Group #14548161 — subgroups

Subgroups — #14548161

Order 2187.

285 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 285 conjugacy classes of subgroups — too many to draw. Only its 50 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	#10624911	C₃	1	1	yes	maximal, characteristic
2	#10624812	C₃	1	1	yes	maximal, characteristic
3	#10624812	C₃	1	1	yes	maximal, characteristic
4	#10624812	C₃	1	1	yes	maximal, characteristic
5	#10624812	C₃	1	1	yes	maximal, characteristic
6	#10624673	C₃	1	1	yes	maximal, characteristic
7	#10624673	C₃	1	1	yes	maximal, characteristic
8	#10624671	C₃	1	1	yes	maximal, characteristic
9	#10624668	C₃	1	1	yes	maximal, characteristic
10	#10624663	C₃	1	1	yes	maximal, characteristic
11	#10624660	C₃	1	1	yes	maximal, characteristic
12	#10624660	C₃	1	1	yes	maximal, characteristic
13	#10624659	C₃	1	1	yes	maximal, characteristic
14	C₃³×C₉	C₃²	1	1	yes	characteristic
15	#6899	—	3	3	no	—
16	#6899	—	3	3	no	—
17	#6897	—	3	3	no	—
18	#6896	—	3	3	no	—
19	#6891	C₃²	1	1	yes	characteristic
20	#6891	C₃²	1	1	yes	characteristic
21	#6891	C₃²	1	1	yes	characteristic
22	#6891	C₃²	1	1	yes	characteristic
23	#6891	C₃²	1	1	yes	characteristic
24	#6891	C₃²	1	1	yes	characteristic
25	#6891	C₃²	1	1	yes	characteristic
26	#6891	C₃²	1	1	yes	characteristic
27	#6891	C₃²	1	1	yes	characteristic
28	#6891	C₃²	1	1	yes	characteristic
29	#6891	C₃²	1	1	yes	characteristic
30	#6891	C₃²	1	1	yes	characteristic
31	3₋¹⁺²⋊C₉	—	3	3	no	—
32	3₋¹⁺²⋊C₉	—	3	3	no	—
33	3₋¹⁺²⋊C₉	—	3	3	no	—
34	3₋¹⁺²⋊C₉	—	3	3	no	—
35	#6861	—	3	3	no	—
36	#6860	—	3	3	no	—
37	#6860	—	3	3	no	—
38	#6860	—	3	3	no	—
39	#6858	—	3	3	no	—
40	#6858	—	3	3	no	—
41	#6858	—	3	3	no	—
42	#6858	—	3	3	no	—
43	C₃⁴	C₃³	1	1	yes	characteristic
44	C₃×3₋¹⁺²	—	3	1	no	—
45	C₃×3₋¹⁺²	—	3	1	no	—
46	C₃×3₋¹⁺²	—	3	1	no	—
47	C₃×3₋¹⁺²	—	3	1	no	—
48	C₃×3₋¹⁺²	—	3	1	no	—
49	C₃×3₋¹⁺²	—	3	1	no	—
50	C₃×3₋¹⁺²	—	3	1	no	—
51	C₃×3₋¹⁺²	—	3	1	no	—
52	C₃×3₋¹⁺²	—	3	1	no	—
53	C₃×3₋¹⁺²	—	3	1	no	—
54	C₃×3₋¹⁺²	H(3,𝔽₃)	1	3	yes	—
55	C₃×3₋¹⁺²	H(3,𝔽₃)	1	3	yes	—
56	C₃²×C₉	—	3	1	no	—
57	C₃²×C₉	—	3	1	no	—
58	C₃²×C₉	—	3	1	no	—
59	C₃²×C₉	—	3	1	no	—
60	C₃²×C₉	—	9	3	no	—
61	C₉⋊C₉	—	3	24	no	—
62	C₃²⋊C₉	—	3	1	no	—
63	C₃²⋊C₉	—	3	1	no	—
64	C₃²⋊C₉	—	3	1	no	—
65	C₃²⋊C₉	—	3	1	no	—
66	C₃²⋊C₉	—	3	1	no	—
67	C₃²⋊C₉	—	3	1	no	—
68	C₃²⋊C₉	—	3	1	no	—
69	C₃²⋊C₉	—	3	1	no	—
70	C₃²⋊C₉	—	3	1	no	—
71	C₃²⋊C₉	—	3	1	no	—
72	C₃²⋊C₉	—	3	1	no	—
73	C₃²⋊C₉	—	3	1	no	—
