Group #34187653 — subgroups

Subgroups — #34187653

Order 20224.

400 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 119 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	#23763531	C₂	1	1	yes	maximal, characteristic, complemented
2	#23761964	C₂	1	2	yes	maximal, complemented
3	#23761960	C₂	1	2	yes	maximal
4	#23761683	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₉×#3131	C₂	1	1	yes	maximal, characteristic
6	#19098616	C₂²	1	2	yes	—
7	#19098429	C₂²	1	2	yes	—
8	#19098326	—	2	2	no	—
9	#19098300	—	2	4	no	—
10	#19098297	—	2	4	no	—
11	#19098284	—	2	1	no	—
12	C₈×C₇₉⋊C₈	—	2	1	no	—
13	#533×C₇₉	C₂²	1	1	yes	characteristic
14	#532×C₇₉	C₂²	1	1	yes	characteristic
15	#488×C₇₉	—	2	1	no	—
16	#488×C₇₉	—	2	1	no	—
17	#487×C₇₉	C₄	1	2	yes	complemented
18	#486×C₇₉	C₄	1	2	yes	complemented
19	#416×C₇₉	—	2	2	no	—
20	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Dic₇₉	—	2	2	no	—
22	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
23	C₂²⋊(C₇₉⋊C₈)	D₄	1	4	yes	—
24	C₄×C₇₉⋊C₈	—	2	2	no	—
25	C₇₉⋊#95	—	2	2	no	—
26	C₇₉⋊#95	—	2	2	no	—
27	C₂×#41×C₇₉	—	2	1	no	—
28	C₂²×C₈×C₇₉	D₄	1	2	yes	—
29	C₄⋊Q₈×C₇₉	C₂×C₄	1	2	yes	—
30	C₄⋊Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
31	#127×C₇₉	C₂×C₄	1	1	yes	characteristic
32	#125×C₇₉	—	2	1	no	—
33	#124×C₇₉	—	2	1	no	—
34	C₂²⋊Q₈×C₇₉	—	2	2	no	—
35	#121×C₇₉	—	2	2	no	—
36	C₄×Q₈×C₇₉	—	2	2	no	—
37	C₄×D₄×C₇₉	—	2	2	no	—
38	#117×C₇₉	—	2	1	no	—
39	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
40	#103×C₇₉	—	2	2	no	—
41	#103×C₇₉	—	2	2	no	—
42	#102×C₇₉	—	2	2	no	—
43	#102×C₇₉	—	2	2	no	—
44	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
46	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
47	C₂²×Dic₇₉	—	4	2	no	—
48	C₂×C₇₉⋊C₈	—	2	4	no	—
49	C₄⋊Dic₇₉	—	4	4	no	—
50	#41×C₇₉	—	4	4	no	—
51	C₂×Q₈×C₇₉	—	2	1	no	—
52	C₂×Q₈×C₇₉	—	2	2	no	—
53	C₂×D₄×C₇₉	—	2	1	no	—
54	C₂×D₄×C₇₉	—	2	2	no	—
55	C₂×D₄×C₇₉	—	4	1	no	—
56	C₂²×C₄×C₇₉	—	4	1	no	—
57	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₇₉	—	2	4	no	—
60	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
62	C₄⋊C₄×C₇₉	—	2	2	no	—
63	C₄⋊C₄×C₇₉	—	2	2	no	—
64	C₄⋊C₄×C₇₉	—	2	2	no	—
65	C₄⋊C₄×C₇₉	—	4	2	no	—
66	C₂²⋊C₄×C₇₉	—	4	1	no	—
67	C₂²⋊C₄×C₇₉	—	4	2	no	—
68	C₄²×C₇₉	—	4	1	no	—
69	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
70	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₇₉	#41	1	2	yes	—
72	C₂³×C₇₉	—	4	1	no	—
73	C₂³×C₇₉	#127	1	1	yes	characteristic
74	Q₈×C₇₉	—	4	2	no	—
75	Q₈×C₇₉	—	8	2	no	—
76	D₄×C₇₉	—	4	2	no	—
77	D₄×C₇₉	—	8	2	no	—
78	D₄×C₇₉	—	8	2	no	—
79	C₂×C₄×C₇₉	—	4	2	no	—
80	C₂×C₄×C₇₉	—	4	2	no	—
81	C₂×C₄×C₇₉	—	4	1	no	—
82	C₂×C₄×C₇₉	—	8	2	no	—
83	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
85	C₂×C₄×C₇₉	#121	1	2	yes	—
86	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
87	C₂×C₄×C₇₉	#135	1	4	yes	—
88	C₂×Dic₇₉	—	8	4	no	—
89	C₂×Dic₇₉	—	8	2	no	—
90	C₈×C₇₉	—	2	2	no	—
91	C₈×C₇₉	—	2	2	no	—
92	C₇₉⋊C₈	—	4	4	no	—
93	C₂²×C₇₉	—	2	2	no	—
94	C₂²×C₇₉	—	8	1	no	—
95	C₂²×C₇₉	—	8	2	no	—
96	C₂²×C₇₉	#390	1	1	yes	characteristic
97	C₂²×C₇₉	#495	1	2	yes	—
98	C₄×C₇₉	—	2	2	no	—
99	C₄×C₇₉	—	2	2	no	—
100	C₄×C₇₉	—	8	1	no	—
101	C₄×C₇₉	—	8	2	no	—
102	C₄×C₇₉	#416	1	2	yes	—
103	C₄×C₇₉	#488	1	2	yes	—
