Group #23764741 — subgroups

Subgroups — #23764741

Order 10112.

250 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 91 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×#14579152	C₂	1	1	yes	maximal, characteristic
2	C₂×#14579095	C₂	1	1	yes	maximal, characteristic
3	C₂×#14579092	C₂	1	1	yes	maximal, characteristic
4	#19098462	C₂	1	1	yes	maximal, characteristic
5	#19098325	C₂	1	1	yes	maximal, characteristic
6	#19098317	C₂	1	1	yes	maximal, characteristic
7	#340×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
9	#14579152	—	2	2	no	—
10	#14579152	—	2	2	no	—
11	C₂×C₄⋊Dic₇₉	—	2	1	no	—
12	C₂×C₄⋊Dic₇₉	—	2	1	no	—
13	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₇₉⋊C₄	—	2	1	no	—
16	C₂×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
17	#14579095	—	2	2	no	—
18	#14579092	—	2	2	no	—
19	C₇₉⋊#95	—	2	1	no	—
20	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
23	C₂²×Dic₇₉	—	2	1	no	—
24	C₂²×Dic₇₉	—	2	1	no	—
25	C₂²×Dic₇₉	D₄	1	1	yes	characteristic
26	C₂²×Dic₇₉	D₄	1	1	yes	characteristic
27	C₂×C₇₉⋊C₈	—	2	1	no	—
28	C₂×C₇₉⋊C₈	—	2	1	no	—
29	C₂×C₇₉⋊C₈	Q₈	1	2	yes	—
30	C₄⋊Dic₇₉	—	2	4	no	—
31	C₄⋊Dic₇₉	—	2	1	no	—
32	C₄⋊Dic₇₉	—	2	1	no	—
33	C₄⋊Dic₇₉	—	2	1	no	—
34	C₄⋊Dic₇₉	—	4	2	no	—
35	C₄⋊Dic₇₉	D₄	1	2	yes	—
36	Dic₇₉⋊C₄	—	2	4	no	—
37	C₄×Dic₇₉	—	2	1	no	—
38	C₄×Dic₇₉	—	2	1	no	—
39	C₂²×C₄×C₇₉	—	2	1	no	—
40	C₂²×C₄×C₇₉	—	2	1	no	—
41	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
42	C₂×C₈×C₇₉	—	2	1	no	—
43	C₂×C₈×C₇₉	—	2	1	no	—
44	C₂×C₈×C₇₉	Q₈	1	2	yes	—
45	C₄⋊C₄×C₇₉	—	2	1	no	—
46	C₄⋊C₄×C₇₉	—	2	1	no	—
47	C₄⋊C₄×C₇₉	D₄	1	2	yes	—
48	C₄⋊C₄×C₇₉	Q₈	1	2	yes	—
49	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₇₉	—	2	2	no	—
51	C₂×C₄×C₇₉	—	2	2	no	—
52	C₂×C₄×C₇₉	—	4	1	no	—
53	C₂×C₄×C₇₉	—	4	1	no	—
54	C₂×C₄×C₇₉	—	4	1	no	—
55	C₂×C₄×C₇₉	—	4	1	no	—
56	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
62	C₂×Dic₇₉	—	2	1	no	—
63	C₂×Dic₇₉	—	2	1	no	—
64	C₂×Dic₇₉	—	2	2	no	—
65	C₂×Dic₇₉	—	2	1	no	—
66	C₂×Dic₇₉	—	2	1	no	—
67	C₂×Dic₇₉	—	2	1	no	—
68	C₂×Dic₇₉	—	2	1	no	—
69	C₂×Dic₇₉	—	4	1	no	—
70	C₂×Dic₇₉	—	4	1	no	—
71	C₂×Dic₇₉	—	4	1	no	—
72	C₂×Dic₇₉	—	4	1	no	—
73	C₂×Dic₇₉	—	4	1	no	—
74	C₈×C₇₉	—	2	2	no	—
75	C₇₉⋊C₈	—	2	2	no	—
76	C₂²×C₇₉	#120	1	1	yes	characteristic
77	C₂²×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
78	C₂²×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
79	C₂²×C₇₉	#135	1	1	yes	characteristic
80	C₂²×C₇₉	#135	1	1	yes	characteristic
81	C₂²×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
82	C₂²×C₇₉	#137	1	1	yes	characteristic
83	C₄×C₇₉	—	4	2	no	—
84	C₄×C₇₉	—	4	2	no	—
85	C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
86	C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
87	C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
88	C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
89	Dic₇₉	—	4	1	no	—
90	Dic₇₉	—	4	2	no	—
91	Dic₇₉	—	4	1	no	—
92	Dic₇₉	—	8	1	no	—
93	C₂×C₇₉	#395	1	1	yes	characteristic
94	C₂×C₇₉	#449	1	1	yes	characteristic
95	C₂×C₇₉	#452	1	1	yes	characteristic
96	C₂×C₇₉	#478	1	1	yes	characteristic
