Group #34203392 — subgroups

Subgroups — #34203392

Order 20224.

382 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 151 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×#19098980	C₂	1	1	yes	maximal, characteristic
2	#23765004	C₂	1	2	yes	maximal, complemented
3	#23763200	C₂	1	1	yes	maximal, characteristic
4	#23761490	C₂	1	2	yes	maximal, complemented
5	C₇₉×#2115	C₂	1	1	yes	maximal, characteristic
6	#19098980	—	2	4	no	—
7	C₂×#14579101	C₄	1	2	yes	complemented
8	C₂×#14579101	C₂²	1	1	yes	characteristic
9	C₂²×C₇₉⋊C₁₆	C₄	1	2	yes	complemented
10	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
11	#19098547	C₄	1	1	yes	characteristic
12	#19098537	C₄	1	1	yes	characteristic
13	#19098518	C₂²	1	2	yes	—
14	#19098340	—	2	4	no	—
15	#19098334	C₄	1	2	yes	—
16	C₂×#131×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×M₃₂×C₇₉	C₂²	1	2	yes	—
18	#349×C₇₉	C₄	1	4	yes	complemented
19	C₂×#11929608	C₂×C₄	1	2	yes	—
20	#14579120	C₂×C₄	1	2	yes	—
21	#14579101	—	2	4	no	—
22	#14579101	D₄	1	4	yes	—
23	C₂×C₇₉⋊C₁₆	—	2	4	no	—
24	C₂×C₇₉⋊C₁₆	—	2	2	no	—
25	C₂×C₇₉⋊C₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂×C₇₉⋊C₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂×C₇₉⋊C₁₆	C₂×C₄	1	2	yes	—
28	#14579080	—	2	1	no	—
29	#14579078	—	2	1	no	—
30	#14579076	—	2	2	no	—
31	#14579071	—	2	2	no	—
32	C₈×Dic₇₉	—	2	2	no	—
33	C₂×#41×C₇₉	C₂×C₄	1	1	yes	characteristic
34	#131×C₇₉	—	2	2	no	—
35	#131×C₇₉	—	2	2	no	—
36	C₂×M₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
37	C₂×M₁₆×C₇₉	C₂×C₄	1	2	yes	—
38	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
40	M₃₂×C₇₉	—	2	2	no	—
41	M₃₂×C₇₉	C₂×C₄	1	4	yes	—
42	C₂×C₁₆×C₇₉	—	2	2	no	—
43	#11929608	—	2	2	no	—
44	#11929608	—	2	2	no	—
45	C₂×C₇₉⋊C₈	—	2	2	no	—
46	C₄⋊Dic₇₉	—	2	2	no	—
47	C₂²⋊Dic₇₉	—	2	2	no	—
48	C₄×Dic₇₉	—	2	2	no	—
49	C₇₉⋊C₁₆	—	2	1	no	—
50	C₇₉⋊C₁₆	—	2	1	no	—
51	C₇₉⋊C₁₆	C₂²⋊C₄	1	4	yes	—
52	#41×C₇₉	—	2	2	no	—
53	#41×C₇₉	—	2	2	no	—
54	C₂×Q₈×C₇₉	M₁₆	1	1	yes	characteristic
55	C₂×D₄×C₇₉	M₁₆	1	2	yes	—
56	C₂×D₄×C₇₉	M₁₆	1	1	yes	characteristic
57	C₂²×C₄×C₇₉	C₄²	1	2	yes	—
