Subgroups — #14554127

Order 2187.

1938 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×H(5,𝔽₃)	C₃	1	1	yes	maximal, characteristic
2	#10624862	C₃	1	2	yes	maximal, complemented
3	C₃×C₉×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
4	#10624698	C₃	1	24	yes	maximal, complemented
5	C₃×H(3,𝔽₃)⋊C₉	C₃	1	12	yes	maximal, complemented
6	H(5,𝔽₃)	—	3	18	no	—
7	H(5,𝔽₃)	C₉	1	27	yes	complemented
8	C₃²×H(3,𝔽₃)	—	3	9	no	—
9	C₃²×H(3,𝔽₃)	C₃²	1	12	yes	—
10	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
11	C₃³×C₉	C₃²	1	4	yes	complemented
12	C₉×H(3,𝔽₃)	—	3	9	no	—
13	C₉×H(3,𝔽₃)	—	3	18	no	—
14	C₃×C₃²⋊C₉	C₃²	1	8	yes	complemented
15	C₃×C₃²⋊C₉	C₃²	1	24	yes	complemented
16	H(3,𝔽₃)⋊C₉	C₃²	1	9	yes	complemented
17	H(3,𝔽₃)⋊C₉	C₃²	1	72	yes	complemented
18	C₃⁴	—	3	12	no	—
19	C₃⁴	C₃³	1	4	yes	—
20	C₃×H(3,𝔽₃)	—	3	24	no	—
21	C₃×H(3,𝔽₃)	—	3	72	no	—
22	C₃×H(3,𝔽₃)	—	3	243	no	—
23	C₃×H(3,𝔽₃)	C₃×C₉	1	9	yes	—
24	C₃×H(3,𝔽₃)	C₃×C₉	1	108	yes	complemented
25	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	9	yes	—
26	C₃×H(3,𝔽₃)	3₋¹⁺²	1	18	yes	—
27	C₃×H(3,𝔽₃)	C₃³	1	9	yes	—
28	C₃²×C₉	—	3	12	no	—
29	C₃²×C₉	—	3	24	no	—
30	C₃²×C₉	—	9	12	no	—
31	C₃²×C₉	C₃³	1	1	yes	characteristic
32	C₃²×C₉	C₃³	1	8	yes	—
33	C₃²⋊C₉	—	3	24	no	—
34	C₃²⋊C₉	—	3	48	no	—
35	C₃²⋊C₉	C₃³	1	18	yes	—
36	C₃³	—	3	9	no	—
37	C₃³	—	3	108	no	—
38	C₃³	—	9	4	no	—
39	C₃³	—	9	4	no	—
40	C₃³	—	9	4	no	—
41	C₃³	—	27	12	no	—
42	C₃³	—	27	12	no	—
43	C₃³	—	27	12	no	—
44	C₃³	C₃²×C₉	1	12	yes	—
45	C₃³	C₃×H(3,𝔽₃)	1	12	yes	—
46	C₃³	C₃×3₋¹⁺²	1	24	yes	—
47	C₃³	C₃⁴	1	1	yes	characteristic
48	H(3,𝔽₃)	—	3	18	no	—
49	H(3,𝔽₃)	—	3	216	no	—
50	H(3,𝔽₃)	C₃²⋊C₉	1	81	yes	—
51	H(3,𝔽₃)	C₃²×C₉	1	27	yes	—
52	C₃×C₉	—	3	1	no	—
53	C₃×C₉	—	3	3	no	—
54	C₃×C₉	—	3	8	no	—
55	C₃×C₉	—	3	18	no	—
56	C₃×C₉	—	9	36	no	—
57	C₃×C₉	—	9	8	no	—
58	C₃×C₉	—	27	24	no	—
59	C₃²	—	3	1	no	—
60	C₃²	—	3	12	no	—
61	C₃²	—	3	1	no	—
62	C₃²	—	3	12	no	—
63	C₃²	—	3	12	no	—
64	C₃²	—	3	1	no	—
65	C₃²	—	3	27	no	—
66	C₃²	—	9	12	no	—
67	C₃²	—	9	24	no	—
68	C₃²	—	9	9	no	—
69	C₃²	—	9	9	no	—
70	C₃²	—	9	9	no	—
71	C₃²	—	27	108	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	3	no	—
77	C₉	—	9	8	no	—
78	C₉	—	9	18	no	—
79	C₃	—	3	1	no	—
80	C₃	—	3	2	no	—
81	C₃	—	3	36	no	—
82	C₃	—	9	27	no	—
83	C₃	C₃²×C₃²⋊C₉	1	1	yes	characteristic
84	C₃	#10624930	1	1	yes	characteristic
85	C₃	#10624931	1	1	yes	characteristic
86	C₃	#10624932	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃²	C₃²×H(3,𝔽₃)
commutator	C₃²	C₃³×C₉
Frattini, cosocle	C₃³	C₃⁴
prime radical	C₃×H(5,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. C₃×C₉×H(3,𝔽₃)	C₃	semidirect
4. #10624698	C₃	semidirect
5. C₃×H(3,𝔽₃)⋊C₉	C₃	semidirect
7. H(5,𝔽₃)	C₉	semidirect
11. C₃³×C₉	C₃²	semidirect
14. C₃×C₃²⋊C₉	C₃²	semidirect
15. C₃×C₃²⋊C₉	C₃²	semidirect
16. H(3,𝔽₃)⋊C₉	C₃²	semidirect
17. H(3,𝔽₃)⋊C₉	C₃²	semidirect
24. 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
C₃×H(5,𝔽₃)	3
#10624862	3
C₃×C₉×H(3,𝔽₃)	3
#10624698	3
C₃×H(3,𝔽₃)⋊C₉	3