Group #34185884 — subgroups

Subgroups — #34185884

Order 20224.

452 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 111 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	#23772502	C₂	1	1	yes	maximal, characteristic, complemented
2	#23762138	C₂	1	1	yes	maximal, characteristic, complemented
3	#23762068	C₂	1	2	yes	maximal
4	#23761290	C₂	1	2	yes	maximal, complemented
5	C₇₉×#3138	C₂	1	1	yes	maximal, characteristic
6	#19098694	—	2	2	no	—
7	Q₈×C₇₉⋊C₈	—	2	2	no	—
8	#19098624	—	2	2	no	—
9	#19098616	C₂²	1	2	yes	—
10	#19098587	C₂²	1	2	yes	—
11	#19098309	—	2	2	no	—
12	#19098307	—	2	2	no	—
13	#19098307	—	2	2	no	—
14	#533×C₇₉	C₂²	1	1	yes	characteristic
15	C₂×#127×C₇₉	C₂²	1	1	yes	characteristic
16	#490×C₇₉	C₄	1	4	yes	—
17	#486×C₇₉	—	2	2	no	—
18	#418×C₇₉	—	2	2	no	—
19	C₄×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
20	C₂×#11929608	—	2	2	no	—
21	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
22	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
23	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
24	C₄⋊(C₇₉⋊C₈)	D₄	1	2	yes	—
25	C₄⋊(C₇₉⋊C₈)	D₄	1	2	yes	—
26	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
27	#14579072	—	2	2	no	—
28	C₄×C₇₉⋊C₈	—	2	2	no	—
29	C₄×C₇₉⋊C₈	—	2	2	no	—
30	C₂²×D₄×C₇₉	—	2	1	no	—
31	C₂²×D₄×C₇₉	—	2	2	no	—
32	C₂×M₁₆×C₇₉	—	2	1	no	—
33	C₄⋊Q₈×C₇₉	D₄	1	2	yes	—
34	#127×C₇₉	—	2	1	no	—
35	#127×C₇₉	—	2	2	no	—
36	#127×C₇₉	C₂×C₄	1	2	yes	—
37	#125×C₇₉	C₂×C₄	1	2	yes	—
38	C₂²⋊Q₈×C₇₉	—	2	2	no	—
39	C₄×Q₈×C₇₉	—	2	2	no	—
40	#117×C₇₉	D₄	1	2	yes	—
41	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
42	#102×C₇₉	—	2	8	no	—
43	#97×C₇₉	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₇₉	—	2	1	no	—
45	#11929608	—	8	2	no	—
46	C₂×C₇₉⋊C₈	—	2	2	no	—
47	C₂×C₇₉⋊C₈	—	2	2	no	—
48	C₂×C₇₉⋊C₈	—	4	2	no	—
49	C₂⁴×C₇₉	—	4	1	no	—
50	C₂×Q₈×C₇₉	—	2	2	no	—
51	C₂×D₄×C₇₉	—	2	2	no	—
52	C₂×D₄×C₇₉	—	2	4	no	—
53	C₂×D₄×C₇₉	—	4	4	no	—
54	C₂×D₄×C₇₉	—	4	1	no	—
55	C₂×D₄×C₇₉	—	4	2	no	—
56	C₂×D₄×C₇₉	—	8	2	no	—
57	C₂×D₄×C₇₉	—	8	2	no	—
58	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₇₉	#41	1	2	yes	—
60	M₁₆×C₇₉	—	4	2	no	—
61	C₂×C₈×C₇₉	—	2	2	no	—
62	C₄⋊C₄×C₇₉	—	2	4	no	—
63	C₄⋊C₄×C₇₉	—	2	2	no	—
64	C₄⋊C₄×C₇₉	C₂²⋊C₄	1	4	yes	—
65	C₂²⋊C₄×C₇₉	—	2	2	no	—
66	C₄²×C₇₉	—	2	2	no	—
67	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₇₉	C₂×D₄	1	2	yes	—
70	C₂³×C₇₉	—	4	2	no	—
71	C₂³×C₇₉	—	8	2	no	—
72	C₂³×C₇₉	—	8	4	no	—
73	C₂³×C₇₉	#123	1	1	yes	characteristic
74	Q₈×C₇₉	—	4	2	no	—
75	Q₈×C₇₉	—	4	2	no	—
76	D₄×C₇₉	—	4	4	no	—
77	D₄×C₇₉	—	8	4	no	—
78	D₄×C₇₉	—	8	2	no	—
79	D₄×C₇₉	—	8	2	no	—
80	D₄×C₇₉	—	8	2	no	—
81	C₂×C₄×C₇₉	—	2	2	no	—
82	C₂×C₄×C₇₉	—	2	2	no	—
83	C₂×C₄×C₇₉	—	2	2	no	—
84	C₂×C₄×C₇₉	—	2	4	no	—
85	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
87	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
88	C₂×C₄×C₇₉	#121	1	2	yes	—
89	C₈×C₇₉	—	4	2	no	—
90	C₇₉⋊C₈	—	4	2	no	—
91	C₇₉⋊C₈	—	4	2	no	—
92	C₇₉⋊C₈	—	8	2	no	—
93	C₂²×C₇₉	—	2	1	no	—
94	C₂²×C₇₉	—	2	2	no	—
95	C₂²×C₇₉	—	8	2	no	—
96	C₂²×C₇₉	—	8	4	no	—
97	C₂²×C₇₉	—	16	4	no	—
98	C₂²×C₇₉	#386	1	1	yes	characteristic
