Group #34207646 — subgroups

Subgroups — #34207646

Order 20224.

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

normal subgroups only

This group has 540 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×#19098376	C₂	1	2	yes	maximal, complemented
2	#23765116	C₂	1	2	yes	maximal, complemented
3	#23764148	C₂	1	1	yes	maximal, characteristic, complemented
4	#23761482	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#493×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	#19098376	—	2	4	no	—
11	#19098364	—	2	4	no	—
12	#19098323	—	2	1	no	—
13	C₂²×D₈×C₇₉	—	2	1	no	—
14	C₂²×D₈×C₇₉	C₂²	1	2	yes	—
15	C₂×#127×C₇₉	—	2	1	no	—
16	#493×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	complemented
25	#14579093	—	2	1	no	—
26	Dic₇₉⋊C₈	—	4	2	no	—
27	C₂²×D₄×C₇₉	—	2	2	no	—
28	C₂²×D₄×C₇₉	—	4	2	no	—
29	C₂×D₈×C₇₉	—	2	4	no	—
30	C₂×D₈×C₇₉	—	2	2	no	—
31	C₂×D₈×C₇₉	—	2	2	no	—
32	C₂×D₈×C₇₉	—	2	2	no	—
33	C₂×D₈×C₇₉	—	2	2	no	—
34	C₂×D₈×C₇₉	—	2	2	no	—
35	C₂×D₈×C₇₉	—	4	2	no	—
36	C₂×D₈×C₇₉	—	4	2	no	—
37	C₂×D₈×C₇₉	D₄	1	4	yes	—
38	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
40	#127×C₇₉	—	2	4	no	—
41	#127×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	C₂⁴×C₇₉	—	4	2	no	—
52	C₂×D₄×C₇₉	—	2	4	no	—
53	C₂×D₄×C₇₉	—	4	8	no	—
54	C₂×D₄×C₇₉	—	4	2	no	—
55	C₂×D₄×C₇₉	—	4	2	no	—
56	C₂×D₄×C₇₉	—	4	2	no	—
57	C₂×D₄×C₇₉	—	4	2	no	—
58	C₂×D₄×C₇₉	—	4	2	no	—
59	C₂×D₄×C₇₉	—	8	8	no	—
60	C₂²×C₄×C₇₉	—	2	1	no	—
61	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
62	D₈×C₇₉	—	2	4	no	—
63	D₈×C₇₉	—	2	4	no	—
64	D₈×C₇₉	—	4	8	no	—
65	D₈×C₇₉	—	4	2	no	—
66	D₈×C₇₉	—	4	2	no	—
67	C₂×C₈×C₇₉	—	2	1	no	—
68	C₂×C₈×C₇₉	—	2	1	no	—
69	C₂×C₈×C₇₉	D₈	1	2	yes	—
70	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
73	C₂×C₈×C₇₉	#41	1	2	yes	—
74	C₄²×C₇₉	—	2	1	no	—
75	C₄²×C₇₉	SD₁₆	1	2	yes	—
76	C₂³×C₇₉	—	4	4	no	—
77	C₂³×C₇₉	—	8	8	no	—
78	C₂³×C₇₉	—	8	2	no	—
79	C₂³×C₇₉	—	8	2	no	—
80	C₂³×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
81	D₄×C₇₉	—	4	4	no	—
82	D₄×C₇₉	—	4	4	no	—
83	D₄×C₇₉	—	8	16	no	—
84	D₄×C₇₉	—	8	2	no	—
85	D₄×C₇₉	—	8	2	no	—
86	C₂×C₄×C₇₉	—	2	4	no	—
87	C₂×C₄×C₇₉	—	2	2	no	—
88	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
89	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
90	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
91	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
93	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
94	C₂×Dic₇₉	—	4	2	no	—
95	C₈×C₇₉	—	2	2	no	—
96	C₈×C₇₉	#121	1	2	yes	—
97	C₈×C₇₉	#123	1	2	yes	—
98	C₇₉⋊C₈	—	8	1	no	—
99	C₂²×C₇₉	—	2	2	no	—
100	C₂²×C₇₉	—	8	8	no	—
101	C₂²×C₇₉	—	8	4	no	—
102	C₂²×C₇₉	—	16	4	no	—
103	C₂²×C₇₉	—	16	4	no	—
104	C₂²×C₇₉	#450	1	1	yes	characteristic
105	C₂²×C₇₉	#459	1	1	yes	characteristic
106	C₂²×C₇₉	#490	1	1	yes	characteristic
107	C₄×C₇₉	—	2	4	no	—
108	C₄×C₇₉	#394	1	1	yes	characteristic
109	C₄×C₇₉	#447	1	1	yes	characteristic
110	C₄×C₇₉	#460	1	1	yes	characteristic
111	C₄×C₇₉	#486	1	1	yes	characteristic
112	Dic₇₉	—	8	2	no	—
113	#13413	—	79	1	no	maximal, Sylow
114	C₂×C₇₉	—	2	1	no	—
115	C₂×C₇₉	—	2	1	no	—
116	C₂×C₇₉	—	16	4	no	—
117	C₂×C₇₉	#2041	1	1	yes	characteristic
118	C₂×C₇₉	#2213	1	2	yes	—
119	C₂×#493	D₇₉	1	1	yes	characteristic
