Group #34193038 — subgroups

Subgroups — #34193038

Order 20224.

330 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 93 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	#23764232	C₂	1	1	yes	maximal, characteristic
2	#23762050	C₂	1	2	yes	maximal
3	#23761297	C₂	1	2	yes	maximal, complemented
4	#23761229	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₉×#1843	C₂	1	1	yes	maximal, characteristic, complemented
6	#19099113	C₂²	1	1	yes	characteristic
7	#19098616	C₂²	1	2	yes	—
8	#19098587	C₂²	1	1	yes	characteristic
9	C₂×#14579089	—	2	1	no	—
10	C₂×#14579089	—	2	1	no	—
11	#19098310	—	2	2	no	—
12	#19098306	—	2	2	no	—
13	C₂×C₄⋊C₈×C₇₉	—	2	1	no	—
14	C₂×C₄⋊C₈×C₇₉	—	2	1	no	—
15	C₂×C₄⋊C₈×C₇₉	C₂²	1	2	yes	—
16	#376×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×#11929624	—	2	1	no	—
18	C₂×#11929608	—	2	1	no	—
19	#14579161	D₄	1	1	yes	characteristic
20	#14579158	D₄	1	1	yes	characteristic
21	#14579149	—	2	1	no	—
22	#14579135	—	2	2	no	—
23	C₄×D₃₁₆	—	2	2	no	—
24	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
25	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
26	#14579089	—	4	2	no	—
27	#14579089	—	4	2	no	—
28	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
29	C₄×C₇₉⋊C₈	—	2	2	no	—
30	C₂²×C₈×C₇₉	—	2	2	no	—
31	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₇₉	—	2	2	no	—
35	C₄⋊C₈×C₇₉	—	2	2	no	—
36	C₄⋊C₈×C₇₉	—	4	2	no	—
37	C₄⋊C₈×C₇₉	—	4	2	no	—
38	#95×C₇₉	—	2	2	no	—
39	#11929624	—	4	1	no	—
40	#11929624	—	4	1	no	—
41	#11929624	—	4	1	no	—
42	#11929624	—	4	1	no	—
43	C₂×Dic₁₅₈	—	2	1	no	—
44	C₂×D₃₁₆	—	2	1	no	—
45	C₂×D₃₁₆	—	2	2	no	—
46	C₂×C₂²⋊D₇₉	—	4	1	no	—
47	C₂×C₄×D₇₉	—	4	1	no	—
48	#11929608	—	4	1	no	—
49	#11929608	—	4	1	no	—
50	C₂×C₇₉⋊C₈	—	4	2	no	—
51	C₂×C₇₉⋊C₈	—	4	1	no	—
52	C₄⋊Dic₇₉	—	2	2	no	—
53	(C₂×D₇₉)⋊C₄	—	4	2	no	—
54	C₂²×C₄×C₇₉	—	2	1	no	—
55	C₂²×C₄×C₇₉	—	2	1	no	—
56	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₇₉	#41	1	2	yes	—
58	C₂×C₈×C₇₉	—	4	2	no	—
59	C₂×C₈×C₇₉	—	4	2	no	—
60	C₂×C₈×C₇₉	—	4	2	no	—
61	C₄²×C₇₉	—	2	1	no	—
62	C₄²×C₇₉	—	2	1	no	—
63	C₄²×C₇₉	D₈	1	2	yes	—
64	C₄²×C₇₉	SD₁₆	1	2	yes	—
65	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₇₉	#41	1	1	yes	characteristic
67	C₄²×C₇₉	#41	1	2	yes	—
68	C₂³×C₇₉	#123	1	1	yes	characteristic
69	C₂²×D₇₉	—	4	1	no	—
70	C₂×C₄×C₇₉	—	2	2	no	—