104	Dic₇₉	—	16	2	no	—
105	#9757	—	79	1	no	maximal, Sylow
106	C₂×C₇₉	—	2	2	no	—
107	C₂×C₇₉	—	16	1	no	—
108	C₂×C₇₉	#1958	1	1	yes	characteristic
109	C₂×C₇₉	#1976	1	2	yes	—
110	#3131	D₇₉	1	1	yes	characteristic
111	#1835	—	79	1	no	—
112	#1518	—	79	2	no	—
113	#1517	—	79	2	no	—
114	#1450	—	79	1	no	—
115	C₇₉	#9757	1	1	yes	characteristic, complemented, Sylow
116	#533	C₂×D₇₉	1	1	yes	characteristic
117	#532	C₂×D₇₉	1	1	yes	characteristic
118	#488	—	2	1	no	—
119	#488	—	2	1	no	—
120	#487	Dic₇₉	1	2	yes	complemented
121	#486	Dic₇₉	1	2	yes	complemented
122	#431	—	79	2	no	—
123	#416	—	2	2	no	—
124	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
125	#382	—	79	2	no	—
126	#340	—	158	2	no	—
127	#328	—	158	4	no	—
128	#327	—	158	4	no	—
129	#322	—	158	1	no	—
130	C₈²	—	158	1	no	—
131	C₂×#41	—	2	1	no	—
132	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
133	C₄⋊Q₈	C₂×Dic₇₉	1	1	yes	characteristic
134	C₄⋊Q₈	C₂×Dic₇₉	1	2	yes	—
135	#127	C₂×Dic₇₉	1	1	yes	characteristic
136	#125	—	2	1	no	—
137	#124	—	2	1	no	—
138	C₂²⋊Q₈	—	2	2	no	—
139	#121	—	2	2	no	—
140	C₄×Q₈	—	2	2	no	—
141	C₄×D₄	—	2	2	no	—
142	#117	—	2	1	no	—
143	C₂×C₄⋊C₄	—	158	2	no	—
144	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
145	C₄⋊C₈	—	158	2	no	—
146	#103	—	2	2	no	—
147	#103	—	2	2	no	—
148	#102	—	2	2	no	—
149	#102	—	2	2	no	—
150	C₂²⋊C₈	—	79	4	no	—
151	C₄×C₈	—	158	2	no	—
152	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
153	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
154	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
155	#95	—	158	2	no	—
156	#95	—	158	2	no	—
157	#41	—	4	4	no	—
158	C₂×Q₈	—	2	1	no	—
159	C₂×Q₈	—	2	2	no	—
160	C₂×D₄	—	2	1	no	—
161	C₂×D₄	—	2	2	no	—
162	C₂×D₄	—	4	1	no	—
163	C₂²×C₄	—	4	1	no	—
164	C₂²×C₄	—	316	2	no	—
165	C₂²×C₄	D₄×D₇₉	1	2	yes	—
166	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
167	C₂×C₈	—	2	4	no	—
168	C₂×C₈	—	158	4	no	—
169	C₂×C₈	C₂²⋊Dic₇₉	1	2	yes	—
170	C₂×C₈	C₂²⋊Dic₇₉	1	2	yes	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	316	4	no	—
176	C₂²⋊C₄	—	4	1	no	—
177	C₂²⋊C₄	—	4	2	no	—
178	C₄²	—	4	1	no	—
179	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
180	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
181	C₄²	#11929623	1	2	yes	—
182	C₂³	—	4	1	no	—
183	C₂³	#14579157	1	1	yes	characteristic
184	Q₈	—	4	2	no	—
185	Q₈	—	8	2	no	—
186	D₄	—	4	2	no	—
187	D₄	—	8	2	no	—
188	D₄	—	8	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	1	no	—
192	C₂×C₄	—	8	2	no	—
193	C₂×C₄	—	632	4	no	—
194	C₂×C₄	—	632	2	no	—
195	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
196	C₂×C₄	D₄×Dic₇₉	1	2	yes	—
197	C₂×C₄	#14579132	1	2	yes	—
198	C₂×C₄	#14579148	1	1	yes	characteristic
199	C₂×C₄	#14579181	1	4	yes	—
200	C₈	—	2	2	no	—
201	C₈	—	2	2	no	—
202	C₈	—	316	4	no	—
203	C₂²	—	2	2	no	—
204	C₂²	—	8	2	no	—
205	C₂²	—	8	1	no	—
206	C₂²	#19098466	1	1	yes	characteristic
207	C₂²	#19098945	1	2	yes	—
208	C₄	—	2	2	no	—
209	C₄	—	2	2	no	—
210	C₄	—	8	2	no	—
211	C₄	—	8	1	no	—
212	C₄	—	1264	2	no	—
213	C₄	#19098559	1	2	yes	—
214	C₄	#19098917	1	2	yes	—
215	C₂	—	2	2	no	—
216	C₂	—	16	1	no	—
217	C₂	#23764212	1	1	yes	characteristic
218	C₂	#23764299	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23763531

