Group #14550414 — subgroups

Subgroups — #14550414

Order 2187.

776 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 776 conjugacy classes of subgroups — too many to draw. Only its 107 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×#6891	C₃	1	1	yes	maximal, characteristic
2	#10624863	C₃	1	1	yes	maximal, characteristic, complemented
3	#10624699	C₃	1	2	yes	maximal, complemented
4	#10624699	C₃	1	2	yes	maximal, complemented
5	#10624699	C₃	1	2	yes	maximal, complemented
6	#10624691	C₃	1	1	yes	maximal, characteristic, complemented
7	#10624690	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃×#6866	C₃	1	2	yes	maximal
9	#10624547	C₃	1	1	yes	maximal, characteristic
10	C₃⁵	C₃²	1	1	yes	characteristic
11	C₃²×3₋¹⁺²	C₃²	1	2	yes	—
12	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
13	#6891	—	3	6	no	—
14	#6891	C₉	1	9	yes	complemented
15	C₉×3₋¹⁺²	—	3	3	no	—
16	C₉×3₋¹⁺²	—	3	6	no	—
17	#6887	—	3	3	no	—
18	C₃×C₃²⋊C₉	—	3	3	no	—
19	C₃×C₃²⋊C₉	—	3	3	no	—
20	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
21	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
22	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
23	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
24	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
25	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
26	3₋¹⁺²⋊C₉	—	3	6	no	—
27	3₋¹⁺²⋊C₉	—	3	6	no	—
28	3₋¹⁺²⋊C₉	—	3	6	no	—
29	#6868	—	3	3	no	—
30	#6867	—	3	3	no	—
31	#6866	—	3	6	no	—
32	C₃⁴	—	3	1	no	—
33	C₃⁴	—	3	1	no	—
34	C₃⁴	—	3	2	no	—
35	C₃⁴	—	3	2	no	—
36	C₃⁴	—	3	2	no	—
37	C₃⁴	—	3	4	no	—
38	C₃⁴	—	9	3	no	—
39	C₃⁴	—	9	6	no	—
40	C₃⁴	C₃×C₉	1	3	yes	—
41	C₃⁴	C₃³	1	1	yes	characteristic
42	C₃×3₋¹⁺²	—	3	1	no	—
43	C₃×3₋¹⁺²	—	3	2	no	—
44	C₃×3₋¹⁺²	—	3	2	no	—
45	C₃×3₋¹⁺²	—	3	4	no	—
46	C₃×3₋¹⁺²	—	3	4	no	—
47	C₃×3₋¹⁺²	—	3	2	no	—
48	C₃×3₋¹⁺²	—	3	2	no	—
49	C₃×3₋¹⁺²	—	3	2	no	—
50	C₃×3₋¹⁺²	—	3	4	no	—
51	C₃×3₋¹⁺²	—	3	4	no	—
52	C₃×3₋¹⁺²	—	3	4	no	—
53	C₃×3₋¹⁺²	C₃×C₉	1	6	yes	—
54	C₃×3₋¹⁺²	C₃×C₉	1	3	yes	—
55	C₃×3₋¹⁺²	3₋¹⁺²	1	6	yes	—
56	C₃²×C₉	—	3	2	no	—
57	C₃²×C₉	—	3	2	no	—
58	C₃²×C₉	—	3	1	no	—
59	C₃²×C₉	—	3	1	no	—
60	C₃²×C₉	—	3	2	no	—
61	C₃²×C₉	—	3	2	no	—
62	C₃²×C₉	H(3,𝔽₃)	1	3	yes	—
63	C₃²×C₉	H(3,𝔽₃)	1	3	yes	—
64	C₉⋊C₉	—	3	18	no	—
65	C₃²⋊C₉	—	3	1	no	—
66	C₃²⋊C₉	—	3	2	no	—
67	C₃²⋊C₉	—	3	2	no	—
68	C₃²⋊C₉	—	3	2	no	—
69	C₃²⋊C₉	—	3	27	no	—
70	C₃²⋊C₉	—	3	2	no	—
71	C₃²⋊C₉	—	3	2	no	—
72	C₃²⋊C₉	—	3	1	no	—
73	C₃²⋊C₉	—	3	2	no	—
74	C₃²⋊C₉	—	3	1	no	—
75	C₃²⋊C₉	—	3	2	no	—
76	C₃²⋊C₉	—	3	2	no	—
77	C₃²⋊C₉	—	3	2	no	—
78	C₃²⋊C₉	—	3	2	no	—
79	C₃²⋊C₉	—	3	2	no	—
80	C₃²⋊C₉	—	9	9	no	—
