Group #23761938 — subgroups

Subgroups — #23761938

Order 10112.

222 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 77 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	#19098620	C₂	1	1	yes	maximal, characteristic, complemented
2	#19098429	C₂	1	1	yes	maximal, characteristic, complemented
3	#19098300	C₂	1	2	yes	maximal, complemented
4	#19098297	C₂	1	2	yes	maximal
5	#529×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₇₉	—	2	1	no	—
7	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
8	C₂²⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
9	#14579072	—	2	1	no	—
10	C₇₉⋊#95	—	2	1	no	—
11	C₇₉⋊#95	C₂²	1	2	yes	—
12	#126×C₇₉	—	2	1	no	—
13	#125×C₇₉	C₄	1	1	yes	characteristic, complemented
14	#124×C₇₉	C₄	1	1	yes	characteristic, complemented
15	#123×C₇₉	—	2	1	no	—
16	C₂²⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
17	#121×C₇₉	C₂²	1	1	yes	characteristic
18	C₄×Q₈×C₇₉	C₄	1	1	yes	characteristic, complemented
19	C₄×D₄×C₇₉	—	2	1	no	—
20	C₄×D₄×C₇₉	C₄	1	1	yes	characteristic, complemented
21	#117×C₇₉	—	2	1	no	—
22	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
23	C₂²×Dic₇₉	—	2	2	no	—
24	C₂×C₇₉⋊C₈	—	2	2	no	—
25	C₄⋊Dic₇₉	—	4	2	no	—
26	C₂×Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
27	C₂×D₄×C₇₉	—	2	1	no	—
28	C₂×D₄×C₇₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₇₉	—	2	1	no	—
30	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₇₉	—	2	1	no	—
34	C₄⋊C₄×C₇₉	—	2	1	no	—
35	C₄⋊C₄×C₇₉	—	2	1	no	—
36	C₄⋊C₄×C₇₉	—	2	1	no	—
37	C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
38	C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
39	C₂²⋊C₄×C₇₉	—	2	1	no	—
40	C₂²⋊C₄×C₇₉	—	2	1	no	—
41	C₂²⋊C₄×C₇₉	—	2	1	no	—
42	C₂²⋊C₄×C₇₉	—	2	1	no	—
43	C₂²⋊C₄×C₇₉	—	2	1	no	—
44	C₄²×C₇₉	—	2	1	no	—
45	C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
46	C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
47	C₄²×C₇₉	D₄	1	1	yes	characteristic
48	C₄²×C₇₉	D₄	1	1	yes	characteristic
49	C₂³×C₇₉	—	2	1	no	—
50	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
51	Q₈×C₇₉	—	4	1	no	—
52	D₄×C₇₉	—	4	1	no	—
53	D₄×C₇₉	—	4	1	no	—
54	D₄×C₇₉	—	4	1	no	—
55	C₂×C₄×C₇₉	—	2	1	no	—
56	C₂×C₄×C₇₉	—	2	1	no	—
57	C₂×C₄×C₇₉	—	2	1	no	—
58	C₂×C₄×C₇₉	—	2	1	no	—
59	C₂×C₄×C₇₉	—	2	1	no	—
60	C₂×C₄×C₇₉	—	4	1	no	—
61	C₂×C₄×C₇₉	—	4	1	no	—
62	C₂×C₄×C₇₉	—	4	2	no	—
63	C₂×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
69	C₂×Dic₇₉	—	4	2	no	—
70	C₂×Dic₇₉	—	4	2	no	—
71	C₂×Dic₇₉	—	4	2	no	—
72	C₇₉⋊C₈	—	4	2	no	—
73	C₂²×C₇₉	—	2	1	no	—
74	C₂²×C₇₉	—	2	1	no	—
75	C₂²×C₇₉	—	4	1	no	—
76	C₂²×C₇₉	—	4	1	no	—
77	C₂²×C₇₉	—	4	1	no	—
78	C₂²×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
80	C₂²×C₇₉	#137	1	1	yes	characteristic
81	C₄×C₇₉	—	2	1	no	—
82	C₄×C₇₉	—	2	1	no	—
83	C₄×C₇₉	—	2	1	no	—
84	C₄×C₇₉	—	2	1	no	—
85	C₄×C₇₉	—	2	1	no	—
86	C₄×C₇₉	—	2	1	no	—
87	C₄×C₇₉	—	4	1	no	—
88	C₄×C₇₉	—	4	1	no	—
89	C₄×C₇₉	—	4	1	no	—
90	Dic₇₉	—	8	2	no	—
91	C₂×C₇₉	—	2	1	no	—
92	C₂×C₇₉	—	2	1	no	—
93	C₂×C₇₉	—	8	1	no	—
94	C₂×C₇₉	#410	1	1	yes	characteristic
95	C₂×C₇₉	#417	1	1	yes	characteristic
96	C₂×C₇₉	#421	1	1	yes	characteristic
