Subgroups — #14552050

Order 2187.

1205 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#6887	C₃	1	2	yes	maximal
2	C₃×#6887	C₃	1	1	yes	maximal, characteristic
3	C₃²×C₉²	C₃	1	1	yes	maximal, characteristic
4	#10624561	C₃	1	18	yes	maximal, complemented
5	#10624556	C₃	1	9	yes	maximal
6	C₉×C₉⋊C₉	C₃	1	9	yes	maximal, complemented
7	C₃³×C₉	C₃²	1	3	yes	—
8	C₃³×C₉	C₃²	1	1	yes	characteristic
9	#6887	—	3	18	no	—
10	#6887	C₉	1	27	yes	complemented
11	C₃×C₉⋊C₉	C₃²	1	27	yes	—
12	C₃×C₉⋊C₉	C₃²	1	12	yes	—
13	C₃×C₉⋊C₉	C₃²	1	6	yes	—
14	C₃×C₃²⋊C₉	C₃²	1	6	yes	—
15	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
16	C₃×C₉²	—	3	9	no	—
17	C₃×C₉²	C₃²	1	6	yes	—
18	C₃×C₉²	C₃²	1	3	yes	—
19	C₃×C₉²	C₃²	1	6	yes	—
20	C₃×C₉²	C₃²	1	3	yes	—
21	#6855	C₃²	1	54	yes	—
22	C₃⁴	C₃³	1	1	yes	characteristic
23	C₃²×C₉	—	3	9	no	—
24	C₃²×C₉	—	3	27	no	—
25	C₃²×C₉	C₃×C₉	1	9	yes	—
26	C₃²×C₉	3₋¹⁺²	1	9	yes	—
27	C₃²×C₉	3₋¹⁺²	1	18	yes	—
28	C₃²×C₉	C₃³	1	18	yes	—
29	C₃²×C₉	C₃³	1	9	yes	—
30	C₃²×C₉	C₃³	1	9	yes	—
31	C₃²×C₉	C₃³	1	2	yes	—
32	C₃²×C₉	C₃³	1	1	yes	characteristic
33	C₉⋊C₉	—	3	36	no	—
34	C₉⋊C₉	—	3	81	no	—
35	C₉⋊C₉	C₃×C₉	1	54	yes	complemented
36	C₃²⋊C₉	—	3	18	no	—
37	C₃²⋊C₉	C₃×C₉	1	27	yes	—
38	C₉²	—	3	18	no	—
39	C₉²	—	3	9	no	—
40	C₉²	—	3	18	no	—
41	C₉²	C₃×C₉	1	27	yes	complemented
42	C₃³	—	3	9	no	—
43	C₃³	C₃²×C₉	1	3	yes	—
44	C₃³	C₃×3₋¹⁺²	1	6	yes	—
45	C₃³	C₃×3₋¹⁺²	1	3	yes	—
46	C₃³	C₃⁴	1	1	yes	characteristic
47	C₃×C₉	—	3	81	no	—
48	C₃×C₉	—	3	9	no	—
49	C₃×C₉	—	3	27	no	—
50	C₃×C₉	—	3	18	no	—
51	C₃×C₉	—	3	18	no	—
52	C₃×C₉	—	3	54	no	—
53	C₃×C₉	C₉⋊C₉	1	81	yes	complemented
54	C₃×C₉	C₃²×C₉	1	27	yes	—
55	C₃×C₉	C₃²×C₉	1	3	yes	—
56	C₃×C₉	C₃²×C₉	1	6	yes	—
57	C₃×C₉	C₃×3₋¹⁺²	1	6	yes	—
58	C₃×C₉	C₃×3₋¹⁺²	1	3	yes	—
59	C₃×C₉	C₃×3₋¹⁺²	1	6	yes	—
60	C₃×C₉	C₃×3₋¹⁺²	1	12	yes	—
61	C₃×C₉	#737	1	27	yes	—
62	C₃²	—	3	9	no	—
63	C₃²	—	3	9	no	—
64	C₃²	—	3	18	no	—
65	C₃²	C₃×C₉⋊C₉	1	9	yes	—
66	C₃²	C₃³×C₉	1	1	yes	characteristic
67	C₃²	C₃²×3₋¹⁺²	1	1	yes	characteristic
68	C₃²	C₃²×3₋¹⁺²	1	2	yes	—
69	C₃²	C₃×#737	1	9	yes	—
70	C₉	—	3	27	no	—
71	C₉	—	3	27	no	—
72	C₉	—	9	18	no	—
73	C₉	C₃×C₉⋊C₉	1	9	yes	—
74	C₉	C₃×C₉⋊C₉	1	18	yes	—
75	C₃	—	3	9	no	—
76	C₃	C₃²×C₉⋊C₉	1	1	yes	characteristic
77	C₃	C₉×#737	1	3	yes	—
78	C₃	#10624872	1	3	yes	—
79	C₃	#10624879	1	6	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
4. #10624561	C₃	semidirect
6. C₉×C₉⋊C₉	C₃	semidirect
10. #6887	C₉	semidirect
35. C₉⋊C₉	C₃×C₉	semidirect
41. C₉²	C₃×C₉	semidirect
53. C₃×C₉	C₉⋊C₉	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
#10624561	3
#10624556	3
C₉×C₉⋊C₉	3