Group #34179409 — subgroups

Subgroups — #34179409

Order 20224.

470 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 199 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	C₄×Dic₇₉⋊C₈	C₂	1	2	yes	maximal
2	C₄×#14579071	C₂	1	1	yes	maximal, characteristic
3	C₂×#19098285	C₂	1	2	yes	maximal
4	C₇₉⋊#1273	C₂	1	1	yes	maximal, characteristic
5	C₇₉×#1273	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₇₉⋊C₈	C₄	1	4	yes	—
7	C₂×#14579071	C₄	1	2	yes	—
8	C₂×#14579071	C₄	1	2	yes	—
9	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₄²×Dic₇₉	C₂²	1	1	yes	characteristic
12	#19098317	C₄	1	4	yes	—
13	#19098285	—	2	4	no	—
14	C₂×C₄×C₈×C₇₉	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
16	C₂²×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
17	C₂²×C₇₉⋊C₈	C₂×C₄	1	4	yes	—
18	C₂×C₄×Dic₇₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₇₉	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₇₉	C₂×C₄	1	2	yes	—
21	C₂×C₄×Dic₇₉	C₂×C₄	1	1	yes	characteristic
22	Dic₇₉⋊C₈	—	2	8	no	—
23	#14579071	—	2	4	no	—
24	#14579071	—	2	4	no	—
25	C₄×C₇₉⋊C₈	—	2	2	no	—
26	C₄×C₇₉⋊C₈	—	2	2	no	—
27	C₄×C₇₉⋊C₈	D₄	1	2	yes	—
28	C₄×C₇₉⋊C₈	Q₈	1	2	yes	—
29	C₂²×C₈×C₇₉	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
32	C₄×C₈×C₇₉	—	2	2	no	—
33	C₄×C₈×C₇₉	—	2	2	no	—
34	C₄×C₈×C₇₉	—	2	2	no	—
35	C₂²×Dic₇₉	C₄²	1	1	yes	characteristic
36	C₂²×Dic₇₉	C₄²	1	1	yes	characteristic
37	C₂²×Dic₇₉	C₄²	1	2	yes	—
38	C₂×C₇₉⋊C₈	—	2	4	no	—
39	C₂×C₇₉⋊C₈	—	2	2	no	—
40	C₂×C₇₉⋊C₈	—	2	2	no	—
41	C₂×C₇₉⋊C₈	—	2	4	no	—
42	C₂×C₇₉⋊C₈	C₄⋊C₄	1	4	yes	—
43	C₂×C₇₉⋊C₈	M₁₆	1	8	yes	—
44	C₄×Dic₇₉	—	2	2	no	—
45	C₄×Dic₇₉	—	2	4	no	—
46	C₄×Dic₇₉	—	2	4	no	—
47	C₄×Dic₇₉	M₁₆	1	4	yes	—
48	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
50	C₂×C₈×C₇₉	—	2	2	no	—
51	C₂×C₈×C₇₉	—	2	4	no	—
52	C₂×C₈×C₇₉	—	2	2	no	—
53	C₂×C₈×C₇₉	—	2	4	no	—
54	C₂×C₈×C₇₉	—	2	4	no	—
55	C₂×C₈×C₇₉	M₁₆	1	4	yes	—
56	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
58	C₄²×C₇₉	#41	1	2	yes	—
59	C₂³×C₇₉	C₂×C₄²	1	1	yes	characteristic
60	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
62	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
63	C₂×C₄×C₇₉	C₄×Q₈	1	2	yes	—
64	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
65	C₂×C₄×C₇₉	C₂×M₁₆	1	4	yes	—
66	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
67	C₂×C₄×C₇₉	#131	1	4	yes	—
68	C₂×Dic₇₉	—	2	2	no	—
69	C₂×Dic₇₉	—	2	2	no	—
70	C₂×Dic₇₉	—	2	1	no	—
71	C₂×Dic₇₉	—	2	1	no	—
72	C₂×Dic₇₉	—	2	2	no	—
73	C₂×Dic₇₉	—	2	2	no	—
74	C₂×Dic₇₉	#97	1	4	yes	—
75	C₈×C₇₉	—	2	4	no	—
76	C₈×C₇₉	—	4	4	no	—
77	C₇₉⋊C₈	—	2	4	no	—
78	C₇₉⋊C₈	—	2	8	no	—
79	C₂²×C₇₉	C₄×C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₇₉	C₂×#97	1	1	yes	characteristic
81	C₂²×C₇₉	C₄×M₁₆	1	1	yes	characteristic
82	C₂²×C₇₉	C₄×M₁₆	1	2	yes	—
83	C₂²×C₇₉	#405	1	2	yes	—
84	C₄×C₇₉	#423	1	2	yes	—
85	C₄×C₇₉	#432	1	2	yes	—
86	C₄×C₇₉	#435	1	4	yes	—
87	C₄×C₇₉	#446	1	4	yes	—
88	Dic₇₉	—	2	4	no	—