71	C₂×C₄×C₇₉	—	2	2	no	—
72	C₂×C₄×C₇₉	—	4	2	no	—
73	C₂×C₄×C₇₉	—	4	2	no	—
74	C₂×C₄×C₇₉	—	4	2	no	—
75	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
76	C₂×C₄×C₇₉	#123	1	2	yes	—
77	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
78	C₂×C₄×C₇₉	#126	1	2	yes	—
79	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
81	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
82	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
83	C₂²⋊D₇₉	—	8	1	no	—
84	C₂²⋊D₇₉	—	8	1	no	—
85	C₂×Dic₇₉	—	4	1	no	—
86	D₃₁₆	—	4	1	no	—
87	D₃₁₆	—	4	1	no	—
88	D₃₁₆	—	8	2	no	—
89	C₄×D₇₉	—	8	1	no	—
90	C₄×D₇₉	—	8	1	no	—
91	Dic₁₅₈	—	4	1	no	—
92	Dic₁₅₈	—	4	1	no	—
93	C₈×C₇₉	—	8	2	no	—
94	C₇₉⋊C₈	—	8	2	no	—
95	C₇₉⋊C₈	—	8	1	no	—
96	C₂²×C₇₉	—	2	2	no	—
97	C₂²×C₇₉	#397	1	1	yes	characteristic
98	C₂²×C₇₉	#480	1	1	yes	characteristic
99	C₂²×C₇₉	#481	1	1	yes	characteristic
100	C₂×D₇₉	—	8	1	no	—
101	C₂×D₇₉	—	8	2	no	—
102	C₄×C₇₉	—	2	2	no	—
103	C₄×C₇₉	—	2	2	no	—
104	C₄×C₇₉	—	4	2	no	—
105	C₄×C₇₉	—	4	2	no	—
106	C₄×C₇₉	#459	1	1	yes	characteristic
107	C₄×C₇₉	#461	1	1	yes	characteristic
108	C₄×C₇₉	#486	1	1	yes	characteristic
109	C₄×C₇₉	#489	1	1	yes	characteristic
110	Dic₇₉	—	8	1	no	—
111	#10627	—	79	1	no	maximal, Sylow
112	C₂×C₇₉	—	2	1	no	—
113	C₂×C₇₉	—	2	1	no	—
114	C₂×C₇₉	#2071	1	1	yes	characteristic
115	C₂×C₇₉	#2080	1	2	yes	—
116	D₇₉	—	16	1	no	—
117	#1963	—	79	1	no	—
118	#1843	D₇₉	1	1	yes	characteristic, complemented
119	#1540	—	79	2	no	—
120	#1301	—	79	2	no	—
121	#1274	—	79	1	no	—
122	C₇₉	#10627	1	1	yes	characteristic, complemented, Sylow
123	#532	—	79	1	no	—
124	#431	—	79	2	no	—
125	#423	—	79	1	no	—
126	C₂×C₄⋊C₈	—	2	1	no	—
127	C₂×C₄⋊C₈	—	2	1	no	—
128	C₂×C₄⋊C₈	C₂×D₇₉	1	2	yes	—
129	C₂×#104	—	158	1	no	—
130	C₂×#104	—	158	1	no	—
131	#376	C₂×D₇₉	1	1	yes	characteristic
132	#332	—	158	2	no	—
133	#331	—	158	2	no	—
134	C₂×#41	—	158	1	no	—
135	C₂×M₁₆	—	158	1	no	—
136	C₂²×C₈	—	2	2	no	—
137	C₄⋊Q₈	—	79	1	no	—
138	#127	—	79	1	no	—
139	#124	—	158	1	no	—
140	#121	—	158	2	no	—
141	C₄×D₄	—	158	2	no	—
142	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
143	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
144	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
145	C₄⋊C₈	—	2	2	no	—
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	—	4	2	no	—
