Group #23764130 — subgroups

Subgroups — #23764130

Order 10112.

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

normal subgroups only

This group has 248 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×#14579094	C₂	1	1	yes	maximal, characteristic
2	C₂×#14579094	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#19098461	C₂	1	1	yes	maximal, characteristic
5	#19098325	C₂	1	1	yes	maximal, characteristic
6	#19098325	C₂	1	1	yes	maximal, characteristic
7	#389×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
12	C₂×Dic₇₉⋊C₄	—	2	1	no	—
13	#14579094	—	2	2	no	—
14	#14579094	C₄	1	4	yes	complemented
15	C₄×C₇₉⋊C₈	—	2	2	no	—
16	C₇₉⋊#95	—	2	1	no	—
17	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
18	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
21	#95×C₇₉	—	2	1	no	—
22	C₂²×Dic₇₉	—	2	1	no	—
23	C₂²×Dic₇₉	—	2	1	no	—
24	C₂×C₇₉⋊C₈	—	2	1	no	—
25	C₂×C₇₉⋊C₈	—	2	1	no	—
26	C₂×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
27	C₂×C₇₉⋊C₈	D₄	1	2	yes	—
28	C₂×C₇₉⋊C₈	Q₈	1	2	yes	—
29	C₂×C₇₉⋊C₈	Q₈	1	2	yes	—
30	C₄⋊Dic₇₉	—	2	1	no	—
31	C₄⋊Dic₇₉	—	2	1	no	—
32	C₄⋊Dic₇₉	—	2	1	no	—
33	C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
34	Dic₇₉⋊C₄	—	4	2	no	—
35	C₂²×C₄×C₇₉	—	2	1	no	—
36	C₂²×C₄×C₇₉	—	2	1	no	—
37	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
38	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
40	C₄⋊C₄×C₇₉	—	2	1	no	—
41	C₄⋊C₄×C₇₉	—	2	1	no	—
42	C₄⋊C₄×C₇₉	—	2	1	no	—
43	C₄⋊C₄×C₇₉	—	4	2	no	—
44	C₄⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
45	C₄²×C₇₉	—	2	1	no	—
46	C₄²×C₇₉	—	2	1	no	—
47	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₇₉	—	2	1	no	—
49	C₂×C₄×C₇₉	—	2	1	no	—
50	C₂×C₄×C₇₉	—	2	1	no	—
51	C₂×C₄×C₇₉	—	2	1	no	—
52	C₂×C₄×C₇₉	—	2	2	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	—	4	1	no	—
57	C₂×C₄×C₇₉	—	4	1	no	—
58	C₂×C₄×C₇₉	D₈	1	2	yes	—
59	C₂×C₄×C₇₉	Q₁₆	1	2	yes	—
60	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
65	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
66	C₂×Dic₇₉	—	2	2	no	—
67	C₂×Dic₇₉	—	4	1	no	—
68	C₂×Dic₇₉	—	4	1	no	—
69	C₂×Dic₇₉	—	4	1	no	—
70	C₂×Dic₇₉	—	4	1	no	—
71	C₂×Dic₇₉	—	4	1	no	—
72	C₇₉⋊C₈	—	2	2	no	—
73	C₇₉⋊C₈	C₄⋊C₄	1	4	yes	complemented
74	C₂²×C₇₉	C₄×D₄	1	1	yes	characteristic
75	C₂²×C₇₉	#121	1	1	yes	characteristic
76	C₂²×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
77	C₂²×C₇₉	C₂×D₈	1	1	yes	characteristic
78	C₂²×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
79	C₂²×C₇₉	#135	1	1	yes	characteristic
80	C₂²×C₇₉	#135	1	1	yes	characteristic
81	C₄×C₇₉	—	2	2	no	—
82	C₄×C₇₉	—	2	2	no	—
83	C₄×C₇₉	—	4	2	no	—
84	C₄×C₇₉	—	8	1	no	—
85	C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₄×C₇₉	C₄×Q₈	1	1	yes	characteristic
87	C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
88	C₄×C₇₉	#125	1	1	yes	characteristic
89	Dic₇₉	—	4	2	no	—
90	Dic₇₉	—	8	1	no	—
91	C₂×C₇₉	#389	1	1	yes	characteristic
92	C₂×C₇₉	C₄×D₈	1	1	yes	characteristic
93	C₂×C₇₉	C₄×Q₁₆	1	1	yes	characteristic
94	C₂×C₇₉	#466	1	1	yes	characteristic
95	C₂×C₇₉	#467	1	1	yes	characteristic
96	C₂×C₇₉	#499	1	1	yes	characteristic
97	C₂×C₇₉	#500	1	1	yes	characteristic
98	#1942	—	79	1	no	maximal, Sylow
99	C₇₉	#1942	1	1	yes	characteristic, complemented, Sylow
100	C₂×#107	—	79	1	no	—
101	C₂×#107	—	79	1	no	—
102	C₂×C₄×C₈	—	79	1	no	—
103	#389	—	79	1	no	—
104	#389	D₇₉	1	1	yes	characteristic
105	#340	—	79	1	no	—
106	#340	—	79	1	no	—
107	C₂²×C₈	—	79	1	no	—
108	C₂²×C₈	—	79	1	no	—
109	C₂×C₄⋊C₄	—	2	1	no	—
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	C₂×D₇₉	1	1	yes	characteristic
114	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
115	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
116	#107	—	79	4	no	—
117	#107	—	158	2	no	—
118	C₄×C₈	—	158	2	no	—
119	#95	—	2	1	no	—
120	#95	—	158	1	no	—
121	C₂²×C₄	—	2	1	no	—
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	—	158	1	no	—
124	C₂²×C₄	—	158	1	no	—
125	C₂²×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
126	C₂²×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
128	C₂×C₈	—	79	2	no	—
129	C₂×C₈	—	79	2	no	—
130	C₂×C₈	—	79	2	no	—
131	C₂×C₈	—	79	2	no	—
132	C₂×C₈	—	158	1	no	—
133	C₂×C₈	—	158	1	no	—
134	C₄⋊C₄	—	2	1	no	—
135	C₄⋊C₄	—	2	1	no	—
136	C₄⋊C₄	—	2	1	no	—
137	C₄⋊C₄	—	4	2	no	—
138	C₄⋊C₄	—	316	2	no	—
139	C₄⋊C₄	—	79	2	no	—
140	C₄⋊C₄	—	158	1	no	—
141	C₄⋊C₄	—	158	1	no	—
142	C₄⋊C₄	—	158	1	no	—
143	C₄⋊C₄	C₄×D₇₉	1	2	yes	—
144	C₄²	—	2	1	no	—
145	C₄²	—	2	1	no	—
146	C₂³	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	1	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	2	no	—
163	C₂×C₄	#11929609	1	2	yes	—
164	C₂×C₄	#11929614	1	2	yes	—
165	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
166	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
167	C₂×C₄	Q₈×D₇₉	1	1	yes	characteristic
168	C₂×C₄	Q₈×D₇₉	1	1	yes	characteristic
169	C₂×C₄	#11929623	1	1	yes	characteristic
170	C₂×C₄	#11929624	1	1	yes	characteristic
171	C₈	—	79	4	no	—
172	C₈	—	158	2	no	—
173	C₂²	C₄×C₂²⋊D₇₉	1	1	yes	characteristic
174	C₂²	#14579133	1	1	yes	characteristic
175	C₂²	#14579159	1	1	yes	characteristic
176	C₂²	C₂×#11929609	1	1	yes	characteristic
177	C₂²	C₂×#11929614	1	1	yes	characteristic
178	C₂²	#14579181	1	1	yes	characteristic
179	C₂²	#14579181	1	1	yes	characteristic
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	4	2	no	—
183	C₄	—	8	1	no	—
184	C₄	—	316	2	no	—
185	C₄	—	632	1	no	—
186	C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
187	C₄	#14579126	1	1	yes	characteristic
188	C₄	#14579139	1	1	yes	characteristic
189	C₄	#14579151	1	1	yes	characteristic
190	C₂	#19098458	1	1	yes	characteristic
191	C₂	C₄×#11929609	1	1	yes	characteristic
192	C₂	C₄×#11929614	1	1	yes	characteristic
193	C₂	#19098797	1	1	yes	characteristic
194	C₂	#19098802	1	1	yes	characteristic
195	C₂	#19098962	1	1	yes	characteristic
196	C₂	#19098966	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579094