58	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
59	M₁₆×C₇₉	—	2	4	no	—
60	M₁₆×C₇₉	—	2	2	no	—
61	C₂×C₈×C₇₉	—	2	2	no	—
62	C₂×C₈×C₇₉	—	2	2	no	—
63	C₂×C₈×C₇₉	C₄²	1	2	yes	—
64	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₇₉	#41	1	2	yes	—
68	C₁₆×C₇₉	—	2	4	no	—
69	C₂³×C₇₉	#97	1	2	yes	—
70	C₂³×C₇₉	C₂×M₁₆	1	1	yes	characteristic
71	Q₈×C₇₉	—	2	2	no	—
72	D₄×C₇₉	—	2	4	no	—
73	D₄×C₇₉	—	2	2	no	—
74	C₂×C₄×C₇₉	—	2	2	no	—
75	C₂×C₄×C₇₉	—	2	2	no	—
76	C₂×C₄×C₇₉	#97	1	2	yes	—
77	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
78	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
79	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
80	C₂×C₄×C₇₉	#131	1	2	yes	—
81	C₂×Dic₇₉	—	4	2	no	—
82	C₈×C₇₉	—	2	2	no	—
83	C₈×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₈×C₇₉	#117	1	1	yes	characteristic
85	C₈×C₇₉	C₄×D₄	1	2	yes	—
86	C₇₉⋊C₈	—	4	2	no	—
87	C₂²×C₇₉	—	2	2	no	—
88	C₂²×C₇₉	—	4	1	no	—
89	C₂²×C₇₉	—	4	2	no	—
90	C₂²×C₇₉	C₂×#97	1	1	yes	characteristic
91	C₂²×C₇₉	#435	1	2	yes	—
92	C₄×C₇₉	—	2	2	no	—
93	C₄×C₇₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
94	C₄×C₇₉	#405	1	1	yes	characteristic
95	C₄×C₇₉	#435	1	2	yes	—
96	Dic₇₉	—	8	2	no	—
97	#12617	—	79	1	no	maximal, Sylow
98	C₂×C₇₉	—	2	2	no	—
99	C₂×C₇₉	—	2	1	no	—
100	C₂×C₇₉	—	4	2	no	—
101	C₂×C₇₉	#1752	1	1	yes	characteristic
102	C₂×#504	—	79	1	no	—
103	#2115	D₇₉	1	1	yes	characteristic
104	#2110	—	79	2	no	—
105	#1757	—	79	1	no	—
106	#1379	—	79	2	no	—
107	C₇₉	#12617	1	1	yes	characteristic, complemented, Sylow
108	C₂×#131	C₂×D₇₉	1	1	yes	characteristic
109	#504	—	158	4	no	—
110	C₂×M₃₂	—	79	1	no	—
111	C₂×M₃₂	—	79	2	no	—
112	C₂×M₃₂	C₂×D₇₉	1	2	yes	—
113	C₂²×C₁₆	—	79	1	no	—
114	C₂²×C₁₆	—	79	2	no	—
115	#413	—	79	1	no	—
116	#410	—	79	1	no	—
117	#405	—	79	2	no	—
118	#349	Dic₇₉	1	4	yes	complemented
119	#347	—	158	4	no	—
120	#344	—	79	2	no	—
121	C₂×#41	C₄×D₇₉	1	1	yes	characteristic
122	#131	—	2	2	no	—
123	#131	—	2	2	no	—
124	C₂×M₁₆	—	79	2	no	—
125	C₂×M₁₆	C₄×D₇₉	1	1	yes	characteristic
126	C₂×M₁₆	C₂×Dic₇₉	1	2	yes	—
127	C₂²×C₈	C₄×D₇₉	1	2	yes	—
128	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
