Group #29599092 — subgroups

Subgroups — #29599092

Order 15616.

382 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 115 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×#16840164	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839566	C₂	1	2	yes	maximal
3	#21443910	C₂	1	2	yes	maximal
4	#21443897	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1969	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519769	—	2	1	no	—
7	C₂²×#11758185	C₂²	1	2	yes	—
8	#16840164	—	2	4	no	—
9	C₂×#14519711	C₂²	1	2	yes	—
10	C₂×#14519709	—	2	1	no	—
11	#16839566	—	2	4	no	—
12	C₂²×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
13	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×#103×C₆₁	—	2	1	no	—
15	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₆₁×#391	—	2	1	no	—
17	#14519769	—	2	4	no	—
18	#14519769	—	4	2	no	—
19	C₂×C₄⋊Dic₆₁	—	2	2	no	—
20	C₂×#11758196	—	4	2	no	—
21	C₂×#11758185	—	2	2	no	—
22	C₂×#11758185	—	2	2	no	—
23	C₂×#11758185	—	2	2	no	—
24	#14519711	—	2	2	no	—
25	#14519711	D₄	1	4	yes	—
26	#14519709	—	2	4	no	—
27	C₂²×Q₈×C₆₁	—	2	1	no	—
28	C₂×Q₁₆×C₆₁	—	2	2	no	—
29	C₂×Q₁₆×C₆₁	—	2	2	no	—
30	C₂×Q₁₆×C₆₁	—	2	1	no	—
31	C₂×Q₁₆×C₆₁	Q₈	1	2	yes	—
32	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
35	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
36	#107×C₆₁	—	2	2	no	—
37	#103×C₆₁	—	4	2	no	—
38	C₄×C₈×C₆₁	—	2	2	no	—
39	#95×C₆₁	—	4	2	no	—
40	C₂×Q₈×C₆₁	—	2	2	no	—
41	C₂×Q₈×C₆₁	—	4	1	no	—
42	C₂×Q₈×C₆₁	—	4	1	no	—
43	C₂×Q₈×C₆₁	—	4	1	no	—
44	C₂×Q₈×C₆₁	—	4	1	no	—
45	C₂×Q₈×C₆₁	—	4	1	no	—
46	C₂²×C₄×C₆₁	—	4	1	no	—
47	C₂²×C₄×C₆₁	—	4	1	no	—
48	C₂²×C₄×C₆₁	D₈	1	2	yes	—
49	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
50	C₂²×Dic₆₁	—	4	2	no	—
51	Q₁₆×C₆₁	—	2	4	no	—
52	Q₁₆×C₆₁	—	4	2	no	—
53	C₂×C₈×C₆₁	—	2	1	no	—
54	C₂×C₈×C₆₁	—	2	1	no	—
55	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
56	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
57	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₈×C₆₁	#41	1	2	yes	—
59	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
60	C₄⋊C₄×C₆₁	—	4	1	no	—
61	C₄⋊C₄×C₆₁	—	4	1	no	—
62	C₄²×C₆₁	—	2	1	no	—
63	C₄²×C₆₁	—	2	1	no	—
64	C₄⋊Dic₆₁	—	2	4	no	—
65	C₄⋊Dic₆₁	—	4	2	no	—
66	#11758196	—	4	8	no	—
67	#11758185	—	4	2	no	—
68	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
69	Q₈×C₆₁	—	4	2	no	—
70	Q₈×C₆₁	—	4	2	no	—
71	Q₈×C₆₁	—	8	1	no	—
72	Q₈×C₆₁	—	8	1	no	—
73	C₂×C₄×C₆₁	—	2	2	no	—
74	C₂×C₄×C₆₁	—	2	2	no	—
75	C₂×C₄×C₆₁	—	2	2	no	—
76	C₂×C₄×C₆₁	—	4	2	no	—
77	C₂×C₄×C₆₁	—	8	1	no	—
78	C₂×C₄×C₆₁	—	8	1	no	—
79	C₂×C₄×C₆₁	—	8	1	no	—
80	C₂×C₄×C₆₁	—	8	2	no	—
81	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
82	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
83	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
84	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
85	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
86	C₂×C₄×C₆₁	#137	1	1	yes	characteristic
87	C₂×Dic₆₁	—	4	4	no	—
88	C₂×Dic₆₁	—	8	2	no	—
89	C₂×Dic₆₁	—	8	2	no	—
90	C₈×C₆₁	—	2	2	no	—
91	C₈×C₆₁	C₂²⋊Q₈	1	2	yes	—
92	C₈×C₆₁	#124	1	2	yes	—
93	#13366	—	61	1	no	maximal, Sylow
94	C₂²×C₆₁	#447	1	1	yes	characteristic
95	C₂²×C₆₁	#474	1	1	yes	characteristic
96	C₂²×C₆₁	#486	1	1	yes	characteristic
97	C₂²×C₆₁	#508	1	2	yes	—
98	C₂²×C₆₁	#510	1	2	yes	—
99	C₄×C₆₁	—	4	2	no	—
100	C₄×C₆₁	—	8	2	no	—
101	C₄×C₆₁	—	16	1	no	—
102	C₄×C₆₁	#393	1	1	yes	characteristic
103	C₄×C₆₁	#448	1	1	yes	characteristic