99	C₄×C₇₉	—	2	1	no	—
100	C₄×C₇₉	—	2	2	no	—
101	C₄×C₇₉	—	4	4	no	—
102	C₄×C₇₉	#454	1	2	yes	—
103	C₄×C₇₉	#471	1	2	yes	—
104	C₄×C₇₉	#472	1	2	yes	—
105	#9438	—	79	1	no	maximal, Sylow
106	C₂×C₇₉	—	4	1	no	—
107	C₂×C₇₉	—	16	2	no	—
108	C₂×C₇₉	#1888	1	1	yes	characteristic
109	C₂×C₇₉	#1907	1	2	yes	—
110	#3138	D₇₉	1	1	yes	characteristic
111	#2986	—	79	1	no	—
112	#1561	—	79	1	no	—
113	#1545	—	79	2	no	—
114	#1299	—	79	2	no	—
115	C₇₉	#9438	1	1	yes	characteristic, complemented, Sylow
116	#533	C₂×D₇₉	1	1	yes	characteristic
117	C₂×#127	C₂×D₇₉	1	1	yes	characteristic
118	#490	Dic₇₉	1	4	yes	—
119	#486	—	2	2	no	—
120	#446	—	158	2	no	—
121	C₈×Q₈	—	158	2	no	—
122	#433	—	158	2	no	—
123	#431	—	79	2	no	—
124	#423	—	79	2	no	—
125	#418	—	2	2	no	—
126	C₄×M₁₆	C₂×D₇₉	1	1	yes	characteristic
127	#332	—	158	2	no	—
128	#331	—	158	2	no	—
129	#331	—	158	2	no	—
130	C₂²×D₄	—	2	1	no	—
131	C₂²×D₄	—	2	2	no	—
132	C₂×M₁₆	—	2	1	no	—
133	C₂×M₁₆	—	158	2	no	—
134	C₄⋊Q₈	C₂²⋊D₇₉	1	2	yes	—
135	#127	—	2	2	no	—
136	#127	—	2	1	no	—
137	#127	C₂×Dic₇₉	1	2	yes	—
138	#125	C₂×Dic₇₉	1	2	yes	—
139	C₂²⋊Q₈	—	2	2	no	—
140	C₄×Q₈	—	2	2	no	—
141	#117	C₂²⋊D₇₉	1	2	yes	—
142	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
143	C₄⋊C₈	—	79	2	no	—
144	C₄⋊C₈	—	79	2	no	—
145	C₄⋊C₈	—	158	4	no	—
146	C₄⋊C₈	—	158	2	no	—
147	C₄⋊C₈	—	158	2	no	—
148	#102	—	2	8	no	—
149	C₂²⋊C₈	—	158	2	no	—
150	#97	—	158	2	no	—
151	#97	C₂×Dic₇₉	1	2	yes	—
152	C₄×C₈	—	2	1	no	—
153	C₄×C₈	—	158	2	no	—
154	C₄×C₈	—	158	2	no	—
155	C₂⁴	—	4	1	no	—
156	C₂×Q₈	—	2	2	no	—
157	C₂×D₄	—	2	2	no	—
158	C₂×D₄	—	2	4	no	—
159	C₂×D₄	—	4	4	no	—
160	C₂×D₄	—	4	1	no	—
161	C₂×D₄	—	4	2	no	—
162	C₂×D₄	—	8	2	no	—
163	C₂×D₄	—	8	2	no	—
164	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
165	C₂²×C₄	#11929623	1	2	yes	—
166	M₁₆	—	4	2	no	—
167	M₁₆	—	632	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	316	2	no	—
170	C₂×C₈	—	158	2	no	—
171	C₂×C₈	—	158	2	no	—
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	C₂²⋊Dic₇₉	1	4	yes	—
175	C₂²⋊C₄	—	2	2	no	—
176	C₄²	—	2	2	no	—
177	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
178	C₄²	D₄×D₇₉	1	2	yes	—
179	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
180	C₂³	—	4	2	no	—
181	C₂³	—	8	2	no	—
182	C₂³	—	8	4	no	—
183	C₂³	#14579142	1	1	yes	characteristic
184	Q₈	—	4	2	no	—
185	Q₈	—	4	2	no	—
186	D₄	—	4	4	no	—
187	D₄	—	8	4	no	—
188	D₄	—	8	2	no	—
189	D₄	—	8	2	no	—
190	D₄	—	8	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
196	C₂×C₄	D₄×Dic₇₉	1	2	yes	—
197	C₂×C₄	#14579129	1	1	yes	characteristic
198	C₂×C₄	#14579133	1	2	yes	—
199	C₈	—	4	2	no	—
200	C₈	—	316	2	no	—
201	C₈	—	316	2	no	—
202	C₈	—	632	2	no	—
203	C₂²	—	2	1	no	—
204	C₂²	—	2	2	no	—
205	C₂²	—	8	2	no	—
206	C₂²	—	8	4	no	—
207	C₂²	—	16	4	no	—
208	C₂²	#19098443	1	1	yes	characteristic
209	C₄	—	2	1	no	—
210	C₄	—	2	2	no	—
211	C₄	—	4	4	no	—
212	C₄	#19098735	1	2	yes	—
213	C₄	#19098824	1	2	yes	—
214	C₄	#19098831	1	2	yes	—
215	C₂	—	4	1	no	—
216	C₂	—	16	2	no	—
217	C₂	#23763804	1	1	yes	characteristic
218	C₂	#23763910	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23772502