89	Dic₇₉	—	4	1	no	—
90	Dic₇₉	—	4	1	no	—
91	#8263	—	79	1	no	maximal, Sylow
92	C₂×C₇₉	#1451	1	2	yes	—
93	C₂×C₇₉	#1453	1	1	yes	characteristic
94	C₂×C₇₉	#1455	1	1	yes	characteristic
95	C₂×C₇₉	#1767	1	1	yes	characteristic
96	C₂×C₇₉	#1771	1	2	yes	—
97	C₄×C₄⋊C₈	—	79	2	no	—
98	C₄×#97	—	79	1	no	—
99	C₂×#322	—	79	2	no	—
100	#1273	—	79	1	no	—
101	#1273	D₇₉	1	1	yes	characteristic
102	C₇₉	#8263	1	1	yes	characteristic, complemented, Sylow
103	C₂×C₄⋊C₈	—	79	4	no	—
104	C₂×#97	—	79	2	no	—
105	C₂×#97	—	79	2	no	—
106	C₂×C₄×C₈	—	79	1	no	—
107	C₂×C₄×C₈	—	79	2	no	—
108	C₂×C₄×C₈	C₂×D₇₉	1	2	yes	—
109	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
110	C₄³	—	79	1	no	—
111	#336	—	79	4	no	—
112	#322	—	158	4	no	—
113	C₂²×C₈	—	79	4	no	—
114	C₂²×C₈	—	79	2	no	—
115	C₂²×C₈	C₄×D₇₉	1	4	yes	—
116	C₂²×C₈	C₄×D₇₉	1	2	yes	—
117	C₂×C₄²	—	79	1	no	—
118	C₂×C₄²	—	79	2	no	—
119	C₂×C₄²	—	79	1	no	—
120	C₂×C₄²	—	79	2	no	—
121	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
122	C₄⋊C₈	—	158	8	no	—
123	#97	—	158	4	no	—
124	#97	—	158	4	no	—
125	C₄×C₈	—	2	2	no	—
126	C₄×C₈	—	2	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	79	2	no	—
129	C₄×C₈	—	79	2	no	—
130	C₄×C₈	—	158	2	no	—
131	C₄×C₈	—	158	2	no	—
132	C₂²×C₄	—	79	2	no	—
133	C₂²×C₄	—	79	1	no	—
134	C₂²×C₄	—	79	1	no	—
135	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
136	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	—	79	4	no	—
143	C₂×C₈	—	79	8	no	—
144	C₂×C₈	—	158	4	no	—
145	C₂×C₈	—	158	2	no	—
146	C₂×C₈	—	158	2	no	—
147	C₂×C₈	—	158	4	no	—
148	C₂×C₈	#11929607	1	4	yes	—
149	C₄²	—	79	4	no	—
150	C₄²	—	158	4	no	—
151	C₄²	—	158	2	no	—
152	C₄²	—	158	4	no	—
153	C₄²	D₄×D₇₉	1	1	yes	characteristic
154	C₄²	Q₈×D₇₉	1	1	yes	characteristic
155	C₄²	#11929623	1	2	yes	—
156	C₂³	C₄²×D₇₉	1	1	yes	characteristic
157	C₂×C₄	—	79	4	no	—
158	C₂×C₄	—	158	1	no	—
159	C₂×C₄	—	158	1	no	—
160	C₂×C₄	—	158	2	no	—
161	C₂×C₄	—	158	2	no	—
162	C₂×C₄	—	158	2	no	—
163	C₂×C₄	—	158	2	no	—
164	C₂×C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
165	C₂×C₄	#14579117	1	1	yes	characteristic
166	C₂×C₄	(C₂²⋊D₇₉)⋊C₄	1	2	yes	—
167	C₂×C₄	#14579126	1	2	yes	—
168	C₂×C₄	M₁₆×D₇₉	1	4	yes	—
169	C₂×C₄	M₁₆×D₇₉	1	2	yes	—
170	C₂×C₄	C₂×#11929607	1	2	yes	—
171	C₂×C₄	#14579167	1	4	yes	—
172	C₈	—	2	4	no	—
173	C₈	—	4	4	no	—
174	C₈	—	158	8	no	—
175	C₈	—	158	4	no	—
176	C₂²	#19098414	1	1	yes	characteristic
177	C₂²	#97×D₇₉	1	1	yes	characteristic
178	C₂²	C₄×#11929607	1	1	yes	characteristic
179	C₂²	#19098515	1	2	yes	—
180	C₂²	#19098517	1	2	yes	—
181	C₄	—	316	1	no	—
182	C₄	—	316	1	no	—
183	C₄	—	158	4	no	—
184	C₄	#19098585	1	2	yes	—
185	C₄	#19098617	1	2	yes	—
186	C₄	#19098637	1	4	yes	—
187	C₄	#19098692	1	4	yes	—
188	C₂	#23761688	1	2	yes	—
189	C₂	#23761694	1	1	yes	characteristic
190	C₂	#23761704	1	1	yes	characteristic
191	C₂	#23763252	1	1	yes	characteristic
192	C₂	#23763273	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×Dic₇₉⋊C₈