104	C₄×C₆₁	#476	1	1	yes	characteristic
105	C₄×C₆₁	#489	1	1	yes	characteristic
106	Dic₆₁	—	8	4	no	—
107	C₂×#499	—	61	1	no	—
108	C₂×#358	—	61	2	no	—
109	#1969	D₆₁	1	1	yes	characteristic
110	#1380	—	61	2	no	—
111	#1374	—	61	1	no	—
112	C₂×C₆₁	#2023	1	1	yes	characteristic
113	C₂×C₆₁	#2189	1	2	yes	—
114	C₂×C₆₁	#2229	1	2	yes	—
115	C₂×C₆₁	#2242	1	1	yes	characteristic
116	C₂×C₆₁	#2245	1	1	yes	characteristic
117	C₂²×Q₁₆	C₂×D₆₁	1	1	yes	characteristic
118	C₂×#125	—	122	1	no	—
119	C₂²×C₁₆	—	61	2	no	—
120	#499	—	122	4	no	—
121	C₂×#107	—	61	2	no	—
122	C₂×#107	C₂×D₆₁	1	1	yes	characteristic
123	C₂×#106	—	122	1	no	—
124	C₂×#103	—	2	1	no	—
125	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
126	#391	—	2	1	no	—
127	#358	—	122	4	no	—
128	C₆₁	#13366	1	1	yes	characteristic, complemented, Sylow
129	C₂²×Q₈	—	2	1	no	—
130	C₂×Q₁₆	—	2	1	no	—
131	C₂×Q₁₆	—	2	2	no	—
132	C₂×Q₁₆	—	2	2	no	—
133	C₂×Q₁₆	Dic₁₂₂	1	2	yes	—
134	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
136	#125	—	122	4	no	—
137	#125	—	244	2	no	—
138	C₂×C₄⋊C₄	—	2	1	no	—
139	C₂×C₄⋊C₄	—	122	2	no	—
140	C₂×C₄⋊C₄	—	244	2	no	—
141	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
142	C₂×C₁₆	—	122	2	no	—
143	C₂×C₁₆	—	122	2	no	—
144	C₂×C₁₆	—	122	2	no	—
145	#107	—	2	2	no	—
146	#107	—	61	4	no	—
147	#107	—	122	2	no	—
148	#106	—	122	4	no	—
149	#103	—	4	2	no	—
150	C₄×C₈	—	2	2	no	—
151	#95	—	4	2	no	—
152	C₂×Q₈	—	2	2	no	—
153	C₂×Q₈	—	4	1	no	—
154	C₂×Q₈	—	4	1	no	—
155	C₂×Q₈	—	4	1	no	—
156	C₂×Q₈	—	4	1	no	—
157	C₂×Q₈	—	4	1	no	—
158	C₂²×C₄	—	4	1	no	—
159	C₂²×C₄	—	4	1	no	—
160	C₂²×C₄	—	244	2	no	—
161	C₂²×C₄	#11758199	1	2	yes	—
162	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
163	Q₁₆	—	2	4	no	—
164	Q₁₆	—	4	2	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	#11758200	1	2	yes	—
168	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
169	C₂×C₈	#11758222	1	1	yes	characteristic
170	C₂×C₈	D₄×D₆₁	1	2	yes	—
171	C₂×C₈	#11758224	1	2	yes	—
172	C₄⋊C₄	—	4	1	no	—
173	C₄⋊C₄	—	4	1	no	—
174	C₄⋊C₄	—	122	4	no	—
175	C₄⋊C₄	—	244	8	no	—
176	C₄⋊C₄	—	244	2	no	—
177	C₄²	—	2	1	no	—
178	C₄²	—	2	1	no	—
179	C₁₆	—	244	2	no	—
180	C₂³	C₂×#11758199	1	1	yes	characteristic
181	Q₈	—	4	2	no	—
182	Q₈	—	4	2	no	—
183	Q₈	—	8	1	no	—
184	Q₈	—	8	1	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	8	1	no	—
190	C₂×C₄	—	8	1	no	—
191	C₂×C₄	—	8	2	no	—
192	C₂×C₄	—	8	1	no	—
193	C₂×C₄	—	488	2	no	—
194	C₂×C₄	—	488	2	no	—
195	C₂×C₄	—	244	4	no	—
196	C₂×C₄	#14519745	1	1	yes	characteristic
197	C₂×C₄	#14519756	1	1	yes	characteristic
198	C₂×C₄	#14519764	1	1	yes	characteristic
199	C₂×C₄	C₂×#11758200	1	1	yes	characteristic
200	C₂×C₄	#14519802	1	1	yes	characteristic
201	C₂×C₄	#14519807	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	#14519754	1	2	yes	—
204	C₈	#14519763	1	2	yes	—
205	C₂²	#16839895	1	1	yes	characteristic
206	C₂²	#16840042	1	1	yes	characteristic
207	C₂²	#16840108	1	1	yes	characteristic
208	C₂²	#16840192	1	2	yes	—
209	C₂²	#16840203	1	2	yes	—
210	C₄	—	4	2	no	—
211	C₄	—	8	2	no	—
212	C₄	—	16	1	no	—
213	C₄	—	488	4	no	—
214	C₄	#16839674	1	1	yes	characteristic
215	C₄	#16839900	1	1	yes	characteristic
216	C₄	#16840052	1	1	yes	characteristic
217	C₄	#16840121	1	1	yes	characteristic
218	C₂	#21446955	1	1	yes	characteristic
219	C₂	#21447757	1	2	yes	—
220	C₂	#21447983	1	2	yes	—
221	C₂	#21448050	1	1	yes	characteristic
222	C₂	#21448064	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16840164