129	#117	—	79	2	no	—
130	M₃₂	—	2	2	no	—
131	M₃₂	—	79	4	no	—
132	M₃₂	—	158	4	no	—
133	M₃₂	C₂×Dic₇₉	1	4	yes	—
134	C₂×C₁₆	—	2	2	no	—
135	C₂×C₁₆	—	79	2	no	—
136	C₂×C₁₆	—	79	1	no	—
137	C₂×C₁₆	—	79	1	no	—
138	C₂×C₁₆	—	158	4	no	—
139	C₂×C₁₆	—	158	2	no	—
140	#101	—	158	1	no	—
141	#100	—	158	1	no	—
142	#99	—	158	2	no	—
143	#97	—	158	2	no	—
144	C₄×C₈	—	158	2	no	—
145	#41	—	2	2	no	—
146	#41	—	2	2	no	—
147	C₂×Q₈	#11929607	1	1	yes	characteristic
148	C₂×D₄	#11929607	1	2	yes	—
149	C₂×D₄	#11929607	1	1	yes	characteristic
150	C₂²×C₄	C₄×Dic₇₉	1	2	yes	—
151	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
152	M₁₆	—	2	2	no	—
153	M₁₆	—	2	4	no	—
154	M₁₆	—	158	2	no	—
155	M₁₆	—	158	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	158	2	no	—
159	C₂×C₈	C₄×Dic₇₉	1	2	yes	—
160	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
161	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
162	C₂×C₈	D₄×D₇₉	1	2	yes	—
163	C₂×C₈	#11929623	1	2	yes	—
164	C₄⋊C₄	—	158	2	no	—
165	C₂²⋊C₄	—	158	2	no	—
166	C₄²	—	158	2	no	—
167	C₁₆	—	2	4	no	—
168	C₁₆	—	79	4	no	—
169	C₁₆	—	158	1	no	—
170	C₁₆	—	158	1	no	—
171	C₂³	#14579071	1	2	yes	—
172	C₂³	C₂×#11929607	1	1	yes	characteristic
173	Q₈	—	2	2	no	—
174	D₄	—	2	4	no	—
175	D₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	316	2	no	—
179	C₂×C₄	#14579071	1	2	yes	—
180	C₂×C₄	C₂×C₄×Dic₇₉	1	1	yes	characteristic
181	C₂×C₄	(C₂²⋊D₇₉)⋊C₄	1	2	yes	—
182	C₂×C₄	C₂×#11929607	1	1	yes	characteristic
183	C₂×C₄	#14579169	1	2	yes	—
184	C₈	—	2	2	no	—
185	C₈	—	316	2	no	—
186	C₈	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
187	C₈	#14579117	1	1	yes	characteristic
188	C₈	D₄×Dic₇₉	1	2	yes	—
189	C₂²	—	2	2	no	—
190	C₂²	—	4	1	no	—
191	C₂²	—	4	2	no	—
192	C₂²	C₂×#14579071	1	1	yes	characteristic
193	C₂²	#19098631	1	2	yes	—
194	C₄	—	2	2	no	—
195	C₄	—	632	2	no	—
196	C₄	C₂²⋊C₄×Dic₇₉	1	1	yes	characteristic
197	C₄	#19098521	1	1	yes	characteristic
198	C₄	#19098631	1	2	yes	—
199	C₂	—	2	2	no	—
200	C₂	—	2	1	no	—
201	C₂	—	4	2	no	—
202	C₂	#23763179	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19098980

