Group #13272031 — subgroups

Subgroups — #13272031

Order 1792.

486 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 159 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	C₂×#89471	C₂	1	1	yes	maximal, characteristic
2	C₇×#1846	C₂	1	1	yes	maximal, characteristic
3	C₂×#89287	C₂	1	4	yes	maximal
4	C₂×#89256	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₈	—	2	1	no	—
6	C₂²×Dic₂₈	C₂²	1	2	yes	—
7	C₂×#6432	—	2	1	no	—
8	C₂×C₇×#107	C₂²	1	2	yes	—
9	C₂×C₇×#106	—	2	1	no	—
10	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₇×#382	—	2	1	no	—
12	C₂²×C₇⋊C₁₆	C₂²	1	2	yes	—
13	#89471	—	2	2	no	—
14	#89471	C₄	1	4	yes	complemented
15	#89287	—	2	8	no	—
16	#89256	—	2	2	no	—
17	#13207	—	7	1	no	maximal, Sylow
18	C₂²×Dic₁₄	—	2	2	no	—
19	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
20	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
21	C₂×C₇×C₄⋊C₄	—	2	2	no	—
22	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
23	C₂×C₄⋊Dic₇	—	4	2	no	—
24	C₂×Dic₂₈	—	2	4	no	—
25	C₂×Dic₂₈	—	2	4	no	—
26	C₂×Dic₂₈	—	2	4	no	—
27	C₂×Dic₂₈	—	2	2	no	—
28	C₂×Dic₂₈	—	4	2	no	—
29	C₂×Dic₂₈	—	4	2	no	—
30	C₂×Dic₂₈	C₂×C₄	1	4	yes	—
31	#6432	—	2	4	no	—
32	#6432	—	4	2	no	—
33	C₇×#107	—	2	4	no	—
34	C₇×#106	—	4	2	no	—
35	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
36	C₄×C₇×C₈	D₄	1	2	yes	—
37	C₇×#95	—	4	2	no	—
38	C₂×C₇⋊C₁₆	—	2	2	no	—
39	C₂×C₇⋊C₁₆	—	2	2	no	—
40	C₂×C₇⋊C₁₆	D₄	1	4	yes	—
41	C₂×#494	—	7	1	no	—
42	C₂×#358	—	7	4	no	—
43	C₂×#346	—	7	1	no	—
44	#1846	D₇	1	1	yes	characteristic
45	C₂²×C₄×C₇	—	4	2	no	—
46	C₂²×C₄×C₇	D₈	1	2	yes	—
47	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×Dic₇	—	4	2	no	—
49	C₂×Dic₁₄	—	2	4	no	—
50	C₂×Dic₁₄	—	4	2	no	—
51	C₂×Dic₁₄	—	4	2	no	—
52	C₂×Dic₁₄	—	4	2	no	—
53	C₂×Dic₁₄	—	4	2	no	—
54	C₂×Dic₁₄	—	4	2	no	—
55	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
56	C₂×C₇×C₈	D₈	1	2	yes	—
57	C₂×C₇×C₈	SD₁₆	1	2	yes	—
58	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
60	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
61	C₂×C₇×C₈	#41	1	2	yes	—
62	C₇×C₄⋊C₄	—	4	2	no	—
63	C₇×C₄⋊C₄	—	4	2	no	—
64	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₇	SD₁₆	1	2	yes	—
66	C₄⋊Dic₇	—	4	8	no	—
67	Dic₂₈	—	2	8	no	—
68	Dic₂₈	—	4	8	no	—
69	Dic₂₈	—	4	4	no	—
70	C₇⋊C₁₆	—	2	4	no	—
71	C₂²×Q₁₆	—	7	2	no	—
72	C₂²×Q₁₆	—	14	1	no	—
73	C₂×C₄⋊Q₈	—	14	1	no	—
74	C₂²×C₁₆	—	7	2	no	—
75	#494	—	7	4	no	—
76	#494	—	14	2	no	—
77	C₂×#107	C₂×D₇	1	2	yes	—
78	C₂×#106	—	2	1	no	—
79	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
80	#382	—	2	1	no	—
81	#358	—	14	8	no	—
82	#346	—	14	2	no	—
83	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₇	—	2	4	no	—
85	C₂×C₄×C₇	—	8	2	no	—
86	C₂×C₄×C₇	—	8	4	no	—
87	C₂×C₄×C₇	#102	1	4	yes	—
88	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₇	#124	1	1	yes	characteristic
90	C₂×C₄×C₇	#127	1	1	yes	characteristic
91	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
93	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
94	C₂×Dic₇	—	4	4	no	—
95	C₂×Dic₇	—	8	2	no	—
96	C₂×Dic₇	—	8	2	no	—
97	Dic₁₄	—	4	4	no	—
98	Dic₁₄	—	4	4	no	—
99	Dic₁₄	—	8	2	no	—
100	Dic₁₄	—	8	2	no	—
101	C₇×C₈	#102	1	4	yes	—
