Group #34183234 — subgroups

Subgroups — #34183234

Order 20224.

398 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 103 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	#23772506	C₂	1	1	yes	maximal, characteristic, complemented
2	#23762065	C₂	1	1	yes	maximal, characteristic, complemented
3	#23761869	C₂	1	2	yes	maximal
4	#23761299	C₂	1	2	yes	maximal, complemented
5	C₇₉×#3135	C₂	1	1	yes	maximal, characteristic
6	#19098694	—	2	4	no	—
7	#19098624	—	2	2	no	—
8	#19098620	C₂²	1	2	yes	—
9	#19098616	C₂²	1	2	yes	—
10	#19098309	—	2	2	no	—
11	D₄⋊(C₇₉⋊C₈)	—	2	4	no	—
12	#533×C₇₉	C₂²	1	1	yes	characteristic
13	#532×C₇₉	C₂²	1	1	yes	characteristic
14	#491×C₇₉	C₄	1	2	yes	—
15	#490×C₇₉	C₄	1	2	yes	—
16	#489×C₇₉	—	2	2	no	—
17	#416×C₇₉	—	2	2	no	—
18	C₂×#97×C₇₉	C₂²	1	1	yes	characteristic
19	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
20	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
21	C₄⋊(C₇₉⋊C₈)	D₄	1	4	yes	—
22	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
23	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
24	#14579072	—	2	2	no	—
25	#14579072	—	2	2	no	—
26	C₄×C₇₉⋊C₈	—	2	2	no	—
27	C₄×C₇₉⋊C₈	—	2	2	no	—
28	C₂×#41×C₇₉	—	2	1	no	—
29	C₂²×C₈×C₇₉	—	2	1	no	—
30	C₄⋊Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
31	C₄⋊Q₈×C₇₉	D₄	1	2	yes	—
32	#127×C₇₉	C₂×C₄	1	1	yes	characteristic
33	#125×C₇₉	C₂×C₄	1	2	yes	—
34	#124×C₇₉	—	2	1	no	—
35	C₂²⋊Q₈×C₇₉	—	2	2	no	—
36	#121×C₇₉	—	2	2	no	—
37	C₄×Q₈×C₇₉	—	2	2	no	—
38	C₄×D₄×C₇₉	—	2	2	no	—
39	#117×C₇₉	D₄	1	2	yes	—
40	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
41	#103×C₇₉	—	2	4	no	—
42	#102×C₇₉	—	2	4	no	—
43	#97×C₇₉	—	2	1	no	—
44	#97×C₇₉	C₂×C₄	1	2	yes	—
45	C₂×C₇₉⋊C₈	—	2	4	no	—
46	C₂×C₇₉⋊C₈	—	4	2	no	—
47	#41×C₇₉	—	4	4	no	—
48	C₂×Q₈×C₇₉	—	2	2	no	—
49	C₂×Q₈×C₇₉	—	2	1	no	—
50	C₂×D₄×C₇₉	—	2	1	no	—
51	C₂×D₄×C₇₉	—	2	2	no	—
52	C₂×D₄×C₇₉	—	4	1	no	—
53	C₂²×C₄×C₇₉	—	4	1	no	—
54	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₇₉	#41	1	2	yes	—
56	C₂×C₈×C₇₉	—	2	2	no	—
57	C₂×C₈×C₇₉	—	4	2	no	—
58	C₄⋊C₄×C₇₉	—	2	4	no	—
59	C₄⋊C₄×C₇₉	—	2	2	no	—
60	C₄⋊C₄×C₇₉	—	2	2	no	—
61	C₄⋊C₄×C₇₉	C₂²⋊C₄	1	4	yes	—
62	C₂²⋊C₄×C₇₉	—	2	2	no	—
63	C₂²⋊C₄×C₇₉	—	4	2	no	—
64	C₄²×C₇₉	—	2	2	no	—
65	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
66	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₇₉	C₂×D₄	1	2	yes	—
68	C₂³×C₇₉	—	4	1	no	—
69	C₂³×C₇₉	#123	1	1	yes	characteristic
70	Q₈×C₇₉	—	4	2	no	—
71	Q₈×C₇₉	—	8	2	no	—
72	D₄×C₇₉	—	4	4	no	—
73	D₄×C₇₉	—	4	2	no	—
74	D₄×C₇₉	—	8	2	no	—
75	C₂×C₄×C₇₉	—	2	2	no	—
76	C₂×C₄×C₇₉	—	2	4	no	—
77	C₂×C₄×C₇₉	—	4	2	no	—
78	C₂×C₄×C₇₉	—	4	1	no	—
79	C₂×C₄×C₇₉	—	8	2	no	—
80	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
82	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
83	C₂×C₄×C₇₉	#121	1	2	yes	—
84	C₈×C₇₉	—	4	2	no	—
85	C₇₉⋊C₈	—	4	4	no	—
86	C₇₉⋊C₈	—	8	2	no	—
87	C₂²×C₇₉	—	2	2	no	—
88	C₂²×C₇₉	—	2	1	no	—
89	C₂²×C₇₉	—	8	2	no	—
90	C₂²×C₇₉	—	8	1	no	—
91	C₂²×C₇₉	#386	1	1	yes	characteristic
92	C₄×C₇₉	—	2	2	no	—
