Subgroups — #14550528

Order 2187.

476 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#6895	C₃	1	1	yes	maximal, characteristic
2	C₃²×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic
3	#10624549	C₃	1	2	yes	maximal
4	#10624466	C₃	1	9	yes	maximal
5	C₃³×C₉	C₃²	1	3	yes	—
6	C₃³×C₉	C₃²	1	1	yes	characteristic
7	#6895	—	3	6	no	—
8	#6895	C₉	1	9	yes	complemented
9	C₃×C₉⋊C₉	—	3	3	no	—
10	C₃×C₉⋊C₉	—	3	3	no	—
11	C₃×C₉⋊C₉	—	3	9	no	—
12	C₃×C₃²⋊C₉	C₃²	1	6	yes	—
13	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
14	#6855	—	3	6	no	—
15	C₃⁴	C₃³	1	1	yes	characteristic
16	C₃²×C₉	—	3	3	no	—
17	C₃²×C₉	—	3	6	no	—
18	C₃²×C₉	—	3	3	no	—
19	C₃²×C₉	—	3	3	no	—
20	C₃²×C₉	—	3	6	no	—
21	C₃²×C₉	—	3	6	no	—
22	C₃²×C₉	—	9	9	no	—
23	C₃²×C₉	C₃×C₉	1	3	yes	—
24	C₃²×C₉	H(3,𝔽₃)	1	3	yes	—
25	C₃²×C₉	3₋¹⁺²	1	6	yes	—
26	C₃²×C₉	3₋¹⁺²	1	9	yes	—
27	C₉⋊C₉	—	3	27	no	—
28	C₉⋊C₉	—	3	27	no	—
29	C₃²⋊C₉	—	3	6	no	—
30	C₃²⋊C₉	—	3	18	no	—
31	C₃²⋊C₉	C₃×C₉	1	9	yes	—
32	C₃³	—	3	9	no	—
33	C₃³	C₃²×C₉	1	1	yes	characteristic
34	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
35	C₃³	C₃×3₋¹⁺²	1	2	yes	—
36	C₃³	C₃×3₋¹⁺²	1	3	yes	—
37	C₃³	#737	1	6	yes	—
38	C₃×C₉	—	3	9	no	—
39	C₃×C₉	—	3	3	no	—
40	C₃×C₉	—	3	6	no	—
41	C₃×C₉	—	3	18	no	—
42	C₃×C₉	—	3	9	no	—
43	C₃×C₉	—	3	9	no	—
44	C₃×C₉	—	9	18	no	—
45	C₃×C₉	—	9	9	no	—
46	C₃×C₉	—	9	9	no	—
47	C₃×C₉	—	9	9	no	—
48	C₃×C₉	—	9	18	no	—
49	C₃×C₉	—	9	6	no	—
50	C₃×C₉	C₃²⋊C₉	1	9	yes	—
51	C₃²	—	3	6	no	—
52	C₃²	—	3	3	no	—
53	C₃²	—	3	18	no	—
54	C₃²	—	3	3	no	—
55	C₃²	—	3	6	no	—
56	C₃²	C₃×C₃²⋊C₉	1	1	yes	characteristic
57	C₃²	C₉×3₋¹⁺²	1	3	yes	—
58	C₃²	#6891	1	1	yes	characteristic
59	C₃²	#6893	1	2	yes	—
60	C₃²	#6894	1	6	yes	—
61	C₃²	#6913	1	9	yes	—
62	C₉	—	3	9	no	—
63	C₉	—	9	9	no	—
64	C₉	—	9	9	no	—
65	C₉	—	27	6	no	—
66	C₃	—	3	3	no	—
67	C₃	—	3	6	no	—
68	C₃	#10624558	1	1	yes	characteristic
69	C₃	#10624700	1	3	yes	—
70	C₃	#10624774	1	3	yes	—
71	C₃	#10624817	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
8. #6895	C₉	semidirect

Maximal subgroups

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

subgroup	index
C₃×#6895	3
C₃²×C₉⋊C₉	3
#10624549	3
#10624466	3