Group #13137264 — subgroups

Subgroups — #13137264

Order 1728.

470 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 117 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×#63436	C₂	1	1	yes	maximal, characteristic
2	C₉×#869	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#88425	C₂	1	4	yes	maximal, complemented
4	C₄×C₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	#10590375	—	3	1	no	maximal
6	#10588770	—	3	1	no	maximal
7	C₂×C₃×D₃₆	—	2	2	no	—
8	C₂×C₃×D₃₆	C₂²	1	2	yes	—
9	C₂×C₉×Dic₆	C₂²	1	2	yes	—
10	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	2	yes	—
12	C₉×C₄⋊Dic₃	—	2	2	no	—
13	#88425	—	2	8	no	—
14	C₃×#874	—	3	1	no	—
15	C₃×#869	D₃	1	1	yes	characteristic
16	C₂×#3715	—	3	4	no	—
17	C₄×C₃⋊(C₃⋊C₈)	—	3	1	no	—
18	C₉×C₄⋊Q₈	—	3	1	no	—
19	C₂×#3675	—	3	4	no	—
20	#63436	D₃	1	1	yes	characteristic
21	C₄×C₉⋊C₈	—	3	1	no	—
22	C₄×C₉⋊C₈	—	6	1	no	—
23	C₂²×C₃×D₉	—	2	2	no	—
24	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
25	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
26	C₂×C₉×Dic₃	—	2	2	no	—
27	C₃×D₃₆	—	2	4	no	—
28	C₃×D₃₆	—	4	4	no	—
29	C₉×Dic₆	—	2	4	no	—
30	C₃⋊(C₉⋊C₈)	D₄	1	4	yes	complemented
31	#5085	—	9	1	no	—
32	#5076	—	9	1	no	—
33	C₂×C₃×D₁₂	—	3	2	no	—
34	C₂×C₃×D₁₂	—	6	2	no	—
35	C₂×C₃×Dic₆	C₂×D₃	1	2	yes	—
36	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
37	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
38	C₃×C₄⋊Dic₃	—	2	2	no	—
39	#3715	—	6	8	no	—
40	C₂×Q₈×C₉	—	3	2	no	—
41	C₂×D₃₆	—	2	2	no	—
42	C₂×D₃₆	C₂×D₃	1	2	yes	—
43	C₉×C₄⋊C₄	—	6	2	no	—
44	C₄²×C₉	—	2	1	no	—
45	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
46	#3675	—	6	8	no	—
47	C₂×C₉⋊C₈	—	3	2	no	—
48	C₂×C₉⋊C₈	—	6	2	no	—
49	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₃×D₉	—	4	2	no	—
51	C₂×C₃×D₉	—	4	4	no	—
52	C₃×C₄×C₉	SD₁₆	1	4	yes	complemented
53	C₃×C₄×C₉	C₂×D₄	1	2	yes	—
54	C₉×Dic₃	—	4	2	no	—
55	C₃×C₄⋊Q₈	—	3	1	no	—
56	C₃×#127	—	9	1	no	—
57	C₂×#215	—	9	4	no	—
58	C₂×#214	—	9	4	no	—
59	#874	—	3	1	no	—
60	#869	D₉	1	1	yes	characteristic, complemented
61	C₄×C₃⋊C₈	—	9	1	no	—
62	C₄×C₃⋊C₈	—	9	1	no	—
63	C₄×C₃⋊C₈	—	18	1	no	—
64	C₂²×C₃×D₃	—	6	2	no	—
65	C₂×C₃²×C₄	#67	1	2	yes	—
66	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
67	C₂×C₃×Dic₃	—	2	2	no	—
68	C₃×D₁₂	—	6	4	no	—
69	C₃×D₁₂	—	12	4	no	—
70	C₃×Dic₆	—	2	4	no	—
71	C₂²×D₉	—	2	2	no	—
72	C₃⋊(C₃⋊C₈)	—	3	4	no	—
73	Q₈×C₉	—	6	4	no	—
74	C₂×C₄×C₉	—	2	2	no	—
75	C₂×C₄×C₉	—	2	1	no	—
76	C₂×C₄×C₉	—	6	2	no	—
77	C₂×C₄×C₉	#67	1	2	yes	—
78	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
79	D₃₆	—	2	4	no	—
80	D₃₆	—	4	4	no	—
81	C₉⋊C₈	—	3	4	no	—
82	C₉⋊C₈	—	6	4	no	—
83	#492	—	27	1	no	Sylow
84	C₂×C₃×C₉	#127	1	1	yes	characteristic
85	C₂×C₃×C₉	C₂×SD₁₆	1	2	yes	—
86	C₃×D₉	—	8	2	no	—
87	C₂×C₃×Q₈	—	3	2	no	—
88	C₂×C₃×D₄	—	9	2	no	—
89	C₂×C₃×D₄	—	18	2	no	—
90	C₂×D₁₂	—	3	2	no	—
91	C₂×D₁₂	—	6	2	no	—
92	C₂×Dic₆	C₂×D₉	1	2	yes	—
93	C₃×C₄⋊C₄	—	6	2	no	—
94	C₃×C₄²	—	2	1	no	—
95	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