93	C₄×C₇₉	—	2	2	no	—
94	C₄×C₇₉	—	2	1	no	—
95	C₄×C₇₉	—	4	4	no	—
96	C₄×C₇₉	—	8	1	no	—
97	C₄×C₇₉	#454	1	2	yes	—
98	#9008	—	79	1	no	maximal, Sylow
99	C₂×C₇₉	—	4	1	no	—
100	C₂×C₇₉	—	16	1	no	—
101	C₂×C₇₉	#1881	1	2	yes	—
102	C₂×C₇₉	#1888	1	1	yes	characteristic
103	#3135	D₇₉	1	1	yes	characteristic
104	#2987	—	79	1	no	—
105	#1544	—	79	1	no	—
106	#1496	—	79	2	no	—
107	#1302	—	79	2	no	—
108	C₇₉	#9008	1	1	yes	characteristic, complemented, Sylow
109	#533	C₂×D₇₉	1	1	yes	characteristic
110	#532	C₂×D₇₉	1	1	yes	characteristic
111	#491	Dic₇₉	1	2	yes	—
112	#490	Dic₇₉	1	2	yes	—
113	#489	—	2	2	no	—
114	#446	—	158	4	no	—
115	#433	—	158	2	no	—
116	#432	—	79	2	no	—
117	#431	—	79	2	no	—
118	#416	—	2	2	no	—
119	C₂×#97	C₂×D₇₉	1	1	yes	characteristic
120	#332	—	158	2	no	—
121	D₄⋊C₈	—	158	4	no	—
122	C₂×#41	—	2	1	no	—
123	C₂²×C₈	—	2	1	no	—
124	C₄⋊Q₈	C₂×Dic₇₉	1	1	yes	characteristic
125	C₄⋊Q₈	C₂²⋊D₇₉	1	2	yes	—
126	#127	C₂×Dic₇₉	1	1	yes	characteristic
127	#125	C₂×Dic₇₉	1	2	yes	—
128	#124	—	2	1	no	—
129	C₂²⋊Q₈	—	2	2	no	—
130	#121	—	2	2	no	—
131	C₄×Q₈	—	2	2	no	—
132	C₄×D₄	—	2	2	no	—
133	#117	C₂²⋊D₇₉	1	2	yes	—
134	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
135	C₄⋊C₈	—	79	4	no	—
136	C₄⋊C₈	—	158	4	no	—
137	C₄⋊C₈	—	158	2	no	—
138	#103	—	2	4	no	—
139	#102	—	2	4	no	—
140	C₂²⋊C₈	—	158	2	no	—
141	C₂²⋊C₈	—	158	2	no	—
142	#97	—	2	1	no	—
143	#97	—	158	2	no	—
144	#97	—	158	2	no	—
145	#97	C₂×Dic₇₉	1	2	yes	—
146	C₄×C₈	—	158	2	no	—
147	C₄×C₈	—	158	2	no	—
148	#41	—	4	4	no	—
149	C₂×Q₈	—	2	2	no	—
150	C₂×Q₈	—	2	1	no	—
151	C₂×D₄	—	2	1	no	—
152	C₂×D₄	—	2	2	no	—
153	C₂×D₄	—	4	1	no	—
154	C₂²×C₄	—	4	1	no	—
155	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
156	C₂²×C₄	#11929623	1	2	yes	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	4	2	no	—
159	C₂×C₈	—	316	2	no	—
160	C₂×C₈	—	158	4	no	—
161	C₄⋊C₄	—	2	4	no	—
162	C₄⋊C₄	—	2	2	no	—
163	C₄⋊C₄	—	2	2	no	—
164	C₄⋊C₄	C₂²⋊Dic₇₉	1	4	yes	—
165	C₂²⋊C₄	—	2	2	no	—
166	C₂²⋊C₄	—	4	2	no	—
167	C₄²	—	2	2	no	—
168	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
169	C₄²	D₄×D₇₉	1	2	yes	—
170	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
171	C₂³	—	4	1	no	—
172	C₂³	#14579142	1	1	yes	characteristic
173	Q₈	—	4	2	no	—
174	Q₈	—	8	2	no	—
175	D₄	—	4	4	no	—
176	D₄	—	4	2	no	—
177	D₄	—	8	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	—	4	1	no	—
182	C₂×C₄	—	8	2	no	—
183	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
184	C₂×C₄	D₄×Dic₇₉	1	2	yes	—
185	C₂×C₄	#14579129	1	1	yes	characteristic
186	C₂×C₄	#14579133	1	2	yes	—
187	C₈	—	4	2	no	—
188	C₈	—	316	4	no	—
189	C₈	—	632	2	no	—
190	C₂²	—	2	2	no	—
191	C₂²	—	2	1	no	—
192	C₂²	—	8	2	no	—
193	C₂²	—	8	1	no	—
194	C₂²	#19098443	1	1	yes	characteristic
195	C₄	—	2	2	no	—
196	C₄	—	2	2	no	—
197	C₄	—	2	1	no	—
198	C₄	—	4	4	no	—
199	C₄	—	8	1	no	—
200	C₄	#19098735	1	2	yes	—
201	C₂	—	4	1	no	—
202	C₂	—	16	1	no	—
203	C₂	#23763764	1	2	yes	—
204	C₂	#23763804	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23772506

