Subgroups — #14546303

Order 2187.

551 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#6891	C₃	1	1	yes	maximal, characteristic
2	C₃×#6866	C₃	1	3	yes	maximal
3	C₃⁵	C₃²	1	1	yes	characteristic
4	C₃²×3₋¹⁺²	C₉	1	3	yes	complemented
5	#6891	—	3	9	no	—
6	C₃×C₃²⋊C₉	—	3	1	no	—
7	C₃×C₃²⋊C₉	—	3	2	no	—
8	C₃×C₃²⋊C₉	—	3	3	no	—
9	#6866	—	3	9	no	—
10	C₃⁴	—	3	3	no	—
11	C₃⁴	—	3	3	no	—
12	C₃⁴	—	3	3	no	—
13	C₃⁴	—	3	3	no	—
14	C₃⁴	—	9	3	no	—
15	C₃⁴	—	9	6	no	—
16	C₃⁴	C₃×C₉	1	1	yes	characteristic
17	C₃⁴	H(3,𝔽₃)	1	1	yes	characteristic
18	C₃⁴	3₋¹⁺²	1	2	yes	—
19	C₃×3₋¹⁺²	—	3	9	no	—
20	C₃×3₋¹⁺²	—	9	9	no	—
21	C₃²×C₉	—	3	3	no	—
22	C₃²×C₉	—	9	3	no	—
23	C₃²⋊C₉	—	3	18	no	—
24	C₃²⋊C₉	—	3	9	no	—
25	C₃²⋊C₉	—	3	9	no	—
26	C₃²⋊C₉	—	9	3	no	—
27	C₃²⋊C₉	—	9	3	no	—
28	C₃³	—	3	3	no	—
29	C₃³	—	3	6	no	—
30	C₃³	—	3	2	no	—
31	C₃³	—	3	1	no	—
32	C₃³	—	3	3	no	—
33	C₃³	—	3	6	no	—
34	C₃³	—	9	1	no	—
35	C₃³	—	9	2	no	—
36	C₃³	—	9	9	no	—
37	C₃³	—	9	18	no	—
38	C₃³	—	9	2	no	—
39	C₃³	—	9	2	no	—
40	C₃³	—	9	2	no	—
41	C₃³	—	9	27	no	—
42	C₃³	—	9	27	no	—
43	C₃³	—	9	27	no	—
44	C₃³	—	9	3	no	—
45	C₃³	—	9	3	no	—
46	C₃³	—	9	3	no	—
47	C₃³	C₃²⋊C₉	1	1	yes	characteristic
48	C₃³	AΣL(1,𝔽₂₇)	1	3	yes	—
49	C₃³	AΣL(1,𝔽₂₇)	1	3	yes	—
50	C₃³	#731	1	3	yes	—
51	C₃³	#733	1	3	yes	—
52	3₋¹⁺²	—	9	27	no	—
53	C₃×C₉	—	3	9	no	—
54	C₃×C₉	—	9	9	no	—
55	C₃×C₉	—	9	9	no	—
56	C₃×C₉	—	27	3	no	—
57	C₃×C₉	—	27	6	no	—
58	C₃²	—	3	1	no	—
59	C₃²	—	3	3	no	—
60	C₃²	—	3	6	no	—
61	C₃²	—	3	2	no	—
62	C₃²	—	3	3	no	—
63	C₃²	—	3	6	no	—
64	C₃²	—	9	3	no	—
65	C₃²	—	9	3	no	—
66	C₃²	—	9	3	no	—
67	C₃²	—	9	9	no	—
68	C₃²	—	9	18	no	—
69	C₃²	—	9	27	no	—
70	C₃²	—	9	27	no	—
71	C₃²	—	9	27	no	—
72	C₃²	—	9	2	no	—
73	C₃²	—	9	1	no	—
74	C₃²	—	9	2	no	—
75	C₃²	—	9	2	no	—
76	C₃²	—	9	2	no	—
77	C₃²	#6857	1	3	yes	—
78	C₃²	#6858	1	3	yes	—
79	C₃²	#6860	1	3	yes	—
80	C₃²	#6866	1	1	yes	characteristic
81	C₃²	3₋¹⁺²⋊C₉	1	3	yes	—
82	C₉	—	9	9	no	—
83	C₉	—	27	9	no	—
84	C₃	—	3	3	no	—
85	C₃	—	3	3	no	—
86	C₃	—	3	3	no	—
87	C₃	—	3	3	no	—
88	C₃	—	9	3	no	—
89	C₃	—	9	6	no	—
90	C₃	#10624452	1	1	yes	characteristic
91	C₃	#10624505	1	1	yes	characteristic
92	C₃	#10624509	1	2	yes	—

	subgroup	G/H
center, socle	C₃²	#6866
commutator	C₃⁴	C₃×C₉
Frattini, prime radical, cosocle	C₃⁵	C₃²
LCS	G ▷ C₃⁴ ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃⁵ ◁ G
LECS	G ▷ C₃⁵ ▷ C₃³ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. C₃²×3₋¹⁺²	C₉	semidirect

Maximal subgroups

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

subgroup	index
C₃×#6891	3
C₃×#6866	3