Group #14550559 — subgroups

Subgroups — #14550559

Order 2187.

521 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 521 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
2	C₃×H(3,𝔽₃)⋊C₉	C₃	1	3	yes	maximal
3	#10624548	C₃	1	2	yes	maximal, complemented
4	C₃×#6855	C₃	1	1	yes	maximal, characteristic, complemented
5	#10624466	C₃	1	6	yes	maximal, complemented
6	C₃²×3₋¹⁺²	—	3	1	no	—
7	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
8	C₃³×C₉	C₃²	1	3	yes	—
9	C₃³×C₉	C₃²	1	1	yes	characteristic
10	C₉⋊H(3,𝔽₃)	—	3	3	no	—
11	C₉⋊H(3,𝔽₃)	—	3	3	no	—
12	C₉⋊H(3,𝔽₃)	C₉	1	9	yes	complemented
13	C₃×C₉⋊C₉	—	3	2	no	—
14	C₃×C₃²⋊C₉	—	3	2	no	—
15	C₃×C₃²⋊C₉	C₃²	1	6	yes	—
16	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
17	H(3,𝔽₃)⋊C₉	—	3	9	no	—
18	#6855	—	3	1	no	—
19	#6855	—	3	2	no	—
20	#6855	—	3	2	no	—
21	#6855	—	3	2	no	—
22	C₃⁴	—	3	1	no	—
23	C₃⁴	C₃³	1	1	yes	characteristic
24	C₃×3₋¹⁺²	—	3	9	no	—
25	C₃×3₋¹⁺²	—	9	3	no	—
26	C₃×3₋¹⁺²	—	9	3	no	—
27	C₃×3₋¹⁺²	—	9	3	no	—
28	C₃×H(3,𝔽₃)	—	3	3	no	—
29	C₃×H(3,𝔽₃)	—	3	6	no	—
30	C₃×H(3,𝔽₃)	—	3	1	no	—
31	C₃×H(3,𝔽₃)	—	3	1	no	—
32	C₃×H(3,𝔽₃)	C₃×C₉	1	3	yes	—
33	C₃²×C₉	—	3	3	no	—
34	C₃²×C₉	—	3	2	no	—
35	C₃²×C₉	—	3	6	no	—
36	C₃²×C₉	—	3	6	no	—
37	C₃²×C₉	—	9	1	no	—
38	C₃²×C₉	—	9	1	no	—
39	C₃²×C₉	—	9	1	no	—
40	C₃²×C₉	—	9	6	no	—
41	C₃²×C₉	—	9	3	no	—
42	C₃²×C₉	C₃×C₉	1	3	yes	—
43	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
44	C₃²×C₉	H(3,𝔽₃)	1	2	yes	—
45	C₃²×C₉	3₋¹⁺²	1	3	yes	—
46	C₃²×C₉	3₋¹⁺²	1	3	yes	—
47	C₃²×C₉	3₋¹⁺²	1	9	yes	complemented
48	C₉⋊C₉	—	9	6	no	—
49	C₃²⋊C₉	—	3	2	no	—
50	C₃²⋊C₉	—	3	2	no	—
51	C₃²⋊C₉	—	3	18	no	—
52	C₃²⋊C₉	—	3	6	no	—
53	C₃²⋊C₉	—	3	12	no	—
54	C₃²⋊C₉	C₃×C₉	1	6	yes	—
55	C₃³	—	3	6	no	—
56	C₃³	—	3	1	no	—
57	C₃³	—	3	1	no	—
58	C₃³	—	3	1	no	—
59	C₃³	—	3	3	no	—
60	C₃³	—	9	6	no	—
61	C₃³	—	9	3	no	—
62	C₃³	—	9	1	no	—
63	C₃³	—	9	1	no	—
64	C₃³	—	9	1	no	—
65	C₃³	C₃²×C₉	1	1	yes	characteristic
