Group #34207658 — subgroups

Subgroups — #34207658

Order 20224.

436 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 101 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×#19098379	C₂	1	2	yes	maximal
2	#23765120	C₂	1	2	yes	maximal
3	#23764148	C₂	1	1	yes	maximal, characteristic
4	#23761482	C₂	1	1	yes	maximal, characteristic
5	C₂×#494×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579101	C₂²	1	2	yes	—
7	C₂×#14579097	C₂²	1	1	yes	characteristic
8	#19098601	C₂²	1	1	yes	characteristic
9	#19098589	—	2	1	no	—
10	#19098379	—	2	4	no	—
11	#19098367	—	2	4	no	—
12	#19098323	—	2	1	no	—
13	C₂²×Q₁₆×C₇₉	—	2	1	no	—
14	C₂²×Q₁₆×C₇₉	C₂²	1	2	yes	—
15	C₂×C₄⋊Q₈×C₇₉	—	2	1	no	—
16	#494×C₇₉	—	2	4	no	—
17	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
18	C₂×#11929608	—	2	1	no	—
19	#14579120	—	2	1	no	—
20	#14579101	—	2	2	no	—
21	#14579101	—	2	2	no	—
22	C₂×C₇₉⋊C₁₆	—	2	2	no	—
23	#14579097	—	2	2	no	—
24	#14579095	D₄	1	2	yes	—
25	#14579093	—	2	1	no	—
26	Dic₇₉⋊C₈	—	4	2	no	—
27	C₂²×Q₈×C₇₉	—	2	2	no	—
28	C₂×Q₁₆×C₇₉	—	2	4	no	—
29	C₂×Q₁₆×C₇₉	—	2	2	no	—
30	C₂×Q₁₆×C₇₉	—	2	2	no	—
31	C₂×Q₁₆×C₇₉	—	2	2	no	—
32	C₂×Q₁₆×C₇₉	—	2	2	no	—
33	C₂×Q₁₆×C₇₉	—	2	2	no	—
34	C₂×Q₁₆×C₇₉	—	4	2	no	—
35	C₂×Q₁₆×C₇₉	—	4	2	no	—
36	C₂×Q₁₆×C₇₉	D₄	1	4	yes	—
37	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
39	C₄⋊Q₈×C₇₉	—	2	4	no	—
40	C₄⋊Q₈×C₇₉	—	4	2	no	—
41	C₂×C₄⋊C₄×C₇₉	—	4	2	no	—
42	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
43	C₄×C₈×C₇₉	—	2	2	no	—
44	#11929608	—	4	1	no	—
45	#11929608	—	4	1	no	—
46	C₂×C₇₉⋊C₈	—	4	1	no	—
47	C₄⋊Dic₇₉	—	2	2	no	—
48	C₂²⋊Dic₇₉	—	4	1	no	—
49	C₄×Dic₇₉	—	4	1	no	—
50	C₇₉⋊C₁₆	—	4	2	no	—
51	C₂×Q₈×C₇₉	—	2	4	no	—
52	C₂×Q₈×C₇₉	—	4	2	no	—
53	C₂×Q₈×C₇₉	—	4	2	no	—
54	C₂×Q₈×C₇₉	—	4	2	no	—
55	C₂×Q₈×C₇₉	—	4	2	no	—
56	C₂×Q₈×C₇₉	—	4	2	no	—
57	C₂²×C₄×C₇₉	—	2	1	no	—
58	C₂²×C₄×C₇₉	—	4	2	no	—
59	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
60	Q₁₆×C₇₉	—	2	4	no	—
61	Q₁₆×C₇₉	—	2	4	no	—
62	Q₁₆×C₇₉	—	4	8	no	—
63	Q₁₆×C₇₉	—	4	2	no	—
64	Q₁₆×C₇₉	—	4	2	no	—
65	C₂×C₈×C₇₉	—	2	1	no	—
66	C₂×C₈×C₇₉	—	2	1	no	—
67	C₂×C₈×C₇₉	D₈	1	2	yes	—
68	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
71	C₂×C₈×C₇₉	#41	1	2	yes	—
72	C₄⋊C₄×C₇₉	—	4	8	no	—
73	C₄²×C₇₉	—	2	1	no	—
