Group #34200210 — subgroups

Subgroups — #34200210

Order 20224.

210 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 75 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×#14579107	C₂	1	1	yes	maximal, characteristic
2	#23762296	C₂	1	1	yes	maximal, characteristic
3	Q₁₆×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#23761473	C₂	1	1	yes	maximal, characteristic
5	#23761365	C₂	1	1	yes	maximal, characteristic
6	#23761319	C₂	1	1	yes	maximal, characteristic
7	C₇₉×#1332	C₂	1	1	yes	maximal, characteristic
8	C₂×#14579107	C₄	1	1	yes	characteristic
9	#19098690	—	2	1	no	—
10	Q₈×C₇₉⋊C₈	—	2	1	no	—
11	C₄×Dic₃₁₆	C₂²	1	1	yes	characteristic
12	#19098388	C₄	1	1	yes	characteristic
13	#19098367	C₄	1	1	yes	characteristic
14	#19098366	C₄	1	1	yes	characteristic
15	C₄×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
16	#19098316	C₂²	1	1	yes	characteristic
17	Q₈⋊(C₇₉⋊C₈)	—	2	1	no	—
18	#19098293	—	2	1	no	—
19	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
20	C₄×Q₁₆×C₇₉	C₂²	1	1	yes	characteristic
21	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
22	#335×C₇₉	C₂²	1	1	yes	characteristic
23	C₂×Dic₃₁₆	C₂×C₄	1	1	yes	characteristic
24	C₄×Dic₁₅₈	—	2	1	no	—
25	#14579107	—	2	2	no	—
26	C₂×C₇₉⋊C₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂×C₇₉⋊C₁₆	C₂×C₄	1	1	yes	characteristic
28	#14579095	C₂×C₄	1	1	yes	characteristic
29	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
30	C₄⋊(C₇₉⋊C₈)	—	4	1	no	—
31	Dic₇₉⋊C₈	—	4	1	no	—
32	#14579086	—	2	1	no	—
33	#14579071	—	4	1	no	—
34	C₄×C₇₉⋊C₈	—	4	1	no	—
35	C₄×C₇₉⋊C₈	D₄	1	1	yes	characteristic
36	C₄×C₇₉⋊C₈	D₄	1	1	yes	characteristic
37	C₂×Q₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
38	C₄×Q₈×C₇₉	—	2	1	no	—
39	C₂×C₁₆×C₇₉	—	2	1	no	—
40	#107×C₇₉	C₂×C₄	1	1	yes	characteristic
41	C₄⋊C₈×C₇₉	—	2	1	no	—
42	#103×C₇₉	—	2	1	no	—
43	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
44	C₂×Dic₁₅₈	—	2	1	no	—
45	Dic₃₁₆	—	2	1	no	—
46	Dic₃₁₆	—	2	1	no	—
47	C₂×C₇₉⋊C₈	—	2	1	no	—
48	C₂×C₇₉⋊C₈	—	4	1	no	—
49	C₂×C₇₉⋊C₈	D₈	1	1	yes	characteristic
50	C₂×C₇₉⋊C₈	D₈	1	1	yes	characteristic
51	C₄⋊Dic₇₉	—	2	1	no	—
52	Dic₇₉⋊C₄	—	4	1	no	—
53	C₄×Dic₇₉	—	4	1	no	—
54	C₇₉⋊C₁₆	—	2	1	no	—
55	C₇₉⋊C₁₆	—	2	1	no	—
56	C₂×Q₈×C₇₉	—	2	1	no	—
57	Q₁₆×C₇₉	—	2	1	no	—
58	Q₁₆×C₇₉	M₁₆	1	2	yes	—
59	C₂×C₈×C₇₉	—	4	1	no	—
60	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
62	C₄⋊C₄×C₇₉	—	2	1	no	—
63	C₄⋊C₄×C₇₉	—	4	1	no	—
64	C₄²×C₇₉	—	4	1	no	—
65	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
66	C₁₆×C₇₉	—	4	1	no	—
67	Q₈×C₇₉	—	2	2	no	—
68	Q₈×C₇₉	—	4	1	no	—
69	C₂×C₄×C₇₉	—	4	1	no	—
70	C₂×C₄×C₇₉	—	4	1	no	—
71	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
74	C₂×Dic₇₉	—	4	1	no	—
75	C₂×Dic₇₉	—	4	1	no	—
76	Dic₁₅₈	—	4	1	no	—
77	Dic₁₅₈	—	4	1	no	—
78	C₈×C₇₉	—	2	1	no	—
79	C₈×C₇₉	—	8	1	no	—
80	C₈×C₇₉	C₂×M₁₆	1	1	yes	characteristic
81	C₈×C₇₉	#131	1	1	yes	characteristic
82	C₇₉⋊C₈	—	2	2	no	—
83	C₇₉⋊C₈	—	2	1	no	—
84	C₇₉⋊C₈	—	2	1	no	—
85	C₇₉⋊C₈	—	8	1	no	—
86	C₂²×C₇₉	C₄×D₈	1	1	yes	characteristic
87	C₄×C₇₉	—	2	1	no	—
88	C₄×C₇₉	—	4	2	no	—
89	C₄×C₇₉	—	8	1	no	—
90	C₄×C₇₉	#435	1	1	yes	characteristic
