Group #34174822 — subgroups

Subgroups — #34174822

Order 20224.

442 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 135 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×C₂²⋊(C₇₉⋊C₈)	C₂	1	2	yes	maximal
2	C₇₉×#1728	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
4	C₂×C₄×C₇₉⋊C₈	—	2	2	no	—
5	#19098318	C₄	1	4	yes	—
6	C₂²×M₁₆×C₇₉	C₄	1	2	yes	—
7	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
8	#336×C₇₉	—	2	2	no	—
9	C₂²×C₇₉⋊C₈	—	2	4	no	—
10	C₂²⋊(C₇₉⋊C₈)	—	2	8	no	—
11	C₄×C₇₉⋊C₈	—	4	4	no	—
12	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
13	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
14	C₂³×C₄×C₇₉	Q₈	1	1	yes	characteristic
15	C₂×M₁₆×C₇₉	—	2	2	no	—
16	C₂×M₁₆×C₇₉	—	4	2	no	—
17	C₂×M₁₆×C₇₉	—	4	2	no	—
18	C₂×M₁₆×C₇₉	C₈	1	4	yes	complemented
19	C₂²×C₈×C₇₉	—	2	2	no	—
20	C₂×C₄²×C₇₉	—	4	2	no	—
21	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₇₉	D₄	1	2	yes	—
23	C₂×C₇₉⋊C₈	—	2	8	no	—
24	C₂×C₇₉⋊C₈	—	4	4	no	—
25	C₂×C₇₉⋊C₈	—	4	4	no	—
26	C₂⁴×C₇₉	C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₄×C₇₉	—	2	2	no	—
28	C₂²×C₄×C₇₉	—	2	1	no	—
29	C₂²×C₄×C₇₉	—	2	1	no	—
30	C₂²×C₄×C₇₉	—	2	4	no	—
31	C₂²×C₄×C₇₉	—	2	4	no	—
32	C₂²×C₄×C₇₉	—	4	1	no	—
33	C₂²×C₄×C₇₉	—	4	1	no	—
34	C₂²×C₄×C₇₉	C₄²	1	1	yes	characteristic
35	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
36	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
38	C₂²×C₄×C₇₉	C₂×C₈	1	2	yes	—
39	C₂²×C₄×C₇₉	M₁₆	1	2	yes	—
40	M₁₆×C₇₉	—	4	4	no	—
41	M₁₆×C₇₉	—	4	4	no	—
42	C₂×C₈×C₇₉	—	2	4	no	—
43	C₂×C₈×C₇₉	—	4	2	no	—
44	C₂×C₈×C₇₉	—	4	2	no	—
45	C₄²×C₇₉	—	2	4	no	—
46	C₄²×C₇₉	—	4	2	no	—
47	C₂³×C₇₉	—	2	2	no	—
48	C₂³×C₇₉	—	2	2	no	—
49	C₂³×C₇₉	—	2	1	no	—
50	C₂³×C₇₉	—	2	1	no	—
51	C₂³×C₇₉	#95	1	1	yes	characteristic
52	C₂³×C₇₉	C₄⋊C₈	1	2	yes	—
53	C₂×C₄×C₇₉	—	2	2	no	—
54	C₂×C₄×C₇₉	—	2	2	no	—
55	C₂×C₄×C₇₉	—	2	2	no	—
56	C₂×C₄×C₇₉	—	2	2	no	—
57	C₂×C₄×C₇₉	—	2	1	no	—
58	C₂×C₄×C₇₉	—	2	1	no	—
59	C₂×C₄×C₇₉	—	2	2	no	—
60	C₂×C₄×C₇₉	—	2	2	no	—
61	C₂×C₄×C₇₉	—	2	2	no	—
62	C₂×C₄×C₇₉	—	2	2	no	—
63	C₂×C₄×C₇₉	—	4	4	no	—
64	C₂×C₄×C₇₉	—	4	4	no	—
65	C₂×C₄×C₇₉	C₄×C₈	1	1	yes	characteristic
66	C₂×C₄×C₇₉	#97	1	1	yes	characteristic
67	C₂×C₄×C₇₉	C₂²⋊C₈	1	2	yes	—
68	C₂×C₄×C₇₉	#99	1	2	yes	—
69	C₂×C₄×C₇₉	#100	1	1	yes	characteristic
70	C₂×C₄×C₇₉	#101	1	1	yes	characteristic
71	C₂×C₄×C₇₉	#104	1	4	yes	—
72	C₂×C₄×C₇₉	#108	1	4	yes	—
73	C₈×C₇₉	—	4	4	no	—
74	C₇₉⋊C₈	—	4	8	no	—
75	C₂²×C₇₉	—	2	2	no	—
76	C₂²×C₇₉	—	2	4	no	—
77	C₂²×C₇₉	—	4	1	no	—
78	C₂²×C₇₉	—	4	1	no	—
79	C₂²×C₇₉	—	4	2	no	—
80	C₂²×C₇₉	#336	1	1	yes	characteristic
81	C₂²×C₇₉	#339	1	2	yes	—
82	C₂²×C₇₉	#341	1	2	yes	—
83	C₂²×C₇₉	#342	1	1	yes	characteristic
84	C₂²×C₇₉	#343	1	1	yes	characteristic
85	C₄×C₇₉	—	2	1	no	—
86	C₄×C₇₉	—	2	1	no	—
87	C₄×C₇₉	—	2	4	no	—
88	C₄×C₇₉	—	2	4	no	—
89	C₄×C₇₉	#323	1	2	yes	—
90	C₄×C₇₉	#324	1	2	yes	—