C₂

yes

maximal, characteristic, complemented

#23761964

C₂

yes

maximal, complemented

#23761960

C₂

yes

maximal

#23761683

C₂

yes

maximal, characteristic, complemented

C₇₉×#3131

C₂

yes

maximal, characteristic

#19098616

C₂²

yes

—

#19098429

C₂²

yes

—

—

—

—

—

C₈×C₇₉⋊C₈

—

#533×C₇₉

C₂²

yes

characteristic

#532×C₇₉

C₂²

yes

characteristic

#488×C₇₉

—

#488×C₇₉

—

#487×C₇₉

C₄

yes

complemented

#486×C₇₉

C₄

yes

complemented

#416×C₇₉

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₉

—

C₄⋊(C₇₉⋊C₈)

—

C₂²⋊(C₇₉⋊C₈)

D₄

yes

—

C₄×C₇₉⋊C₈

—

C₇₉⋊#95

—

C₇₉⋊#95

—

C₂×#41×C₇₉

—

C₂²×C₈×C₇₉

D₄

yes

—

C₄⋊Q₈×C₇₉

C₂×C₄

yes

—

C₄⋊Q₈×C₇₉

C₂×C₄

yes

characteristic

#127×C₇₉

C₂×C₄

yes

characteristic

#125×C₇₉

—

#124×C₇₉

—

C₂²⋊Q₈×C₇₉

—

#121×C₇₉

—

C₄×Q₈×C₇₉

—

C₄×D₄×C₇₉

—

#117×C₇₉

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

#103×C₇₉

—

#103×C₇₉

—

#102×C₇₉

—

#102×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

C₂²×Dic₇₉

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

#41×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

#41

yes

—

C₂³×C₇₉

—

C₂³×C₇₉

#127

yes

characteristic

Q₈×C₇₉

—

Q₈×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄×D₄

yes

—

C₂×C₄×C₇₉

#121

yes

—

C₂×C₄×C₇₉

#124

yes

characteristic

C₂×C₄×C₇₉

#135

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#390

yes

characteristic

C₂²×C₇₉

#495

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

100

C₄×C₇₉

—

101

C₄×C₇₉

—

102

C₄×C₇₉

#416

yes

—

103

C₄×C₇₉

#488

yes

—

104

Dic₇₉

—

105

#9757

—

maximal, Sylow

106

C₂×C₇₉

—

107

C₂×C₇₉

—

108

C₂×C₇₉

#1958

yes

characteristic

109

C₂×C₇₉

#1976

yes

—

110

#3131

D₇₉

yes

characteristic

111

#1835

—

112

#1518

—

113

#1517

—

114

#1450

—

115

C₇₉

#9757

yes

characteristic, complemented, Sylow

116

#533

C₂×D₇₉

yes

characteristic

117

#532

C₂×D₇₉

yes

characteristic

118

#488

—

119

#488

—

120

#487

Dic₇₉

yes

complemented

121

#486

Dic₇₉

yes

complemented

122

#431

—

123

#416

—

124

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

125

#382

—

126

#340

—

158

—

127

#328

—

158

—

128

#327

—

158

—

129

#322

—

158

—

130

C₈²

—

158

—

131

C₂×#41

—

132

C₂²×C₈

C₂²⋊D₇₉

yes

—

133

C₄⋊Q₈

C₂×Dic₇₉

yes

characteristic

134

C₄⋊Q₈

C₂×Dic₇₉

yes

—

135

#127

C₂×Dic₇₉

yes

characteristic

136

#125

—

137

#124

—

138

C₂²⋊Q₈

—

139

#121

—

140

C₄×Q₈

—

141

C₄×D₄

—

142

#117

—

143

C₂×C₄⋊C₄

—

158

—

144

C₂×C₄²

C₂²×D₇₉

yes

characteristic

145

C₄⋊C₈

—

158

—

146

#103

—

147

#103

—

148

#102

—

149

#102

—

150

C₂²⋊C₈

—

151

C₄×C₈

—

158

—

