Group #23764162 — subgroups

Subgroups — #23764162

Order 10112.

256 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 107 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×#14579093	C₂	1	1	yes	maximal, characteristic
2	C₂×#14579092	C₂	1	1	yes	maximal, characteristic
3	C₂×Dic₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#19098432	C₂	1	1	yes	maximal, characteristic
5	C₄⋊C₄×Dic₇₉	C₂	1	1	yes	maximal, characteristic
6	#19098325	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	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
11	C₂×Dic₇₉⋊C₄	—	2	1	no	—
12	C₂×Dic₇₉⋊C₄	—	2	1	no	—
13	C₂×C₄×Dic₇₉	—	2	1	no	—
14	C₂×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
15	#14579093	—	2	2	no	—
16	#14579092	—	2	2	no	—
17	Dic₇₉⋊C₈	C₄	1	4	yes	complemented
18	C₇₉⋊#95	—	2	1	no	—
19	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
22	C₂²×Dic₇₉	—	2	1	no	—
23	C₂²×Dic₇₉	—	2	1	no	—
24	C₂²×Dic₇₉	D₄	1	1	yes	characteristic
25	C₂²×Dic₇₉	D₄	1	1	yes	characteristic
26	C₂×C₇₉⋊C₈	—	2	1	no	—
27	C₂×C₇₉⋊C₈	—	2	1	no	—
28	C₂×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
29	C₄⋊Dic₇₉	—	2	1	no	—
30	C₄⋊Dic₇₉	—	2	1	no	—
31	C₄⋊Dic₇₉	—	2	1	no	—
32	C₄⋊Dic₇₉	D₄	1	2	yes	—
33	Dic₇₉⋊C₄	—	4	2	no	—
34	Dic₇₉⋊C₄	—	4	2	no	—
35	C₄×Dic₇₉	—	4	2	no	—
36	C₄×Dic₇₉	C₂×C₄	1	2	yes	—
37	C₄×Dic₇₉	Q₈	1	2	yes	—
38	C₂²×C₄×C₇₉	—	2	1	no	—
39	C₂²×C₄×C₇₉	—	2	1	no	—
40	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
41	C₂×C₈×C₇₉	—	2	1	no	—
42	C₂×C₈×C₇₉	—	2	1	no	—
43	C₂×C₈×C₇₉	C₂×C₄	1	2	yes	—
44	C₄⋊C₄×C₇₉	—	2	1	no	—
45	C₄⋊C₄×C₇₉	—	2	1	no	—
46	C₄⋊C₄×C₇₉	—	2	1	no	—
47	C₄⋊C₄×C₇₉	Q₈	1	2	yes	—
48	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₇₉	—	2	2	no	—
50	C₂×C₄×C₇₉	—	4	1	no	—
51	C₂×C₄×C₇₉	—	4	1	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	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
60	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
61	C₂×Dic₇₉	—	2	1	no	—
62	C₂×Dic₇₉	—	2	1	no	—
63	C₂×Dic₇₉	—	2	2	no	—
64	C₂×Dic₇₉	—	4	1	no	—
65	C₂×Dic₇₉	—	4	1	no	—
66	C₂×Dic₇₉	—	4	1	no	—
67	C₂×Dic₇₉	—	4	1	no	—
68	C₂×Dic₇₉	—	4	1	no	—
69	C₂×Dic₇₉	C₄⋊C₄	1	2	yes	—
70	C₂×Dic₇₉	C₄⋊C₄	1	2	yes	—
71	C₂×Dic₇₉	SD₁₆	1	2	yes	—
72	C₂×Dic₇₉	SD₁₆	1	2	yes	—
73	C₈×C₇₉	—	2	2	no	—
74	C₇₉⋊C₈	—	2	2	no	—
75	C₂²×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₇₉	#121	1	1	yes	characteristic
77	C₂²×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
78	C₂²×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
79	C₂²×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
80	C₂²×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₂²×C₇₉	#137	1	1	yes	characteristic
82	C₄×C₇₉	—	4	2	no	—
83	C₄×C₇₉	—	8	1	no	—
84	C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
85	C₄×C₇₉	C₄×Q₈	1	1	yes	characteristic
86	C₄×C₇₉	#125	1	1	yes	characteristic
87	C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
88	Dic₇₉	—	2	2	no	—
89	Dic₇₉	—	4	2	no	—
90	Dic₇₉	—	8	1	no	—
91	Dic₇₉	#106	1	4	yes	complemented
92	C₂×C₇₉	#389	1	1	yes	characteristic
93	C₂×C₇₉	C₂×#106	1	1	yes	characteristic
94	C₂×C₇₉	#428	1	1	yes	characteristic