91	#7026	—	79	1	no	maximal, Sylow
92	C₂×C₇₉	—	4	2	no	—
93	C₂×C₇₉	#1278	1	2	yes	—
94	C₂×C₇₉	#1280	1	1	yes	characteristic
95	C₂×C₇₉	#1297	1	1	yes	characteristic
96	C₂×C₇₉	#1298	1	1	yes	characteristic
97	C₂×C₇₉	#1299	1	2	yes	—
98	C₄×C₂²⋊C₈	—	79	2	no	—
99	#1728	D₇₉	1	1	yes	characteristic
100	C₇₉	#7026	1	1	yes	characteristic, complemented, Sylow
101	C₂²×M₁₆	Dic₇₉	1	2	yes	—
102	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
103	C₂×C₂²⋊C₈	—	158	2	no	—
104	C₂×C₄×C₈	—	158	2	no	—
105	#336	—	2	2	no	—
106	#336	—	79	4	no	—
107	C₂³×C₄	Dic₁₅₈	1	1	yes	characteristic
108	C₂³×C₄	D₃₁₆	1	1	yes	characteristic
109	C₂³×C₄	C₂×Dic₇₉	1	1	yes	characteristic
110	C₂×M₁₆	—	2	2	no	—
111	C₂×M₁₆	—	4	2	no	—
112	C₂×M₁₆	—	4	2	no	—
113	C₂×M₁₆	C₇₉⋊C₈	1	4	yes	complemented
114	C₂²×C₈	—	2	2	no	—
115	C₂²×C₈	—	158	4	no	—
116	C₂×C₄²	—	4	2	no	—
117	C₂×C₄²	C₄×D₇₉	1	2	yes	—
118	C₂×C₄²	C₂²⋊D₇₉	1	2	yes	—
119	C₂²⋊C₈	—	158	8	no	—
120	C₄×C₈	—	316	4	no	—
121	C₂⁴	C₄⋊Dic₇₉	1	1	yes	characteristic
122	C₂²×C₄	—	2	2	no	—
123	C₂²×C₄	—	2	1	no	—
124	C₂²×C₄	—	2	1	no	—
125	C₂²×C₄	—	2	4	no	—
126	C₂²×C₄	—	2	4	no	—
127	C₂²×C₄	—	4	1	no	—
128	C₂²×C₄	—	4	1	no	—
129	C₂²×C₄	C₄×Dic₇₉	1	1	yes	characteristic
130	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	2	yes	—
131	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
132	C₂²×C₄	Dic₇₉⋊C₄	1	2	yes	—
133	C₂²×C₄	C₂×C₇₉⋊C₈	1	2	yes	—
134	C₂²×C₄	#11929608	1	2	yes	—
135	M₁₆	—	4	4	no	—
136	M₁₆	—	4	4	no	—
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	—	316	4	no	—
141	C₂×C₈	—	316	4	no	—
142	C₂×C₈	—	158	8	no	—
143	C₄²	—	2	4	no	—
144	C₄²	—	4	2	no	—
145	C₂³	—	2	2	no	—
146	C₂³	—	2	2	no	—
147	C₂³	—	2	1	no	—
148	C₂³	—	2	1	no	—
149	C₂³	C₇₉⋊#95	1	1	yes	characteristic
150	C₂³	C₄⋊(C₇₉⋊C₈)	1	2	yes	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	C₄×C₇₉⋊C₈	1	1	yes	characteristic
164	C₂×C₄	#14579072	1	1	yes	characteristic
165	C₂×C₄	C₂²⋊(C₇₉⋊C₈)	1	2	yes	—
166	C₂×C₄	#14579076	1	2	yes	—
167	C₂×C₄	#14579078	1	1	yes	characteristic
168	C₂×C₄	#14579080	1	1	yes	characteristic
169	C₂×C₄	#14579087	1	4	yes	—
170	C₂×C₄	#14579096	1	4	yes	—
171	C₈	—	4	4	no	—
172	C₈	—	316	8	no	—
173	C₂²	—	2	4	no	—
174	C₂²	—	2	2	no	—
175	C₂²	—	4	1	no	—
176	C₂²	—	4	1	no	—
177	C₂²	—	4	2	no	—
178	C₂²	#19098318	1	1	yes	characteristic
179	C₂²	#19098324	1	2	yes	—
180	C₂²	#19098327	1	2	yes	—
181	C₂²	#19098330	1	1	yes	characteristic
182	C₂²	#19098332	1	1	yes	characteristic
183	C₄	—	2	1	no	—
184	C₄	—	2	1	no	—
185	C₄	—	2	4	no	—
186	C₄	—	2	4	no	—
187	C₄	#19098287	1	2	yes	—
188	C₄	#19098289	1	2	yes	—
189	C₂	—	4	2	no	—
190	C₂	#23761238	1	2	yes	—
191	C₂	#23761243	1	1	yes	characteristic
192	C₂	#23761285	1	1	yes	characteristic
193	C₂	#23761287	1	1	yes	characteristic
194	C₂	#23761289	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₂²⋊(C₇₉⋊C₈)

