Subgroups — #14552615

Order 2187.

3140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⁴×C₉	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×#6893	C₃	1	9	yes	maximal, complemented
3	C₃×#6891	C₃	1	2	yes	maximal, complemented
4	C₃×C₃²⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
5	#10624555	C₃	1	27	yes	maximal, complemented
6	C₃⁵	C₃²	1	1	yes	characteristic
7	C₃²×3₋¹⁺²	C₃²	1	6	yes	—
8	C₃²×H(3,𝔽₃)	C₃²	1	9	yes	—
9	C₃²×H(3,𝔽₃)	C₃²	1	3	yes	—
10	C₃³×C₉	—	3	9	no	—
11	C₃³×C₉	—	3	27	no	—
12	C₃³×C₉	C₃²	1	9	yes	—
13	C₃³×C₉	C₃²	1	3	yes	—
14	#6893	—	3	81	no	—
15	#6891	—	3	18	no	—
16	C₃²⋊H(3,𝔽₃)	—	3	9	no	—
17	C₃×C₃²⋊C₉	C₃²	1	18	yes	—
18	C₃×C₃²⋊C₉	C₃²	1	81	yes	complemented
19	C₃⁴	—	3	9	no	—
20	C₃⁴	—	3	27	no	—
21	C₃⁴	H(3,𝔽₃)	1	9	yes	—
22	C₃⁴	C₃³	1	9	yes	—
23	C₃⁴	C₃³	1	1	yes	characteristic
24	C₃⁴	C₃³	1	3	yes	—
25	C₃×3₋¹⁺²	—	3	18	no	—
26	C₃×3₋¹⁺²	—	3	54	no	—
27	C₃×H(3,𝔽₃)	—	3	9	no	—
28	C₃×H(3,𝔽₃)	—	3	27	no	—
29	C₃×H(3,𝔽₃)	—	3	27	no	—
30	C₃×H(3,𝔽₃)	—	3	81	no	—
31	C₃²×C₉	—	3	243	no	—
32	C₃²×C₉	—	3	27	no	—
33	C₃²×C₉	—	3	9	no	—
34	C₃²×C₉	—	3	18	no	—
35	C₃²×C₉	—	3	81	no	—
36	C₃²×C₉	H(3,𝔽₃)	1	27	yes	complemented
37	C₃²×C₉	C₃³	1	18	yes	—
38	C₃²×C₉	C₃³	1	9	yes	—
39	C₃²⋊C₉	—	3	54	no	—
40	C₃²⋊C₉	—	3	243	no	—
41	C₃³	—	3	243	no	—
42	C₃³	—	3	9	no	—
43	C₃³	—	3	9	no	—
44	C₃³	—	3	18	no	—
45	C₃³	—	3	27	no	—
46	C₃³	—	3	81	no	—
47	C₃³	—	3	27	no	—
48	C₃³	—	3	9	no	—
49	C₃³	—	3	81	no	—
50	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
51	C₃³	C₃×H(3,𝔽₃)	1	9	yes	—
52	C₃³	C₃×H(3,𝔽₃)	1	9	yes	—
53	C₃³	#737	1	27	yes	—
54	C₃³	C₃⁴	1	1	yes	characteristic
55	3₋¹⁺²	—	9	54	no	—
56	H(3,𝔽₃)	—	9	27	no	—
57	H(3,𝔽₃)	—	9	81	no	—
58	C₃×C₉	—	3	243	no	—
59	C₃×C₉	—	3	9	no	—
60	C₃×C₉	—	3	18	no	—
61	C₃×C₉	—	3	81	no	—
62	C₃×C₉	—	3	18	no	—
63	C₃×C₉	—	3	54	no	—
64	C₃×C₉	C₃×H(3,𝔽₃)	1	27	yes	—
65	C₃²	—	3	243	no	—
66	C₃²	—	3	9	no	—
67	C₃²	—	3	9	no	—
68	C₃²	—	3	18	no	—
69	C₃²	—	3	27	no	—
70	C₃²	—	3	81	no	—
71	C₃²	—	9	9	no	—
72	C₃²	—	9	27	no	—
73	C₃²	—	9	81	no	—
74	C₃²	C₃²⋊H(3,𝔽₃)	1	9	yes	—
75	C₃²	#6893	1	27	yes	complemented
76	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
77	C₃²	C₃²×H(3,𝔽₃)	1	3	yes	—
78	C₃²	C₃×#737	1	9	yes	—
79	C₉	—	3	27	no	—
80	C₉	—	9	18	no	—
81	C₃	—	3	9	no	—
82	C₃	—	3	27	no	—
83	C₃	—	27	9	no	—
84	C₃	C₃×C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
85	C₃	C₃×#6893	1	3	yes	—
86	C₃	#10624869	1	9	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₃⁴×C₉	C₃	semidirect
2. C₃×#6893	C₃	semidirect
3. C₃×#6891	C₃	semidirect
5. #10624555	C₃	semidirect
18. C₃×C₃²⋊C₉	C₃²	semidirect
36. C₃²×C₉	H(3,𝔽₃)	semidirect
75. C₃²	#6893	semidirect

Maximal subgroups

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

subgroup	index
C₃⁴×C₉	3
C₃×#6893	3
C₃×#6891	3
C₃×C₃²⋊H(3,𝔽₃)	3
#10624555	3