Subgroups — #14554137

Order 2187.

2681 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
2	#10624862	C₃	1	3	yes	maximal, complemented
3	#10624689	C₃	1	24	yes	maximal, complemented
4	C₃×#6866	C₃	1	12	yes	maximal, complemented
5	C₃⁵	C₃²	1	4	yes	—
6	C₃²×H(3,𝔽₃)	—	3	9	no	—
7	C₃²×H(3,𝔽₃)	—	3	18	no	—
8	C₃²×H(3,𝔽₃)	C₉	1	27	yes	complemented
9	C₃²×H(3,𝔽₃)	C₃²	1	8	yes	—
10	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
11	C₉×H(3,𝔽₃)	—	3	27	no	—
12	C₃×C₃²⋊C₉	C₃²	1	36	yes	complemented
13	#6866	C₃²	1	9	yes	—
14	#6866	C₃²	1	72	yes	complemented
15	C₃⁴	—	3	12	no	—
16	C₃⁴	—	3	24	no	—
17	C₃⁴	—	9	36	no	—
18	C₃⁴	C₃×C₉	1	36	yes	complemented
19	C₃⁴	C₃³	1	1	yes	characteristic
20	C₃⁴	C₃³	1	8	yes	—
21	C₃⁴	C₃³	1	4	yes	—
22	C₃×H(3,𝔽₃)	—	3	24	no	—
23	C₃×H(3,𝔽₃)	—	3	48	no	—
24	C₃×H(3,𝔽₃)	—	3	243	no	—
25	C₃×H(3,𝔽₃)	C₃×C₉	1	9	yes	—
26	C₃×H(3,𝔽₃)	C₃×C₉	1	72	yes	complemented
27	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	9	yes	—
28	C₃×H(3,𝔽₃)	3₋¹⁺²	1	18	yes	—
29	C₃²×C₉	—	3	36	no	—
30	C₃²⋊C₉	—	3	72	no	—
31	C₃²⋊C₉	C₃³	1	27	yes	—
32	C₃³	—	3	9	no	—
33	C₃³	—	3	1	no	—
34	C₃³	—	3	2	no	—
35	C₃³	—	3	8	no	—
36	C₃³	—	3	16	no	—
37	C₃³	—	3	108	no	—
38	C₃³	—	9	12	no	—
39	C₃³	—	9	108	no	—
40	C₃³	—	9	108	no	—
41	C₃³	—	9	24	no	—
42	C₃³	—	9	216	no	—
43	C₃³	C₃²×C₉	1	3	yes	—
44	C₃³	C₃²×C₉	1	24	yes	—
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	H(3,𝔽₃)	—	3	216	no	—
50	H(3,𝔽₃)	C₃²⋊C₉	1	81	yes	—
51	C₃×C₉	—	3	27	no	—
52	C₃×C₉	—	27	36	no	—
53	C₃²	—	3	9	no	—
54	C₃²	—	3	18	no	—
55	C₃²	—	3	3	no	—
56	C₃²	—	3	36	no	—
57	C₃²	—	3	9	no	—
58	C₃²	—	3	3	no	—
59	C₃²	—	3	24	no	—
60	C₃²	—	9	12	no	—
61	C₃²	—	9	24	no	—
62	C₃²	—	9	324	no	—
63	C₃²	—	9	24	no	—
64	C₃²	—	9	48	no	—
65	C₃²	—	9	24	no	—
66	C₃²	C₃×C₃²⋊C₉	1	36	yes	—
67	C₃²	C₃³×C₉	1	1	yes	characteristic
68	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
69	C₃²	C₃²×3₋¹⁺²	1	2	yes	—
70	C₉	—	9	27	no	—
71	C₃	—	3	9	no	—
72	C₃	—	3	1	no	—
73	C₃	—	3	2	no	—
74	C₃	—	3	36	no	—
75	C₃	—	9	24	no	—
76	C₃	C₃²×C₃²⋊C₉	1	1	yes	characteristic
77	C₃	#10624930	1	3	yes	—

	subgroup	G/H
center, socle	C₃²	C₃²×H(3,𝔽₃)
commutator	C₃²	C₃³×C₉
Frattini, cosocle	C₃³	C₃⁴
prime radical	C₃³×H(3,𝔽₃)	C₃
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
2. #10624862	C₃	semidirect
3. #10624689	C₃	semidirect
4. C₃×#6866	C₃	semidirect
8. C₃²×H(3,𝔽₃)	C₉	semidirect
12. C₃×C₃²⋊C₉	C₃²	semidirect
14. #6866	C₃²	semidirect
18. C₃⁴	C₃×C₉	semidirect
26. C₃×H(3,𝔽₃)	C₃×C₉	semidirect

Maximal subgroups

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

subgroup	index
C₃³×H(3,𝔽₃)	3
#10624862	3
#10624689	3
C₃×#6866	3