97	#1512	—	79	1	no	maximal, Sylow
98	C₇₉	#1512	1	1	yes	characteristic, complemented, Sylow
99	#529	D₇₉	1	1	yes	characteristic
100	#432	—	79	1	no	—
101	#382	—	79	1	no	—
102	#328	—	79	2	no	—
103	#327	—	79	2	no	—
104	#126	—	2	1	no	—
105	#125	Dic₇₉	1	1	yes	characteristic, complemented
106	#124	Dic₇₉	1	1	yes	characteristic, complemented
107	#123	—	2	1	no	—
108	C₂²⋊Q₈	C₂×D₇₉	1	1	yes	characteristic
109	#121	C₂×D₇₉	1	1	yes	characteristic
110	C₄×Q₈	Dic₇₉	1	1	yes	characteristic, complemented
111	C₄×D₄	—	2	1	no	—
112	C₄×D₄	Dic₇₉	1	1	yes	characteristic, complemented
113	#117	—	2	1	no	—
114	C₂×C₄⋊C₄	—	158	1	no	—
115	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
116	C₄⋊C₈	—	158	1	no	—
117	C₂²⋊C₈	—	79	2	no	—
118	#97	—	158	1	no	—
119	#95	—	79	2	no	—
120	#95	—	158	1	no	—
121	C₂×Q₈	C₂×Dic₇₉	1	1	yes	characteristic
122	C₂×D₄	—	2	1	no	—
123	C₂×D₄	C₂×Dic₇₉	1	1	yes	characteristic
124	C₂²×C₄	—	2	1	no	—
125	C₂²×C₄	—	158	2	no	—
126	C₂²×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
127	C₂²×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
128	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
129	C₂×C₈	—	158	2	no	—
130	C₄⋊C₄	—	2	1	no	—
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	—	2	1	no	—
133	C₄⋊C₄	—	2	1	no	—
134	C₄⋊C₄	—	316	2	no	—
135	C₄⋊C₄	C₂×Dic₇₉	1	1	yes	characteristic
136	C₄⋊C₄	C₂×Dic₇₉	1	1	yes	characteristic
137	C₂²⋊C₄	—	2	1	no	—
138	C₂²⋊C₄	—	2	1	no	—
139	C₂²⋊C₄	—	2	1	no	—
140	C₂²⋊C₄	—	2	1	no	—
141	C₂²⋊C₄	—	2	1	no	—
142	C₄²	—	2	1	no	—
143	C₄²	C₂×Dic₇₉	1	1	yes	characteristic
144	C₄²	C₂×Dic₇₉	1	1	yes	characteristic
145	C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
146	C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
147	C₂³	—	2	1	no	—
148	C₂³	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
149	Q₈	—	4	1	no	—
150	D₄	—	4	1	no	—
151	D₄	—	4	1	no	—
152	D₄	—	4	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	4	1	no	—
159	C₂×C₄	—	4	1	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	316	2	no	—
162	C₂×C₄	—	316	2	no	—
163	C₂×C₄	—	316	2	no	—
164	C₂×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
165	C₂×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
166	C₂×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
167	C₂×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
168	C₂×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
169	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
170	C₈	—	316	2	no	—
171	C₂²	—	2	1	no	—
172	C₂²	—	2	1	no	—
173	C₂²	—	4	1	no	—
174	C₂²	—	4	1	no	—
175	C₂²	—	4	1	no	—
176	C₂²	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
177	C₂²	#14579186	1	1	yes	characteristic
178	C₂²	#14579191	1	1	yes	characteristic
179	C₄	—	2	1	no	—
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	2	1	no	—
185	C₄	—	4	1	no	—
186	C₄	—	4	1	no	—
187	C₄	—	4	1	no	—
188	C₄	—	632	2	no	—
189	C₂	—	2	1	no	—
190	C₂	—	2	1	no	—
191	C₂	—	8	1	no	—
192	C₂	#19098537	1	1	yes	characteristic
193	C₂	#19098564	1	1	yes	characteristic
194	C₂	#19098580	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19098620

