Subgroups — #14550311

Order 2187.

464 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic
2	C₃×#6868	C₃	1	3	yes	maximal
3	#10624548	C₃	1	3	yes	maximal
4	#10624466	C₃	1	6	yes	maximal
5	C₃³×C₉	C₃²	1	1	yes	characteristic
6	C₃³×C₉	C₃²	1	2	yes	—
7	C₃³×C₉	C₃²	1	1	yes	characteristic
8	C₃×C₉⋊C₉	—	3	3	no	—
9	C₃×C₉⋊C₉	—	3	6	no	—
10	C₃×C₉⋊C₉	—	3	3	no	—
11	C₃×C₉⋊C₉	C₉	1	9	yes	complemented
12	C₃×C₃²⋊C₉	C₃²	1	9	yes	—
13	#6868	—	3	9	no	—
14	#6855	—	3	6	no	—
15	C₃⁴	C₃³	1	1	yes	characteristic
16	C₃²×C₉	—	3	1	no	—
17	C₃²×C₉	—	3	2	no	—
18	C₃²×C₉	—	3	6	no	—
19	C₃²×C₉	—	3	2	no	—
20	C₃²×C₉	—	3	4	no	—
21	C₃²×C₉	—	3	9	no	—
22	C₃²×C₉	—	9	2	no	—
23	C₃²×C₉	—	9	1	no	—
24	C₃²×C₉	—	9	6	no	—
25	C₃²×C₉	C₃×C₉	1	3	yes	—
26	C₃²×C₉	C₃×C₉	1	6	yes	—
27	C₃²×C₉	C₃×C₉	1	3	yes	—
28	C₃²×C₉	H(3,𝔽₃)	1	3	yes	—
29	C₃²×C₉	3₋¹⁺²	1	9	yes	—
30	C₃²×C₉	3₋¹⁺²	1	6	yes	—
31	C₉⋊C₉	—	3	27	no	—
32	C₉⋊C₉	—	9	9	no	—
33	C₃²⋊C₉	—	3	9	no	—
34	C₃²⋊C₉	—	3	18	no	—
35	C₃³	—	3	2	no	—
36	C₃³	—	3	6	no	—
37	C₃³	—	3	1	no	—
38	C₃³	C₃²×C₉	1	1	yes	characteristic
39	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
40	C₃³	C₃×3₋¹⁺²	1	2	yes	—
41	C₃³	C₃×3₋¹⁺²	1	3	yes	—
42	C₃³	#737	1	6	yes	—
43	C₃×C₉	—	3	6	no	—
44	C₃×C₉	—	3	3	no	—
45	C₃×C₉	—	3	6	no	—
46	C₃×C₉	—	3	9	no	—
47	C₃×C₉	—	3	18	no	—
48	C₃×C₉	—	3	3	no	—
49	C₃×C₉	—	3	6	no	—
50	C₃×C₉	—	9	3	no	—
51	C₃×C₉	—	9	6	no	—
52	C₃×C₉	—	9	12	no	—
53	C₃×C₉	—	9	6	no	—
54	C₃×C₉	—	9	6	no	—
55	C₃×C₉	—	9	18	no	—
56	C₃×C₉	—	9	9	no	—
57	C₃×C₉	—	9	9	no	—
58	C₃×C₉	C₃²⋊C₉	1	9	yes	—
59	C₃×C₉	C₉⋊C₉	1	9	yes	complemented
60	C₃²	—	3	3	no	—
61	C₃²	—	3	6	no	—
62	C₃²	—	3	6	no	—
63	C₃²	—	3	12	no	—
64	C₃²	—	3	1	no	—
65	C₃²	—	3	2	no	—
66	C₃²	—	3	2	no	—
67	C₃²	—	3	4	no	—
68	C₃²	C₃×C₃²⋊C₉	1	1	yes	characteristic
69	C₃²	C₃×C₉⋊C₉	1	1	yes	characteristic
70	C₃²	#6887	1	2	yes	—
71	C₃²	C₉⋊H(3,𝔽₃)	1	3	yes	—
72	C₃²	#6894	1	6	yes	—
73	C₃²	#6911	1	1	yes	characteristic
74	C₃²	#6912	1	2	yes	—
75	C₃²	#6913	1	6	yes	—
76	C₉	—	3	9	no	—
77	C₉	—	3	9	no	—
78	C₉	—	9	6	no	—
79	C₉	—	27	9	no	—
80	C₃	—	3	1	no	—
81	C₃	—	3	2	no	—
82	C₃	—	3	6	no	—
83	C₃	#10624548	1	1	yes	characteristic
84	C₃	#10624691	1	1	yes	characteristic
85	C₃	#10624692	1	2	yes	—
86	C₃	#10624790	1	3	yes	—
87	C₃	#10624791	1	6	yes	—

	subgroup	G/H
center, socle	C₃³	C₃×H(3,𝔽₃)
commutator	C₃³	C₃²×C₉
Frattini, prime radical, cosocle	C₃⁴	C₃³
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃³ ◁ C₃³×C₉ ◁ G
LECS	G ▷ C₃⁴ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
11. C₃×C₉⋊C₉	C₉	semidirect
59. 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
C₃²×C₉⋊C₉	3
C₃×#6868	3
#10624548	3
#10624466	3