91	C₄×C₇₉	#444	1	1	yes	characteristic
92	C₄×C₇₉	#510	1	1	yes	characteristic
93	C₄×C₇₉	#510	1	1	yes	characteristic
94	Dic₇₉	—	8	1	no	—
95	Dic₇₉	—	8	1	no	—
96	#12076	—	79	1	no	maximal, Sylow
97	C₂×C₇₉	#1581	1	1	yes	characteristic
98	C₂×C₇₉	#2176	1	1	yes	characteristic
99	C₂×C₇₉	#2178	1	1	yes	characteristic
100	C₄×Q₃₂	—	79	1	no	—
101	#1591	—	79	1	no	—
102	C₈×Q₁₆	—	79	1	no	—
103	#1372	—	79	1	no	—
104	#1332	—	79	1	no	—
105	#1332	D₇₉	1	1	yes	characteristic
106	#1312	—	79	1	no	—
107	C₇₉	#12076	1	1	yes	characteristic, complemented, Sylow
108	C₂×Q₃₂	—	79	1	no	—
109	#446	—	158	1	no	—
110	C₈×Q₈	—	158	1	no	—
111	C₄×Q₁₆	—	79	1	no	—
112	C₄×Q₁₆	C₂×D₇₉	1	1	yes	characteristic
113	#366	—	79	1	no	—
114	#358	—	79	1	no	—
115	#358	—	79	1	no	—
116	C₄×C₁₆	—	79	1	no	—
117	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
118	#335	—	79	1	no	—
119	#335	C₂×D₇₉	1	1	yes	characteristic
120	Q₈⋊C₈	—	158	1	no	—
121	Q₈⋊C₈	—	158	1	no	—
122	C₈²	—	79	1	no	—
123	C₂×Q₁₆	—	79	1	no	—
124	C₂×Q₁₆	C₄×D₇₉	1	1	yes	characteristic
125	C₄×Q₈	—	2	1	no	—
126	C₄×Q₈	—	158	1	no	—
127	Q₃₂	—	158	2	no	—
128	C₂×C₁₆	—	2	1	no	—
129	C₂×C₁₆	—	79	1	no	—
130	C₂×C₁₆	—	79	1	no	—
131	#107	—	79	1	no	—
132	#107	C₄×D₇₉	1	1	yes	characteristic
133	C₄⋊C₈	—	2	1	no	—
134	C₄⋊C₈	—	316	1	no	—
135	C₄⋊C₈	—	316	1	no	—
136	C₄⋊C₈	—	158	1	no	—
137	#103	—	2	1	no	—
138	#103	—	158	1	no	—
139	#97	—	316	1	no	—
140	C₄×C₈	—	316	1	no	—
141	C₄×C₈	—	79	1	no	—
142	C₄×C₈	—	79	1	no	—
143	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
144	C₂×Q₈	—	2	1	no	—
145	C₂×Q₈	—	158	1	no	—
146	Q₁₆	—	2	1	no	—
147	Q₁₆	—	158	1	no	—
148	Q₁₆	—	158	1	no	—
149	Q₁₆	#11929607	1	2	yes	—
150	C₂×C₈	—	4	1	no	—
151	C₂×C₈	—	316	1	no	—
152	C₂×C₈	—	79	1	no	—
153	C₂×C₈	—	79	1	no	—
154	C₂×C₈	—	158	1	no	—
155	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
156	C₂×C₈	#11929623	1	1	yes	characteristic
157	C₄⋊C₄	—	2	1	no	—
158	C₄⋊C₄	—	4	1	no	—
159	C₄⋊C₄	—	316	1	no	—
160	C₄⋊C₄	—	158	1	no	—
161	C₄²	—	4	1	no	—
162	C₄²	—	316	1	no	—
163	C₄²	D₄×D₇₉	1	1	yes	characteristic
164	C₁₆	—	4	1	no	—
165	C₁₆	—	158	1	no	—
166	C₁₆	—	158	1	no	—
167	Q₈	—	2	2	no	—
168	Q₈	—	4	1	no	—
169	Q₈	—	316	1	no	—
170	Q₈	—	316	1	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	4	1	no	—
173	C₂×C₄	—	316	1	no	—
174	C₂×C₄	—	316	1	no	—
175	C₂×C₄	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
176	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
177	C₂×C₄	#14579182	1	1	yes	characteristic
178	C₈	—	2	1	no	—
179	C₈	—	8	1	no	—
180	C₈	—	632	1	no	—
181	C₈	—	158	2	no	—
182	C₈	—	158	1	no	—
183	C₈	—	158	1	no	—
184	C₈	C₂×#11929607	1	1	yes	characteristic
185	C₈	#14579169	1	1	yes	characteristic
186	C₂²	#19098643	1	1	yes	characteristic
187	C₄	—	2	1	no	—
188	C₄	—	4	2	no	—
189	C₄	—	8	1	no	—
190	C₄	—	632	1	no	—
191	C₄	—	632	1	no	—
192	C₄	#19098631	1	1	yes	characteristic
193	C₄	#19098682	1	1	yes	characteristic
194	C₄	#19099002	1	1	yes	characteristic
195	C₄	#19099005	1	1	yes	characteristic
196	C₂	#23762226	1	1	yes	characteristic
197	C₂	#23765268	1	1	yes	characteristic
198	C₂	#23765278	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14579107

