. C₃×3₋¹⁺²	C₃×C₉	semidirect
26. C₃²×C₉	C₃×C₉	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#737	3
#10624862	3
#10624690	3
C₃×#6867	3