Subgroups — #14552053

Order 2187.

1205 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#6887	C₃	1	3	yes	maximal
2	C₃²×C₉²	C₃	1	1	yes	maximal, characteristic
3	#10624554	C₃	1	3	yes	maximal, complemented
4	#10624553	C₃	1	6	yes	maximal, complemented
5	#10624550	C₃	1	9	yes	maximal, complemented
6	#10624548	C₃	1	9	yes	maximal
7	C₉×C₉⋊C₉	C₃	1	9	yes	maximal, complemented
8	C₃³×C₉	C₃²	1	3	yes	—
9	C₃³×C₉	C₃²	1	1	yes	characteristic
10	#6887	—	3	27	no	—
11	C₃×C₉⋊C₉	C₃²	1	27	yes	—
12	C₃×C₉⋊C₉	C₃²	1	9	yes	—
13	C₃×C₉⋊C₉	C₃²	1	9	yes	—
14	C₃×C₃²⋊C₉	C₃²	1	9	yes	—
15	C₃×C₉²	C₉	1	27	yes	complemented
16	C₃×C₉²	C₃²	1	9	yes	—
17	C₃×C₉²	C₃²	1	3	yes	—
18	C₃×C₉²	C₃²	1	2	yes	—
19	C₃×C₉²	C₃²	1	1	yes	characteristic
20	C₃×C₉²	C₃²	1	3	yes	—
21	#6855	C₃²	1	54	yes	—
22	C₃⁴	C₃³	1	1	yes	characteristic
23	C₃²×C₉	—	3	27	no	—
24	C₃²×C₉	C₃×C₉	1	27	yes	—
25	C₃²×C₉	C₃×C₉	1	9	yes	—
26	C₃²×C₉	3₋¹⁺²	1	27	yes	—
27	C₃²×C₉	C₃³	1	27	yes	—
28	C₃²×C₉	C₃³	1	6	yes	—
29	C₃²×C₉	C₃³	1	1	yes	characteristic
30	C₃²×C₉	C₃³	1	1	yes	characteristic
31	C₃²×C₉	C₃³	1	3	yes	—
32	C₃²×C₉	C₃³	1	1	yes	characteristic
33	C₉⋊C₉	—	3	27	no	—
34	C₉⋊C₉	—	3	27	no	—
35	C₉⋊C₉	—	3	81	no	—
36	C₃²⋊C₉	—	3	27	no	—
37	C₉²	—	3	27	no	—
38	C₉²	C₃×C₉	1	18	yes	—
39	C₉²	C₃×C₉	1	27	yes	—
40	C₉²	C₃×C₉	1	27	yes	—
41	C₉²	C₃×C₉	1	9	yes	—
42	C₃³	—	3	9	no	—
43	C₃³	C₃²×C₉	1	3	yes	—
44	C₃³	C₃×3₋¹⁺²	1	9	yes	—
45	C₃³	C₃⁴	1	1	yes	characteristic
46	C₃×C₉	—	3	18	no	—
47	C₃×C₉	—	3	81	no	—
48	C₃×C₉	—	3	27	no	—
49	C₃×C₉	—	3	81	no	—
50	C₃×C₉	C₉⋊C₉	1	81	yes	complemented
51	C₃×C₉	C₃²×C₉	1	18	yes	—
52	C₃×C₉	C₃²×C₉	1	3	yes	—
53	C₃×C₉	C₃²×C₉	1	3	yes	—
54	C₃×C₉	C₃²×C₉	1	3	yes	—
55	C₃×C₉	C₃²×C₉	1	9	yes	—
56	C₃×C₉	C₃×3₋¹⁺²	1	9	yes	—
57	C₃×C₉	C₃×3₋¹⁺²	1	9	yes	—
58	C₃×C₉	C₃×3₋¹⁺²	1	9	yes	—
59	C₃×C₉	#737	1	27	yes	—
60	C₃²	—	3	9	no	—
61	C₃²	—	3	27	no	—
62	C₃²	C₃×C₉⋊C₉	1	9	yes	—
63	C₃²	C₃³×C₉	1	1	yes	characteristic
64	C₃²	C₃²×3₋¹⁺²	1	3	yes	—
65	C₃²	C₃×#737	1	9	yes	—
66	C₉	—	3	18	no	—
67	C₉	—	9	27	no	—
68	C₉	C₃×C₉⋊C₉	1	9	yes	—
69	C₉	C₃×C₉⋊C₉	1	9	yes	—
70	C₉	C₃×C₉⋊C₉	1	9	yes	—
71	C₉	#6887	1	27	yes	complemented
72	C₃	—	3	9	no	—
73	C₃	C₃²×C₉⋊C₉	1	1	yes	characteristic
74	C₃	C₃×#6887	1	3	yes	—
75	C₃	#10624879	1	9	yes	—

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

Complements

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

N	H ≅ G/N	type
3. #10624554	C₃	semidirect
4. #10624553	C₃	semidirect
5. #10624550	C₃	semidirect
7. C₉×C₉⋊C₉	C₃	semidirect
15. C₃×C₉²	C₉	semidirect
50. C₃×C₉	C₉⋊C₉	semidirect
71. C₉	#6887	semidirect

Maximal subgroups

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

subgroup	index
C₃×#6887	3
C₃²×C₉²	3
#10624554	3
#10624553	3
#10624550	3
#10624548	3
C₉×C₉⋊C₉	3