74	C₄²×C₇₉	SD₁₆	1	2	yes	—
75	C₂³×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
76	Q₈×C₇₉	—	4	4	no	—
77	Q₈×C₇₉	—	4	4	no	—
78	Q₈×C₇₉	—	8	2	no	—
79	Q₈×C₇₉	—	8	2	no	—
80	C₂×C₄×C₇₉	—	2	4	no	—
81	C₂×C₄×C₇₉	—	2	2	no	—
82	C₂×C₄×C₇₉	—	4	4	no	—
83	C₂×C₄×C₇₉	—	8	2	no	—
84	C₂×C₄×C₇₉	—	8	2	no	—
85	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
86	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
87	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
88	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
90	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
91	C₂×Dic₇₉	—	4	2	no	—
92	C₈×C₇₉	—	2	2	no	—
93	C₈×C₇₉	#121	1	2	yes	—
94	C₈×C₇₉	#123	1	2	yes	—
95	C₇₉⋊C₈	—	8	1	no	—
96	C₂²×C₇₉	—	2	2	no	—
97	C₂²×C₇₉	#450	1	1	yes	characteristic
98	C₂²×C₇₉	#459	1	1	yes	characteristic
99	C₂²×C₇₉	#490	1	1	yes	characteristic
100	C₄×C₇₉	—	2	4	no	—
101	C₄×C₇₉	—	8	4	no	—
102	C₄×C₇₉	#394	1	1	yes	characteristic
103	C₄×C₇₉	#447	1	1	yes	characteristic
104	C₄×C₇₉	#460	1	1	yes	characteristic
105	C₄×C₇₉	#486	1	1	yes	characteristic
106	Dic₇₉	—	8	2	no	—
107	#13415	—	79	1	no	maximal, Sylow
108	C₂×C₇₉	—	2	1	no	—
109	C₂×C₇₉	—	2	1	no	—
110	C₂×C₇₉	#2041	1	1	yes	characteristic
111	C₂×C₇₉	#2214	1	2	yes	—
112	C₂×#494	D₇₉	1	1	yes	characteristic
113	C₂×#362	—	79	2	no	—
114	#2141	—	79	2	no	—
115	#1945	—	79	1	no	—
116	#1376	—	79	1	no	—
117	C₇₉	#13415	1	1	yes	characteristic, complemented, Sylow
118	C₂²×Q₁₆	—	2	1	no	—
119	C₂²×Q₁₆	C₂×D₇₉	1	2	yes	—
120	C₂×C₄⋊Q₈	—	2	1	no	—
121	C₂×M₃₂	—	79	2	no	—
122	#494	—	2	4	no	—
123	C₂×#108	—	79	1	no	—
124	#427	—	79	1	no	—
125	#424	—	158	1	no	—
126	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
127	#362	—	158	4	no	—
128	#358	—	158	4	no	—
129	#339	—	158	1	no	—
130	C₂²×Q₈	—	2	2	no	—
131	C₂×Q₁₆	—	2	4	no	—
132	C₂×Q₁₆	—	2	2	no	—
133	C₂×Q₁₆	—	2	2	no	—
134	C₂×Q₁₆	—	2	2	no	—
135	C₂×Q₁₆	—	2	2	no	—
136	C₂×Q₁₆	—	2	2	no	—
137	C₂×Q₁₆	—	4	2	no	—
138	C₂×Q₁₆	—	4	2	no	—
139	C₂×Q₁₆	C₂²⋊D₇₉	1	4	yes	—
140	C₂×M₁₆	—	158	1	no	—
141	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
142	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
143	C₄⋊Q₈	—	2	4	no	—
144	C₄⋊Q₈	—	4	2	no	—
145	#117	—	158	1	no	—
146	C₂×C₄⋊C₄	—	4	2	no	—