C₂

yes

maximal, characteristic

#23765004

C₂

yes

maximal, complemented

#23763200

C₂

yes

maximal, characteristic

#23761490

C₂

yes

maximal, complemented

C₇₉×#2115

C₂

yes

maximal, characteristic

#19098980

—

C₂×#14579101

C₄

yes

complemented

C₂×#14579101

C₂²

yes

characteristic

C₂²×C₇₉⋊C₁₆

C₄

yes

complemented

C₂²×C₇₉⋊C₁₆

C₂²

yes

characteristic

#19098547

C₄

yes

characteristic

#19098537

C₄

yes

characteristic

#19098518

C₂²

yes

—

#19098340

—

#19098334

C₄

yes

—

C₂×#131×C₇₉

C₂²

yes

characteristic

C₂×M₃₂×C₇₉

C₂²

yes

—

#349×C₇₉

C₄

yes

complemented

C₂×#11929608

C₂×C₄

yes

—

#14579120

C₂×C₄

yes

—

#14579101

—

#14579101

D₄

yes

—

C₂×C₇₉⋊C₁₆

—

C₂×C₇₉⋊C₁₆

—

C₂×C₇₉⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₇₉⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₇₉⋊C₁₆

C₂×C₄

yes

—

—

—

—

—

C₈×Dic₇₉

—

C₂×#41×C₇₉

C₂×C₄

yes

characteristic

#131×C₇₉

—

#131×C₇₉

—

C₂×M₁₆×C₇₉

C₂×C₄

yes

characteristic

C₂×M₁₆×C₇₉

C₂×C₄

yes

—

C₂²×C₈×C₇₉

C₂×C₄

yes

—

C₂²×C₈×C₇₉

C₂³

yes

characteristic

M₃₂×C₇₉

—

M₃₂×C₇₉

C₂×C₄

yes

—

C₂×C₁₆×C₇₉

—

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₇₉⋊C₁₆

—

C₇₉⋊C₁₆

—

C₇₉⋊C₁₆

C₂²⋊C₄

yes

—

#41×C₇₉

—

#41×C₇₉

—

C₂×Q₈×C₇₉

M₁₆

yes

characteristic

C₂×D₄×C₇₉

M₁₆

yes

—

C₂×D₄×C₇₉

M₁₆

yes

characteristic

C₂²×C₄×C₇₉

C₄²

yes

—

C₂²×C₄×C₇₉

C₂²×C₄

yes

characteristic

M₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₄²

yes

—

C₂×C₈×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₉

C₂×D₄

yes

—

C₂×C₈×C₇₉

#41

yes

—

C₁₆×C₇₉

—

C₂³×C₇₉

#97

yes

—

C₂³×C₇₉

C₂×M₁₆

yes

characteristic

Q₈×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#97

yes

—

C₂×C₄×C₇₉

C₂×C₄²

yes

characteristic

C₂×C₄×C₇₉

C₄×D₄

yes

—

C₂×C₄×C₇₉

C₂×M₁₆

yes

characteristic

C₂×C₄×C₇₉

#131

yes

—

C₂×Dic₇₉

—

C₈×C₇₉

—

C₈×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₇₉

#117

yes

characteristic

C₈×C₇₉

C₄×D₄

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×#97

yes

characteristic

C₂²×C₇₉

#435

yes

—

C₄×C₇₉

—

C₄×C₇₉

C₄×C₂²⋊C₄

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

Dic₇₉

—

#12617

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

100

C₂×C₇₉

—

101

C₂×C₇₉

#1752

yes

characteristic

102

C₂×#504

—

103

#2115

D₇₉

yes

characteristic

104

#2110

—

105

#1757

—

106

#1379

—

107

C₇₉

#12617

yes

characteristic, complemented, Sylow

108

C₂×#131

C₂×D₇₉

yes

characteristic

109

#504

—

158

—

110

C₂×M₃₂

—

111

C₂×M₃₂

—

112

C₂×M₃₂

C₂×D₇₉

yes

—

113

C₂²×C₁₆

—

114

C₂²×C₁₆

—

115

#413

—

116

#410

—

117

#405

—

118

#349

Dic₇₉

yes

complemented

119

#347

—

158

—

120

#344

—

121

C₂×#41

C₄×D₇₉

yes

characteristic

122

#131

—

123

#131

—

124

C₂×M₁₆

—

125

C₂×M₁₆

C₄×D₇₉

yes

characteristic

126

C₂×M₁₆

C₂×Dic₇₉

yes

—

127

C₂²×C₈

C₄×D₇₉

yes

—

128

C₂²×C₈

C₂²×D₇₉

yes

characteristic

129

#117

—

130

M₃₂

—

131

M₃₂

—

132

M₃₂