96	C₃×C₄²	D₃²	1	1	yes	characteristic
97	#215	—	18	8	no	—
98	#214	—	18	8	no	—
99	C₄⋊Dic₃	—	2	2	no	—
100	C₂×C₃⋊C₈	—	9	2	no	—
101	C₂×C₃⋊C₈	—	9	2	no	—
102	C₂×C₃⋊C₈	—	18	2	no	—
103	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
104	C₂×C₃×D₃	—	12	2	no	—
105	C₂×C₃×D₃	—	12	4	no	—
106	C₃²×C₄	#215	1	4	yes	—
107	C₃²×C₄	D₃×D₄	1	2	yes	—
108	C₃×Dic₃	—	4	2	no	—
109	C₂²×C₉	—	2	1	no	—
110	C₂²×C₉	C₂×#67	1	1	yes	characteristic
111	C₂×D₉	—	4	4	no	—
112	C₂×D₉	—	4	2	no	—
113	C₄×C₉	—	2	4	no	—
114	C₄×C₉	—	2	2	no	—
115	C₄×C₉	—	12	2	no	—
116	C₄×C₉	#214	1	4	yes	complemented
117	C₄×C₉	D₃×D₄	1	2	yes	—
118	C₂×SD₁₆	—	27	4	no	—
119	C₄⋊Q₈	—	3	1	no	—
120	#127	—	9	1	no	—
121	C₄×C₈	—	27	1	no	—
122	C₃×C₉	#492	1	1	yes	characteristic, complemented, Sylow
123	C₂³×C₃	—	18	2	no	—
124	C₂²×D₃	—	6	2	no	—
125	C₃×Q₈	—	6	4	no	—
126	C₃×D₄	—	18	4	no	—
127	C₃×D₄	—	36	4	no	—
128	C₂×C₃×C₄	—	2	2	no	—
129	C₂×C₃×C₄	—	2	1	no	—
130	C₂×C₃×C₄	—	6	2	no	—
131	C₂×C₃×C₄	#611	1	2	yes	—
132	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
133	C₂×C₃×C₄	#625	1	2	yes	—
134	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
135	C₂×Dic₃	—	2	2	no	—
136	D₁₂	—	6	4	no	—
137	D₁₂	—	12	4	no	—
138	Dic₆	—	2	4	no	—
139	C₃⋊C₈	—	9	4	no	—
140	C₃⋊C₈	—	9	4	no	—
141	C₃⋊C₈	—	18	4	no	—
142	C₂×C₃²	#940	1	1	yes	characteristic
143	C₂×C₃²	C₂×#215	1	2	yes	—
144	C₃×D₃	—	24	2	no	—
145	C₂×C₉	—	2	2	no	—
146	C₂×C₉	—	2	1	no	—
147	C₂×C₉	C₂×#214	1	2	yes	—
148	C₂×C₉	#940	1	1	yes	characteristic
149	D₉	—	8	2	no	—
150	C₂×Q₈	—	3	2	no	—
151	C₂×D₄	—	9	2	no	—
152	C₂×D₄	—	18	2	no	—
153	SD₁₆	—	54	8	no	—
154	C₂×C₈	—	27	2	no	—
155	C₄⋊C₄	—	6	2	no	—
156	C₄²	D₃×D₉	1	1	yes	characteristic
157	C₂²×C₃	—	2	1	no	—
158	C₂²×C₃	—	36	2	no	—
159	C₂²×C₃	—	36	4	no	—
160	C₂²×C₃	C₂×#611	1	1	yes	characteristic
161	C₂²×C₃	C₂×#625	1	1	yes	characteristic
162	C₂×D₃	—	12	4	no	—
163	C₂×D₃	—	12	2	no	—
164	C₃×C₄	—	2	2	no	—
165	C₃×C₄	—	2	4	no	—
166	C₃×C₄	—	12	2	no	—
167	C₃×C₄	#3675	1	4	yes	complemented
168	C₃×C₄	D₄×D₉	1	2	yes	—
169	C₃×C₄	#3715	1	4	yes	—
170	C₃×C₄	#3802	1	2	yes	—
171	Dic₃	—	4	2	no	—
172	C₃²	#5085	1	1	yes	characteristic
173	C₉	—	2	1	no	—
174	C₉	#5076	1	1	yes	characteristic, complemented
175	C₂³	—	18	2	no	—
176	Q₈	—	6	4	no	—
177	D₄	—	18	4	no	—
178	D₄	—	36	4	no	—
179	C₂×C₄	—	6	2	no	—
180	C₂×C₄	#6194	1	2	yes	—
181	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
182	C₈	—	27	4	no	—
183	C₂×C₃	—	2	1	no	—
184	C₂×C₃	—	2	2	no	—
185	C₂×C₃	—	72	2	no	—
186	C₂×C₃	#63502	1	1	yes	characteristic
187	C₂×C₃	C₂×#3675	1	2	yes	—
188	C₂×C₃	C₂×#3715	1	2	yes	—
189	C₂×C₃	#63914	1	1	yes	characteristic
190	D₃	—	24	2	no	—
191	C₂²	—	36	2	no	—
192	C₂²	—	36	4	no	—
193	C₂²	C₂×#6194	1	1	yes	characteristic
194	C₄	—	12	2	no	—
195	C₄	#88425	1	4	yes	complemented
196	C₄	#88646	1	2	yes	—
197	C₃	—	2	1	no	—
198	C₃	#10588770	1	1	yes	characteristic, complemented
199	C₃	#10590375	1	1	yes	characteristic
200	C₂	—	72	2	no	—
201	C₂	C₂×#88425	1	2	yes	—
202	C₂	#11721043	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#63436