C₂

yes

maximal, characteristic, complemented

#23762065

C₂

yes

maximal, characteristic, complemented

#23761869

C₂

yes

maximal

#23761299

C₂

yes

maximal, complemented

C₇₉×#3135

C₂

yes

maximal, characteristic

#19098694

—

#19098624

—

#19098620

C₂²

yes

—

#19098616

C₂²

yes

—

#19098309

—

D₄⋊(C₇₉⋊C₈)

—

#533×C₇₉

C₂²

yes

characteristic

#532×C₇₉

C₂²

yes

characteristic

#491×C₇₉

C₄

yes

—

#490×C₇₉

C₄

yes

—

#489×C₇₉

—

#416×C₇₉

—

C₂×#97×C₇₉

C₂²

yes

characteristic

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

D₄

yes

—

C₂²⋊(C₇₉⋊C₈)

—

C₂²⋊(C₇₉⋊C₈)

—

#14579072

—

#14579072

—

C₄×C₇₉⋊C₈

—

C₄×C₇₉⋊C₈

—

C₂×#41×C₇₉

—

C₂²×C₈×C₇₉

—

C₄⋊Q₈×C₇₉

C₂×C₄

yes

characteristic

C₄⋊Q₈×C₇₉

D₄

yes

—

#127×C₇₉

C₂×C₄

yes

characteristic

#125×C₇₉

C₂×C₄

yes

—

#124×C₇₉

—

C₂²⋊Q₈×C₇₉

—

#121×C₇₉

—

C₄×Q₈×C₇₉

—

C₄×D₄×C₇₉

—

#117×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

#103×C₇₉

—

#102×C₇₉

—

#97×C₇₉

—

#97×C₇₉

C₂×C₄

yes

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

#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

characteristic

C₂²×C₄×C₇₉

#41

yes

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

C₂²⋊C₄

yes

—

C₂²⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

—

C₂³×C₇₉

—

C₂³×C₇₉

#123

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×C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄×D₄