102	C₇×C₈	C₄×D₄	1	2	yes	—
103	C₇×C₈	#121	1	2	yes	—
104	C₂²×Q₈	—	14	2	no	—
105	C₂×Q₁₆	—	7	4	no	—
106	C₂×Q₁₆	—	14	4	no	—
107	C₂×Q₁₆	—	14	2	no	—
108	C₂×Q₁₆	—	14	4	no	—
109	C₂×Q₁₆	—	14	4	no	—
110	C₂×Q₁₆	—	28	2	no	—
111	C₂×Q₁₆	—	28	2	no	—
112	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
113	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
114	C₄⋊Q₈	—	14	4	no	—
115	C₄⋊Q₈	—	28	2	no	—
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄⋊C₄	—	28	2	no	—
118	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
119	C₂×C₁₆	—	7	4	no	—
120	C₂×C₁₆	—	14	2	no	—
121	C₂×C₁₆	—	14	2	no	—
122	#107	—	2	4	no	—
123	#106	—	4	2	no	—
124	C₄×C₈	C₄×D₇	1	2	yes	—
125	C₄×C₈	D₂₈	1	2	yes	—
126	#95	—	4	2	no	—
127	C₂²×C₇	C₂×#102	1	1	yes	characteristic
128	C₂²×C₇	#486	1	1	yes	characteristic
129	C₂²×C₇	#493	1	1	yes	characteristic
130	C₂²×C₇	#510	1	4	yes	—
131	C₄×C₇	—	2	4	no	—
132	C₄×C₇	—	16	2	no	—
133	C₄×C₇	#390	1	1	yes	characteristic
134	C₄×C₇	C₂×#102	1	1	yes	characteristic
135	C₄×C₇	#486	1	1	yes	characteristic
136	C₄×C₇	#492	1	1	yes	characteristic
137	Dic₇	—	8	4	no	—
138	C₂×Q₈	—	14	4	no	—
139	C₂×Q₈	—	28	2	no	—
140	C₂×Q₈	—	28	2	no	—
141	C₂×Q₈	—	28	2	no	—
142	C₂×Q₈	—	28	2	no	—
143	C₂×Q₈	—	28	2	no	—
144	C₂²×C₄	—	4	2	no	—
145	C₂²×C₄	—	28	2	no	—
146	C₂²×C₄	#1141	1	2	yes	—
147	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
148	Q₁₆	—	14	8	no	—
149	Q₁₆	—	28	4	no	—
150	Q₁₆	—	28	8	no	—
151	C₂×C₈	(C₂×D₇)⋊C₄	1	2	yes	—
152	C₂×C₈	#1141	1	2	yes	—
153	C₂×C₈	#1143	1	2	yes	—
154	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
155	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
156	C₂×C₈	D₄×D₇	1	2	yes	—
157	C₂×C₈	#1161	1	2	yes	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	28	8	no	—
161	C₄²	(C₂×D₇)⋊C₄	1	2	yes	—
162	C₄²	#1143	1	2	yes	—
163	C₁₆	—	14	4	no	—
164	C₂×C₇	#1957	1	1	yes	characteristic
165	C₂×C₇	#2141	1	2	yes	—
166	C₂×C₇	#2245	1	2	yes	—
167	C₂×C₇	#2251	1	2	yes	—
168	C₂³	C₂×#1141	1	1	yes	characteristic
169	Q₈	—	28	4	no	—
170	Q₈	—	28	4	no	—
171	Q₈	—	56	2	no	—
172	Q₈	—	56	2	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	8	4	no	—
175	C₂×C₄	—	8	2	no	—
176	C₂×C₄	—	28	4	no	—
177	C₂×C₄	—	56	2	no	—
178	C₂×C₄	—	56	2	no	—
179	C₂×C₄	#6383	1	4	yes	—
180	C₂×C₄	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
181	C₂×C₄	C₂×#1141	1	1	yes	characteristic
182	C₂×C₄	#6503	1	1	yes	characteristic
183	C₂×C₄	C₂×#1143	1	1	yes	characteristic
184	C₂×C₄	C₂×#1143	1	1	yes	characteristic
185	C₂×C₄	#6510	1	1	yes	characteristic
186	C₈	#6383	1	4	yes	—
187	C₈	#6456	1	2	yes	—
188	C₈	#6458	1	2	yes	—
189	C₇	#13207	1	1	yes	characteristic, complemented, Sylow
190	C₂²	C₂×#6383	1	1	yes	characteristic
191	C₂²	#89845	1	1	yes	characteristic
192	C₂²	#89860	1	1	yes	characteristic
193	C₂²	#89967	1	4	yes	—
194	C₄	—	2	4	no	—
195	C₄	—	16	2	no	—
196	C₄	—	56	4	no	—
197	C₄	C₂×#6383	1	1	yes	characteristic
198	C₄	#89760	1	1	yes	characteristic
199	C₄	#89857	1	1	yes	characteristic
200	C₄	#89860	1	1	yes	characteristic
201	C₂	#11727334	1	1	yes	characteristic
202	C₂	#11728408	1	2	yes	—
203	C₂	#11729076	1	2	yes	—
204	C₂	#11729107	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#89471