C₂

yes

maximal, characteristic, complemented

#19098429

C₂

yes

maximal, characteristic, complemented

#19098300

C₂

yes

maximal, complemented

#19098297

C₂

yes

maximal

#529×C₇₉

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₇₉

—

C₄⋊(C₇₉⋊C₈)

—

C₂²⋊(C₇₉⋊C₈)

C₂²

yes

—

#14579072

—

C₇₉⋊#95

—

C₇₉⋊#95

C₂²

yes

—

#126×C₇₉

—

#125×C₇₉

C₄

yes

characteristic, complemented

#124×C₇₉

C₄

yes

characteristic, complemented

#123×C₇₉

—

C₂²⋊Q₈×C₇₉

C₂²

yes

characteristic

#121×C₇₉

C₂²

yes

characteristic

C₄×Q₈×C₇₉

C₄

yes

characteristic, complemented

C₄×D₄×C₇₉

—

C₄×D₄×C₇₉

C₄

yes

characteristic, complemented

#117×C₇₉

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₂²×Dic₇₉

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₂×Q₈×C₇₉

C₂×C₄

yes

characteristic

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

C₂×C₄

yes

characteristic

C₄⋊C₄×C₇₉

C₂×C₄

yes

characteristic

C₂²⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂×C₄

yes

characteristic

C₄²×C₇₉

C₂×C₄

yes

characteristic

C₄²×C₇₉

D₄

yes

characteristic

C₄²×C₇₉

D₄

yes

characteristic

C₂³×C₇₉

—

C₂³×C₇₉

C₂×D₄

yes

characteristic

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×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×C₂²⋊C₄

