Group #34193018 — subgroups

Subgroups — #34193018

Order 20224.

350 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 123 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	#23763528	C₂	1	1	yes	maximal, characteristic
2	#23763405	C₂	1	2	yes	maximal
3	#23761284	C₂	1	2	yes	maximal
4	#23761235	C₂	1	1	yes	maximal, characteristic
5	C₇₉×#1984	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×#14579085	—	2	2	no	—
8	#19098433	C₂²	1	2	yes	—
9	#19098326	—	2	2	no	—
10	#19098325	—	2	2	no	—
11	C₂×C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
14	#336×C₇₉	—	2	2	no	—
15	C₂²×C₇₉⋊C₈	—	2	2	no	—
16	C₂×C₄⋊Dic₇₉	—	2	2	no	—
17	C₂×C₄⋊Dic₇₉	—	2	2	no	—
18	C₂×C₄⋊Dic₇₉	—	2	2	no	—
19	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
20	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
21	#14579085	—	4	4	no	—
22	C₂²×Q₈×C₇₉	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₇₉	—	2	1	no	—
24	C₂²×C₈×C₇₉	D₄	1	2	yes	—
25	C₄×Q₈×C₇₉	—	2	2	no	—
26	C₄×Q₈×C₇₉	—	2	2	no	—
27	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
28	C₂×C₄⋊C₄×C₇₉	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₇₉	—	2	1	no	—
30	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₇₉	—	2	2	no	—
32	C₄×C₈×C₇₉	—	2	2	no	—
33	C₂²×Dic₇₉	—	4	2	no	—
34	C₂×C₇₉⋊C₈	—	4	2	no	—
35	C₂×C₇₉⋊C₈	—	4	2	no	—
36	C₂×C₇₉⋊C₈	—	4	2	no	—
37	C₄⋊Dic₇₉	—	4	2	no	—
38	C₄⋊Dic₇₉	—	4	2	no	—
39	C₄⋊Dic₇₉	—	4	2	no	—
40	C₄⋊Dic₇₉	—	4	2	no	—
41	C₄⋊Dic₇₉	—	8	2	no	—
42	C₂×Q₈×C₇₉	—	2	1	no	—
43	C₂×Q₈×C₇₉	—	2	2	no	—
44	C₂×Q₈×C₇₉	—	2	2	no	—
45	C₂×Q₈×C₇₉	Q₁₆	1	2	yes	—
46	C₂²×C₄×C₇₉	—	2	1	no	—
47	C₂²×C₄×C₇₉	—	2	1	no	—
48	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
51	C₂×C₈×C₇₉	—	2	2	no	—
52	C₂×C₈×C₇₉	—	2	2	no	—
53	C₂×C₈×C₇₉	—	2	2	no	—
54	C₂×C₈×C₇₉	—	4	1	no	—
55	C₂×C₈×C₇₉	—	4	2	no	—
56	C₄⋊C₄×C₇₉	—	2	2	no	—
57	C₄⋊C₄×C₇₉	—	4	2	no	—
58	C₄²×C₇₉	—	4	1	no	—
59	C₄²×C₇₉	—	4	1	no	—
60	C₄²×C₇₉	#41	1	2	yes	—
61	C₄²×C₇₉	#41	1	2	yes	—
62	C₂³×C₇₉	#121	1	1	yes	characteristic
63	Q₈×C₇₉	—	2	4	no	—
64	Q₈×C₇₉	—	4	2	no	—
65	C₂×C₄×C₇₉	—	2	2	no	—
66	C₂×C₄×C₇₉	—	2	2	no	—
67	C₂×C₄×C₇₉	—	4	1	no	—
68	C₂×C₄×C₇₉	—	4	1	no	—
69	C₂×C₄×C₇₉	—	4	1	no	—
70	C₂×C₄×C₇₉	—	4	2	no	—
71	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
72	C₂×C₄×C₇₉	#123	1	2	yes	—
73	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
74	C₂×C₄×C₇₉	#126	1	2	yes	—
75	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
76	C₂×C₄×C₇₉	#135	1	2	yes	—
77	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
78	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
79	C₂×C₄×C₇₉	#137	1	2	yes	—
80	C₂×Dic₇₉	—	8	2	no	—
81	C₂×Dic₇₉	—	8	2	no	—
82	C₂×Dic₇₉	—	8	2	no	—
83	C₈×C₇₉	—	4	2	no	—
84	C₈×C₇₉	—	8	1	no	—
85	C₇₉⋊C₈	—	8	2	no	—
86	C₂²×C₇₉	#397	1	1	yes	characteristic
87	C₂²×C₇₉	#464	1	2	yes	—
88	C₂²×C₇₉	#467	1	1	yes	characteristic
89	C₂²×C₇₉	#469	1	1	yes	characteristic
90	C₂²×C₇₉	#471	1	2	yes	—
91	C₄×C₇₉	—	2	2	no	—
92	C₄×C₇₉	—	4	2	no	—
93	C₄×C₇₉	—	8	1	no	—
94	C₄×C₇₉	#482	1	2	yes	—
95	C₄×C₇₉	#484	1	2	yes	—
96	C₄×C₇₉	#485	1	2	yes	—
97	C₄×C₇₉	#488	1	1	yes	characteristic
98	C₄×C₇₉	#491	1	1	yes	characteristic
99	Dic₇₉	—	16	2	no	—
100	#10624	—	79	1	no	maximal, Sylow
101	C₂×C₇₉	#1684	1	2	yes	—
102	C₂×C₇₉	#1702	1	2	yes	—