147	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
148	M₃₂	—	158	2	no	—
149	M₃₂	—	158	2	no	—
150	C₂×C₁₆	—	158	2	no	—
151	#108	—	158	2	no	—
152	#107	—	79	2	no	—
153	#106	—	158	1	no	—
154	C₄⋊C₈	—	316	2	no	—
155	C₄×C₈	—	2	2	no	—
156	C₂×Q₈	—	2	4	no	—
157	C₂×Q₈	—	4	2	no	—
158	C₂×Q₈	—	4	2	no	—
159	C₂×Q₈	—	4	2	no	—
160	C₂×Q₈	—	4	2	no	—
161	C₂×Q₈	—	4	2	no	—
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	—	4	2	no	—
164	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
165	Q₁₆	—	2	4	no	—
166	Q₁₆	—	2	4	no	—
167	Q₁₆	—	4	8	no	—
168	Q₁₆	—	4	2	no	—
169	Q₁₆	—	4	2	no	—
170	M₁₆	—	316	1	no	—
171	M₁₆	—	316	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	316	1	no	—
175	C₂×C₈	#11929609	1	2	yes	—
176	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
177	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
178	C₂×C₈	#11929623	1	2	yes	—
179	C₂×C₈	#11929623	1	1	yes	characteristic
180	C₄⋊C₄	—	4	8	no	—
181	C₄⋊C₄	—	158	2	no	—
182	C₂²⋊C₄	—	316	1	no	—
183	C₄²	—	2	1	no	—
184	C₄²	—	316	1	no	—
185	C₄²	#11929612	1	2	yes	—
186	C₁₆	—	316	2	no	—
187	C₂³	#14579187	1	1	yes	characteristic
188	Q₈	—	4	4	no	—
189	Q₈	—	4	4	no	—
190	Q₈	—	8	2	no	—
191	Q₈	—	8	2	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	4	4	no	—
195	C₂×C₄	—	8	2	no	—
196	C₂×C₄	—	8	2	no	—
197	C₂×C₄	—	316	2	no	—
198	C₂×C₄	#14579130	1	1	yes	characteristic
199	C₂×C₄	#14579132	1	1	yes	characteristic
200	C₂×C₄	#14579148	1	1	yes	characteristic
201	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
202	C₂×C₄	C₂×#11929612	1	1	yes	characteristic
203	C₂×C₄	#14579187	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	—	632	1	no	—
206	C₈	#14579133	1	2	yes	—
207	C₈	#14579142	1	2	yes	—
208	C₂²	—	2	2	no	—
209	C₂²	#19098714	1	1	yes	characteristic
210	C₂²	#19098755	1	1	yes	characteristic
211	C₂²	#19098927	1	1	yes	characteristic
212	C₄	—	2	4	no	—
213	C₄	—	8	4	no	—
214	C₄	—	632	2	no	—
215	C₄	#19098477	1	1	yes	characteristic
216	C₄	#19098697	1	1	yes	characteristic
217	C₄	#19098760	1	1	yes	characteristic
218	C₄	#19098909	1	1	yes	characteristic
219	C₂	—	2	1	no	—
220	C₂	—	2	1	no	—
221	C₂	#23764631	1	1	yes	characteristic
222	C₂	#23765478	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19098379