81	C₃²⋊C₉	H(3,𝔽₃)	1	6	yes	—
82	C₉²	—	3	9	no	—
83	C₃³	—	3	3	no	—
84	C₃³	—	3	3	no	—
85	C₃³	—	3	6	no	—
86	C₃³	—	3	6	no	—
87	C₃³	—	3	3	no	—
88	C₃³	—	3	3	no	—
89	C₃³	—	3	6	no	—
90	C₃³	—	9	1	no	—
91	C₃³	—	9	2	no	—
92	C₃³	—	9	9	no	—
93	C₃³	—	9	6	no	—
94	C₃³	—	9	6	no	—
95	C₃³	—	9	6	no	—
96	C₃³	—	9	2	no	—
97	C₃³	—	9	2	no	—
98	C₃³	—	9	4	no	—
99	C₃³	—	9	4	no	—
100	C₃³	—	9	18	no	—
101	C₃³	—	9	18	no	—
102	C₃³	—	9	36	no	—
103	C₃³	—	9	1	no	—
104	C₃³	—	9	2	no	—
105	C₃³	—	9	4	no	—
106	C₃³	—	9	2	no	—
107	C₃³	C₃²×C₉	1	1	yes	characteristic
108	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
109	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
110	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
111	C₃³	C₃×3₋¹⁺²	1	2	yes	—
112	C₃³	#737	1	2	yes	—
113	C₃³	#737	1	4	yes	—
114	3₋¹⁺²	—	3	3	no	—
115	3₋¹⁺²	—	3	6	no	—
116	3₋¹⁺²	—	3	6	no	—
117	3₋¹⁺²	—	3	12	no	—
118	3₋¹⁺²	—	3	6	no	—
119	3₋¹⁺²	—	3	6	no	—
120	3₋¹⁺²	—	3	6	no	—
121	3₋¹⁺²	—	9	4	no	—
122	3₋¹⁺²	—	9	4	no	—
123	3₋¹⁺²	—	9	4	no	—
124	C₃×C₉	—	3	3	no	—
125	C₃×C₉	—	3	6	no	—
126	C₃×C₉	—	3	6	no	—
127	C₃×C₉	—	3	3	no	—
128	C₃×C₉	—	3	6	no	—
129	C₃×C₉	—	9	2	no	—
130	C₃×C₉	—	9	4	no	—
131	C₃×C₉	—	9	3	no	—
132	C₃×C₉	—	9	3	no	—
133	C₃×C₉	—	9	3	no	—
134	C₃×C₉	—	9	1	no	—
135	C₃×C₉	—	9	1	no	—
136	C₃×C₉	—	9	2	no	—
137	C₃×C₉	—	9	2	no	—
138	C₃×C₉	—	27	2	no	—
139	C₃×C₉	—	27	2	no	—
140	C₃×C₉	—	27	2	no	—
141	C₃×C₉	C₃²⋊C₉	1	9	yes	complemented
142	C₃²	—	3	1	no	—
143	C₃²	—	3	2	no	—
144	C₃²	—	3	1	no	—
145	C₃²	—	3	2	no	—
146	C₃²	—	3	2	no	—
147	C₃²	—	3	2	no	—
148	C₃²	—	3	2	no	—
149	C₃²	—	3	9	no	—
150	C₃²	—	3	9	no	—
151	C₃²	—	9	1	no	—
152	C₃²	—	9	2	no	—
153	C₃²	—	9	2	no	—
154	C₃²	—	9	4	no	—
155	C₃²	—	9	18	no	—
156	C₃²	—	9	6	no	—
157	C₃²	—	9	6	no	—
158	C₃²	—	9	6	no	—
159	C₃²	—	9	12	no	—
160	C₃²	—	9	6	no	—
161	C₃²	—	9	12	no	—
162	C₃²	—	9	36	no	—
163	C₃²	—	9	3	no	—
164	C₃²	—	9	3	no	—
165	C₃²	—	9	6	no	—
166	C₃²	C₃×C₃²⋊C₉	1	1	yes	characteristic
167	C₃²	C₉×H(3,𝔽₃)	1	1	yes	characteristic
168	C₃²	C₉×H(3,𝔽₃)	1	2	yes	—
169	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
170	C₃²	C₉⋊H(3,𝔽₃)	1	2	yes	—
171	C₃²	C₉⋊H(3,𝔽₃)	1	4	yes	—
172	C₃²	#6893	1	2	yes	—
173	C₉	—	3	9	no	—
174	C₉	—	9	6	no	—
175	C₉	—	9	3	no	—
176	C₉	—	9	6	no	—
177	C₉	—	27	1	no	—
178	C₉	—	27	1	no	—
179	C₉	—	27	2	no	—
180	C₉	—	27	2	no	—
181	C₃	—	3	1	no	—
182	C₃	—	3	1	no	—
183	C₃	—	3	2	no	—
184	C₃	—	3	2	no	—
185	C₃	—	3	4	no	—
186	C₃	—	3	2	no	—
187	C₃	—	9	9	no	—
188	C₃	#10624555	1	1	yes	characteristic
189	C₃	#10624749	1	1	yes	characteristic
190	C₃	#10624752	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#6891