148	C₄⋊C₈	—	4	2	no	—
149	C₄⋊C₈	—	158	2	no	—
150	C₄⋊C₈	—	158	2	no	—
151	#104	—	316	2	no	—
152	#104	—	316	2	no	—
153	C₂²⋊C₈	—	158	2	no	—
154	C₄×C₈	—	158	2	no	—
155	#95	—	2	2	no	—
156	#41	—	316	1	no	—
157	#41	—	316	1	no	—
158	#41	—	316	1	no	—
159	#41	—	316	1	no	—
160	C₂×Q₈	—	158	1	no	—
161	C₂×D₄	—	316	1	no	—
162	C₂×D₄	—	158	1	no	—
163	C₂×D₄	—	158	2	no	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	316	1	no	—
167	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
168	C₂²×C₄	#11929624	1	2	yes	—
169	M₁₆	—	316	1	no	—
170	M₁₆	—	316	1	no	—
171	C₂×C₈	—	4	2	no	—
172	C₂×C₈	—	4	2	no	—
173	C₂×C₈	—	4	2	no	—
174	C₂×C₈	—	316	2	no	—
175	C₂×C₈	—	316	1	no	—
176	C₄⋊C₄	—	158	2	no	—
177	C₂²⋊C₄	—	316	2	no	—
178	C₄²	—	2	1	no	—
179	C₄²	—	2	1	no	—
180	C₄²	#11929609	1	2	yes	—
181	C₄²	#11929611	1	2	yes	—
182	C₄²	D₄×D₇₉	1	1	yes	characteristic
183	C₄²	#11929623	1	1	yes	characteristic
184	C₄²	#11929625	1	2	yes	—
185	C₂³	—	316	1	no	—
186	C₂³	#14579144	1	1	yes	characteristic
187	Q₈	—	316	1	no	—
188	Q₈	—	316	1	no	—
189	D₄	—	316	1	no	—
190	D₄	—	316	1	no	—
191	D₄	—	632	2	no	—
192	D₄	—	632	1	no	—
193	D₄	—	632	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	316	1	no	—
200	C₂×C₄	—	632	1	no	—
201	C₂×C₄	—	632	1	no	—
202	C₂×C₄	#14579132	1	1	yes	characteristic
203	C₂×C₄	#14579146	1	2	yes	—
204	C₂×C₄	#14579150	1	1	yes	characteristic
205	C₂×C₄	#14579156	1	2	yes	—
206	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
207	C₂×C₄	C₂×#11929611	1	1	yes	characteristic
208	C₂×C₄	#14579187	1	1	yes	characteristic
209	C₂×C₄	#14579192	1	1	yes	characteristic
210	C₈	—	8	2	no	—
211	C₈	—	632	2	no	—
212	C₈	—	632	1	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	632	1	no	—
215	C₂²	—	632	2	no	—
216	C₂²	#19098489	1	1	yes	characteristic
217	C₂²	#19098876	1	1	yes	characteristic
218	C₂²	#19098881	1	1	yes	characteristic
219	C₄	—	2	2	no	—
220	C₄	—	2	2	no	—
221	C₄	—	4	2	no	—
222	C₄	—	4	2	no	—
223	C₄	—	632	1	no	—
224	C₄	#19098755	1	1	yes	characteristic
225	C₄	#19098765	1	1	yes	characteristic
226	C₄	#19098911	1	1	yes	characteristic
227	C₄	#19098925	1	1	yes	characteristic
228	C₂	—	2	1	no	—
229	C₂	—	2	1	no	—
230	C₂	—	1264	1	no	—
231	C₂	#23764804	1	1	yes	characteristic
232	C₂	#23764851	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23764232