C₂

yes

maximal

#23765120

C₂

yes

maximal

#23764148

C₂

yes

maximal, characteristic

#23761482

C₂

yes

maximal, characteristic

C₂×#494×C₇₉

C₂

yes

maximal, characteristic

C₂×#14579101

C₂²

yes

—

C₂×#14579097

C₂²

yes

characteristic

#19098601

C₂²

yes

—

—

—

—

C₂²×Q₁₆×C₇₉

—

C₂²×Q₁₆×C₇₉

C₂²

yes

—

C₂×C₄⋊Q₈×C₇₉

—

#494×C₇₉

—

C₂×C₄×C₈×C₇₉

C₂²

yes

—

—

—

—

C₂×C₇₉⋊C₁₆

—

#14579097

—

#14579095

D₄

yes

—

#14579093

—

Dic₇₉⋊C₈

—

C₂²×Q₈×C₇₉

—

C₂×Q₁₆×C₇₉

—

C₂×Q₁₆×C₇₉

—

C₂×Q₁₆×C₇₉

—

C₂×Q₁₆×C₇₉

—

C₂×Q₁₆×C₇₉

—

C₂×Q₁₆×C₇₉

—

C₂×Q₁₆×C₇₉

—

C₂×Q₁₆×C₇₉

—

C₂×Q₁₆×C₇₉

D₄

yes

—

C₂²×C₈×C₇₉

D₄

yes

characteristic

C₂²×C₈×C₇₉

C₂³

yes

characteristic

C₄⋊Q₈×C₇₉

—

C₄⋊Q₈×C₇₉

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄²×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

—

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₇₉⋊C₁₆

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

Q₁₆×C₇₉

—

Q₁₆×C₇₉

—

Q₁₆×C₇₉

—

Q₁₆×C₇₉

—

Q₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

D₈

yes

—

C₂×C₈×C₇₉

C₂×D₄

yes

—

C₂×C₈×C₇₉

C₂×D₄

yes

C₂×C₈×C₇₉

yes

C₂×C₈×C₇₉

yes

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

SD₁₆

yes

—

C₂³×C₇₉

AGL(1,ℤ/8)