C₂

yes

maximal, characteristic, complemented

#23762138

C₂

yes

maximal, characteristic, complemented

#23762068

C₂

yes

maximal

#23761290

C₂

yes

maximal, complemented

C₇₉×#3138

C₂

yes

maximal, characteristic

#19098694

—

Q₈×C₇₉⋊C₈

—

#19098624

—

#19098616

C₂²

yes

—

#19098587

C₂²

yes

—

#19098309

—

#19098307

—

#19098307

—

#533×C₇₉

C₂²

yes

characteristic

C₂×#127×C₇₉

C₂²

yes

characteristic

#490×C₇₉

C₄

yes

—

#486×C₇₉

—

#418×C₇₉

—

C₄×M₁₆×C₇₉

C₂²

yes

characteristic

C₂×#11929608

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

D₄

yes

—

C₄⋊(C₇₉⋊C₈)

D₄

yes

—

C₂²⋊(C₇₉⋊C₈)

—

#14579072

—

C₄×C₇₉⋊C₈

—

C₄×C₇₉⋊C₈

—

C₂²×D₄×C₇₉

—

C₂²×D₄×C₇₉

—

C₂×M₁₆×C₇₉

—

C₄⋊Q₈×C₇₉

D₄

yes

—

#127×C₇₉

—

#127×C₇₉

—

#127×C₇₉

C₂×C₄

yes

—

#125×C₇₉

C₂×C₄

yes

—

C₂²⋊Q₈×C₇₉

—

C₄×Q₈×C₇₉

—

#117×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

#102×C₇₉

—

#97×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂⁴×C₇₉

—

C₂×Q₈×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

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

#41

yes

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

C₂²⋊C₄

yes

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

#123

yes

characteristic

Q₈×C₇₉

—

Q₈×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄×D₄

yes

—

C₂×C₄×C₇₉

#120

yes

characteristic

C₂×C₄×C₇₉

#121

yes

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#386

yes

characteristic

C₄×C₇₉

—

100

C₄×C₇₉

—

101

C₄×C₇₉

—

102

C₄×C₇₉

#454

yes

—

103

C₄×C₇₉

#471

yes

—

104

C₄×C₇₉

#472

yes

—

105

#9438

—

maximal, Sylow

106

C₂×C₇₉

—

107

C₂×C₇₉

—

108

C₂×C₇₉

#1888

yes

characteristic

109

C₂×C₇₉

#1907

yes

—

110

#3138

D₇₉

yes

characteristic

111

#2986

—

112

#1561

—

113

#1545

—

114

#1299

—

115

C₇₉

#9438

yes

characteristic, complemented, Sylow

116

#533

C₂×D₇₉

yes

characteristic

117

C₂×#127

C₂×D₇₉

yes

characteristic

118

#490

Dic₇₉

yes

—

119

#486

—

120

#446

—

158

—

121

C₈×Q₈

—

158

—

122

#433

—

158

—

123

#431

—

124

#423

—

125

#418

—

126

C₄×M₁₆

C₂×D₇₉

yes

characteristic

127

#332

—

158

—

128

#331

—

158

—

129

#331

—

158

—

130

C₂²×D₄

—

131

C₂²×D₄

—

132

C₂×M₁₆

—

133

C₂×M₁₆

—

158

—

134

C₄⋊Q₈

C₂²⋊D₇₉

yes

—

135

#127

—

136

#127

—

137

#127

C₂×Dic₇₉

yes

—

138

#125

C₂×Dic₇₉

yes

—

139

C₂²⋊Q₈

—

140

C₄×Q₈

—

141

#117

C₂²⋊D₇₉

yes

—

142

C₂×C₄²

C₂²×D₇₉

yes

characteristic

143

C₄⋊C₈

—

144