97	C₂×C₇₉	#479	1	1	yes	characteristic
98	C₂×C₇₉	#499	1	1	yes	characteristic
99	C₂×C₇₉	#501	1	1	yes	characteristic
100	#2059	—	79	1	no	maximal, Sylow
101	C₇₉	#2059	1	1	yes	characteristic, complemented, Sylow
102	C₂×#125	—	79	1	no	—
103	C₂×#107	—	79	1	no	—
104	C₂×#106	—	79	1	no	—
105	#389	—	79	1	no	—
106	#340	—	79	1	no	—
107	#340	D₇₉	1	1	yes	characteristic
108	#336	—	79	1	no	—
109	C₂²×C₈	—	79	1	no	—
110	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
111	#125	—	158	2	no	—
112	#125	—	158	2	no	—
113	C₂×C₄⋊C₄	—	79	1	no	—
114	C₂×C₄⋊C₄	—	79	1	no	—
115	C₂×C₄⋊C₄	—	158	1	no	—
116	C₂×C₄⋊C₄	—	158	1	no	—
117	C₂×C₄⋊C₄	—	158	1	no	—
118	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
119	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
120	C₂×C₄²	—	79	1	no	—
121	#107	—	158	2	no	—
122	#106	—	158	2	no	—
123	#95	—	158	1	no	—
124	C₂²×C₄	—	2	1	no	—
125	C₂²×C₄	—	2	1	no	—
126	C₂²×C₄	—	79	1	no	—
127	C₂²×C₄	—	79	1	no	—
128	C₂²×C₄	—	158	1	no	—
129	C₂²×C₄	—	158	1	no	—
130	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	79	2	no	—
134	C₂×C₈	—	158	1	no	—
135	C₂×C₈	—	158	1	no	—
136	C₂×C₈	Dic₁₅₈	1	2	yes	—
137	C₄⋊C₄	—	2	1	no	—
138	C₄⋊C₄	—	2	1	no	—
139	C₄⋊C₄	—	316	2	no	—
140	C₄⋊C₄	—	79	2	no	—
141	C₄⋊C₄	—	158	4	no	—
142	C₄⋊C₄	—	158	4	no	—
143	C₄⋊C₄	—	158	1	no	—
144	C₄⋊C₄	—	158	1	no	—
145	C₄⋊C₄	—	158	1	no	—
146	C₄⋊C₄	Dic₁₅₈	1	2	yes	—
147	C₄⋊C₄	D₃₁₆	1	2	yes	—
148	C₄²	—	158	1	no	—
149	C₄²	—	158	1	no	—
150	C₂³	D₄×D₇₉	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	316	1	no	—
158	C₂×C₄	—	316	1	no	—
159	C₂×C₄	—	316	1	no	—
160	C₂×C₄	—	316	1	no	—
161	C₂×C₄	—	316	1	no	—
162	C₂×C₄	—	158	1	no	—
163	C₂×C₄	—	158	1	no	—
164	C₂×C₄	—	158	2	no	—
165	C₂×C₄	—	158	1	no	—
166	C₂×C₄	—	158	1	no	—
167	C₂×C₄	—	158	1	no	—
168	C₂×C₄	—	158	1	no	—
169	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
170	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
171	C₂×C₄	Q₈×D₇₉	1	1	yes	characteristic
172	C₂×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
173	C₂×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
174	C₂×C₄	#11929623	1	1	yes	characteristic
175	C₈	—	2	2	no	—
176	C₈	—	158	2	no	—
177	C₂²	#14579131	1	1	yes	characteristic
178	C₂²	#14579138	1	1	yes	characteristic
179	C₂²	#14579160	1	1	yes	characteristic
180	C₂²	#14579180	1	1	yes	characteristic
181	C₂²	#14579184	1	1	yes	characteristic
182	C₂²	#14579188	1	1	yes	characteristic
183	C₂²	#14579190	1	1	yes	characteristic
184	C₄	—	4	2	no	—
185	C₄	—	4	2	no	—
186	C₄	—	316	2	no	—
187	C₄	—	316	1	no	—
188	C₄	—	316	1	no	—
189	C₄	—	632	1	no	—
190	C₄	#14579138	1	1	yes	characteristic
191	C₄	#14579141	1	1	yes	characteristic
192	C₄	#14579160	1	1	yes	characteristic
193	C₄	#14579161	1	1	yes	characteristic
194	C₂	#19098484	1	1	yes	characteristic
195	C₂	#19098711	1	1	yes	characteristic
196	C₂	#19098726	1	1	yes	characteristic
197	C₂	#19098861	1	1	yes	characteristic
198	C₂	#19098872	1	1	yes	characteristic
199	C₂	#19098964	1	1	yes	characteristic
200	C₂	#19098970	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579152