66	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
67	C₃³	C₃×3₋¹⁺²	1	1	yes	characteristic
68	C₃³	C₃×3₋¹⁺²	1	1	yes	characteristic
69	C₃³	C₃×3₋¹⁺²	1	3	yes	—
70	C₃³	#737	1	6	yes	—
71	3₋¹⁺²	—	9	27	no	—
72	H(3,𝔽₃)	—	3	9	no	—
73	H(3,𝔽₃)	—	9	3	no	—
74	H(3,𝔽₃)	—	9	2	no	—
75	H(3,𝔽₃)	—	9	1	no	—
76	C₃×C₉	—	3	2	no	—
77	C₃×C₉	—	3	1	no	—
78	C₃×C₉	—	3	2	no	—
79	C₃×C₉	—	3	1	no	—
80	C₃×C₉	—	3	2	no	—
81	C₃×C₉	—	3	1	no	—
82	C₃×C₉	—	3	6	no	—
83	C₃×C₉	—	3	9	no	—
84	C₃×C₉	—	9	3	no	—
85	C₃×C₉	—	9	3	no	—
86	C₃×C₉	—	9	3	no	—
87	C₃×C₉	—	9	18	no	—
88	C₃×C₉	—	9	6	no	—
89	C₃×C₉	—	9	3	no	—
90	C₃×C₉	—	9	2	no	—
91	C₃×C₉	—	9	2	no	—
92	C₃×C₉	—	9	2	no	—
93	C₃×C₉	—	9	9	no	—
94	C₃×C₉	—	9	12	no	—
95	C₃×C₉	—	9	6	no	—
96	C₃×C₉	—	9	6	no	—
97	C₃×C₉	C₃²⋊C₉	1	6	yes	—
98	C₃×C₉	C₃²⋊C₉	1	3	yes	—
99	C₃²	—	3	3	no	—
100	C₃²	—	3	6	no	—
101	C₃²	—	3	3	no	—
102	C₃²	—	3	6	no	—
103	C₃²	—	3	3	no	—
104	C₃²	—	3	6	no	—
105	C₃²	—	3	1	no	—
106	C₃²	—	3	1	no	—
107	C₃²	—	3	1	no	—
108	C₃²	—	3	2	no	—
109	C₃²	—	3	2	no	—
110	C₃²	—	3	2	no	—
111	C₃²	—	9	3	no	—
112	C₃²	—	9	3	no	—
113	C₃²	—	9	6	no	—
114	C₃²	—	27	3	no	—
115	C₃²	—	27	3	no	—
116	C₃²	—	27	3	no	—
117	C₃²	C₃×C₃²⋊C₉	1	1	yes	characteristic
118	C₃²	C₉×3₋¹⁺²	1	3	yes	—
119	C₃²	C₉⋊H(3,𝔽₃)	1	3	yes	—
120	C₃²	#6896	1	3	yes	—
121	C₃²	#6899	1	2	yes	—
122	C₃²	#6900	1	1	yes	characteristic
123	C₃²	#6909	1	1	yes	characteristic
124	C₃²	#6910	1	1	yes	characteristic
125	C₃²	#6911	1	1	yes	characteristic
126	C₃²	#6913	1	6	yes	—
127	C₉	—	3	6	no	—
128	C₉	—	3	3	no	—
129	C₉	—	9	9	no	—
130	C₉	—	9	6	no	—
131	C₉	—	27	6	no	—
132	C₃	—	3	3	no	—
133	C₃	—	3	3	no	—
134	C₃	—	3	2	no	—
135	C₃	—	3	1	no	—
136	C₃	—	27	3	no	—
137	C₃	#10624560	1	1	yes	characteristic
138	C₃	#10624698	1	1	yes	characteristic
139	C₃	#10624700	1	2	yes	—
140	C₃	#10624788	1	3	yes	—
141	C₃	#10624789	1	3	yes	—
142	C₃	#10624790	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₉⋊H(3,𝔽₃)