C₂

yes

maximal

C₄×#14579071

C₂

yes

maximal, characteristic

C₂×#19098285

C₂

yes

maximal

C₇₉⋊#1273

C₂

yes

maximal, characteristic

C₇₉×#1273

C₂

yes

maximal, characteristic

C₂×Dic₇₉⋊C₈

C₄

yes

—

C₂×#14579071

C₄

yes

—

C₂×#14579071

C₄

yes

—

C₂×C₄×C₇₉⋊C₈

C₂²

yes

—

C₂×C₄×C₇₉⋊C₈

C₂²

yes

characteristic

C₄²×Dic₇₉

C₂²

yes

characteristic

#19098317

C₄

yes

—

#19098285

—

C₂×C₄×C₈×C₇₉

C₂²

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂²×C₇₉⋊C₈

C₂×C₄

yes

—

C₂²×C₇₉⋊C₈

C₂×C₄

yes

—

C₂×C₄×Dic₇₉

C₂×C₄

yes

characteristic

C₂×C₄×Dic₇₉

C₂×C₄

yes

—

C₂×C₄×Dic₇₉

C₂×C₄

yes

—

C₂×C₄×Dic₇₉

C₂×C₄

yes

characteristic

Dic₇₉⋊C₈

—

#14579071

—

#14579071

—

C₄×C₇₉⋊C₈

—

C₄×C₇₉⋊C₈

—

C₄×C₇₉⋊C₈

D₄

yes

—

C₄×C₇₉⋊C₈

Q₈

yes

—

C₂²×C₈×C₇₉

C₂×C₄

yes

—

C₂²×C₈×C₇₉

C₂×C₄

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄×C₈×C₇₉

—

C₄×C₈×C₇₉

—

C₄×C₈×C₇₉

—

C₂²×Dic₇₉

C₄²

yes

characteristic

C₂²×Dic₇₉

C₄²

yes

characteristic

C₂²×Dic₇₉

C₄²

yes

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

C₄⋊C₄

yes

—

C₂×C₇₉⋊C₈

M₁₆

yes

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₄×Dic₇₉

M₁₆

yes

—

C₂²×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₉

C₂²×C₄

yes

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

M₁₆

yes

—

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

C₂×Q₈

yes

characteristic

C₄²×C₇₉

#41

yes

—

C₂³×C₇₉

C₂×C₄²

yes

characteristic

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

#117

yes

characteristic

C₂×C₄×C₇₉

C₄×D₄

yes

—

C₂×C₄×C₇₉

C₄×Q₈

yes

—

C₂×C₄×C₇₉

C₂×M₁₆

yes

—

C₂×C₄×C₇₉

C₂×M₁₆

yes

—

C₂×C₄×C₇₉

C₂×M₁₆

yes

—

C₂×C₄×C₇₉

#131

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

#97

yes

—

C₈×C₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

C₄×C₄⋊C₄

yes

characteristic

C₂²×C₇₉

C₂×#97

yes

characteristic

C₂²×C₇₉

C₄×M₁₆

yes

characteristic

C₂²×C₇₉

C₄×M₁₆

yes

—

C₂²×C₇₉

#405

yes

—

C₄×C₇₉

#423

yes

—

C₄×C₇₉

#432

yes

—

C₄×C₇₉

#435

yes

—

C₄×C₇₉

#446

yes

—

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

#8263

—

maximal, Sylow

C₂×C₇₉

#1451

yes

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₄×C₄⋊C₈

—

C₄×#97

—

C₂×#322

—

100

#1273

—

101

#1273

D₇₉

yes

characteristic

102

C₇₉

#8263

yes

characteristic, complemented, Sylow

103

C₂×C₄⋊C₈

—

104

C₂×#97

—

105

C₂×#97

—

106

C₂×C₄×C₈

—

107

C₂×C₄×C₈

—

108

C₂×C₄×C₈

C₂×D₇₉

yes

—

109

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

110

C₄³

—

111

#336

—

112

#322

—

158

—

113

C₂²×C₈

—

114

C₂²×C₈

—

115

C₂²×C₈

C₄×D₇₉

yes

—

116

C₂²×C₈

C₄×D₇₉

yes

—

117

C₂×C₄²