C₂

yes

maximal, characteristic

C₂×#16839566

C₂

yes

maximal

#21443910

C₂

yes

maximal

#21443897

C₂

yes

maximal, characteristic

C₆₁×#1969

C₂

yes

maximal, characteristic

C₂×#14519769

—

C₂²×#11758185

C₂²

yes

—

#16840164

—

C₂×#14519711

C₂²

yes

—

C₂×#14519709

—

#16839566

—

C₂²×Q₁₆×C₆₁

C₂²

yes

characteristic

C₂×#107×C₆₁

C₂²

yes

characteristic

C₂×#103×C₆₁

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₆₁×#391

—

#14519769

—

#14519769

—

C₂×C₄⋊Dic₆₁

—

—

—

—

—

—

D₄

yes

—

#14519709

—

C₂²×Q₈×C₆₁

—

C₂×Q₁₆×C₆₁

—

C₂×Q₁₆×C₆₁

—

C₂×Q₁₆×C₆₁

—

C₂×Q₁₆×C₆₁

Q₈

yes

—

C₂²×C₈×C₆₁

D₄

yes

characteristic

C₂²×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄²×C₆₁

D₄

yes

characteristic

#107×C₆₁

—

#103×C₆₁

—

C₄×C₈×C₆₁

—

#95×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

D₈

yes

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×Dic₆₁

—

Q₁₆×C₆₁

—

Q₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

SD₁₆

yes

—

C₂×C₈×C₆₁

C₂×D₄

yes

—

C₂×C₈×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₁

#41

yes

—

C₂×C₈×C₆₁

#41

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

#11758196

—

#11758185

—

C₂³×C₆₁

C₂×D₈

yes

characteristic

Q₈×C₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#120

yes

characteristic

C₂×C₄×C₆₁

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₆₁

#124

yes

characteristic

C₂×C₄×C₆₁

C₂×SD₁₆

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₈×C₆₁

—

C₈×C₆₁

C₂²⋊Q₈

yes

—

C₈×C₆₁

#124

yes

—

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂²×C₆₁

#510

yes

—

C₄×C₆₁

—

100

C₄×C₆₁

—

101

C₄×C₆₁

—

102

C₄×C₆₁

#393

yes

characteristic

103

C₄×C₆₁

#448

yes

characteristic

104

C₄×C₆₁

#476

yes

characteristic

105

C₄×C₆₁

#489

yes

characteristic

106

Dic₆₁

—

107

C₂×#499

—

108

C₂×#358

—

109

#1969

D₆₁

yes

characteristic

110

#1380

—

111

#1374

—

112

C₂×C₆₁

#2023

yes

characteristic

113

C₂×C₆₁

#2189

yes

—

114

C₂×C₆₁

#2229

yes

—

115

C₂×C₆₁

#2242

yes

characteristic

116

C₂×C₆₁

#2245

yes

characteristic

117

C₂²×Q₁₆

C₂×D₆₁

yes

characteristic

118

C₂×#125

—

122

—

119

C₂²×C₁₆

—

120

#499

—

122

—

121

C₂×#107

—

122

C₂×#107

C₂×D₆₁

yes

characteristic

123

C₂×#106

—

122

—

124

C₂×#103

—

125

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

126

#391

—

127

#358

—

122

—

128

C₆₁

#13366

yes

characteristic, complemented, Sylow

129

C₂²×Q₈

—

130

C₂×Q₁₆

—

131

C₂×Q₁₆

—

132

C₂×Q₁₆

—

133

C₂×Q₁₆

Dic₁₂₂

yes

—

134

C₂²×C₈

C₂²⋊D₆₁

yes

characteristic

135

C₂²×C₈

C₂²×D₆₁

yes

characteristic

136

#125

—

122

—

137

#125

—

244

—

138

C₂×C₄⋊C₄

—

139

C₂×C₄⋊C₄

—

122

—

140

C₂×C₄⋊C₄

—

244

—

141

C₂×C₄²

C₂²⋊D₆₁

yes

characteristic