74	C₃²⋊C₉	—	3	1	no	—
75	C₃²⋊C₉	—	3	1	no	—
76	C₃²⋊C₉	—	3	1	no	—
77	C₃²⋊C₉	—	3	1	no	—
78	C₃²⋊C₉	—	3	1	no	—
79	C₃²⋊C₉	—	3	1	no	—
80	C₃²⋊C₉	—	3	1	no	—
81	C₃²⋊C₉	—	3	1	no	—
82	C₃²⋊C₉	—	3	1	no	—
83	C₃²⋊C₉	—	3	1	no	—
84	C₃²⋊C₉	—	3	1	no	—
85	C₃²⋊C₉	—	3	1	no	—
86	C₃²⋊C₉	—	3	1	no	—
87	C₃²⋊C₉	—	3	1	no	—
88	C₃²⋊C₉	—	3	1	no	—
89	C₃²⋊C₉	—	3	1	no	—
90	C₃²⋊C₉	—	3	1	no	—
91	C₃²⋊C₉	—	3	1	no	—
92	C₃²⋊C₉	—	3	1	no	—
93	C₃²⋊C₉	—	3	1	no	—
94	C₃²⋊C₉	—	3	1	no	—
95	C₃²⋊C₉	—	3	1	no	—
96	C₃²⋊C₉	H(3,𝔽₃)	1	3	yes	—
97	C₃²⋊C₉	H(3,𝔽₃)	1	3	yes	—
98	C₉²	—	3	3	no	—
99	C₃³	—	9	1	no	—
100	C₃³	—	9	1	no	—
101	C₃³	—	9	1	no	—
102	C₃³	—	9	1	no	—
103	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
104	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
105	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
106	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
107	3₋¹⁺²	—	9	3	no	—
108	3₋¹⁺²	—	9	3	no	—
109	3₋¹⁺²	—	9	3	no	—
110	3₋¹⁺²	—	9	3	no	—
111	3₋¹⁺²	—	27	1	no	—
112	3₋¹⁺²	—	27	1	no	—
113	3₋¹⁺²	—	27	1	no	—
114	3₋¹⁺²	—	27	1	no	—
115	3₋¹⁺²	—	27	1	no	—
116	3₋¹⁺²	—	27	1	no	—
117	3₋¹⁺²	—	27	1	no	—
118	3₋¹⁺²	—	27	1	no	—
119	C₃×C₉	—	3	3	no	—
120	C₃×C₉	—	3	3	no	—
121	C₃×C₉	—	3	3	no	—
122	C₃×C₉	—	3	3	no	—
123	C₃×C₉	—	9	1	no	—
124	C₃×C₉	—	9	3	no	—
125	C₃×C₉	—	9	3	no	—
126	C₃×C₉	—	9	3	no	—
127	C₃×C₉	—	9	3	no	—
128	C₃×C₉	—	9	1	no	—
129	C₃×C₉	—	9	1	no	—
130	C₃×C₉	—	9	1	no	—
131	C₃×C₉	—	9	1	no	—
132	C₃×C₉	—	9	1	no	—
133	C₃×C₉	—	9	1	no	—
134	C₃×C₉	—	9	1	no	—
135	C₃×C₉	—	9	1	no	—
136	C₃²	—	3	1	no	—
137	C₃²	—	3	1	no	—
138	C₃²	—	3	1	no	—
139	C₃²	—	3	1	no	—
140	C₃²	—	3	1	no	—
141	C₃²	—	3	1	no	—
142	C₃²	—	3	1	no	—
143	C₃²	—	3	1	no	—
144	C₃²	—	3	1	no	—
145	C₃²	—	3	1	no	—
146	C₃²	—	3	1	no	—
147	C₃²	—	3	1	no	—
148	C₃²	—	3	1	no	—
149	C₃²	—	3	1	no	—
150	C₃²	—	3	1	no	—
151	C₃²	—	3	1	no	—
152	C₃²	—	9	9	no	—
153	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
154	C₉	—	9	3	no	—
155	C₉	—	9	3	no	—
156	C₉	—	9	3	no	—
157	C₉	—	9	3	no	—
158	C₉	—	9	3	no	—
159	C₉	—	27	1	no	—
160	C₉	—	27	1	no	—
161	C₉	—	27	1	no	—
162	C₉	—	27	1	no	—
163	C₉	—	27	1	no	—
164	C₉	—	27	1	no	—
165	C₉	—	27	1	no	—
166	C₉	—	27	1	no	—
167	C₃	—	9	1	no	—
168	C₃	—	9	1	no	—
169	C₃	—	9	1	no	—
170	C₃	—	9	1	no	—
171	C₃	#10624750	1	1	yes	characteristic
172	C₃	#10624750	1	1	yes	characteristic
173	C₃	#10624762	1	1	yes	characteristic
174	C₃	#10624764	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10624911