95	C₂×C₇₉	#460	1	1	yes	characteristic
96	C₂×C₇₉	#461	1	1	yes	characteristic
97	C₂×C₇₉	#475	1	1	yes	characteristic
98	C₂×C₇₉	#476	1	1	yes	characteristic
99	#1948	—	79	1	no	maximal, Sylow
100	C₇₉	#1948	1	1	yes	characteristic, complemented, Sylow
101	C₂×#106	—	79	1	no	—
102	C₂×#106	—	79	1	no	—
103	C₂×C₄⋊C₈	—	79	1	no	—
104	#384	—	79	1	no	—
105	C₄×C₄⋊C₄	—	79	1	no	—
106	#340	—	79	1	no	—
107	#340	D₇₉	1	1	yes	characteristic
108	C₂²×C₈	—	79	1	no	—
109	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
110	C₂×C₄⋊C₄	—	79	1	no	—
111	C₂×C₄⋊C₄	—	79	1	no	—
112	C₂×C₄⋊C₄	—	158	1	no	—
113	C₂×C₄⋊C₄	—	158	1	no	—
114	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
115	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
116	C₂×C₄²	—	79	1	no	—
117	C₂×C₄²	—	158	1	no	—
118	#106	—	158	2	no	—
119	#106	—	158	2	no	—
120	C₄⋊C₈	—	79	4	no	—
121	#95	—	158	1	no	—
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	—	2	1	no	—
124	C₂²×C₄	—	79	1	no	—
125	C₂²×C₄	—	79	1	no	—
126	C₂²×C₄	—	158	1	no	—
127	C₂²×C₄	—	158	1	no	—
128	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	79	2	no	—
132	C₂×C₈	—	158	1	no	—
133	C₂×C₈	—	158	1	no	—
134	C₂×C₈	C₄×D₇₉	1	2	yes	—
135	C₄⋊C₄	—	2	1	no	—
136	C₄⋊C₄	—	2	1	no	—
137	C₄⋊C₄	—	2	1	no	—
138	C₄⋊C₄	—	316	2	no	—
139	C₄⋊C₄	—	316	2	no	—
140	C₄⋊C₄	—	79	2	no	—
141	C₄⋊C₄	—	158	1	no	—
142	C₄⋊C₄	—	158	1	no	—
143	C₄⋊C₄	—	158	1	no	—
144	C₄⋊C₄	Dic₁₅₈	1	2	yes	—
145	C₄²	—	316	2	no	—
146	C₄²	—	79	2	no	—
147	C₄²	—	79	2	no	—
148	C₂³	D₄×D₇₉	1	1	yes	characteristic
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	316	1	no	—
156	C₂×C₄	—	316	1	no	—
157	C₂×C₄	—	316	1	no	—
158	C₂×C₄	—	316	1	no	—
159	C₂×C₄	—	316	1	no	—
160	C₂×C₄	—	79	2	no	—
161	C₂×C₄	—	79	2	no	—
162	C₂×C₄	—	79	2	no	—
163	C₂×C₄	—	79	2	no	—
164	C₂×C₄	—	158	2	no	—
165	C₂×C₄	—	158	1	no	—
166	C₂×C₄	—	158	1	no	—
167	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
168	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
169	C₂×C₄	Q₈×D₇₉	1	1	yes	characteristic
170	C₂×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
171	C₂×C₄	#11929623	1	1	yes	characteristic
172	C₂×C₄	#11929624	1	1	yes	characteristic
173	C₈	—	2	2	no	—
174	C₈	—	158	2	no	—
175	C₂²	C₄⋊C₄×D₇₉	1	1	yes	characteristic
176	C₂²	#14579134	1	1	yes	characteristic
177	C₂²	#14579138	1	1	yes	characteristic
178	C₂²	SD₁₆×D₇₉	1	1	yes	characteristic
179	C₂²	SD₁₆×D₇₉	1	1	yes	characteristic
180	C₂²	#14579185	1	1	yes	characteristic
181	C₂²	#14579193	1	1	yes	characteristic
182	C₄	—	4	2	no	—
183	C₄	—	8	1	no	—
184	C₄	—	316	2	no	—
185	C₄	—	79	4	no	—
186	C₄	—	632	1	no	—
187	C₄	—	158	2	no	—
188	C₄	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
189	C₄	C₄×Dic₁₅₈	1	1	yes	characteristic
190	C₄	#14579151	1	1	yes	characteristic
191	C₄	#14579160	1	1	yes	characteristic
192	C₂	#19098460	1	1	yes	characteristic
193	C₂	#106×D₇₉	1	1	yes	characteristic
194	C₂	#19098604	1	1	yes	characteristic
195	C₂	#19098762	1	1	yes	characteristic
196	C₂	#19098767	1	1	yes	characteristic
197	C₂	#19098846	1	1	yes	characteristic
198	C₂	#19098851	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579093