C₃

yes

maximal, characteristic

#10624863

C₃

yes

maximal, characteristic, complemented

#10624699

C₃

yes

maximal, complemented

#10624699

C₃

yes

maximal, complemented

#10624699

C₃

yes

maximal, complemented

#10624691

C₃

yes

maximal, characteristic, complemented

#10624690

C₃

yes

maximal, characteristic, complemented

C₃×#6866

C₃

yes

maximal

#10624547

C₃

yes

maximal, characteristic

C₃⁵

C₃²

yes

characteristic

C₃²×3₋¹⁺²

C₃²

yes

—

C₃²×3₋¹⁺²

C₃²

yes

characteristic

#6891

—

#6891

C₉

yes

complemented

C₉×3₋¹⁺²

—

C₉×3₋¹⁺²

—

#6887

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

—

C₃×C₃²⋊C₉

C₃²

yes

characteristic

C₃×C₃²⋊C₉

C₃²

yes

characteristic

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

—

C₃×C₃²⋊C₉

C₃²

yes

characteristic

C₃×C₃²⋊C₉

C₃²

yes

—

3₋¹⁺²⋊C₉

—

3₋¹⁺²⋊C₉

—

3₋¹⁺²⋊C₉

—

#6868

—

#6867

—

#6866

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

C₃×C₉

yes

—

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

—

C₃×3₋¹⁺²

C₃×C₉

yes

—

C₃×3₋¹⁺²

C₃×C₉

yes

—

C₃×3₋¹⁺²

3₋¹⁺²

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

H(3,𝔽₃)

yes

—

C₃²×C₉

H(3,𝔽₃)

yes

—

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

H(3,𝔽₃)

yes

—

C₉²

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

100

C₃³

—

101

C₃³

—

102

C₃³

—

103

C₃³

—

104

C₃³

—

105

C₃³

—

106

C₃³

—

107

C₃³

C₃²×C₉

yes

characteristic

108

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

109

C₃³

C₃×H(3,𝔽₃)

yes

characteristic

110

C₃³

C₃×H(3,𝔽₃)