120	C₂×#361	—	79	2	no	—
121	#2140	—	79	2	no	—
122	#1945	—	79	1	no	—
123	#1376	—	79	1	no	—
124	C₇₉	#13413	1	1	yes	characteristic, complemented, Sylow
125	C₂²×D₈	—	2	1	no	—
126	C₂²×D₈	C₂×D₇₉	1	2	yes	—
127	C₂×#127	—	2	1	no	—
128	C₂×M₃₂	—	79	2	no	—
129	#493	—	2	4	no	—
130	C₂×#108	—	79	1	no	—
131	#427	—	79	1	no	—
132	#424	—	158	1	no	—
133	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
134	#361	—	158	4	no	—
135	#357	—	158	4	no	—
136	#339	—	158	1	no	—
137	C₂²×D₄	—	2	2	no	—
138	C₂²×D₄	—	4	2	no	—
139	C₂×D₈	—	2	4	no	—
140	C₂×D₈	—	2	2	no	—
141	C₂×D₈	—	2	2	no	—
142	C₂×D₈	—	2	2	no	—
143	C₂×D₈	—	2	2	no	—
144	C₂×D₈	—	2	2	no	—
145	C₂×D₈	—	4	2	no	—
146	C₂×D₈	—	4	2	no	—
147	C₂×D₈	C₂²⋊D₇₉	1	4	yes	—
148	C₂×M₁₆	—	158	1	no	—
149	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
150	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
151	#127	—	2	4	no	—
152	#127	—	4	2	no	—
153	#117	—	158	1	no	—
154	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
155	M₃₂	—	158	2	no	—
156	M₃₂	—	158	2	no	—
157	C₂×C₁₆	—	158	2	no	—
158	#108	—	158	2	no	—
159	#107	—	79	2	no	—
160	#106	—	158	1	no	—
161	C₄⋊C₈	—	316	2	no	—
162	C₄×C₈	—	2	2	no	—
163	C₂⁴	—	4	2	no	—
164	C₂×D₄	—	2	4	no	—
165	C₂×D₄	—	4	8	no	—
166	C₂×D₄	—	4	2	no	—
167	C₂×D₄	—	4	2	no	—
168	C₂×D₄	—	4	2	no	—
169	C₂×D₄	—	4	2	no	—
170	C₂×D₄	—	4	2	no	—
171	C₂×D₄	—	8	8	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
174	D₈	—	2	4	no	—
175	D₈	—	2	4	no	—
176	D₈	—	4	8	no	—
177	D₈	—	4	2	no	—
178	D₈	—	4	2	no	—
179	M₁₆	—	316	1	no	—
180	M₁₆	—	316	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	316	1	no	—
184	C₂×C₈	#11929609	1	2	yes	—
185	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
186	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
187	C₂×C₈	#11929623	1	2	yes	—
188	C₂×C₈	#11929623	1	1	yes	characteristic
189	C₄⋊C₄	—	158	2	no	—
190	C₂²⋊C₄	—	316	1	no	—
191	C₄²	—	2	1	no	—
192	C₄²	—	316	1	no	—
193	C₄²	#11929612	1	2	yes	—
194	C₁₆	—	316	2	no	—
195	C₂³	—	4	4	no	—
196	C₂³	—	8	8	no	—
197	C₂³	—	8	2	no	—
198	C₂³	—	8	2	no	—
199	C₂³	#14579187	1	1	yes	characteristic
200	D₄	—	4	4	no	—
201	D₄	—	4	4	no	—
202	D₄	—	8	16	no	—
203	D₄	—	8	2	no	—
204	D₄	—	8	2	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	316	2	no	—
208	C₂×C₄	#14579130	1	1	yes	characteristic
209	C₂×C₄	#14579132	1	1	yes	characteristic
210	C₂×C₄	#14579148	1	1	yes	characteristic
211	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
212	C₂×C₄	C₂×#11929612	1	1	yes	characteristic
213	C₂×C₄	#14579187	1	1	yes	characteristic
214	C₈	—	2	2	no	—
215	C₈	—	632	1	no	—
216	C₈	#14579133	1	2	yes	—
217	C₈	#14579142	1	2	yes	—
218	C₂²	—	2	2	no	—
219	C₂²	—	8	8	no	—
220	C₂²	—	8	4	no	—
221	C₂²	—	16	4	no	—
222	C₂²	—	16	4	no	—
223	C₂²	#19098714	1	1	yes	characteristic
224	C₂²	#19098755	1	1	yes	characteristic
225	C₂²	#19098927	1	1	yes	characteristic
226	C₄	—	2	4	no	—
227	C₄	—	632	2	no	—
228	C₄	#19098477	1	1	yes	characteristic
229	C₄	#19098697	1	1	yes	characteristic
230	C₄	#19098760	1	1	yes	characteristic
231	C₄	#19098909	1	1	yes	characteristic
232	C₂	—	2	1	no	—
233	C₂	—	2	1	no	—
234	C₂	—	16	4	no	—
235	C₂	#23764631	1	1	yes	characteristic
236	C₂	#23765473	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19098376