103	C₂×C₇₉	#2074	1	1	yes	characteristic
104	C₂×C₇₉	#2076	1	1	yes	characteristic
105	C₂×C₇₉	#2079	1	1	yes	characteristic
106	#1984	D₇₉	1	1	yes	characteristic
107	#1834	—	79	1	no	—
108	#1803	—	79	2	no	—
109	#1296	—	79	2	no	—
110	#1277	—	79	1	no	—
111	C₇₉	#10624	1	1	yes	characteristic, complemented, Sylow
112	C₂×C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
113	C₂×C₄⋊C₈	—	79	2	no	—
114	C₂×C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
115	C₂×#103	—	158	2	no	—
116	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
117	#384	—	79	2	no	—
118	#340	—	158	2	no	—
119	#340	—	158	2	no	—
120	#336	—	2	2	no	—
121	C₂²×Q₈	D₃₁₆	1	1	yes	characteristic
122	C₂²×C₈	—	2	1	no	—
123	C₂²×C₈	—	158	2	no	—
124	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
125	C₄×Q₈	—	2	2	no	—
126	C₄×Q₈	—	2	2	no	—
127	C₂×C₄⋊C₄	—	2	1	no	—
128	C₂×C₄⋊C₄	—	158	2	no	—
129	C₂×C₄⋊C₄	—	158	2	no	—
130	C₂×C₄⋊C₄	—	158	2	no	—
131	C₂×C₄⋊C₄	D₃₁₆	1	1	yes	characteristic
132	C₂×C₄²	—	2	1	no	—
133	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
134	C₄⋊C₈	—	2	2	no	—
135	C₄⋊C₈	—	158	2	no	—
136	C₄⋊C₈	—	158	2	no	—
137	#103	—	316	4	no	—
138	C₄×C₈	—	2	2	no	—
139	C₂×Q₈	—	2	1	no	—
140	C₂×Q₈	—	2	2	no	—
141	C₂×Q₈	—	2	2	no	—
142	C₂×Q₈	Dic₃₁₆	1	2	yes	—
143	C₂²×C₄	—	2	1	no	—
144	C₂²×C₄	—	2	1	no	—
145	C₂²×C₄	—	316	2	no	—
146	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
147	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
148	C₂²×C₄	#11929624	1	1	yes	characteristic
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	4	1	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	—	316	2	no	—
155	C₂×C₈	—	316	2	no	—
156	C₂×C₈	—	316	2	no	—
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	316	2	no	—
160	C₄⋊C₄	—	316	2	no	—
161	C₄⋊C₄	—	316	2	no	—
162	C₄⋊C₄	—	316	2	no	—
163	C₄⋊C₄	—	632	2	no	—
164	C₄²	—	4	1	no	—
165	C₄²	—	4	1	no	—
166	C₄²	#11929623	1	2	yes	—
167	C₄²	#11929623	1	2	yes	—
168	C₂³	#14579135	1	1	yes	characteristic
169	Q₈	—	2	4	no	—
170	Q₈	—	4	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	4	1	no	—
174	C₂×C₄	—	4	1	no	—
175	C₂×C₄	—	4	1	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	632	2	no	—
178	C₂×C₄	—	632	2	no	—
179	C₂×C₄	—	632	2	no	—
180	C₂×C₄	#14579142	1	1	yes	characteristic
181	C₂×C₄	#14579145	1	2	yes	—
182	C₂×C₄	#14579148	1	1	yes	characteristic
183	C₂×C₄	#14579154	1	2	yes	—
184	C₂×C₄	C₂×Dic₃₁₆	1	1	yes	characteristic
185	C₂×C₄	#14579181	1	2	yes	—
186	C₂×C₄	#14579183	1	1	yes	characteristic
187	C₂×C₄	#14579189	1	2	yes	—
188	C₂×C₄	#14579192	1	2	yes	—
189	C₈	—	4	2	no	—
190	C₈	—	8	1	no	—
191	C₈	—	632	2	no	—
192	C₂²	#19098488	1	1	yes	characteristic
193	C₂²	#19098785	1	2	yes	—
194	C₂²	#19098804	1	1	yes	characteristic
195	C₂²	#19098816	1	1	yes	characteristic
196	C₂²	#19098826	1	2	yes	—
197	C₄	—	2	2	no	—
198	C₄	—	4	2	no	—
199	C₄	—	8	1	no	—
200	C₄	—	1264	2	no	—
201	C₄	#19098888	1	2	yes	—
202	C₄	#19098899	1	2	yes	—
203	C₄	#19098901	1	2	yes	—
204	C₄	#19098917	1	1	yes	characteristic
205	C₄	#19098931	1	1	yes	characteristic
206	C₂	#23762858	1	2	yes	—
207	C₂	#23762968	1	2	yes	—
208	C₂	#23764818	1	1	yes	characteristic
209	C₂	#23764828	1	1	yes	characteristic
210	C₂	#23764845	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23763528