C₂

yes

maximal, characteristic

C₂×#14579094

C₂

yes

maximal, characteristic

C₂×C₄×C₇₉⋊C₈

C₂

yes

maximal, characteristic

#19098461

C₂

yes

maximal, characteristic

#19098325

C₂

yes

maximal, characteristic

#19098325

C₂

yes

maximal, characteristic

#389×C₇₉

C₂

yes

maximal, characteristic

C₂²×C₇₉⋊C₈

C₂²

yes

characteristic

C₂²×C₇₉⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₉

C₂²

yes

characteristic

C₂×Dic₇₉⋊C₄

—

#14579094

—

#14579094

C₄

yes

complemented

C₄×C₇₉⋊C₈

—

C₇₉⋊#95

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₉

C₂²

yes

characteristic

C₂×C₄²×C₇₉

C₂²

yes

characteristic

#95×C₇₉

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

C₂×C₄

yes

—

C₂×C₇₉⋊C₈

D₄

yes

—

C₂×C₇₉⋊C₈

Q₈

yes

—

C₂×C₇₉⋊C₈

Q₈

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

C₂×C₄

yes

—

Dic₇₉⋊C₄

—

C₂²×C₄×C₇₉

—

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

—

C₄²×C₇₉

—

C₄²×C₇₉

—

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×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