—

158

—

133

M₃₂

C₂×Dic₇₉

yes

—

134

C₂×C₁₆

—

135

C₂×C₁₆

—

136

C₂×C₁₆

—

137

C₂×C₁₆

—

138

C₂×C₁₆

—

158

—

139

C₂×C₁₆

—

158

—

140

#101

—

158

—

141

#100

—

158

—

142

#99

—

158

—

143

#97

—

158

—

144

C₄×C₈

—

158

—

145

#41

—

146

#41

—

147

C₂×Q₈

#11929607

yes

characteristic

148

C₂×D₄

#11929607

yes

—

149

C₂×D₄

#11929607

yes

characteristic

150

C₂²×C₄

C₄×Dic₇₉

yes

—

151

C₂²×C₄

C₂×C₄×D₇₉

yes

characteristic

152

M₁₆

—

153

M₁₆

—

154

M₁₆

—

158

—

155

M₁₆

—

158

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

158

—

159

C₂×C₈

C₄×Dic₇₉

yes

—

160

C₂×C₈

C₂×C₄×D₇₉

yes

characteristic

161

C₂×C₈

C₂²×Dic₇₉

yes

characteristic

162

C₂×C₈

D₄×D₇₉

yes

—

163

C₂×C₈

#11929623

yes

—

164

C₄⋊C₄

—

158

—

165

C₂²⋊C₄

—

158

—

166

C₄²

—

158

—

167

C₁₆

—

168

C₁₆

—

169

C₁₆

—

158

—

170

C₁₆

—

158

—

171

C₂³

#14579071

yes

—

172

C₂³

C₂×#11929607

yes

characteristic

173

Q₈

—

174

D₄

—

175

D₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

316

—

179

C₂×C₄

#14579071

yes

—

180

C₂×C₄

C₂×C₄×Dic₇₉

yes

characteristic

181

C₂×C₄

(C₂²⋊D₇₉)⋊C₄

yes

—

182

C₂×C₄

C₂×#11929607

yes

characteristic

183

C₂×C₄

#14579169

yes

—

184

C₈

—

185

C₈

—

316

—

186

C₈

C₂²⋊C₄×D₇₉

yes

characteristic

187

C₈

#14579117

yes

characteristic

188

C₈

D₄×Dic₇₉

yes

—

189

C₂²

—

190

C₂²

—

191

C₂²

—

192

C₂²

C₂×#14579071

yes

characteristic

193

C₂²

#19098631

yes

—

194

C₄

—

195

C₄

—

632

—

196

C₄

C₂²⋊C₄×Dic₇₉

yes

characteristic

197

C₄

#19098521

yes

characteristic

198

C₄

#19098631

yes

—

199

C₂

—

200

C₂

—

201

C₂

—

202

C₂

#23763179

yes

characteristic

	subgroup	G/H
center	C₈	#14579117
commutator	C₂×C₄×C₇₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#2115	C₂
socle	C₂×C₇₉	#1752
prime radical	C₂×#41×C₇₉	C₂×C₄
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#2115	D₇₉
hypocenter	C₇₉	#12617
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2115

subgroup

G/H

center

C₈

#14579117

commutator

C₂×C₄×C₇₉

C₂×C₄²

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₇₉×#2115

C₂

socle

C₂×C₇₉

#1752

prime radical

C₂×#41×C₇₉

C₂×C₄

cosocle

C₂²×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2115

N	H ≅ G/N	type
2. #23765004	C₂	semidirect
4. #23761490	C₂	semidirect
7. C₂×#14579101	C₄	semidirect
9. C₂²×C₇₉⋊C₁₆	C₄	semidirect
18. #349×C₇₉	C₄	semidirect
107. C₇₉	#12617	semidirect
118. #349	Dic₇₉	semidirect

H ≅ G/N

type

2. #23765004

C₂

semidirect

4. #23761490

C₂

semidirect

7. C₂×#14579101

C₄

semidirect

9. C₂²×C₇₉⋊C₁₆

C₄

semidirect

18. #349×C₇₉

C₄

semidirect

107. C₇₉

#12617

semidirect

118. #349

Dic₇₉

semidirect

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

subgroup	index
C₂×#19098980	2
#23765004	2
#23763200	2
#23761490	2
C₇₉×#2115	2
#12617	79

subgroup

index

C₇₉×#2115

Subgroups — #34203392

Conjugacy classes of subgroups

Complements

Maximal subgroups