C₂

yes

maximal

C₇₉×#1728

C₂

yes

maximal, characteristic

C₂×C₂²⋊(C₇₉⋊C₈)

—

C₂×C₄×C₇₉⋊C₈

—

#19098318

C₄

yes

—

C₂²×M₁₆×C₇₉

C₄

yes

—

C₂²×C₄²×C₇₉

C₂²

yes

characteristic

#336×C₇₉

—

C₂²×C₇₉⋊C₈

—

C₂²⋊(C₇₉⋊C₈)

—

C₄×C₇₉⋊C₈

—

C₂³×C₄×C₇₉

C₂×C₄

yes

characteristic

C₂³×C₄×C₇₉

D₄

yes

characteristic

C₂³×C₄×C₇₉

Q₈

yes

characteristic

C₂×M₁₆×C₇₉

—

C₂×M₁₆×C₇₉

—

C₂×M₁₆×C₇₉

—

C₂×M₁₆×C₇₉

C₈

yes

complemented

C₂²×C₈×C₇₉

—

C₂×C₄²×C₇₉

—

C₂×C₄²×C₇₉

C₂×C₄

yes

—

C₂×C₄²×C₇₉

D₄

yes

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂⁴×C₇₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₄²

yes

characteristic

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₉

C₄⋊C₄

yes

—

C₂²×C₄×C₇₉

C₂×C₈

yes

—

C₂²×C₄×C₇₉

M₁₆

yes

—

M₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

#95

yes

characteristic

C₂³×C₇₉

C₄⋊C₈

yes

—

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

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₄×C₈

yes

characteristic

C₂×C₄×C₇₉

#97

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊C₈

yes

—

C₂×C₄×C₇₉

#99

yes

—

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

—

C₂×C₄×C₇₉

#108

yes

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#336

yes

characteristic

C₂²×C₇₉

#339

yes

—

C₂²×C₇₉

#341

yes

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#323

yes

—

C₄×C₇₉

#324

yes

—

#7026

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

#1278

yes

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₄×C₂²⋊C₈

—

#1728

D₇₉

yes

characteristic

100

C₇₉

#7026

yes

characteristic, complemented, Sylow

101

C₂²×M₁₆

Dic₇₉

yes

—

102

C₂²×C₄²

C₂×D₇₉

yes

characteristic

103

C₂×C₂²⋊C₈

—

158

—

104

C₂×C₄×C₈

—

158

—

105

#336

—

106

#336

—

107

C₂³×C₄

Dic₁₅₈

yes

characteristic

108

C₂³×C₄

D₃₁₆

yes

characteristic

109

C₂³×C₄

C₂×Dic₇₉

yes

characteristic

110

C₂×M₁₆

—

111

C₂×M₁₆

—

112

C₂×M₁₆

—

113

C₂×M₁₆

C₇₉⋊C₈

yes

complemented

114

C₂²×C₈

—

115

C₂²×C₈

—

158

—

116

C₂×C₄²

—

117

C₂×C₄²

C₄×D₇₉

yes

—

118

C₂×C₄²

C₂²⋊D₇₉

yes

—

119

C₂²⋊C₈

—

158

—

120

C₄×C₈

—

316

—

121

C₂⁴

C₄⋊Dic₇₉

yes

characteristic

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

C₄×Dic₇₉

yes

characteristic

130

C₂²×C₄

(C₂×D₇₉)⋊C₄

yes

—

131

C₂²×C₄

C₂²⋊Dic₇₉

yes