yes

characteristic

Q₈×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₉

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₇₉

—

C₈×C₇₉

—

C₈×C₇₉

#121

yes

—

C₈×C₇₉

#123

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

100

C₄×C₇₉

—

101

C₄×C₇₉

—

102

C₄×C₇₉

#394

yes

characteristic

103

C₄×C₇₉

#447

yes

characteristic

104

C₄×C₇₉

#460

yes

characteristic

105

C₄×C₇₉

#486

yes

characteristic

106

Dic₇₉

—

107

#13415

—

maximal, Sylow

108

C₂×C₇₉

—

109

C₂×C₇₉

—

110

C₂×C₇₉

#2041

yes

characteristic

111

C₂×C₇₉

#2214

yes

—

112

C₂×#494

D₇₉

yes

characteristic

113

C₂×#362

—

114

#2141

—

115

#1945

—

116

#1376

—

117

C₇₉

#13415

yes

characteristic, complemented, Sylow

118

C₂²×Q₁₆

—

119

C₂²×Q₁₆

C₂×D₇₉

yes

—

120

C₂×C₄⋊Q₈

—

121

C₂×M₃₂

—

122

#494

—

123

C₂×#108

—

124

#427

—

125

#424

—

158

—

126

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

127

#362

—

158

—

128

#358

—

158

—

129

#339

—

158

—

130

C₂²×Q₈

—

131

C₂×Q₁₆

—

132

C₂×Q₁₆

—

133

C₂×Q₁₆

—

134

C₂×Q₁₆

—

135

C₂×Q₁₆

—

136

C₂×Q₁₆

—

137

C₂×Q₁₆

—

138

C₂×Q₁₆

—

139

C₂×Q₁₆

C₂²⋊D₇₉

yes

—

140

C₂×M₁₆

—

158

—

141

C₂²×C₈

C₂²⋊D₇₉

yes

characteristic

142

C₂²×C₈

C₂²×D₇₉

yes

characteristic

143

C₄⋊Q₈

—

144

C₄⋊Q₈

—

145

#117

—

158

—

146

C₂×C₄⋊C₄

—

147

C₂×C₄²

C₂²⋊D₇₉

yes

characteristic

148

M₃₂

—

158

—

149

M₃₂

—

158

—

150

C₂×C₁₆

—

158

—

151

#108

—

158

—

152

#107

—

153

#106

—

158

—

154

C₄⋊C₈

—

316

—

155

C₄×C₈

—

156

C₂×Q₈

—

157

C₂×Q₈

—

158

C₂×Q₈

—

159

C₂×Q₈

—

160

C₂×Q₈

—

161

C₂×Q₈

—

162

C₂²×C₄

—

163

C₂²×C₄

—

164

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

165

Q₁₆

—

166

Q₁₆

—

167

Q₁₆

—

168

Q₁₆

—

169

Q₁₆

—

170

M₁₆

—

316

—

171

M₁₆

—

316

—

172

C₂×C₈

—

173

C₂×C₈

—

174

C₂×C₈

—

316

—

175

C₂×C₈

#11929609

yes

—

176

C₂×C₈

D₄×D₇₉

yes

characteristic

177

C₂×C₈

C₂×C₂²⋊D₇₉

yes

—

178

C₂×C₈

#11929623

yes

—

179

C₂×C₈

#11929623

yes

characteristic

180

C₄⋊C₄

—

181

C₄⋊C₄

—

158

—

182

C₂²⋊C₄

—

316

—

183

C₄²

—

184

C₄²

—

316

—

185

C₄²

#11929612

yes

—

186

C₁₆

—

316

—

187

C₂³

#14579187

yes

characteristic

188

Q₈

—

189

Q₈

—

190

Q₈

—

191

Q₈

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

316

—

198

C₂×C₄

#14579130

yes

characteristic

199

C₂×C₄

#14579132

yes

characteristic

200

C₂×C₄

#14579148

yes

characteristic

201

C₂×C₄

C₂×#11929609

yes

characteristic

202

C₂×C₄

C₂×#11929612

yes

characteristic

203

C₂×C₄

#14579187

yes

characteristic

204

C₈

—

205

C₈

—

632

—

206

C₈

#14579133

yes

—

207

C₈

#14579142

yes

—

208

C₂²

—

209

C₂²

#19098714

yes

characteristic

210

C₂²

#19098755

yes

characteristic

211

C₂²

#19098927

yes

characteristic

212

C₄

—

213

C₄

—

214

C₄

—

632

—

215

C₄

#19098477

yes

characteristic

216

C₄

#19098697

yes

characteristic

217

C₄

#19098760

yes

characteristic

218

C₄

#19098909

yes

characteristic

219

C₂

—

220

C₂

—

221

C₂

#23764631

yes

characteristic

222

C₂

#23765478

yes

—

	subgroup	G/H
center	C₂²	#19098755
commutator, cosocle	C₂²×C₈×C₇₉	C₂³
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×#494×C₇₉	C₂
socle	C₂²×C₇₉	#459
prime radical	C₂³×C₇₉	AGL(1,ℤ/8)
hypercenter	C₂×#494	D₇₉
hypocenter	C₇₉	#13415
LCS	G ▷ C₂²×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#494

subgroup

G/H

center

C₂²

#19098755

commutator, cosocle

C₂²×C₈×C₇₉

C₂³

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₂×#494×C₇₉

C₂

C₂²×C₇₉

C₂³×C₇₉

D₇₉

C₇₉

G ▷ C₂²×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#494

N	H ≅ G/N	type
117. C₇₉	#13415	semidirect

H ≅ G/N

type

117. C₇₉

#13415

semidirect

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

subgroup	index
C₂×#19098379	2
#23765120	2
#23764148	2
#23761482	2
C₂×#494×C₇₉	2
#13415	79

subgroup

index

C₂×#494×C₇₉

Subgroups — #34207658

Conjugacy classes of subgroups

Complements

Maximal subgroups