C₂

yes

maximal, complemented

#23765116

C₂

yes

maximal, complemented

#23764148

C₂

yes

maximal, characteristic, complemented

#23761482

C₂

yes

maximal, characteristic, complemented

C₂×#493×C₇₉

C₂

yes

maximal, characteristic

C₂×#14579101

C₂²

yes

—

C₂×#14579097

C₂²

yes

characteristic

#19098601

C₂²

yes

—

—

—

—

C₂²×D₈×C₇₉

—

C₂²×D₈×C₇₉

C₂²

yes

—

C₂×#127×C₇₉

—

#493×C₇₉

—

C₂×C₄×C₈×C₇₉

C₂²

yes

—

—

—

—

C₂×C₇₉⋊C₁₆

—

#14579097

—

#14579095

D₄

yes

complemented

#14579093

—

Dic₇₉⋊C₈

—

C₂²×D₄×C₇₉

—

C₂²×D₄×C₇₉

—

C₂×D₈×C₇₉

—

C₂×D₈×C₇₉

—

C₂×D₈×C₇₉

—

C₂×D₈×C₇₉

—

C₂×D₈×C₇₉

—

C₂×D₈×C₇₉

—

C₂×D₈×C₇₉

—

C₂×D₈×C₇₉

—

C₂×D₈×C₇₉

D₄

yes

—

C₂²×C₈×C₇₉

D₄

yes

characteristic

C₂²×C₈×C₇₉

C₂³

yes

characteristic

#127×C₇₉

—

#127×C₇₉

—

C₂×C₄²×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

—

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₇₉⋊C₁₆

—

C₂⁴×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

D₈×C₇₉

—

D₈×C₇₉

—

D₈×C₇₉

—

D₈×C₇₉

—

D₈×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

—

C₄²×C₇₉

—

C₄²×C₇₉

SD₁₆

yes

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

AGL(1,ℤ/8)

yes

characteristic

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×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

