Group #23772926 — subgroups

Subgroups — #23772926

Order 10112.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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×#14579191	C₂	1	1	yes	maximal, characteristic
2	C₂²×#11929614	C₂	1	1	yes	maximal, characteristic, complemented
3	#19099040	C₂	1	1	yes	maximal, characteristic, complemented
4	#19098782	C₂	1	4	yes	maximal, complemented
5	#19098770	C₂	1	4	yes	maximal, complemented
6	#19098590	C₂	1	1	yes	maximal, characteristic, complemented
7	#19098572	C₂	1	1	yes	maximal, characteristic, complemented
8	#19098559	C₂	1	1	yes	maximal, characteristic
9	#536×C₇₉	C₂	1	1	yes	maximal, characteristic
10	C₂²×Dic₁₅₈	C₂²	1	1	yes	characteristic
11	#14579191	—	2	4	no	—
12	C₂×#11929614	—	2	4	no	—
13	C₂×#11929614	C₂²	1	2	yes	—
14	C₂×#11929614	C₂²	1	4	yes	—
15	C₂×#11929612	C₂²	1	2	yes	—
16	C₂×#11929608	C₂²	1	1	yes	characteristic
17	C₂²×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
18	C₄×Dic₁₅₈	C₂²	1	4	yes	—
19	#14579126	—	2	2	no	—
20	#14579120	C₂²	1	1	yes	characteristic
21	#14579117	—	2	2	no	—
22	C₄⋊(C₇₉⋊C₈)	C₂²	1	4	yes	—
23	#14579085	C₂²	1	2	yes	—
24	#14579085	C₂²	1	4	yes	—
25	#14579082	C₂²	1	2	yes	—
26	C₂×#41×C₇₉	C₂²	1	1	yes	characteristic
27	C₂²×Q₈×C₇₉	C₂²	1	1	yes	characteristic
28	C₄⋊Q₈×C₇₉	C₂²	1	2	yes	—
29	#124×C₇₉	C₂²	1	2	yes	—
30	#123×C₇₉	—	2	2	no	—
31	C₂²⋊Q₈×C₇₉	—	2	2	no	—
32	#117×C₇₉	C₂²	1	1	yes	characteristic
33	C₂×Dic₁₅₈	—	2	1	no	—
34	C₂×Dic₁₅₈	—	2	2	no	—
35	C₂×Dic₁₅₈	D₄	1	4	yes	—
36	C₂×Dic₁₅₈	C₂³	1	2	yes	—
37	C₂²×Dic₇₉	—	2	1	no	—
38	#11929614	—	2	8	no	—
39	#11929614	—	2	4	no	—
40	#11929612	—	2	4	no	—
41	#11929608	—	2	2	no	—
42	C₂×C₇₉⋊C₈	—	2	2	no	—
43	C₂×C₇₉⋊C₈	C₂³	1	2	yes	—
44	C₂×C₇₉⋊C₈	C₂³	1	2	yes	—
45	C₄⋊Dic₇₉	C₂³	1	2	yes	—
46	Dic₇₉⋊C₄	—	2	4	no	—
47	C₂²⋊Dic₇₉	—	2	1	no	—
48	C₄×Dic₇₉	—	2	4	no	—
49	C₄×Dic₇₉	—	2	1	no	—
50	#41×C₇₉	—	2	2	no	—
51	#41×C₇₉	—	2	2	no	—
52	C₂×Q₈×C₇₉	—	2	1	no	—
53	C₂×Q₈×C₇₉	—	2	2	no	—
54	C₂×Q₈×C₇₉	—	2	2	no	—
55	C₂×Q₈×C₇₉	C₂³	1	2	yes	—
56	C₂×Q₈×C₇₉	C₂³	1	1	yes	characteristic
57	C₂×D₄×C₇₉	—	2	1	no	—
58	C₂×D₄×C₇₉	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₇₉	—	2	1	no	—
60	C₂²×C₄×C₇₉	—	2	1	no	—
61	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