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

#529

D₇₉

yes

characteristic

100

#432

—

101

#382

—

102

#328

—

103

#327

—

104

#126

—

105

#125

Dic₇₉

yes

characteristic, complemented

106

#124

Dic₇₉

yes

characteristic, complemented

107

#123

—

108

C₂²⋊Q₈

C₂×D₇₉

yes

characteristic

109

#121

C₂×D₇₉

yes

characteristic

110

C₄×Q₈

Dic₇₉

yes

characteristic, complemented

111

C₄×D₄

—

112

C₄×D₄

Dic₇₉

yes

characteristic, complemented

113

#117

—

114

C₂×C₄⋊C₄

—

158

—

115

C₂×C₄²

C₂×D₇₉

yes

characteristic

116

C₄⋊C₈

—

158

—

117

C₂²⋊C₈

—

118

#97

—

158

—

119

#95

—

120

#95

—

158

—

121

C₂×Q₈

C₂×Dic₇₉

yes

characteristic

122

C₂×D₄

—

123

C₂×D₄

C₂×Dic₇₉

yes

characteristic

124

C₂²×C₄

—

125

C₂²×C₄

—

158

—

126

C₂²×C₄

C₂²⋊D₇₉

yes

characteristic

127

C₂²×C₄

C₂²⋊D₇₉

yes

characteristic

128

C₂²×C₄

C₂²×D₇₉

yes

characteristic

129

C₂×C₈

—

158

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

316

—

135

C₄⋊C₄

C₂×Dic₇₉

yes

characteristic

136

C₄⋊C₄

C₂×Dic₇₉

yes

characteristic

137

C₂²⋊C₄

—

138

C₂²⋊C₄

—

139

C₂²⋊C₄

—

140

C₂²⋊C₄

—

141

C₂²⋊C₄

—

142

C₄²

—

143

C₄²

C₂×Dic₇₉

yes

characteristic

144

C₄²

C₂×Dic₇₉

yes

characteristic

145

C₄²

C₂²⋊D₇₉

yes

characteristic

146

C₄²

C₂²⋊D₇₉

yes

characteristic

147

C₂³

—

148

C₂³

C₂×C₂²⋊D₇₉

yes

characteristic

149

Q₈

—

150

D₄

—

151

D₄

—

152

D₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

316

—

162

C₂×C₄

—

316

—

163

C₂×C₄

—

316

—

164

C₂×C₄

C₂²⋊Dic₇₉

yes

characteristic

165

C₂×C₄

C₂²⋊Dic₇₉

yes

characteristic

166

C₂×C₄

C₂²⋊Dic₇₉

yes

characteristic

167

C₂×C₄

C₂²⋊Dic₇₉

yes

characteristic

168

C₂×C₄

C₂²×Dic₇₉

yes

characteristic

169

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

170

C₈

—

316

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

175

C₂²

—

176

C₂²

C₂×C₂²⋊Dic₇₉

yes

characteristic

177

C₂²

#14579186

yes

characteristic

178

C₂²

#14579191

yes

characteristic

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

—

632

—

189

C₂

—

190

C₂

—

191

C₂

—

192

C₂

#19098537

yes

characteristic

193

C₂

#19098564

yes

characteristic

194

C₂

#19098580

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₇₉
commutator	C₂×C₄×C₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	#529×C₇₉	C₂
socle	C₂²×C₇₉	C₂×C₂²⋊C₄
prime radical	#121×C₇₉	C₂²
cosocle	C₂²×C₄×C₇₉	C₂³
hypercenter	#529	D₇₉
hypocenter	C₇₉	#1512
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #529

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₇₉

commutator

C₂×C₄×C₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

#529×C₇₉

C₂

socle

C₂²×C₇₉

C₂×C₂²⋊C₄

prime radical

#121×C₇₉

C₂²

cosocle

C₂²×C₄×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #529

N	H ≅ G/N	type
1. #19098620	C₂	semidirect
2. #19098429	C₂	semidirect
3. #19098300	C₂	semidirect
13. #125×C₇₉	C₄	semidirect
14. #124×C₇₉	C₄	semidirect
18. C₄×Q₈×C₇₉	C₄	semidirect
20. C₄×D₄×C₇₉	C₄	semidirect
98. C₇₉	#1512	semidirect
105. #125	Dic₇₉	semidirect
106. #124	Dic₇₉	semidirect
110. C₄×Q₈	Dic₇₉	semidirect
112. C₄×D₄	Dic₇₉	semidirect

H ≅ G/N

type

1. #19098620

C₂

semidirect

2. #19098429

C₂

semidirect

3. #19098300

C₂

semidirect

13. #125×C₇₉

C₄

semidirect

14. #124×C₇₉

C₄

semidirect

18. C₄×Q₈×C₇₉

C₄

semidirect

20. C₄×D₄×C₇₉

C₄

semidirect

98. C₇₉

#1512

semidirect

105. #125

Dic₇₉

semidirect

106. #124

Dic₇₉

semidirect

110. C₄×Q₈

Dic₇₉

semidirect

112. C₄×D₄

Dic₇₉

semidirect

Subgroups — #23761938

Conjugacy classes of subgroups

Complements