C₂

yes

maximal, characteristic

C₇×#1846

C₂

yes

maximal, characteristic

C₂×#89287

C₂

yes

maximal

C₂×#89256

C₂

yes

maximal, characteristic

C₂²×Dic₂₈

—

C₂²×Dic₂₈

C₂²

yes

—

C₂×#6432

—

C₂×C₇×#107

C₂²

yes

—

C₂×C₇×#106

—

C₂×C₄×C₇×C₈

C₂²

yes

characteristic

C₇×#382

—

C₂²×C₇⋊C₁₆

C₂²

yes

—

#89471

—

#89471

C₄

yes

—

—

—

C₂²×Dic₁₄

—

C₂²×C₇×C₈

D₄

yes

characteristic

C₂²×C₇×C₈

C₂³

yes

characteristic

C₂×C₇×C₄⋊C₄

—

C₂×C₄²×C₇

D₄

yes

characteristic

C₂×C₄⋊Dic₇

—

C₂×Dic₂₈

—

C₂×Dic₂₈

—

C₂×Dic₂₈

—

C₂×Dic₂₈

—

C₂×Dic₂₈

—

C₂×Dic₂₈

—

C₂×Dic₂₈

C₂×C₄

yes

—

—

—

—

—

C₄×C₇×C₈

C₂×C₄

yes

—

C₄×C₇×C₈

D₄

yes

—

C₇×#95

—

C₂×C₇⋊C₁₆

—

C₂×C₇⋊C₁₆

—

C₂×C₇⋊C₁₆

D₄

yes

—

—

—

—

D₇

yes

characteristic

C₂²×C₄×C₇

—

C₂²×C₄×C₇

D₈

yes

—

C₂²×C₄×C₇

C₂×D₄

yes

characteristic

C₂²×Dic₇

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×C₇×C₈

C₂²⋊C₄

yes

—

C₂×C₇×C₈

D₈

yes

—

C₂×C₇×C₈

SD₁₆

yes

—

C₂×C₇×C₈

C₂²×C₄

yes

characteristic

C₂×C₇×C₈

C₂×D₄

yes

—

C₂×C₇×C₈

C₂×D₄

yes

characteristic

C₂×C₇×C₈

#41

yes

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₄²×C₇

C₂²⋊C₄

yes

—

C₄²×C₇

SD₁₆

yes

—

C₄⋊Dic₇

—

Dic₂₈

—

Dic₂₈

—

Dic₂₈

—

C₇⋊C₁₆

—

C₂²×Q₁₆

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

C₂²×C₁₆

—

#494

—

#494

—

C₂×#107

C₂×D₇

yes

—

C₂×#106

—

C₂×C₄×C₈

C₂×D₇

yes

—

—

—

C₂³×C₇

C₂×D₈

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

#102

yes

—

C₂×C₄×C₇

C₂×C₂²⋊C₄

yes

C₂×C₄×C₇

yes

C₂×C₄×C₇

yes

C₂×C₄×C₇

C₂×D₈

yes

characteristic

C₂×C₄×C₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇

C₂×SD₁₆

yes

characteristic

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

Dic₁₄

—

Dic₁₄

—

Dic₁₄

—

100

Dic₁₄

—

101

C₇×C₈

#102

yes

—

102

C₇×C₈

C₄×D₄

yes

—

103

C₇×C₈

#121

yes

—

104

C₂²×Q₈

—

105

C₂×Q₁₆

—

106

C₂×Q₁₆

—

107

C₂×Q₁₆

—

108

C₂×Q₁₆

—

109

C₂×Q₁₆

—

110

C₂×Q₁₆

—

111

C₂×Q₁₆

—

112

C₂²×C₈

C₂²⋊D₇

yes

characteristic

113

C₂²×C₈

C₂²×D₇

yes

characteristic

114

C₄⋊Q₈

—

115

C₄⋊Q₈

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄²

C₂²⋊D₇

yes

characteristic

119

C₂×C₁₆

—

120

C₂×C₁₆

—

121

C₂×C₁₆

—

122

#107

—

123

#106

—

124

C₄×C₈

C₄×D₇

yes

—

125

C₄×C₈

D₂₈

yes

—

126

#95

—

127

C₂²×C₇

C₂×#102

yes

characteristic

128

C₂²×C₇

#486

yes

characteristic

129

C₂²×C₇

#493

yes

characteristic

130

C₂²×C₇

#510

yes

—

131

C₄×C₇

—

132

C₄×C₇

—

133

C₄×C₇

#390

yes

characteristic

134

C₄×C₇

C₂×#102

yes

characteristic

135

C₄×C₇

#486

yes

characteristic

136

C₄×C₇