C₄⋊C₈

—

145

C₄⋊C₈

—

158

—

146

C₄⋊C₈

—

158

—

147

C₄⋊C₈

—

158

—

148

#102

—

149

C₂²⋊C₈

—

158

—

150

#97

—

158

—

151

#97

C₂×Dic₇₉

yes

—

152

C₄×C₈

—

153

C₄×C₈

—

158

—

154

C₄×C₈

—

158

—

155

C₂⁴

—

156

C₂×Q₈

—

157

C₂×D₄

—

158

C₂×D₄

—

159

C₂×D₄

—

160

C₂×D₄

—

161

C₂×D₄

—

162

C₂×D₄

—

163

C₂×D₄

—

164

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

165

C₂²×C₄

#11929623

yes

—

166

M₁₆

—

167

M₁₆

—

632

—

168

C₂×C₈

—

169

C₂×C₈

—

316

—

170

C₂×C₈

—

158

—

171

C₂×C₈

—

158

—

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₄⋊C₄

C₂²⋊Dic₇₉

yes

—

175

C₂²⋊C₄

—

176

C₄²

—

177

C₄²

C₂²×Dic₇₉

yes

characteristic

178

C₄²

D₄×D₇₉

yes

—

179

C₄²

C₂×C₂²⋊D₇₉

yes

characteristic

180

C₂³

—

181

C₂³

—

182

C₂³

—

183

C₂³

#14579142

yes

characteristic

184

Q₈

—

185

Q₈

—

186

D₄

—

187

D₄

—

188

D₄

—

189

D₄

—

190

D₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

C₂×C₂²⋊Dic₇₉

yes

characteristic

196

C₂×C₄

D₄×Dic₇₉

yes

—

197

C₂×C₄

#14579129

yes

characteristic

198

C₂×C₄

#14579133

yes

—

199

C₈

—

200

C₈

—

316

—

201

C₈

—

316

—

202

C₈

—

632

—

203

C₂²

—

204

C₂²

—

205

C₂²

—

206

C₂²

—

207

C₂²

—

208

C₂²

#19098443

yes

characteristic

209

C₄

—

210

C₄

—

211

C₄

—

212

C₄

#19098735

yes

—

213

C₄

#19098824

yes

—

214

C₄

#19098831

yes

—

215

C₂

—

216

C₂

—

217

C₂

#23763804

yes

characteristic

218

C₂

#23763910

yes

—

	subgroup	G/H
center	C₂²	#19098443
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#3138	C₂
socle	C₂²×C₇₉	#386
prime radical	C₂×#127×C₇₉	C₂²
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#3138	D₇₉
hypocenter	C₇₉	#9438
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3138

subgroup

G/H

center

C₂²

#19098443

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#3138

C₂

socle

C₂²×C₇₉

#386

prime radical

C₂×#127×C₇₉

C₂²

cosocle

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3138

N	H ≅ G/N	type
1. #23772502	C₂	semidirect
2. #23762138	C₂	semidirect
4. #23761290	C₂	semidirect
115. C₇₉	#9438	semidirect

H ≅ G/N

type

1. #23772502

C₂

semidirect

2. #23762138

C₂

semidirect

4. #23761290

C₂

semidirect

115. C₇₉

#9438

semidirect

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

subgroup	index
#23772502	2
#23762138	2
#23762068	2
#23761290	2
C₇₉×#3138	2
#9438	79

subgroup

index

C₇₉×#3138

Subgroups — #34185884

Conjugacy classes of subgroups

Complements

Maximal subgroups