C₂

yes

maximal, characteristic

C₂×#14579095

C₂

yes

maximal, characteristic

C₂×#14579092

C₂

yes

maximal, characteristic

#19098462

C₂

yes

maximal, characteristic

#19098325

C₂

yes

maximal, characteristic

#19098317

C₂

yes

maximal, characteristic

#340×C₇₉

C₂

yes

maximal, characteristic

C₂²×C₇₉⋊C₈

C₂²

yes

characteristic

#14579152

—

#14579152

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₉

C₂²

yes

characteristic

C₂×Dic₇₉⋊C₄

—

C₂×C₄×Dic₇₉

C₂²

yes

characteristic

#14579095

—

#14579092

—

C₇₉⋊#95

—

C₂²×C₈×C₇₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₉

C₂²

yes

characteristic

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

D₄

yes

characteristic

C₂²×Dic₇₉

D₄

yes

characteristic

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

Q₈

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

D₄

yes

—

Dic₇₉⋊C₄

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

Q₈

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

D₄

yes

—

C₄⋊C₄×C₇₉

Q₈

yes

—

C₂³×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₉

#41

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

#120

yes

characteristic

C₂²×C₇₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₇₉

C₄⋊Q₈

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂²⋊Q₈

yes

characteristic

C₄×C₇₉

C₂²⋊Q₈

yes

characteristic

C₄×C₇₉

C₄⋊Q₈

yes

characteristic

C₄×C₇₉

C₄⋊Q₈

yes

characteristic

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

100

—

101

C₇₉

#2059

yes

characteristic, complemented, Sylow

102

C₂×#125

—

103

C₂×#107

—

104

C₂×#106

—

105

#389

—

106

#340

—

107

#340

D₇₉

yes

characteristic

108

#336

—

109

C₂²×C₈

—

110

C₂²×C₈

C₂×D₇₉

yes

characteristic

111

#125

—

158

—

112

#125

—

158

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

158

—

116

C₂×C₄⋊C₄

—

158

—

117

C₂×C₄⋊C₄

—

158

—

118

C₂×C₄⋊C₄

C₂×D₇₉

yes

characteristic

119

C₂×C₄⋊C₄

C₂×D₇₉

yes

characteristic

120

C₂×C₄²

—

121

#107

—

158

—

122

#106

—

158

—

123

#95

—

158

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂²×C₄

—

158

—

129

C₂²×C₄

—

158

—

130

C₂²×C₄

C₂²×D₇₉

yes

characteristic

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

158

—

135

C₂×C₈

—

158

—

136

C₂×C₈

Dic₁₅₈

yes

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

316

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

158

—

142

C₄⋊C₄

—

158

—

143

C₄⋊C₄

—

158

—

144

C₄⋊C₄

—

158

—

145

C₄⋊C₄

—

158

—

146

C₄⋊C₄

Dic₁₅₈

yes

—

147

C₄⋊C₄

D₃₁₆

yes

—

148

C₄²

—

158

—

149

C₄²

—

158

—

150

C₂³

D₄×D₇₉

yes

characteristic

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

316

—

158

C₂×C₄

—

316

—

159

C₂×C₄

—

316

—

160

C₂×C₄

—

316

—

161

C₂×C₄

—

316

—

162

C₂×C₄

—

158

—

163

C₂×C₄

—

158

—

164

C₂×C₄

—

158

—

165

C₂×C₄

—

158

—

166

C₂×C₄

—

158

—

167

C₂×C₄

—

158

—

168

C₂×C₄

—

158

—

169

C₂×C₄

D₄×D₇₉

yes

characteristic

170

C₂×C₄

C₂×D₃₁₆

yes

characteristic

171

C₂×C₄

Q₈×D₇₉

yes

characteristic

172

C₂×C₄

C₂×Dic₁₅₈

yes

characteristic

173

C₂×C₄

C₂×Dic₁₅₈

yes

characteristic

174

C₂×C₄

#11929623

yes

characteristic

175

C₈

—

176

C₈

—

158

—

177

C₂²

#14579131

yes

characteristic

178

C₂²

#14579138

yes

characteristic

179

C₂²

#14579160

yes

characteristic

180

C₂²

#14579180

yes

characteristic

181

C₂²

#14579184

yes

characteristic

182

C₂²

#14579188

yes

characteristic

183

C₂²

#14579190

yes

characteristic

184

C₄

—

185

C₄

—

186

C₄

—

316

—

187

C₄

—

316

—

188

C₄

—

316

—

189

C₄

—

632

—

190

C₄

#14579138

yes

characteristic

191

C₄

#14579141

yes

characteristic

192

C₄

#14579160

yes

characteristic

193

C₄

#14579161

yes

characteristic

194

C₂

#19098484

yes

characteristic

195

C₂

#19098711

yes

characteristic

196

C₂

#19098726

yes

characteristic

197

C₂

#19098861

yes

characteristic

198

C₂

#19098872

yes

characteristic

199

C₂

#19098964

yes

characteristic

200

C₂

#19098970

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₇₉
commutator, cosocle	C₂²×C₄×C₇₉	C₂³
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	#340×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×D₄
hypercenter	#340	D₇₉
hypocenter	C₇₉	#2059
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₇₉

commutator, cosocle

C₂²×C₄×C₇₉

C₂³

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

#340×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

C₂×D₄

D₇₉

C₇₉

G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
101. C₇₉	#2059	semidirect

H ≅ G/N

type

101. C₇₉

#2059

semidirect

Subgroups — #23764741

Conjugacy classes of subgroups

Complements