#492

yes

characteristic

137

Dic₇

—

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂×Q₈

—

144

C₂²×C₄

—

145

C₂²×C₄

—

146

C₂²×C₄

#1141

yes

—

147

C₂²×C₄

C₂×C₂²⋊D₇

yes

characteristic

148

Q₁₆

—

149

Q₁₆

—

150

Q₁₆

—

151

C₂×C₈

(C₂×D₇)⋊C₄

yes

—

152

C₂×C₈

#1141

yes

—

153

C₂×C₈

#1143

yes

—

154

C₂×C₈

C₂×C₄×D₇

yes

characteristic

155

C₂×C₈

C₂×D₂₈

yes

characteristic

156

C₂×C₈

D₄×D₇

yes

—

157

C₂×C₈

#1161

yes

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄²

(C₂×D₇)⋊C₄

yes

—

162

C₄²

#1143

yes

—

163

C₁₆

—

164

C₂×C₇

#1957

yes

characteristic

165

C₂×C₇

#2141

yes

—

166

C₂×C₇

#2245

yes

—

167

C₂×C₇

#2251

yes

—

168

C₂³

C₂×#1141

yes

characteristic

169

Q₈

—

170

Q₈

—

171

Q₈

—

172

Q₈

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

#6383

yes

—

180

C₂×C₄

C₂×(C₂×D₇)⋊C₄

yes

characteristic

181

C₂×C₄

C₂×#1141

yes

characteristic

182

C₂×C₄

#6503

yes

characteristic

183

C₂×C₄

C₂×#1143

yes

characteristic

184

C₂×C₄

C₂×#1143

yes

characteristic

185

C₂×C₄

#6510

yes

characteristic

186

C₈

#6383

yes

—

187

C₈

#6456

yes

—

188

C₈

#6458

yes

—

189

C₇

#13207

yes

characteristic, complemented, Sylow

190

C₂²

C₂×#6383

yes

characteristic

191

C₂²

#89845

yes

characteristic

192

C₂²

#89860

yes

characteristic

193

C₂²

#89967

yes

—

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

C₂×#6383

yes

characteristic

198

C₄

#89760

yes

characteristic

199

C₄

#89857

yes

characteristic

200

C₄

#89860

yes

characteristic

201

C₂

#11727334

yes

characteristic

202

C₂

#11728408

yes

—

203

C₂

#11729076

yes

—

204

C₂

#11729107

yes

—

	subgroup	G/H
center	C₂³	C₂×#1141
commutator	C₂×C₇×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₇×#1846	C₂
socle, prime radical	C₂³×C₇	C₂×D₈
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	#1846	D₇
hypocenter	C₇	#13207
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

subgroup

G/H

center

C₂³

C₂×#1141

commutator

C₂×C₇×C₈

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇

Fitting

C₇×#1846

C₂

socle, prime radical

C₂³×C₇

C₂×D₈

cosocle

C₂²×C₇×C₈

C₂³

D₇

C₇

G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

N	H ≅ G/N	type
14. #89471	C₄	semidirect
189. C₇	#13207	semidirect

H ≅ G/N

type

14. #89471

C₄

semidirect

189. C₇

#13207

semidirect

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

subgroup	index
C₂×#89471	2
C₇×#1846	2
C₂×#89287	2
C₂×#89256	2
#13207	7

subgroup

index

Subgroups — #13272031

Conjugacy classes of subgroups

Complements

Maximal subgroups