C₂

yes

maximal, characteristic

C₉×#869

C₂

yes

maximal, characteristic, complemented

C₂×#88425

C₂

yes

maximal, complemented

C₄×C₃⋊(C₉⋊C₈)

C₂

yes

maximal, characteristic, complemented

—

—

C₂×C₃×D₃₆

—

C₂×C₃×D₃₆

C₂²

yes

—

C₂×C₉×Dic₆

C₂²

yes

—

C₃×C₄²×C₉

C₂²

yes

characteristic

C₂×C₃⋊(C₉⋊C₈)

C₂²

yes

—

C₉×C₄⋊Dic₃

—

#88425

—

C₃×#874

—

C₃×#869

D₃

yes

characteristic

C₂×#3715

—

C₄×C₃⋊(C₃⋊C₈)

—

C₉×C₄⋊Q₈

—

C₂×#3675

—

#63436

D₃

yes

characteristic

C₄×C₉⋊C₈

—

C₄×C₉⋊C₈

—

C₂²×C₃×D₉

—

C₂×C₃×C₄×C₉

D₄

yes

—

C₂×C₃×C₄×C₉

C₂³

yes

characteristic

C₂×C₉×Dic₃

—

C₃×D₃₆

—

C₃×D₃₆

—

C₉×Dic₆

—

C₃⋊(C₉⋊C₈)

D₄

yes

complemented

#5085

—

#5076

—

C₂×C₃×D₁₂

—

C₂×C₃×D₁₂

—

C₂×C₃×Dic₆

C₂×D₃

yes

—

C₃²×C₄²

C₂×D₃

yes

characteristic

C₂×C₃⋊(C₃⋊C₈)