C₂

yes

maximal, characteristic

#23762296

C₂

yes

maximal, characteristic

Q₁₆×C₇₉⋊C₈

C₂

yes

maximal, characteristic

#23761473

C₂

yes

maximal, characteristic

#23761365

C₂

yes

maximal, characteristic

#23761319

C₂

yes

maximal, characteristic

C₇₉×#1332

C₂

yes

maximal, characteristic

C₂×#14579107

C₄

yes

characteristic

#19098690

—

Q₈×C₇₉⋊C₈

—

C₄×Dic₃₁₆

C₂²

yes

characteristic

#19098388

C₄

yes

characteristic

#19098367

C₄

yes

characteristic

#19098366

C₄

yes

characteristic

C₄×C₇₉⋊C₁₆

C₂²

yes

characteristic

#19098316

C₂²

yes

characteristic

Q₈⋊(C₇₉⋊C₈)

—

#19098293

—

C₈×C₇₉⋊C₈

C₂²

yes

characteristic

C₄×Q₁₆×C₇₉

C₂²

yes

characteristic

C₄×C₁₆×C₇₉

C₂²

yes

characteristic

#335×C₇₉

C₂²

yes

characteristic

C₂×Dic₃₁₆

C₂×C₄

yes

characteristic

C₄×Dic₁₅₈

—

#14579107

—

C₂×C₇₉⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₇₉⋊C₁₆

C₂×C₄

yes

characteristic

#14579095

C₂×C₄

yes

characteristic

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

Dic₇₉⋊C₈

—

#14579086

—

#14579071

—

C₄×C₇₉⋊C₈

—

C₄×C₇₉⋊C₈

D₄

yes

characteristic

C₄×C₇₉⋊C₈

D₄

yes

characteristic

C₂×Q₁₆×C₇₉

C₂×C₄

yes

characteristic

C₄×Q₈×C₇₉

—

C₂×C₁₆×C₇₉

—

#107×C₇₉

C₂×C₄

yes

characteristic

C₄⋊C₈×C₇₉

—

#103×C₇₉

—

C₄×C₈×C₇₉

C₂³

yes

characteristic

C₂×Dic₁₅₈

—

Dic₃₁₆

—

Dic₃₁₆

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

D₈

yes

characteristic

C₂×C₇₉⋊C₈

D₈

yes

characteristic

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

C₄×Dic₇₉

—

C₇₉⋊C₁₆

—

C₇₉⋊C₁₆

—

C₂×Q₈×C₇₉

—

Q₁₆×C₇₉

—

Q₁₆×C₇₉

M₁₆

yes

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₉

#41

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₁₆×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₄×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