yes

—

C₂×C₄×C₇₉

#120

yes

characteristic

C₂×C₄×C₇₉

#121

yes

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#386

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#454

yes

—

#9008

—

maximal, Sylow

C₂×C₇₉

—

100

C₂×C₇₉

—

101

C₂×C₇₉

#1881

yes

—

102

C₂×C₇₉

#1888

yes

characteristic

103

#3135

D₇₉

yes

characteristic

104

#2987

—

105

#1544

—

106

#1496

—

107

#1302

—

108

C₇₉

#9008

yes

characteristic, complemented, Sylow

109

#533

C₂×D₇₉

yes

characteristic

110

#532

C₂×D₇₉

yes

characteristic

111

#491

Dic₇₉

yes

—

112

#490

Dic₇₉

yes

—

113

#489

—

114

#446

—

158

—

115

#433

—

158

—

116

#432

—

117

#431

—

118

#416

—

119

C₂×#97

C₂×D₇₉

yes

characteristic

120

#332

—

158

—

121

D₄⋊C₈

—

158

—

122

C₂×#41

—

123

C₂²×C₈

—

124

C₄⋊Q₈

C₂×Dic₇₉

yes

characteristic

125

C₄⋊Q₈

C₂²⋊D₇₉

yes

—

126

#127

C₂×Dic₇₉

yes

characteristic

127

#125

C₂×Dic₇₉

yes

—

128

#124

—

129

C₂²⋊Q₈

—

130

#121

—

131

C₄×Q₈

—

132

C₄×D₄

—

133

#117

C₂²⋊D₇₉

yes

—

134

C₂×C₄²

C₂²×D₇₉

yes

characteristic

135

C₄⋊C₈

—

136

C₄⋊C₈

—

158

—

137

C₄⋊C₈

—

158

—

138

#103

—

139

#102

—

140

C₂²⋊C₈

—

158

—

141

C₂²⋊C₈

—

158

—

142

#97

—

143

#97

—

158

—

144

#97

—

158

—

145

#97

C₂×Dic₇₉

yes

—

146

C₄×C₈

—

158

—

147

C₄×C₈

—

158

—

148

#41

—

149

C₂×Q₈

—

150

C₂×Q₈

—

151

C₂×D₄

—

152

C₂×D₄

—

153

C₂×D₄

—

154

C₂²×C₄

—

155

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

156

C₂²×C₄

#11929623

yes

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

316

—

160

C₂×C₈

—

158

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄⋊C₄

C₂²⋊Dic₇₉

yes

—

165

C₂²⋊C₄

—

166

C₂²⋊C₄

—

167

C₄²

—

168

C₄²

C₂²×Dic₇₉

yes

characteristic

169

C₄²

D₄×D₇₉

yes

—

170

C₄²

C₂×C₂²⋊D₇₉

yes

characteristic

171

C₂³

—

172

C₂³

#14579142

yes

characteristic

173

Q₈

—

174

Q₈

—

175

D₄

—

176

D₄

—

177

D₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

C₂×C₂²⋊Dic₇₉

yes

characteristic

184

C₂×C₄

D₄×Dic₇₉

yes

—

185

C₂×C₄

#14579129

yes

characteristic

186

C₂×C₄

#14579133

yes

—

187

C₈

—

188

C₈

—

316

—

189

C₈

—

632

—

190

C₂²

—

191

C₂²

—

192

C₂²

—

193

C₂²

—

194

C₂²

#19098443

yes

characteristic

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

#19098735

yes

—

201

C₂

—

202

C₂

—

203

C₂

#23763764

yes

—

204

C₂

#23763804

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14579129
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#3135	C₂
socle	C₂²×C₇₉	#386
prime radical	#532×C₇₉	C₂²
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#3135	D₇₉
hypocenter	C₇₉	#9008
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3135

subgroup

G/H

center

C₂×C₄

#14579129

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#3135

C₂

socle

C₂²×C₇₉

#386

prime radical

#532×C₇₉

C₂²

cosocle

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3135

N	H ≅ G/N	type
1. #23772506	C₂	semidirect
2. #23762065	C₂	semidirect
4. #23761299	C₂	semidirect
108. C₇₉	#9008	semidirect

H ≅ G/N

type

1. #23772506

C₂

semidirect

2. #23762065

C₂

semidirect

4. #23761299

C₂

semidirect

108. C₇₉

#9008

semidirect

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

subgroup	index
#23772506	2
#23762065	2
#23761869	2
#23761299	2
C₇₉×#3135	2
#9008	79

subgroup

index

C₇₉×#3135

Subgroups — #34183234

Conjugacy classes of subgroups

Complements

Maximal subgroups