62	C₄⋊C₄×C₇₉	—	2	4	no	—
63	C₄⋊C₄×C₇₉	D₄	1	2	yes	—
64	C₂²⋊C₄×C₇₉	—	2	4	no	—
65	C₂²⋊C₄×C₇₉	D₄	1	2	yes	—
66	C₄²×C₇₉	C₂³	1	2	yes	—
67	C₂³×C₇₉	—	2	1	no	—
68	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
69	Q₈×C₇₉	—	2	2	no	—
70	Q₈×C₇₉	—	2	1	no	—
71	Q₈×C₇₉	—	2	2	no	—
72	Q₈×C₇₉	—	2	1	no	—
73	Q₈×C₇₉	—	4	2	no	—
74	D₄×C₇₉	—	2	1	no	—
75	D₄×C₇₉	—	2	1	no	—
76	D₄×C₇₉	—	4	2	no	—
77	C₂×C₄×C₇₉	—	2	2	no	—
78	C₂×C₄×C₇₉	—	2	1	no	—
79	C₂×C₄×C₇₉	—	4	2	no	—
80	C₂×C₄×C₇₉	—	4	2	no	—
81	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
82	C₂×C₄×C₇₉	C₂×D₄	1	4	yes	—
83	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
84	C₂×C₄×C₇₉	#41	1	2	yes	—
85	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
86	C₂×Dic₇₉	—	2	4	no	—
87	C₂×Dic₇₉	—	2	2	no	—
88	C₂×Dic₇₉	—	2	2	no	—
89	Dic₁₅₈	—	2	4	no	—
90	Dic₁₅₈	—	2	2	no	—
91	Dic₁₅₈	C₂×D₄	1	4	yes	—
92	C₇₉⋊C₈	—	2	2	no	—
93	C₇₉⋊C₈	—	2	2	no	—
94	C₂²×C₇₉	—	2	2	no	—
95	C₂²×C₇₉	—	4	1	no	—
96	C₂²×C₇₉	—	4	2	no	—
97	C₂²×C₇₉	#121	1	2	yes	—
98	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
99	C₄×C₇₉	—	2	4	no	—
100	C₄×C₇₉	—	4	1	no	—
101	C₄×C₇₉	—	4	2	no	—
102	C₄×C₇₉	#121	1	2	yes	—
103	C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
104	C₄×C₇₉	C₂×#41	1	1	yes	characteristic
105	Dic₇₉	—	2	4	no	—
106	Dic₇₉	—	4	2	no	—
107	C₂×C₇₉	—	2	2	no	—
108	C₂×C₇₉	—	8	1	no	—
109	C₂×C₇₉	C₂×#121	1	1	yes	characteristic
110	C₂×C₇₉	#578	1	2	yes	—
111	#3044	—	79	1	no	maximal, Sylow
112	C₇₉	#3044	1	1	yes	characteristic, complemented, Sylow
113	C₂×#137	—	79	1	no	—
114	C₂²×Q₁₆	—	79	1	no	—
115	#536	D₇₉	1	1	yes	characteristic
116	#519	—	79	1	no	—
117	#464	—	79	4	no	—
118	#462	—	79	4	no	—
119	#424	—	79	1	no	—
120	#419	—	79	1	no	—
121	#416	—	79	1	no	—
122	C₂×#41	C₂×D₇₉	1	1	yes	characteristic
123	C₂²×Q₈	—	79	1	no	—
124	C₂²×Q₈	C₂×D₇₉	1	1	yes	characteristic
125	#137	—	158	4	no	—
126	C₂×Q₁₆	—	79	4	no	—
127	C₂×Q₁₆	—	79	2	no	—
128	C₂×Q₁₆	—	158	4	no	—
129	C₂×SD₁₆	—	79	2	no	—
130	C₂×M₁₆	—	79	1	no	—
131	C₂²×C₈	—	79	1	no	—
132	C₄⋊Q₈	C₂×D₇₉	1	2	yes	—
133	#124	C₂×D₇₉	1	2	yes	—
134	#123	—	2	2	no	—
135	C₂²⋊Q₈	—	2	2	no	—
136	C₄×Q₈	—	79	4	no	—
137	C₄×Q₈	—	158	2	no	—
138	#117	—	79	1	no	—
139	#117	—	158	2	no	—