C₂

yes

maximal, characteristic

#23763405

C₂

yes

maximal

#23761284

C₂

yes

maximal

#23761235

C₂

yes

maximal, characteristic

C₇₉×#1984

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₇₉⋊C₈)

C₂²

yes

—

C₂×#14579085

—

#19098433

C₂²

yes

—

#19098326

—

#19098325

—

C₂×C₄×Q₈×C₇₉

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₇₉

C₂²

yes

characteristic

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

#336×C₇₉

—

C₂²×C₇₉⋊C₈

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

#14579085

—

C₂²×Q₈×C₇₉

D₄

yes

characteristic

C₂²×C₈×C₇₉

—

C₂²×C₈×C₇₉

D₄

yes

—

C₄×Q₈×C₇₉

—

C₄×Q₈×C₇₉

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

D₄

yes

characteristic

C₂×C₄²×C₇₉

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄⋊C₈×C₇₉

—

C₄×C₈×C₇₉

—

C₂²×Dic₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

Q₁₆

yes

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

#41

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

#41

yes

—

C₄²×C₇₉

#41

yes

—

C₂³×C₇₉

#121

yes

characteristic

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#123

yes

characteristic

C₂×C₄×C₇₉

#123

yes

—

C₂×C₄×C₇₉

#124

yes

characteristic

C₂×C₄×C₇₉

#126

yes

—

C₂×C₄×C₇₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₉

#135

yes

—

C₂×C₄×C₇₉

#135

yes

characteristic

C₂×C₄×C₇₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₇₉

#137

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

#397

yes

characteristic

C₂²×C₇₉

#464

yes

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#482

yes

—

C₄×C₇₉

#484

yes

—

C₄×C₇₉

#485

yes

—

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

100

#10624

—

maximal, Sylow

101

C₂×C₇₉

#1684

yes

—

102

C₂×C₇₉

#1702

yes

—

103

C₂×C₇₉

#2074

yes

characteristic

104

C₂×C₇₉

#2076

yes

characteristic

105

C₂×C₇₉

#2079

yes

characteristic

106

#1984

D₇₉

yes

characteristic

107

#1834

—

108

#1803

—

109

#1296

—

110

#1277

—

111

C₇₉

#10624

yes

characteristic, complemented, Sylow

112

C₂×C₄×Q₈

C₂×D₇₉

yes

characteristic

113

C₂×C₄⋊C₈

—

114

C₂×C₄⋊C₈

C₂×D₇₉

yes

characteristic

115

C₂×#103

—

158

—

116

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

117

#384

—

118

#340

—

158

—

119

#340

—

158

—

120

#336

—

121

C₂²×Q₈

D₃₁₆

yes

characteristic

122

C₂²×C₈

—

123

C₂²×C₈

—

158

—

124

C₂²×C₈

C₂²⋊D₇₉

yes

—

125

C₄×Q₈

—

126

C₄×Q₈

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄⋊C₄

—

158

—

129

C₂×C₄⋊C₄

—

158

—

130

C₂×C₄⋊C₄

—

158

—

131

C₂×C₄⋊C₄

D₃₁₆

yes

characteristic

132

C₂×C₄²

—

133

C₂×C₄²

C₂²×D₇₉

yes

characteristic

134

C₄⋊C₈

—

135