yes

—

111

C₃³

C₃×3₋¹⁺²

yes

—

112

C₃³

#737

yes

—

113

C₃³

#737

yes

—

114

3₋¹⁺²

—

115

3₋¹⁺²

—

116

3₋¹⁺²

—

117

3₋¹⁺²

—

118

3₋¹⁺²

—

119

3₋¹⁺²

—

120

3₋¹⁺²

—

121

3₋¹⁺²

—

122

3₋¹⁺²

—

123

3₋¹⁺²

—

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

C₃×C₉

—

137

C₃×C₉

—

138

C₃×C₉

—

139

C₃×C₉

—

140

C₃×C₉

—

141

C₃×C₉

C₃²⋊C₉

yes

complemented

142

C₃²

—

143

C₃²

—

144

C₃²

—

145

C₃²

—

146

C₃²

—

147

C₃²

—

148

C₃²

—

149

C₃²

—

150

C₃²

—

151

C₃²

—

152

C₃²

—

153

C₃²

—

154

C₃²

—

155

C₃²

—

156

C₃²

—

157

C₃²

—

158

C₃²

—

159

C₃²

—

160

C₃²

—

161

C₃²

—

162

C₃²

—

163

C₃²

—

164

C₃²

—

165

C₃²

—

166

C₃²

C₃×C₃²⋊C₉

yes

characteristic

167

C₃²

C₉×H(3,𝔽₃)

yes

characteristic

168

C₃²

C₉×H(3,𝔽₃)

yes

—

169

C₃²

C₃²⋊H(3,𝔽₃)

yes

characteristic

170

C₃²

C₉⋊H(3,𝔽₃)

yes

—

171

C₃²

C₉⋊H(3,𝔽₃)

yes

—

172

C₃²

#6893

yes

—

173

C₉

—

174

C₉

—

175

C₉

—

176

C₉

—

177

C₉

—

178

C₉

—

179

C₉

—

180

C₉

—

181

C₃

—

182

C₃

—

183

C₃

—

184

C₃

—

185

C₃

—

186

C₃

—

187

C₃

—

188

C₃

#10624555

yes

characteristic

189

C₃

#10624749

yes

characteristic

190

C₃

#10624752

yes

—

	subgroup	G/H
center, socle	C₃²	C₃²⋊H(3,𝔽₃)
commutator	C₃³	C₃²×C₉
Frattini, cosocle	C₃⁴	C₃³
prime radical	C₃⁵	C₃²
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ G
LECS	G ▷ C₃⁴ ▷ C₃ ▷ C₁

subgroup

G/H

center, socle

C₃²

C₃²⋊H(3,𝔽₃)

commutator

C₃³

C₃²×C₉

Frattini, cosocle

C₃⁴

C₃³

prime radical

C₃⁵

C₃²

LCS

G ▷ C₃³ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ G

LECS

G ▷ C₃⁴ ▷ C₃ ▷ C₁

N	H ≅ G/N	type
2. #10624863	C₃	semidirect
3. #10624699	C₃	semidirect
4. #10624699	C₃	semidirect
5. #10624699	C₃	semidirect
6. #10624691	C₃	semidirect
7. #10624690	C₃	semidirect
14. #6891	C₉	semidirect
141. C₃×C₉	C₃²⋊C₉	semidirect

H ≅ G/N

type

2. #10624863

C₃

semidirect

3. #10624699

C₃

semidirect

4. #10624699

C₃

semidirect

5. #10624699

C₃

semidirect

6. #10624691

C₃

semidirect

7. #10624690

C₃

semidirect

14. #6891

C₉

semidirect

141. C₃×C₉

C₃²⋊C₉

semidirect

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

subgroup	index
C₃×#6891	3
#10624863	3
#10624699	3
#10624691	3
#10624690	3
C₃×#6866	3
#10624547	3

subgroup

index

Subgroups — #14550414

Conjugacy classes of subgroups

Complements

Maximal subgroups