140	#117	C₂×D₇₉	1	1	yes	characteristic
141	C₄⋊C₈	—	79	4	no	—
142	#103	—	79	2	no	—
143	#103	—	79	4	no	—
144	#102	—	79	2	no	—
145	#41	—	2	2	no	—
146	#41	—	2	2	no	—
147	C₂×Q₈	—	2	1	no	—
148	C₂×Q₈	—	2	2	no	—
149	C₂×Q₈	—	2	2	no	—
150	C₂×Q₈	—	79	4	no	—
151	C₂×Q₈	—	79	2	no	—
152	C₂×Q₈	—	158	1	no	—
153	C₂×Q₈	—	158	2	no	—
154	C₂×Q₈	C₂²×D₇₉	1	2	yes	—
155	C₂×Q₈	C₂²×D₇₉	1	1	yes	characteristic
156	C₂×D₄	—	2	1	no	—
157	C₂×D₄	C₂²×D₇₉	1	1	yes	characteristic
158	C₂²×C₄	—	2	1	no	—
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	—	158	1	no	—
161	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
162	Q₁₆	—	158	8	no	—
163	Q₁₆	—	158	4	no	—
164	SD₁₆	—	158	4	no	—
165	M₁₆	—	158	2	no	—
166	C₂×C₈	—	79	2	no	—
167	C₂×C₈	—	79	2	no	—
168	C₂×C₈	—	158	2	no	—
169	C₄⋊C₄	—	2	4	no	—
170	C₄⋊C₄	—	79	2	no	—
171	C₄⋊C₄	—	158	4	no	—
172	C₄⋊C₄	C₂²⋊D₇₉	1	2	yes	—
173	C₂²⋊C₄	—	2	4	no	—
174	C₂²⋊C₄	—	158	1	no	—
175	C₂²⋊C₄	C₂²⋊D₇₉	1	2	yes	—
176	C₄²	—	158	1	no	—
177	C₄²	—	158	4	no	—
178	C₄²	C₂²×D₇₉	1	2	yes	—
179	C₂³	—	2	1	no	—
180	C₂³	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
181	Q₈	—	2	2	no	—
182	Q₈	—	2	1	no	—
183	Q₈	—	2	2	no	—
184	Q₈	—	2	1	no	—
185	Q₈	—	4	2	no	—
186	Q₈	—	79	4	no	—
187	Q₈	—	158	2	no	—
188	Q₈	—	158	4	no	—
189	D₄	—	2	1	no	—
190	D₄	—	2	1	no	—
191	D₄	—	4	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	158	2	no	—
197	C₂×C₄	—	158	4	no	—
198	C₂×C₄	—	158	2	no	—
199	C₂×C₄	D₄×D₇₉	1	2	yes	—
200	C₂×C₄	C₂×C₂²⋊D₇₉	1	4	yes	—
201	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
202	C₂×C₄	#11929623	1	2	yes	—
203	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
204	C₈	—	158	2	no	—
205	C₈	—	158	2	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	4	1	no	—
208	C₂²	—	4	2	no	—
209	C₂²	#14579132	1	2	yes	—
210	C₂²	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
211	C₄	—	2	4	no	—
212	C₄	—	4	1	no	—
213	C₄	—	4	2	no	—
214	C₄	—	316	2	no	—
215	C₄	—	158	4	no	—
216	C₄	#14579132	1	2	yes	—
217	C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
218	C₄	C₂×#11929623	1	1	yes	characteristic
219	C₂	—	2	2	no	—
220	C₂	—	8	1	no	—
221	C₂	C₂×#14579132	1	1	yes	characteristic
222	C₂	#19099378	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579191