D₈

yes

—

C₂×C₄×C₇₉

Q₁₆

yes

—

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

—

C₇₉⋊C₈

—

C₇₉⋊C₈

C₄⋊C₄

yes

complemented

C₂²×C₇₉

C₄×D₄

yes

characteristic

C₂²×C₇₉

#121

yes

characteristic

C₂²×C₇₉

C₄⋊Q₈

yes

characteristic

C₂²×C₇₉

C₂×D₈

yes

characteristic

C₂²×C₇₉

C₂×Q₁₆

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₇₉

C₄×Q₈

yes

characteristic

C₄×C₇₉

C₂²⋊Q₈

yes

characteristic

C₄×C₇₉

#125

yes

characteristic

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

#389

yes

characteristic

C₂×C₇₉

C₄×D₈

yes

characteristic

C₂×C₇₉

C₄×Q₁₆

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

100

C₂×#107

—

101

C₂×#107

—

102

C₂×C₄×C₈

—

103

#389

—

104

#389

D₇₉

yes

characteristic

105

#340

—

106

#340

—

107

C₂²×C₈

—

108

C₂²×C₈

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

158

—

113

C₂×C₄⋊C₄

C₂×D₇₉

yes

characteristic

114

C₂×C₄⋊C₄

C₂×D₇₉

yes

characteristic

115

C₂×C₄²

C₂×D₇₉

yes

characteristic

116

#107

—

117

#107

—

158

—

118

C₄×C₈

—

158

—

119

#95

—

120

#95

—

158

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

158

—

124

C₂²×C₄

—

158

—

125

C₂²×C₄

C₂²⋊D₇₉

yes

characteristic

126

C₂²×C₄

C₂²⋊D₇₉

yes

characteristic

127

C₂²×C₄

C₂²×D₇₉

yes

characteristic

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

158

—

133

C₂×C₈

—

158

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

316

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

158

—

141

C₄⋊C₄

—

158

—

142

C₄⋊C₄

—

158

—

143

C₄⋊C₄

C₄×D₇₉

yes

—

144

C₄²

—

145

C₄²

—

146

C₂³

C₂×C₂²⋊D₇₉

yes

characteristic

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

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

#11929609

yes

—

164

C₂×C₄

#11929614

yes

—

165

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

166

C₂×C₄

D₄×D₇₉

yes

characteristic

167

C₂×C₄

Q₈×D₇₉

yes

characteristic

168

C₂×C₄

Q₈×D₇₉

yes

characteristic

169

C₂×C₄

#11929623

yes

characteristic

170

C₂×C₄

#11929624

yes

characteristic

171

C₈

—

172

C₈

—

158

—

173

C₂²

C₄×C₂²⋊D₇₉

yes

characteristic

174

C₂²

#14579133

yes

characteristic

175

C₂²

#14579159

yes

characteristic

176

C₂²

C₂×#11929609

yes

characteristic

177

C₂²

C₂×#11929614

yes

characteristic

178

C₂²

#14579181

yes

characteristic

179

C₂²

#14579181

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

316

—

185

C₄

—

632

—

186

C₄

C₄⋊C₄×D₇₉

yes

characteristic

187

C₄

#14579126

yes

characteristic

188

C₄

#14579139

yes

characteristic

189

C₄

#14579151

yes

characteristic

190

C₂

#19098458

yes

characteristic

191

C₂

C₄×#11929609

yes

characteristic

192

C₂

C₄×#11929614

yes

characteristic

193

C₂

#19098797

yes

characteristic

194

C₂

#19098802

yes

characteristic

195

C₂

#19098962

yes

characteristic

196

C₂

#19098966

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₇₉
commutator	C₂×C₄×C₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	#389×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×D₄
cosocle	C₂²×C₄×C₇₉	C₂³
hypercenter	#389	D₇₉
hypocenter	C₇₉	#1942
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇₉

commutator

C₂×C₄×C₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

#389×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 ◁ C₂×C₄² ◁ #389

N	H ≅ G/N	type
14. #14579094	C₄	semidirect
73. C₇₉⋊C₈	C₄⋊C₄	semidirect
99. C₇₉	#1942	semidirect

H ≅ G/N

type

14. #14579094

C₄

semidirect

73. C₇₉⋊C₈

C₄⋊C₄

semidirect

99. C₇₉

#1942

semidirect

Subgroups — #23764130

Conjugacy classes of subgroups

Complements