—

100

C₂²×C₇₉

—

101

C₂²×C₇₉

—

102

C₂²×C₇₉

—

103

C₂²×C₇₉

—

104

C₂²×C₇₉

#450

yes

characteristic

105

C₂²×C₇₉

#459

yes

characteristic

106

C₂²×C₇₉

#490

yes

characteristic

107

C₄×C₇₉

—

108

C₄×C₇₉

#394

yes

characteristic

109

C₄×C₇₉

#447

yes

characteristic

110

C₄×C₇₉

#460

yes

characteristic

111

C₄×C₇₉

#486

yes

characteristic

112

Dic₇₉

—

113

#13413

—

maximal, Sylow

114

C₂×C₇₉

—

115

C₂×C₇₉

—

116

C₂×C₇₉

—

117

C₂×C₇₉

#2041

yes

characteristic

118

C₂×C₇₉

#2213

yes

—

119

C₂×#493

D₇₉

yes

characteristic

120

C₂×#361

—

121

#2140

—

122

#1945

—

123

#1376

—

124

C₇₉

#13413

yes

characteristic, complemented, Sylow

125

C₂²×D₈

—

126

C₂²×D₈

C₂×D₇₉

yes

—

127

C₂×#127

—

128

C₂×M₃₂

—

129

#493

—

130

C₂×#108

—

131

#427

—

132

#424

—

158

—

133

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

134

#361

—

158

—

135

#357

—

158

—

136

#339

—

158

—

137

C₂²×D₄

—

138

C₂²×D₄

—

139

C₂×D₈

—

140

C₂×D₈

—

141

C₂×D₈

—

142

C₂×D₈

—

143

C₂×D₈

—

144

C₂×D₈

—

145

C₂×D₈

—

146

C₂×D₈

—

147

C₂×D₈

C₂²⋊D₇₉

yes

—

148

C₂×M₁₆

—

158

—

149

C₂²×C₈

C₂²⋊D₇₉

yes

characteristic

150

C₂²×C₈

C₂²×D₇₉

yes

characteristic

151

#127

—

152

#127

—

153

#117

—

158

—

154

C₂×C₄²

C₂²⋊D₇₉

yes

characteristic

155

M₃₂

—

158

—

156

M₃₂

—

158

—

157

C₂×C₁₆

—

158

—

158

#108

—

158

—

159

#107

—

160

#106

—

158

—

161

C₄⋊C₈

—

316

—

162

C₄×C₈

—

163

C₂⁴

—

164

C₂×D₄

—

165

C₂×D₄

—

166

C₂×D₄

—

167

C₂×D₄

—

168

C₂×D₄

—

169

C₂×D₄

—

170

C₂×D₄

—

171

C₂×D₄

—

172

C₂²×C₄

—

173

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

174

D₈

—

175

D₈

—

176

D₈

—

177

D₈

—

178

D₈

—

179

M₁₆

—

316

—

180

M₁₆

—

316

—

181

C₂×C₈

—

182

C₂×C₈

—

183

C₂×C₈

—

316

—

184

C₂×C₈

#11929609

yes

—

185

C₂×C₈

D₄×D₇₉

yes

characteristic

186

C₂×C₈

C₂×C₂²⋊D₇₉

yes

—

187

C₂×C₈

#11929623

yes

—

188

C₂×C₈

#11929623

yes

characteristic

189

C₄⋊C₄

—

158

—

190

C₂²⋊C₄

—

316

—

191

C₄²

—

192

C₄²

—

316

—

193

C₄²

#11929612

yes

—

194

C₁₆

—

316

—

195

C₂³

—

196

C₂³

—

197

C₂³

—

198

C₂³

—

199

C₂³

#14579187

yes

characteristic

200

D₄

—

201

D₄

—

202

D₄

—

203

D₄

—

204

D₄

—

205

C₂×C₄

—

206

C₂×C₄

—

207

C₂×C₄

—

316

—

208

C₂×C₄

#14579130

yes

characteristic

209

C₂×C₄

#14579132

yes

characteristic

210

C₂×C₄

#14579148

yes

characteristic

211

C₂×C₄

C₂×#11929609

yes

characteristic

212

C₂×C₄

C₂×#11929612

yes

characteristic

213

C₂×C₄

#14579187

yes

characteristic

214

C₈

—

215

C₈

—

632

—

216

C₈

#14579133

yes

—

217

C₈

#14579142

yes

—

218

C₂²

—

219

C₂²

—

220

C₂²

—

221

C₂²

—

222

C₂²

—

223

C₂²

#19098714

yes

characteristic

224

C₂²

#19098755

yes

characteristic

225

C₂²

#19098927

yes

characteristic

226

C₄

—

227

C₄

—

632

—

228

C₄

#19098477

yes

characteristic

229

C₄

#19098697

yes

characteristic

230

C₄

#19098760

yes

characteristic

231

C₄

#19098909

yes

characteristic

232

C₂

—

233

C₂

—

234

C₂

—

235

C₂

#23764631

yes

characteristic

236

C₂

#23765473

yes

—

	subgroup	G/H
center	C₂²	#19098755
commutator, cosocle	C₂²×C₈×C₇₉	C₂³
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting, prime radical	C₂×#493×C₇₉	C₂
socle	C₂²×C₇₉	#459
hypercenter	C₂×#493	D₇₉
hypocenter	C₇₉	#13413
LCS	G ▷ C₂²×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#493

subgroup

G/H

center

C₂²

#19098755

commutator, cosocle

C₂²×C₈×C₇₉

C₂³

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting, prime radical

C₂×#493×C₇₉

C₂

C₂²×C₇₉

D₇₉

C₇₉

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#493

N	H ≅ G/N	type
1. C₂×#19098376	C₂	semidirect
2. #23765116	C₂	semidirect
3. #23764148	C₂	semidirect
4. #23761482	C₂	semidirect
24. #14579095	D₄	semidirect
124. C₇₉	#13413	semidirect

H ≅ G/N

type

1. C₂×#19098376

C₂

semidirect

2. #23765116

C₂

semidirect

3. #23764148

C₂

semidirect

4. #23761482

C₂

semidirect

24. #14579095

D₄

semidirect

124. C₇₉

#13413

semidirect

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

subgroup	index
C₂×#19098376	2
#23765116	2
#23764148	2
#23761482	2
C₂×#493×C₇₉	2
#13413	79

subgroup

index

C₂×#493×C₇₉

Subgroups — #34207646

Conjugacy classes of subgroups

Complements

Maximal subgroups