C₂

yes

maximal, characteristic

C₂²×#11929614

C₂

yes

maximal, characteristic, complemented

#19099040

C₂

yes

maximal, characteristic, complemented

#19098782

C₂

yes

maximal, complemented

#19098770

C₂

yes

maximal, complemented

#19098590

C₂

yes

maximal, characteristic, complemented

#19098572

C₂

yes

maximal, characteristic, complemented

#19098559

C₂

yes

maximal, characteristic

#536×C₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₅₈

C₂²

yes

—

—

C₂²

yes

—

C₂×#11929614

C₂²

yes

—

C₂×#11929612

C₂²

yes

—

C₂×#11929608

C₂²

yes

characteristic

C₂²×C₇₉⋊C₈

C₂²

yes

characteristic

C₄×Dic₁₅₈

C₂²

yes

—

#14579126

—

#14579120

C₂²

yes

characteristic

#14579117

—

C₄⋊(C₇₉⋊C₈)

C₂²

yes

—

#14579085

C₂²

yes

—

#14579085

C₂²

yes

—

#14579082

C₂²

yes

—

C₂×#41×C₇₉

C₂²

yes

characteristic

C₂²×Q₈×C₇₉

C₂²

yes

characteristic

C₄⋊Q₈×C₇₉

C₂²

yes

—

#124×C₇₉

C₂²

yes

—

#123×C₇₉

—

C₂²⋊Q₈×C₇₉

—

#117×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

D₄

yes

—

C₂×Dic₁₅₈

C₂³

yes

—

C₂²×Dic₇₉

—

—

—

—

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

C₂³

yes

—

C₂×C₇₉⋊C₈

C₂³

yes

—

C₄⋊Dic₇₉

C₂³

yes

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

#41×C₇₉

—

#41×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

C₂³

yes

—

C₂×Q₈×C₇₉

C₂³

yes

characteristic

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

C₂³

yes

characteristic

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

D₄

yes

—

C₂²⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

D₄

yes

—

C₄²×C₇₉

C₂³

yes

—

C₂³×C₇₉

—

C₂³×C₇₉

C₂×D₄

yes

characteristic

Q₈×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

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×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