C₂

yes

maximal, characteristic

#23762050

C₂

yes

maximal

#23761297

C₂

yes

maximal, complemented

#23761229

C₂

yes

maximal, characteristic, complemented

C₇₉×#1843

C₂

yes

maximal, characteristic, complemented

#19099113

C₂²

yes

characteristic

#19098616

C₂²

yes

—

#19098587

C₂²

yes

—

—

—

—

C₂×C₄⋊C₈×C₇₉

—

C₂×C₄⋊C₈×C₇₉

—

C₂×C₄⋊C₈×C₇₉

C₂²

yes

—

#376×C₇₉

C₂²

yes

—

—

D₄

yes

characteristic

#14579158

D₄

yes

characteristic

#14579149

—

#14579135

—

C₄×D₃₁₆

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

#14579089

—

#14579089

—

C₂²⋊(C₇₉⋊C₈)

—

C₄×C₇₉⋊C₈

—

C₂²×C₈×C₇₉

—

C₂×C₄²×C₇₉

D₄

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

characteristic

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

—

#95×C₇₉

—

—

—

—

—

C₂×Dic₁₅₈

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×C₂²⋊D₇₉

—

C₂×C₄×D₇₉

—

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

(C₂×D₇₉)⋊C₄

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

#41

yes

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

D₈

yes

—

C₄²×C₇₉

SD₁₆

yes

—

C₄²×C₇₉

C₂×D₄

yes

C₄²×C₇₉

yes

C₄²×C₇₉

yes

—

C₂³×C₇₉

#123

yes

characteristic

C₂²×D₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#121

yes

characteristic

C₂×C₄×C₇₉

#123

yes

—

C₂×C₄×C₇₉

#124

yes

characteristic

C₂×C₄×C₇₉

#126

yes

—

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

C₂×SD₁₆

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂²⋊D₇₉

—

C₂²⋊D₇₉

—

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

C₄×D₇₉

—

C₄×D₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

100

C₂×D₇₉

—

101

C₂×D₇₉

—

102

C₄×C₇₉

—

103

C₄×C₇₉

—

104

C₄×C₇₉

—

105

C₄×C₇₉

—

106

C₄×C₇₉

#459

yes

characteristic

107

C₄×C₇₉

#461

yes

characteristic

108

C₄×C₇₉

#486

yes

characteristic

109

C₄×C₇₉

#489

yes

characteristic

110

Dic₇₉

—

111

#10627

—

maximal, Sylow

112

C₂×C₇₉

—

113

C₂×C₇₉

—

114

C₂×C₇₉

#2071

yes

characteristic

115

C₂×C₇₉

#2080

yes

—

116

D₇₉

—

117

#1963

—

118

#1843

D₇₉

yes

characteristic, complemented

119

#1540

—

120

#1301

—

121

#1274

—

122

C₇₉

#10627

yes

characteristic, complemented, Sylow

123

#532

—

124

#431

—

125

#423

—

126

C₂×C₄⋊C₈

—

127

C₂×C₄⋊C₈

—

128

C₂×C₄⋊C₈

C₂×D₇₉

yes

—

129

C₂×#104

—

158

—

130

C₂×#104

—

158

—

131

#376

C₂×D₇₉

yes

characteristic

132

#332

—

158

—

133

#331

—

158

—

134

C₂×#41

—

158

—

135

C₂×M₁₆

—

158

—

136

C₂²×C₈

—

137

C₄⋊Q₈

—

138

#127

—

139

#124

—

158

—

140

#121

—

158

—

141

C₄×D₄

—

158

—

142

C₂×C₄²

C₂²⋊D₇₉

yes

characteristic

143

C₂×C₄²

C₂²⋊D₇₉

yes

characteristic

144

C₂×C₄²

C₂²×D₇₉

yes

characteristic

145

C₄⋊C₈

—

146

C₄⋊C₈

—

147

C₄⋊C₈

—

148

C₄⋊C₈

—

149

C₄⋊C₈

—

158

—

150

C₄⋊C₈

—

158

—

151

#104

—

316

—

152

#104

—

316

—

153

C₂²⋊C₈

—

158

—

154

C₄×C₈

—

158

—

155

#95

—

156

#41

—

316

—

157

#41

—