152

C₄×C₈

C₂×Dic₇₉

yes

—

153

C₄×C₈

C₂²⋊D₇₉

yes

characteristic

154

C₄×C₈

C₂²⋊D₇₉

yes

characteristic

155

#95

—

158

—

156

#95

—

158

—

157

#41

—

158

C₂×Q₈

—

159

C₂×Q₈

—

160

C₂×D₄

—

161

C₂×D₄

—

162

C₂×D₄

—

163

C₂²×C₄

—

164

C₂²×C₄

—

316

—

165

C₂²×C₄

D₄×D₇₉

yes

—

166

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

167

C₂×C₈

—

168

C₂×C₈

—

158

—

169

C₂×C₈

C₂²⋊Dic₇₉

yes

—

170

C₂×C₈

C₂²⋊Dic₇₉

yes

—

171

C₄⋊C₄

—

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₄⋊C₄

—

175

C₄⋊C₄

—

316

—

176

C₂²⋊C₄

—

177

C₂²⋊C₄

—

178

C₄²

—

179

C₄²

C₂²×Dic₇₉

yes

characteristic

180

C₄²

C₂×C₂²⋊D₇₉

yes

characteristic

181

C₄²

#11929623

yes

—

182

C₂³

—

183

C₂³

#14579157

yes

characteristic

184

Q₈

—

185

Q₈

—

186

D₄

—

187

D₄

—

188

D₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

632

—

194

C₂×C₄

—

632

—

195

C₂×C₄

C₂×C₂²⋊Dic₇₉

yes

characteristic

196

C₂×C₄

D₄×Dic₇₉

yes

—

197

C₂×C₄

#14579132

yes

—

198

C₂×C₄

#14579148

yes

characteristic

199

C₂×C₄

#14579181

yes

—

200

C₈

—

201

C₈

—

202

C₈

—

316

—

203

C₂²

—

204

C₂²

—

205

C₂²

—

206

C₂²

#19098466

yes

characteristic

207

C₂²

#19098945

yes

—

208

C₄

—

209

C₄

—

210

C₄

—

211

C₄

—

212

C₄

—

1264

—

213

C₄

#19098559

yes

—

214

C₄

#19098917

yes

—

215

C₂

—

216

C₂

—

217

C₂

#23764212

yes

characteristic

218

C₂

#23764299

yes

—

	subgroup	G/H
center	C₂²	#19098466
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#3131	C₂
socle	C₂²×C₇₉	#390
prime radical	#532×C₇₉	C₂²
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#3131	D₇₉
hypocenter	C₇₉	#9757
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3131

subgroup

G/H

center

C₂²

#19098466

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#3131

C₂

socle

C₂²×C₇₉

#390

prime radical

#532×C₇₉

C₂²

cosocle

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3131

N	H ≅ G/N	type
1. #23763531	C₂	semidirect
2. #23761964	C₂	semidirect
4. #23761683	C₂	semidirect
17. #487×C₇₉	C₄	semidirect
18. #486×C₇₉	C₄	semidirect
115. C₇₉	#9757	semidirect
120. #487	Dic₇₉	semidirect
121. #486	Dic₇₉	semidirect

H ≅ G/N

type

1. #23763531

C₂

semidirect

2. #23761964

C₂

semidirect

4. #23761683

C₂

semidirect

17. #487×C₇₉

C₄

semidirect

18. #486×C₇₉

C₄

semidirect

115. C₇₉

#9757

semidirect

120. #487

Dic₇₉

semidirect

121. #486

Dic₇₉

semidirect

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

subgroup	index
#23763531	2
#23761964	2
#23761960	2
#23761683	2
C₇₉×#3131	2
#9757	79

subgroup

index

C₇₉×#3131

Subgroups — #34187653

Conjugacy classes of subgroups

Complements

Maximal subgroups