—

C₃×C₄⋊Dic₃

—

#3715

—

C₂×Q₈×C₉

—

C₂×D₃₆

—

C₂×D₃₆

C₂×D₃

yes

—

C₉×C₄⋊C₄

—

C₄²×C₉

—

C₄²×C₉

C₂×D₃

yes

characteristic

#3675

—

C₂×C₉⋊C₈

—

C₂×C₉⋊C₈

—

C₂²×C₃×C₉

C₂×D₄

yes

characteristic

C₂×C₃×D₉

—

C₂×C₃×D₉

—

C₃×C₄×C₉

SD₁₆

yes

complemented

C₃×C₄×C₉

C₂×D₄

yes

—

C₉×Dic₃

—

C₃×C₄⋊Q₈

—

—

—

—

—

D₉

yes

characteristic, complemented

C₄×C₃⋊C₈

—

C₄×C₃⋊C₈

—

C₄×C₃⋊C₈

—

C₂²×C₃×D₃

—

C₂×C₃²×C₄

#67

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

characteristic

C₂×C₃×Dic₃

—

C₃×D₁₂

—

C₃×D₁₂

—

C₃×Dic₆

—

C₂²×D₉

—

C₃⋊(C₃⋊C₈)

—

Q₈×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

#67

yes

—

C₂×C₄×C₉

C₂²×D₃

yes

characteristic

D₃₆

—

D₃₆

—

C₉⋊C₈

—

C₉⋊C₈

—

—

C₂×C₃×C₉

yes

C₂×C₃×C₉

C₂×SD₁₆

yes

—

C₃×D₉

—

C₂×C₃×Q₈

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×D₁₂

—

C₂×D₁₂

—

C₂×Dic₆

C₂×D₉

yes

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₃×C₄²

C₂×D₉

yes

characteristic

C₃×C₄²

D₃²

yes

characteristic

#215

—

#214

—

C₄⋊Dic₃

—

100

C₂×C₃⋊C₈

—

101

C₂×C₃⋊C₈

—

102

C₂×C₃⋊C₈

—

103

C₂²×C₃²

C₂×#67

yes

characteristic

104

C₂×C₃×D₃

—

105

C₂×C₃×D₃

—

106

C₃²×C₄

#215

yes

—

107

C₃²×C₄

D₃×D₄

yes

—

108

C₃×Dic₃

—

109

C₂²×C₉

—

110

C₂²×C₉

C₂×#67

yes

characteristic

111

C₂×D₉

—

112

C₂×D₉

—

113

C₄×C₉

—

114

C₄×C₉

—

115

C₄×C₉

—

116

C₄×C₉

#214

yes

complemented

117

C₄×C₉

D₃×D₄

yes

—

118

C₂×SD₁₆

—

119

C₄⋊Q₈

—

120

#127

—

121

C₄×C₈

—

122

C₃×C₉

#492

yes

characteristic, complemented, Sylow

123

C₂³×C₃

—

124

C₂²×D₃

—

125

C₃×Q₈

—

126

C₃×D₄

—

127

C₃×D₄

—

128

C₂×C₃×C₄

—

129

C₂×C₃×C₄

—

130

C₂×C₃×C₄

—

131

C₂×C₃×C₄

#611

yes

—

132

C₂×C₃×C₄

C₂²×D₉

yes

characteristic

133

C₂×C₃×C₄

#625

yes

—

134

C₂×C₃×C₄

C₂×D₃²

yes

characteristic

135

C₂×Dic₃

—

136

D₁₂

—

137

D₁₂

—

138

Dic₆

—

139

C₃⋊C₈

—

140

C₃⋊C₈

—

141

C₃⋊C₈

—

142

C₂×C₃²

#940

yes

characteristic

143

C₂×C₃²

C₂×#215

yes

—

144

C₃×D₃

—

145

C₂×C₉

—

146

C₂×C₉

—

147

C₂×C₉

C₂×#214

yes

—

148

C₂×C₉

#940

yes

characteristic

149

D₉

—

150

C₂×Q₈

—

151

C₂×D₄

—

152

C₂×D₄

—

153

SD₁₆

—

154

C₂×C₈

—

155

C₄⋊C₄

—

156

C₄²

D₃×D₉

yes

characteristic

157

C₂²×C₃

—

158

C₂²×C₃

—

159

C₂²×C₃

—

160

C₂²×C₃

C₂×#611

yes

characteristic

161

C₂²×C₃

C₂×#625

yes

characteristic

162

C₂×D₃

—

163

C₂×D₃

—

164

C₃×C₄

—

165

C₃×C₄

—

166

C₃×C₄

—

167

C₃×C₄