C₃

yes

maximal, characteristic

#10624812

C₃

yes

maximal, characteristic

#10624812

C₃

yes

maximal, characteristic

#10624812

C₃

yes

maximal, characteristic

#10624812

C₃

yes

maximal, characteristic

#10624673

C₃

yes

maximal, characteristic

#10624673

C₃

yes

maximal, characteristic

#10624671

C₃

yes

maximal, characteristic

#10624668

C₃

yes

maximal, characteristic

#10624663

C₃

yes

maximal, characteristic

#10624660

C₃

yes

maximal, characteristic

#10624660

C₃

yes

maximal, characteristic

#10624659

C₃

yes

maximal, characteristic

C₃³×C₉

C₃²

yes

—

—

—

—

C₃²

yes

characteristic

#6891

C₃²

yes

characteristic

#6891

C₃²

yes

characteristic

#6891

C₃²

yes

characteristic

#6891

C₃²

yes

characteristic

#6891

C₃²

yes

characteristic

#6891

C₃²

yes

characteristic

#6891

C₃²

yes

characteristic

#6891

C₃²

yes

characteristic

#6891

C₃²

yes

characteristic

#6891

C₃²

yes

characteristic

#6891

C₃²

yes

characteristic

3₋¹⁺²⋊C₉

—

3₋¹⁺²⋊C₉

—

3₋¹⁺²⋊C₉

—

3₋¹⁺²⋊C₉

—

—

—

—

—

—

—

—

—

C₃⁴

C₃³

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

H(3,𝔽₃)

yes

—

C₃×3₋¹⁺²

H(3,𝔽₃)

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

H(3,𝔽₃)

yes

—

C₃²⋊C₉

H(3,𝔽₃)

yes

—

C₉²

—

C₃³

—

100

C₃³

—

101

C₃³

—

102

C₃³

—

103

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

104

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

105

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

106

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

107

3₋¹⁺²

—

108

3₋¹⁺²

—

109

3₋¹⁺²

—

110

3₋¹⁺²

—

111

3₋¹⁺²

—

112

3₋¹⁺²

—

113

3₋¹⁺²

—

114

3₋¹⁺²

—

115

3₋¹⁺²

—

116

3₋¹⁺²

—

117

3₋¹⁺²

—

118

3₋¹⁺²

—

119

C₃×C₉

—

120

C₃×C₉

—

121

C₃×C₉

—

122

C₃×C₉

—

123

C₃×C₉

—

124

C₃×C₉

—

125

C₃×C₉

—

126

C₃×C₉

—

127

C₃×C₉

—

128

C₃×C₉

—

129

C₃×C₉

—

130

C₃×C₉

—

131

C₃×C₉

—

132

C₃×C₉

—

133

C₃×C₉

—

134

C₃×C₉

—

135

C₃×C₉

—

136

C₃²

—

137

C₃²

—

138

C₃²

—

139

C₃²

—

140

C₃²

—

141

C₃²

—

142

C₃²

—

143

C₃²

—

144

C₃²

—

145

C₃²

—

146

C₃²

—

147

C₃²

—

148

C₃²

—

149

C₃²

—

150

C₃²

—

151

C₃²

—

152

C₃²

—

153

C₃²

C₃²⋊H(3,𝔽₃)

yes

characteristic

154

C₉

—

155

C₉

—

156

C₉

—

157

C₉

—

158

C₉

—

159

C₉

—

160

C₉

—

161

C₉

—

162

C₉

—

163

C₉

—

164

C₉

—

165

C₉

—

166

C₉

—

167

C₃

—

168

C₃

—

169

C₃

—

170

C₃

—

171

C₃

#10624750

yes

characteristic

172

C₃

#10624750

yes

characteristic

173

C₃

#10624762

yes

characteristic

174

C₃

#10624764

yes

characteristic

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

subgroup

G/H

center, socle

C₃²

C₃²⋊H(3,𝔽₃)

commutator, Frattini, prime radical, cosocle

C₃⁴

C₃³

LCS

G ▷ C₃⁴ ▷ C₃² ▷ C₁

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ G

LECS

G ▷ C₃⁴ ▷ C₃² ▷ C₁

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

subgroup	index
#10624911	3
#10624812	3
#10624673	3
#10624671	3
#10624668	3
#10624663	3
#10624660	3
#10624659	3

subgroup

index

Subgroups — #14548161

Conjugacy classes of subgroups

Maximal subgroups