#41

yes

—

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

C₂×D₄

yes

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#121

yes

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₄×C₇₉

—

100

C₄×C₇₉

—

101

C₄×C₇₉

—

102

C₄×C₇₉

#121

yes

—

103

C₄×C₇₉

C₂²×D₄

yes

characteristic

104

C₄×C₇₉

C₂×#41

yes

characteristic

105

Dic₇₉

—

106

Dic₇₉

—

107

C₂×C₇₉

—

108

C₂×C₇₉

—

109

C₂×C₇₉

C₂×#121

yes

characteristic

110

C₂×C₇₉

#578

yes

—

111

#3044

—

maximal, Sylow

112

C₇₉

#3044

yes

characteristic, complemented, Sylow

113

C₂×#137

—

114

C₂²×Q₁₆

—

115

#536

D₇₉

yes

characteristic

116

#519

—

117

#464

—

118

#462

—

119

#424

—

120

#419

—

121

#416

—

122

C₂×#41

C₂×D₇₉

yes

characteristic

123

C₂²×Q₈

—

124

C₂²×Q₈

C₂×D₇₉

yes

characteristic

125

#137

—

158

—

126

C₂×Q₁₆

—

127

C₂×Q₁₆

—

128

C₂×Q₁₆

—

158

—

129

C₂×SD₁₆

—

130

C₂×M₁₆

—

131

C₂²×C₈

—

132

C₄⋊Q₈

C₂×D₇₉

yes

—

133

#124

C₂×D₇₉

yes

—

134

#123

—

135

C₂²⋊Q₈

—

136

C₄×Q₈

—

137

C₄×Q₈

—

158

—

138

#117

—

139

#117

—

158

—

140

#117

C₂×D₇₉

yes

characteristic

141

C₄⋊C₈

—

142

#103

—

143

#103

—

144

#102

—

145

#41

—

146

#41

—

147

C₂×Q₈

—

148

C₂×Q₈

—

149

C₂×Q₈

—

150

C₂×Q₈

—

151

C₂×Q₈

—

152

C₂×Q₈

—

158

—

153

C₂×Q₈

—

158

—

154

C₂×Q₈

C₂²×D₇₉

yes

—

155

C₂×Q₈

C₂²×D₇₉

yes

characteristic

156

C₂×D₄

—

157

C₂×D₄

C₂²×D₇₉

yes

characteristic

158

C₂²×C₄

—

159

C₂²×C₄

—

160

C₂²×C₄

—

158

—

161

C₂²×C₄

C₂²×D₇₉

yes

characteristic

162

Q₁₆

—

158

—

163

Q₁₆

—

158

—

164

SD₁₆

—

158

—

165

M₁₆

—

158

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

—

158

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

171

C₄⋊C₄

—

158

—

172

C₄⋊C₄

C₂²⋊D₇₉

yes

—

173

C₂²⋊C₄

—

174

C₂²⋊C₄

—

158

—

175

C₂²⋊C₄

C₂²⋊D₇₉

yes

—

176

C₄²

—

158

—

177

C₄²

—

158

—

178

C₄²

C₂²×D₇₉

yes

—

179

C₂³

—

180

C₂³

C₂×C₂²⋊D₇₉

yes

characteristic

181

Q₈

—

182

Q₈

—

183

Q₈

—

184

Q₈

—

185

Q₈

—

186

Q₈

—

187

Q₈

—

158

—

188

Q₈

—

158

—

189

D₄

—

190

D₄

—

191

D₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

158

—

197

C₂×C₄

—

158

—

198

C₂×C₄

—

158

—

199

C₂×C₄

D₄×D₇₉

yes

—

200

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

201

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

202

C₂×C₄

#11929623

yes

—

203

C₂×C₄

C₂³×D₇₉

yes

characteristic

204

C₈

—

158

—

205

C₈

—

158

—

206

C₂²

—

207

C₂²

—

208

C₂²

—

209

C₂²

#14579132

yes

—

210

C₂²

C₂²×C₂²⋊D₇₉

yes

characteristic

211

C₄

—

212

C₄

—

213

C₄

—

214

C₄

—

316

—

215

C₄

—

158

—

216

C₄

#14579132

yes

—

217

C₄

C₂×D₄×D₇₉

yes

characteristic

218

C₄

C₂×#11929623

yes

characteristic

219

C₂

—

220

C₂

—

221

C₂

C₂×#14579132

yes

characteristic

222

C₂

#19099378

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₇₉
commutator, cosocle	C₂×C₄×C₇₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	#536×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₄
prime radical	C₂×#41×C₇₉	C₂²
hypercenter	#536	D₇₉
hypocenter	C₇₉	#3044
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #536

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₇₉

commutator, cosocle

C₂×C₄×C₇₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

#536×C₇₉

C₂

socle

C₂²×C₇₉

C₂²×D₄

prime radical

C₂×#41×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ #117 ◁ #536

N	H ≅ G/N	type
2. C₂²×#11929614	C₂	semidirect
3. #19099040	C₂	semidirect
4. #19098782	C₂	semidirect
5. #19098770	C₂	semidirect
6. #19098590	C₂	semidirect
7. #19098572	C₂	semidirect
112. C₇₉	#3044	semidirect

H ≅ G/N

type

2. C₂²×#11929614

C₂

semidirect

3. #19099040

C₂

semidirect

4. #19098782

C₂

semidirect

5. #19098770

C₂

semidirect

6. #19098590

C₂

semidirect

7. #19098572

C₂

semidirect

112. C₇₉

#3044

semidirect

Subgroups — #23772926

Conjugacy classes of subgroups

Complements