characteristic

132

C₂²×C₄

Dic₇₉⋊C₄

yes

—

133

C₂²×C₄

C₂×C₇₉⋊C₈

yes

—

134

C₂²×C₄

#11929608

yes

—

135

M₁₆

—

136

M₁₆

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

316

—

141

C₂×C₈

—

316

—

142

C₂×C₈

—

158

—

143

C₄²

—

144

C₄²

—

145

C₂³

—

146

C₂³

—

147

C₂³

—

148

C₂³

—

149

C₂³

C₇₉⋊#95

yes

characteristic

150

C₂³

C₄⋊(C₇₉⋊C₈)

yes

—

151

C₂×C₄

—

152

C₂×C₄

—

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

—

162

C₂×C₄

—

163

C₂×C₄

C₄×C₇₉⋊C₈

yes

characteristic

164

C₂×C₄

#14579072

yes

characteristic

165

C₂×C₄

C₂²⋊(C₇₉⋊C₈)

yes

—

166

C₂×C₄

#14579076

yes

—

167

C₂×C₄

#14579078

yes

characteristic

168

C₂×C₄

#14579080

yes

characteristic

169

C₂×C₄

#14579087

yes

—

170

C₂×C₄

#14579096

yes

—

171

C₈

—

172

C₈

—

316

—

173

C₂²

—

174

C₂²

—

175

C₂²

—

176

C₂²

—

177

C₂²

—

178

C₂²

#19098318

yes

characteristic

179

C₂²

#19098324

yes

—

180

C₂²

#19098327

yes

—

181

C₂²

#19098330

yes

characteristic

182

C₂²

#19098332

yes

characteristic

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

#19098287

yes

—

188

C₄

#19098289

yes

—

189

C₂

—

190

C₂

#23761238

yes

—

191

C₂

#23761243

yes

characteristic

192

C₂

#23761285

yes

characteristic

193

C₂

#23761287

yes

characteristic

194

C₂

#23761289

yes

—

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₇₉
commutator	C₂×C₄×C₇₉	C₄×C₈
Frattini	C₂²×C₄²	C₂×D₇₉
Fitting	C₇₉×#1728	C₂
socle	C₂³×C₇₉	#95
prime radical	C₂⁴×C₇₉	C₄⋊C₄
cosocle	C₂²×C₄²×C₇₉	C₂²
hypercenter	#1728	D₇₉
hypocenter	C₇₉	#7026
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #1728

subgroup

G/H

center

C₂²×C₄

C₂²⋊Dic₇₉

commutator

C₂×C₄×C₇₉

C₄×C₈

Frattini

C₂²×C₄²

C₂×D₇₉

Fitting

C₇₉×#1728

C₂

socle

C₂³×C₇₉

#95

prime radical

C₂⁴×C₇₉

C₄⋊C₄

cosocle

C₂²×C₄²×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #1728

N	H ≅ G/N	type
18. C₂×M₁₆×C₇₉	C₈	semidirect
100. C₇₉	#7026	semidirect
113. C₂×M₁₆	C₇₉⋊C₈	semidirect

H ≅ G/N

type

18. C₂×M₁₆×C₇₉

C₈

semidirect

100. C₇₉

#7026

semidirect

113. C₂×M₁₆

C₇₉⋊C₈

semidirect

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

subgroup	index
C₄×C₂²⋊(C₇₉⋊C₈)	2
C₇₉×#1728	2
#7026	79

subgroup

index

C₄×C₂²⋊(C₇₉⋊C₈)

C₇₉×#1728

#7026

Subgroups — #34174822

Conjugacy classes of subgroups

Complements

Maximal subgroups