Subgroups — #14548098

Order 2187.

395 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#6887	C₃	1	1	yes	maximal, characteristic, complemented
2	#10624731	C₃	1	3	yes	maximal, complemented
3	#10624637	C₃	1	3	yes	maximal, complemented
4	#10624636	C₃	1	3	yes	maximal, complemented
5	#10624635	C₃	1	3	yes	maximal, complemented
6	C₃³×C₉	C₃²	1	1	yes	characteristic
7	C₃×#731	—	3	3	no	—
8	C₃×#731	—	3	6	no	—
9	#6891	C₃²	1	3	yes	—
10	#6891	C₃²	1	3	yes	—
11	C₃²⋊H(3,𝔽₃)	C₃²	1	3	yes	—
12	#6887	—	3	9	no	—
13	C₃×C₉⋊C₉	—	3	2	no	—
14	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
15	C₃×C₉²	—	3	1	no	—
16	#6861	—	3	9	no	—
17	#6859	—	3	9	no	—
18	#6856	—	3	9	no	—
19	C₃⁴	C₃³	1	1	yes	characteristic
20	C₃×3₋¹⁺²	—	3	3	no	—
21	C₃×3₋¹⁺²	—	3	3	no	—
22	C₃×H(3,𝔽₃)	—	3	3	no	—
23	C₃×H(3,𝔽₃)	—	3	6	no	—
24	C₃×H(3,𝔽₃)	—	3	3	no	—
25	C₃²×C₉	—	3	3	no	—
26	C₃²×C₉	—	3	6	no	—
27	C₃²×C₉	—	3	3	no	—
28	C₃²×C₉	—	3	3	no	—
29	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
30	C₃²×C₉	H(3,𝔽₃)	1	2	yes	—
31	#731	—	9	9	no	—
32	#731	—	9	18	no	—
33	C₉⋊C₉	—	3	18	no	—
34	C₃²⋊C₉	—	3	6	no	—
35	C₃²⋊C₉	—	3	3	no	—
36	C₃²⋊C₉	—	3	6	no	—
37	C₃²⋊C₉	—	3	3	no	—
38	C₃²⋊C₉	—	3	6	no	—
39	C₃²⋊C₉	H(3,𝔽₃)	1	9	yes	—
40	C₉²	—	3	9	no	—
41	C₃³	—	3	2	no	—
42	C₃³	—	3	1	no	—
43	C₃³	—	9	3	no	—
44	C₃³	—	9	3	no	—
45	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
46	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
47	3₋¹⁺²	—	9	6	no	—
48	3₋¹⁺²	—	9	3	no	—
49	3₋¹⁺²	—	9	6	no	—
50	3₋¹⁺²	—	9	3	no	—
51	H(3,𝔽₃)	—	9	6	no	—
52	H(3,𝔽₃)	—	9	3	no	—
53	H(3,𝔽₃)	—	27	6	no	—
54	H(3,𝔽₃)	—	27	3	no	—
55	C₃×C₉	—	3	27	no	—
56	C₃×C₉	—	3	1	no	—
57	C₃×C₉	—	3	2	no	—
58	C₃×C₉	—	3	2	no	—
59	C₃×C₉	—	3	2	no	—
60	C₃×C₉	—	3	2	no	—
61	C₃×C₉	—	3	9	no	—
62	C₃×C₉	—	9	9	no	—
63	C₃×C₉	—	9	3	no	—
64	C₃×C₉	—	9	3	no	—
65	C₃×C₉	#732	1	3	yes	—
66	C₃×C₉	#732	1	6	yes	—
67	C₃²	—	3	3	no	—
68	C₃²	—	3	3	no	—
69	C₃²	—	3	6	no	—
70	C₃²	—	3	3	no	—
71	C₃²	—	9	6	no	—
72	C₃²	—	9	3	no	—
73	C₃²	—	27	3	no	—
74	C₃²	—	27	6	no	—
75	C₃²	—	27	3	no	—
76	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
77	C₃²	C₃×#732	1	2	yes	—
78	C₃²	C₃×#732	1	1	yes	characteristic
79	C₉	—	3	3	no	—
80	C₉	—	3	6	no	—
81	C₉	—	9	9	no	—
82	C₉	—	27	3	no	—
83	C₉	—	27	3	no	—
84	C₃	—	3	2	no	—
85	C₃	—	3	1	no	—
86	C₃	—	9	3	no	—
87	C₃	—	81	3	no	—
88	C₃	#10624732	1	1	yes	characteristic
89	C₃	#10624756	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₃×#6887	C₃	semidirect
2. #10624731	C₃	semidirect
3. #10624637	C₃	semidirect
4. #10624636	C₃	semidirect
5. #10624635	C₃	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
#10624731	3
#10624637	3
#10624636	3
#10624635	3