C₃

yes

maximal, characteristic

C₃×H(3,𝔽₃)⋊C₉

C₃

yes

maximal

#10624548

C₃

yes

maximal, complemented

C₃×#6855

C₃

yes

maximal, characteristic, complemented

#10624466

C₃

yes

maximal, complemented

C₃²×3₋¹⁺²

—

C₃²×H(3,𝔽₃)

C₃²

yes

characteristic

C₃³×C₉

C₃²

yes

—

C₃³×C₉

C₃²

yes

characteristic

C₉⋊H(3,𝔽₃)

—

C₉⋊H(3,𝔽₃)

—

C₉⋊H(3,𝔽₃)

C₉

yes

complemented

C₃×C₉⋊C₉

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

—

H(3,𝔽₃)⋊C₉

—

—

—

—

—

C₃⁴

—

C₃⁴

C₃³

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃×C₉

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×C₉

yes

—

C₃²×C₉

H(3,𝔽₃)

yes

characteristic

C₃²×C₉

H(3,𝔽₃)

yes

—

C₃²×C₉

3₋¹⁺²

yes

—

C₃²×C₉

3₋¹⁺²

yes

—

C₃²×C₉

3₋¹⁺²

yes

complemented

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×C₉

yes

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃²×C₉

yes

characteristic

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

C₃³

C₃×3₋¹⁺²

yes

characteristic

C₃³

C₃×3₋¹⁺²

yes

characteristic

C₃³

C₃×3₋¹⁺²

yes

—

C₃³

#737

yes

—

3₋¹⁺²

—

—

—

—

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₃²⋊C₉

yes

—

C₃×C₉

C₃²⋊C₉

yes

—

C₃²

—

100

C₃²

—

101

C₃²

—

102

C₃²

—

103

C₃²

—

104

C₃²

—

105

C₃²

—

106

C₃²

—

107

C₃²

—

108

C₃²

—

109

C₃²

—

110

C₃²

—

111

C₃²

—

112

C₃²

—

113

C₃²

—

114

C₃²

—

115

C₃²

—

116

C₃²

—

117

C₃²

C₃×C₃²⋊C₉

yes

characteristic

118

C₃²

C₉×3₋¹⁺²

yes

—

119

C₃²

C₉⋊H(3,𝔽₃)

yes

—

120

C₃²

#6896

yes

—

121

C₃²

#6899

yes

—

122

C₃²

#6900

yes

characteristic

123

C₃²

#6909

yes

characteristic

124

C₃²

#6910

yes

characteristic

125

C₃²

#6911

yes

characteristic

126

C₃²

#6913

yes

—

127

C₉

—

128

C₉

—

129

C₉

—

130

C₉

—

131

C₉

—

132

C₃

—

133

C₃

—

134

C₃

—

135

C₃

—

136

C₃

—

137

C₃

#10624560

yes

characteristic

138

C₃

#10624698

yes

characteristic

139

C₃

#10624700

yes

—

140

C₃

#10624788

yes

—

141

C₃

#10624789

yes

—

142

C₃

#10624790

yes

—

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

subgroup

G/H

center, socle

C₃³

C₃×H(3,𝔽₃)

commutator

C₃³

C₃²×C₉

Frattini, cosocle

C₃⁴

C₃³

prime radical

C₃²×H(3,𝔽₃)

C₃²

LCS

G ▷ C₃³ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃³ ◁ C₃³×C₉ ◁ G

LECS

G ▷ C₃⁴ ▷ C₃ ▷ C₁

N	H ≅ G/N	type
3. #10624548	C₃	semidirect
4. C₃×#6855	C₃	semidirect
5. #10624466	C₃	semidirect
12. C₉⋊H(3,𝔽₃)	C₉	semidirect
47. C₃²×C₉	3₋¹⁺²	semidirect

H ≅ G/N

type

3. #10624548

C₃

semidirect

4. C₃×#6855

C₃

semidirect

5. #10624466

C₃

semidirect

12. C₉⋊H(3,𝔽₃)

C₉

semidirect

47. C₃²×C₉

3₋¹⁺²

semidirect

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

subgroup	index
C₃×C₉⋊H(3,𝔽₃)	3
C₃×H(3,𝔽₃)⋊C₉	3
#10624548	3
C₃×#6855	3
#10624466	3

subgroup

index

C₃×C₉⋊H(3,𝔽₃)

C₃×H(3,𝔽₃)⋊C₉

#10624548

C₃×#6855

#10624466

Subgroups — #14550559

Conjugacy classes of subgroups

Complements

Maximal subgroups