142

C₂×C₁₆

—

122

—

143

C₂×C₁₆

—

122

—

144

C₂×C₁₆

—

122

—

145

#107

—

146

#107

—

147

#107

—

122

—

148

#106

—

122

—

149

#103

—

150

C₄×C₈

—

151

#95

—

152

C₂×Q₈

—

153

C₂×Q₈

—

154

C₂×Q₈

—

155

C₂×Q₈

—

156

C₂×Q₈

—

157

C₂×Q₈

—

158

C₂²×C₄

—

159

C₂²×C₄

—

160

C₂²×C₄

—

244

—

161

C₂²×C₄

#11758199

yes

—

162

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

163

Q₁₆

—

164

Q₁₆

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

#11758200

yes

—

168

C₂×C₈

C₂×Dic₁₂₂

yes

characteristic

169

C₂×C₈

#11758222

yes

characteristic

170

C₂×C₈

D₄×D₆₁

yes

—

171

C₂×C₈

#11758224

yes

—

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₄⋊C₄

—

122

—

175

C₄⋊C₄

—

244

—

176

C₄⋊C₄

—

244

—

177

C₄²

—

178

C₄²

—

179

C₁₆

—

244

—

180

C₂³

C₂×#11758199

yes

characteristic

181

Q₈

—

182

Q₈

—

183

Q₈

—

184

Q₈

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

488

—

194

C₂×C₄

—

488

—

195

C₂×C₄

—

244

—

196

C₂×C₄

#14519745

yes

characteristic

197

C₂×C₄

#14519756

yes

characteristic

198

C₂×C₄

#14519764

yes

characteristic

199

C₂×C₄

C₂×#11758200

yes

characteristic

200

C₂×C₄

#14519802

yes

characteristic

201

C₂×C₄

#14519807

yes

characteristic

202

C₈

—

203

C₈

#14519754

yes

—

204

C₈

#14519763

yes

—

205

C₂²

#16839895

yes

characteristic

206

C₂²

#16840042

yes

characteristic

207

C₂²

#16840108

yes

characteristic

208

C₂²

#16840192

yes

—

209

C₂²

#16840203

yes

—

210

C₄

—

211

C₄

—

212

C₄

—

213

C₄

—

488

—

214

C₄

#16839674

yes

characteristic

215

C₄

#16839900

yes

characteristic

216

C₄

#16840052

yes

characteristic

217

C₄

#16840121

yes

characteristic

218

C₂

#21446955

yes

characteristic

219

C₂

#21447757

yes

—

220

C₂

#21447983

yes

—

221

C₂

#21448050

yes

characteristic

222

C₂

#21448064

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×#11758199
commutator, cosocle	C₂²×C₈×C₆₁	C₂³
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#1969	C₂
socle, prime radical	C₂³×C₆₁	C₂×D₈
hypercenter	#1969	D₆₁
hypocenter	C₆₁	#13366
LCS	G ▷ C₂²×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1969

subgroup

G/H

center

C₂³

C₂×#11758199

commutator, cosocle

C₂²×C₈×C₆₁

C₂³

Frattini

C₂²×C₈

C₂²×D₆₁

Fitting

C₆₁×#1969

C₂

socle, prime radical

C₂³×C₆₁

C₂×D₈

D₆₁

C₆₁

G ▷ C₂²×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1969

N	H ≅ G/N	type
128. C₆₁	#13366	semidirect

H ≅ G/N

type

128. C₆₁

#13366

semidirect

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

subgroup	index
C₂×#16840164	2
C₂×#16839566	2
#21443910	2
#21443897	2
C₆₁×#1969	2
#13366	61

subgroup

index

C₆₁×#1969

Subgroups — #29599092

Conjugacy classes of subgroups

Complements

Maximal subgroups