—

118

C₂×C₄²

—

119

C₂×C₄²

—

120

C₂×C₄²

—

121

C₂×C₄²

C₂²×D₇₉

yes

characteristic

122

C₄⋊C₈

—

158

—

123

#97

—

158

—

124

#97

—

158

—

125

C₄×C₈

—

126

C₄×C₈

—

127

C₄×C₈

—

128

C₄×C₈

—

129

C₄×C₈

—

130

C₄×C₈

—

158

—

131

C₄×C₈

—

158

—

132

C₂²×C₄

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

C₂×C₄×D₇₉

yes

characteristic

136

C₂²×C₄

C₂×C₄×D₇₉

yes

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

158

—

145

C₂×C₈

—

158

—

146

C₂×C₈

—

158

—

147

C₂×C₈

—

158

—

148

C₂×C₈

#11929607

yes

—

149

C₄²

—

150

C₄²

—

158

—

151

C₄²

—

158

—

152

C₄²

—

158

—

153

C₄²

D₄×D₇₉

yes

characteristic

154

C₄²

Q₈×D₇₉

yes

characteristic

155

C₄²

#11929623

yes

—

156

C₂³

C₄²×D₇₉

yes

characteristic

157

C₂×C₄

—

158

C₂×C₄

—

158

—

159

C₂×C₄

—

158

—

160

C₂×C₄

—

158

—

161

C₂×C₄

—

158

—

162

C₂×C₄

—

158

—

163

C₂×C₄

—

158

—

164

C₂×C₄

C₄⋊C₄×D₇₉

yes

characteristic

165

C₂×C₄

#14579117

yes

characteristic

166

C₂×C₄

(C₂²⋊D₇₉)⋊C₄

yes

—

167

C₂×C₄

#14579126

yes

—

168

C₂×C₄

M₁₆×D₇₉

yes

—

169

C₂×C₄

M₁₆×D₇₉

yes

—

170

C₂×C₄

C₂×#11929607

yes

—

171

C₂×C₄

#14579167

yes

—

172

C₈

—

173

C₈

—

174

C₈

—

158

—

175

C₈

—

158

—

176

C₂²

#19098414

yes

characteristic

177

C₂²

#97×D₇₉

yes

characteristic

178

C₂²

C₄×#11929607

yes

characteristic

179

C₂²

#19098515

yes

—

180

C₂²

#19098517

yes

—

181

C₄

—

316

—

182

C₄

—

316

—

183

C₄

—

158

—

184

C₄

#19098585

yes

—

185

C₄

#19098617

yes

—

186

C₄

#19098637

yes

—

187

C₄

#19098692

yes

—

188

C₂

#23761688

yes

—

189

C₂

#23761694

yes

characteristic

190

C₂

#23761704

yes

characteristic

191

C₂

#23763252

yes

characteristic

192

C₂

#23763273

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₇₉
commutator, socle, prime radical	C₂³×C₇₉	C₂×C₄²
Fitting	C₇₉×#1273	C₂
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1273	D₇₉
hypocenter	C₇₉	#8263
LCS	G ▷ C₂³×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄² ◁ #1273

subgroup

G/H

center, Frattini

C₂×C₄²

C₂²×D₇₉

commutator, socle, prime radical

C₂³×C₇₉

C₂×C₄²

Fitting

C₇₉×#1273

C₂

cosocle

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂³×C₇₉ ▷ C₇₉

C₁ ◁ C₂×C₄² ◁ #1273

N	H ≅ G/N	type
102. C₇₉	#8263	semidirect

H ≅ G/N

type

102. C₇₉

#8263

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×Dic₇₉⋊C₈	2
C₄×#14579071	2
C₂×#19098285	2
C₇₉⋊#1273	2
C₇₉×#1273	2
#8263	79

subgroup

index

C₄×Dic₇₉⋊C₈

C₄×#14579071

C₂×#19098285

C₇₉⋊#1273

C₇₉×#1273

#8263

Subgroups — #34179409

Conjugacy classes of subgroups

Complements

Maximal subgroups