#135

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

C₂×M₁₆

yes

characteristic

C₈×C₇₉

#131

yes

characteristic

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

C₄×D₈

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

100

C₄×Q₃₂

—

101

#1591

—

102

C₈×Q₁₆

—

103

#1372

—

104

#1332

—

105

#1332

D₇₉

yes

characteristic

106

#1312

—

107

C₇₉

#12076

yes

characteristic, complemented, Sylow

108

C₂×Q₃₂

—

109

#446

—

158

—

110

C₈×Q₈

—

158

—

111

C₄×Q₁₆

—

112

C₄×Q₁₆

C₂×D₇₉

yes

characteristic

113

#366

—

114

#358

—

115

#358

—

116

C₄×C₁₆

—

117

C₄×C₁₆

C₂×D₇₉

yes

characteristic

118

#335

—

119

#335

C₂×D₇₉

yes

characteristic

120

Q₈⋊C₈

—

158

—

121

Q₈⋊C₈

—

158

—

122

C₈²

—

123

C₂×Q₁₆

—

124

C₂×Q₁₆

C₄×D₇₉

yes

characteristic

125

C₄×Q₈

—

126

C₄×Q₈

—

158

—

127

Q₃₂

—

158

—

128

C₂×C₁₆

—

129

C₂×C₁₆

—

130

C₂×C₁₆

—

131

#107

—

132

#107

C₄×D₇₉

yes

characteristic

133

C₄⋊C₈

—

134

C₄⋊C₈

—

316

—

135

C₄⋊C₈

—

316

—

136

C₄⋊C₈

—

158

—

137

#103

—

138

#103

—

158

—

139

#97

—

316

—

140

C₄×C₈

—

316

—

141

C₄×C₈

—

142

C₄×C₈

—

143

C₄×C₈

C₂²×D₇₉

yes

characteristic

144

C₂×Q₈

—

145

C₂×Q₈

—

158

—

146

Q₁₆

—

147

Q₁₆

—

158

—

148

Q₁₆

—

158

—

149

Q₁₆

#11929607

yes

—

150

C₂×C₈

—

151

C₂×C₈

—

316

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

158

—

155

C₂×C₈

C₂×C₄×D₇₉

yes

characteristic

156

C₂×C₈

#11929623

yes

characteristic

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

316

—

160

C₄⋊C₄

—

158

—

161

C₄²

—

162

C₄²

—

316

—

163

C₄²

D₄×D₇₉

yes

characteristic

164

C₁₆

—

165

C₁₆

—

158

—

166

C₁₆

—

158

—

167

Q₈

—

168

Q₈

—

169

Q₈

—

316

—

170

Q₈

—

316

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

316

—

174

C₂×C₄

—

316

—

175

C₂×C₄

(C₂²⋊D₇₉)⋊C₄

yes

characteristic

176

C₂×C₄

D₈×D₇₉

yes

characteristic

177

C₂×C₄

#14579182

yes

characteristic

178

C₈

—

179

C₈

—

180

C₈

—

632

—

181

C₈

—

158

—

182

C₈

—

158

—

183

C₈

—

158

—

184

C₈

C₂×#11929607

yes

characteristic

185

C₈

#14579169

yes

characteristic

186

C₂²

#19098643

yes

characteristic

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

—

632

—

191

C₄

—

632

—

192

C₄

#19098631

yes

characteristic

193

C₄

#19098682

yes

characteristic

194

C₄

#19099002

yes

characteristic

195

C₄

#19099005

yes

characteristic

196

C₂

#23762226

yes

characteristic

197

C₂

#23765268

yes

characteristic

198

C₂

#23765278

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₇₉
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₇₉×#1332	C₂
socle, prime radical	C₂²×C₇₉	C₄×D₈
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	#1332	D₇₉
hypocenter	C₇₉	#12076
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #1332

subgroup

G/H

center

C₂×C₄

D₈×D₇₉

commutator

C₂×C₈×C₇₉

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₇₉

Fitting

C₇₉×#1332

C₂

socle, prime radical

C₂²×C₇₉

C₄×D₈

cosocle

C₄×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #1332

N	H ≅ G/N	type
107. C₇₉	#12076	semidirect

H ≅ G/N

type

107. C₇₉

#12076

semidirect

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

subgroup	index
C₄×#14579107	2
#23762296	2
Q₁₆×C₇₉⋊C₈	2
#23761473	2
#23761365	2
#23761319	2
C₇₉×#1332	2
#12076	79

subgroup

index

Q₁₆×C₇₉⋊C₈

C₇₉×#1332

Subgroups — #34200210

Conjugacy classes of subgroups

Complements

Maximal subgroups