316

—

158

#41

—

316

—

159

#41

—

316

—

160

C₂×Q₈

—

158

—

161

C₂×D₄

—

316

—

162

C₂×D₄

—

158

—

163

C₂×D₄

—

158

—

164

C₂²×C₄

—

165

C₂²×C₄

—

166

C₂²×C₄

—

316

—

167

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

168

C₂²×C₄

#11929624

yes

—

169

M₁₆

—

316

—

170

M₁₆

—

316

—

171

C₂×C₈

—

172

C₂×C₈

—

173

C₂×C₈

—

174

C₂×C₈

—

316

—

175

C₂×C₈

—

316

—

176

C₄⋊C₄

—

158

—

177

C₂²⋊C₄

—

316

—

178

C₄²

—

179

C₄²

—

180

C₄²

#11929609

yes

—

181

C₄²

#11929611

yes

—

182

C₄²

D₄×D₇₉

yes

characteristic

183

C₄²

#11929623

yes

characteristic

184

C₄²

#11929625

yes

—

185

C₂³

—

316

—

186

C₂³

#14579144

yes

characteristic

187

Q₈

—

316

—

188

Q₈

—

316

—

189

D₄

—

316

—

190

D₄

—

316

—

191

D₄

—

632

—

192

D₄

—

632

—

193

D₄

—

632

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

316

—

200

C₂×C₄

—

632

—

201

C₂×C₄

—

632

—

202

C₂×C₄

#14579132

yes

characteristic

203

C₂×C₄

#14579146

yes

—

204

C₂×C₄

#14579150

yes

characteristic

205

C₂×C₄

#14579156

yes

—

206

C₂×C₄

C₂×#11929609

yes

characteristic

207

C₂×C₄

C₂×#11929611

yes

characteristic

208

C₂×C₄

#14579187

yes

characteristic

209

C₂×C₄

#14579192

yes

characteristic

210

C₈

—

211

C₈

—

632

—

212

C₈

—

632

—

213

C₂²

—

214

C₂²

—

632

—

215

C₂²

—

632

—

216

C₂²

#19098489

yes

characteristic

217

C₂²

#19098876

yes

characteristic

218

C₂²

#19098881

yes

characteristic

219

C₄

—

220

C₄

—

221

C₄

—

222

C₄

—

223

C₄

—

632

—

224

C₄

#19098755

yes

characteristic

225

C₄

#19098765

yes

characteristic

226

C₄

#19098911

yes

characteristic

227

C₄

#19098925

yes

characteristic

228

C₂

—

229

C₂

—

230

C₂

—

1264

—

231

C₂

#23764804

yes

characteristic

232

C₂

#23764851

yes

—

	subgroup	G/H
center	C₂×C₄	#14579150
commutator, cosocle	C₂×C₄²×C₇₉	C₂³
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1843	C₂
socle	C₂²×C₇₉	#397
prime radical	#19099113	C₂²
hypercenter	#1843	D₇₉
hypocenter	C₇₉	#10627
LCS	G ▷ C₂×C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1843

subgroup

G/H

center

C₂×C₄

#14579150

commutator, cosocle

C₂×C₄²×C₇₉

C₂³

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#1843

C₂

C₂²×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₂×C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1843

N	H ≅ G/N	type
3. #23761297	C₂	semidirect
4. #23761229	C₂	semidirect
5. C₇₉×#1843	C₂	semidirect
118. #1843	D₇₉	semidirect
122. C₇₉	#10627	semidirect

H ≅ G/N

type

3. #23761297

C₂

semidirect

4. #23761229

C₂

semidirect

5. C₇₉×#1843

C₂

semidirect

118. #1843

D₇₉

semidirect

122. C₇₉

#10627

semidirect

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

subgroup	index
#23764232	2
#23762050	2
#23761297	2
#23761229	2
C₇₉×#1843	2
#10627	79

subgroup

index

C₇₉×#1843

Subgroups — #34193038

Conjugacy classes of subgroups

Complements

Maximal subgroups