#3675

yes

complemented

168

C₃×C₄

D₄×D₉

yes

—

169

C₃×C₄

#3715

yes

—

170

C₃×C₄

#3802

yes

—

171

Dic₃

—

172

C₃²

#5085

yes

characteristic

173

C₉

—

174

C₉

#5076

yes

characteristic, complemented

175

C₂³

—

176

Q₈

—

177

D₄

—

178

D₄

—

179

C₂×C₄

—

180

C₂×C₄

#6194

yes

—

181

C₂×C₄

C₂×D₃×D₉

yes

characteristic

182

C₈

—

183

C₂×C₃

—

184

C₂×C₃

—

185

C₂×C₃

—

186

C₂×C₃

#63502

yes

characteristic

187

C₂×C₃

C₂×#3675

yes

—

188

C₂×C₃

C₂×#3715

yes

—

189

C₂×C₃

#63914

yes

characteristic

190

D₃

—

191

C₂²

—

192

C₂²

—

193

C₂²

C₂×#6194

yes

characteristic

194

C₄

—

195

C₄

#88425

yes

complemented

196

C₄

#88646

yes

—

197

C₃

—

198

C₃

#10588770

yes

characteristic, complemented

199

C₃

#10590375

yes

characteristic

200

C₂

—

201

C₂

C₂×#88425

yes

—

202

C₂

#11721043

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#6194
commutator, cosocle	C₂×C₃×C₄×C₉	C₂³
Frattini	C₂×C₃×C₄	C₂×D₃²
Fitting	C₃×C₄²×C₉	C₂²
socle	C₂²×C₃²	C₂×#67
prime radical	C₃×#63436	C₂
hypercenter	C₄²	D₃×D₉
hypocenter	C₃×C₉	#492
LCS	G ▷ C₂×C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂² ◁ C₄²

subgroup

G/H

center

C₂²

C₂×#6194

commutator, cosocle

C₂×C₃×C₄×C₉

C₂³

Frattini

C₂×C₃×C₄

C₂×D₃²

Fitting

C₃×C₄²×C₉

C₂²

socle

C₂²×C₃²

C₂×#67

prime radical

C₃×#63436

C₂

hypercenter

C₄²

D₃×D₉

hypocenter

C₃×C₉

#492

LCS

G ▷ C₂×C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉

UCS

C₁ ◁ C₂² ◁ C₄²

N	H ≅ G/N	type
2. C₉×#869	C₂	semidirect
3. C₂×#88425	C₂	semidirect
4. C₄×C₃⋊(C₉⋊C₈)	C₂	semidirect
30. C₃⋊(C₉⋊C₈)	D₄	semidirect
52. C₃×C₄×C₉	SD₁₆	semidirect
60. #869	D₉	semidirect
116. C₄×C₉	#214	semidirect
122. C₃×C₉	#492	semidirect
167. C₃×C₄	#3675	semidirect
174. C₉	#5076	semidirect
195. C₄	#88425	semidirect
198. C₃	#10588770	semidirect

H ≅ G/N

type

2. C₉×#869

C₂

semidirect

3. C₂×#88425

C₂

semidirect

4. C₄×C₃⋊(C₉⋊C₈)

C₂

semidirect

30. C₃⋊(C₉⋊C₈)

D₄

semidirect

52. C₃×C₄×C₉

SD₁₆

semidirect

60. #869

D₉

semidirect

116. C₄×C₉

#214

semidirect

122. C₃×C₉

#492

semidirect

167. C₃×C₄

#3675

semidirect

174. C₉

#5076

semidirect

195. C₄

#88425

semidirect

198. C₃

#10588770

semidirect

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

subgroup	index
C₃×#63436	2
C₉×#869	2
C₂×#88425	2
C₄×C₃⋊(C₉⋊C₈)	2
#10590375	3
#10588770	3

subgroup

index

C₄×C₃⋊(C₉⋊C₈)

Subgroups — #13137264

Conjugacy classes of subgroups

Complements

Maximal subgroups