C₄⋊C₈

—

158

—

136

C₄⋊C₈

—

158

—

137

#103

—

316

—

138

C₄×C₈

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂×Q₈

—

142

C₂×Q₈

Dic₃₁₆

yes

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

—

316

—

146

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

147

C₂²×C₄

C₂×D₃₁₆

yes

characteristic

148

C₂²×C₄

#11929624

yes

characteristic

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

316

—

155

C₂×C₈

—

316

—

156

C₂×C₈

—

316

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

316

—

160

C₄⋊C₄

—

316

—

161

C₄⋊C₄

—

316

—

162

C₄⋊C₄

—

316

—

163

C₄⋊C₄

—

632

—

164

C₄²

—

165

C₄²

—

166

C₄²

#11929623

yes

—

167

C₄²

#11929623

yes

—

168

C₂³

#14579135

yes

characteristic

169

Q₈

—

170

Q₈

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

632

—

178

C₂×C₄

—

632

—

179

C₂×C₄

—

632

—

180

C₂×C₄

#14579142

yes

characteristic

181

C₂×C₄

#14579145

yes

—

182

C₂×C₄

#14579148

yes

characteristic

183

C₂×C₄

#14579154

yes

—

184

C₂×C₄

C₂×Dic₃₁₆

yes

characteristic

185

C₂×C₄

#14579181

yes

—

186

C₂×C₄

#14579183

yes

characteristic

187

C₂×C₄

#14579189

yes

—

188

C₂×C₄

#14579192

yes

—

189

C₈

—

190

C₈

—

191

C₈

—

632

—

192

C₂²

#19098488

yes

characteristic

193

C₂²

#19098785

yes

—

194

C₂²

#19098804

yes

characteristic

195

C₂²

#19098816

yes

characteristic

196

C₂²

#19098826

yes

—

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

—

1264

—

201

C₄

#19098888

yes

—

202

C₄

#19098899

yes

—

203

C₄

#19098901

yes

—

204

C₄

#19098917

yes

characteristic

205

C₄

#19098931

yes

characteristic

206

C₂

#23762858

yes

—

207

C₂

#23762968

yes

—

208

C₂

#23764818

yes

characteristic

209

C₂

#23764828

yes

characteristic

210

C₂

#23764845

yes

characteristic

	subgroup	G/H
center	C₂³	#14579135
commutator, cosocle	C₂×C₄²×C₇₉	C₂³
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1984	C₂
socle, prime radical	C₂³×C₇₉	#121
hypercenter	#1984	D₇₉
hypocenter	C₇₉	#10624
LCS	G ▷ C₂×C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1984

subgroup

G/H

center

C₂³

#14579135

commutator, cosocle

C₂×C₄²×C₇₉

C₂³

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#1984

C₂

C₂³×C₇₉

D₇₉

C₇₉

G ▷ C₂×C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1984

N	H ≅ G/N	type
111. C₇₉	#10624	semidirect

H ≅ G/N

type

111. C₇₉

#10624

semidirect

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

subgroup	index
#23763528	2
#23763405	2
#23761284	2
#23761235	2
C₇₉×#1984	2
#10624	79

subgroup

index

C₇₉×#1984

Subgroups — #34193018

Conjugacy classes of subgroups

Complements

Maximal subgroups