C₂

yes

maximal, characteristic

C₂×#14579092

C₂

yes

maximal, characteristic

C₂×Dic₇₉⋊C₈

C₂

yes

maximal, characteristic

#19098432

C₂

yes

maximal, characteristic

C₄⋊C₄×Dic₇₉

C₂

yes

maximal, characteristic

#19098325

C₂

yes

maximal, characteristic

#340×C₇₉

C₂

yes

maximal, characteristic

C₂²×C₇₉⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₉

C₂²

yes

characteristic

C₂×Dic₇₉⋊C₄

—

C₂×Dic₇₉⋊C₄

—

C₂×C₄×Dic₇₉

—

C₂×C₄×Dic₇₉

C₂²

yes

characteristic

#14579093

—

#14579092

—

Dic₇₉⋊C₈

C₄

yes

complemented

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

C₂×C₄

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

D₄

yes

—

Dic₇₉⋊C₄

—

Dic₇₉⋊C₄

—

C₄×Dic₇₉

—

C₄×Dic₇₉

C₂×C₄

yes

—

C₄×Dic₇₉

Q₈

yes

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂×C₄

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

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

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₈

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

C₄⋊C₄

yes

—

C₂×Dic₇₉

C₄⋊C₄

yes

—

C₂×Dic₇₉

SD₁₆

yes

—

C₂×Dic₇₉

SD₁₆

yes

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₇₉

#121

yes

characteristic

C₂²×C₇₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₇₉

C₂×SD₁₆

yes

characteristic

C₂²×C₇₉

C₂×SD₁₆

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₄×D₄

yes

characteristic

C₄×C₇₉

C₄×Q₈

yes

characteristic

C₄×C₇₉

#125

yes

characteristic

C₄×C₇₉

C₄⋊Q₈

yes

characteristic

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

Dic₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

100

C₇₉

#1948

yes

characteristic, complemented, Sylow

101

C₂×#106

—

102

C₂×#106

—

103

C₂×C₄⋊C₈

—

104

#384

—

105

C₄×C₄⋊C₄

—

106

#340

—

107

#340

D₇₉

yes

characteristic

108

C₂²×C₈

—

109

C₂²×C₈

C₂×D₇₉

yes

characteristic

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

158

—

113

C₂×C₄⋊C₄

—

158

—

114

C₂×C₄⋊C₄

C₂×D₇₉

yes

characteristic

115

C₂×C₄⋊C₄

C₂×D₇₉

yes

characteristic

116

C₂×C₄²

—

117

C₂×C₄²

—

158

—

118

#106

—

158

—

119

#106

—

158

—

120

C₄⋊C₈

—

121

#95

—

158

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

158

—

127

C₂²×C₄

—

158

—

128

C₂²×C₄

C₂²×D₇₉

yes

characteristic

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

158

—

133

C₂×C₈

—

158

—

134

C₂×C₈

C₄×D₇₉

yes

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

316

—

139

C₄⋊C₄

—

316

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

158

—

142

C₄⋊C₄

—

158

—

143

C₄⋊C₄

—

158

—

144

C₄⋊C₄

Dic₁₅₈

yes

—

145

C₄²

—

316

—

146

C₄²

—

147

C₄²

—

148

C₂³

D₄×D₇₉

yes

characteristic

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

316

—

156

C₂×C₄

—

316

—

157

C₂×C₄

—

316

—

158

C₂×C₄

—

316

—

159

C₂×C₄

—

316

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

158

—

165

C₂×C₄

—

158

—

166

C₂×C₄

—

158

—

167

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

168

C₂×C₄

D₄×D₇₉

yes

characteristic

169

C₂×C₄

Q₈×D₇₉

yes

characteristic

170

C₂×C₄

C₂×Dic₁₅₈

yes

characteristic

171

C₂×C₄

#11929623

yes

characteristic

172

C₂×C₄

#11929624

yes

characteristic

173

C₈

—

174

C₈

—

158

—

175

C₂²

C₄⋊C₄×D₇₉

yes

characteristic

176

C₂²

#14579134

yes

characteristic

177

C₂²

#14579138

yes

characteristic

178

C₂²

SD₁₆×D₇₉

yes

characteristic

179

C₂²

SD₁₆×D₇₉

yes

characteristic

180

C₂²

#14579185

yes

characteristic

181

C₂²

#14579193

yes

characteristic

182

C₄

—

183

C₄

—

184

C₄

—

316

—

185

C₄

—

186

C₄

—

632

—

187

C₄

—

158

—

188

C₄

(C₂²⋊D₇₉)⋊C₄

yes

characteristic

189

C₄

C₄×Dic₁₅₈

yes

characteristic

190

C₄

#14579151

yes

characteristic

191

C₄

#14579160

yes

characteristic

192

C₂

#19098460

yes

characteristic

193

C₂

#106×D₇₉

yes

characteristic

194

C₂

#19098604

yes

characteristic

195

C₂

#19098762

yes

characteristic

196

C₂

#19098767

yes

characteristic

197

C₂

#19098846

yes

characteristic

198

C₂

#19098851

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	#340×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×D₄
cosocle	C₂²×C₄×C₇₉	C₂³
hypercenter	#340	D₇₉
hypocenter	C₇₉	#1948
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₇₉

commutator

C₂×C₄×C₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

#340×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

C₂×D₄

cosocle

C₂²×C₄×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
17. Dic₇₉⋊C₈	C₄	semidirect
91. Dic₇₉	#106	semidirect
100. C₇₉	#1948	semidirect

H ≅ G/N

type

17. Dic₇₉⋊C₈

C₄

semidirect

91. Dic₇₉

#106

semidirect

100. C₇₉

#1948

semidirect

Subgroups — #23764162

Conjugacy classes of subgroups

Complements