Group #34181225 — subgroups

Subgroups — #34181225

Order 20224.

362 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 113 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×Q₈×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#23762161	C₂	1	2	yes	maximal
3	#23761744	C₂	1	1	yes	maximal, characteristic
4	#23761269	C₂	1	2	yes	maximal
5	C₇₉×#1467	C₂	1	1	yes	maximal, characteristic
6	Q₈×C₇₉⋊C₈	—	2	2	no	—
7	Q₈×C₇₉⋊C₈	—	2	2	no	—
8	#19098620	C₂²	1	2	yes	—
9	C₂×C₄⋊(C₇₉⋊C₈)	C₄	1	2	yes	—
10	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₇₉⋊C₈	C₄	1	2	yes	—
12	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
13	#19098315	—	2	2	no	—
14	#19098313	—	2	2	no	—
15	#19098289	—	2	2	no	—
16	C₂×C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
17	#432×C₇₉	C₂²	1	2	yes	—
18	#324×C₇₉	—	2	2	no	—
19	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
21	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
22	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
23	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
24	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
25	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
26	#14579072	—	2	2	no	—
27	C₄×C₇₉⋊C₈	—	2	2	no	—
28	C₄×C₇₉⋊C₈	—	2	2	no	—
29	C₄×C₇₉⋊C₈	D₄	1	2	yes	—
30	C₄×C₇₉⋊C₈	D₄	1	2	yes	—
31	C₂²×Q₈×C₇₉	Q₈	1	1	yes	characteristic
32	C₄×Q₈×C₇₉	—	2	2	no	—
33	C₄×Q₈×C₇₉	—	2	2	no	—
34	C₂×C₄⋊C₄×C₇₉	D₄	1	2	yes	—
35	C₂×C₄⋊C₄×C₇₉	Q₈	1	1	yes	characteristic
36	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
38	C₄⋊C₈×C₇₉	—	2	2	no	—
39	C₂²⋊C₈×C₇₉	—	2	2	no	—
40	#97×C₇₉	—	2	2	no	—
41	C₂×C₇₉⋊C₈	—	2	2	no	—
42	C₂×C₇₉⋊C₈	—	2	2	no	—
43	C₂×C₇₉⋊C₈	—	4	1	no	—
44	C₂×C₇₉⋊C₈	—	4	2	no	—
45	C₂×C₇₉⋊C₈	—	4	2	no	—
46	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	2	yes	—
48	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×Q₈×C₇₉	—	2	4	no	—
50	C₂×Q₈×C₇₉	—	2	2	no	—
51	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₈×C₇₉	—	4	2	no	—
57	C₄⋊C₄×C₇₉	—	2	4	no	—
58	C₄⋊C₄×C₇₉	—	2	2	no	—
59	C₄²×C₇₉	—	2	2	no	—
60	C₄²×C₇₉	—	2	2	no	—
61	C₄²×C₇₉	C₂×D₄	1	2	yes	—
62	C₄²×C₇₉	#41	1	2	yes	—
63	C₂³×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
64	Q₈×C₇₉	—	2	8	no	—
65	C₂×C₄×C₇₉	—	2	2	no	—
66	C₂×C₄×C₇₉	—	2	2	no	—
67	C₂×C₄×C₇₉	—	2	2	no	—
68	C₂×C₄×C₇₉	—	2	1	no	—
69	C₂×C₄×C₇₉	—	2	1	no	—
70	C₂×C₄×C₇₉	—	2	2	no	—
71	C₂×C₄×C₇₉	—	4	1	no	—
72	C₂×C₄×C₇₉	—	4	2	no	—
73	C₂×C₄×C₇₉	#108	1	4	yes	—
74	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
76	C₂×C₄×C₇₉	#121	1	2	yes	—
77	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
78	C₈×C₇₉	—	8	2	no	—
79	C₇₉⋊C₈	—	2	1	no	—
80	C₇₉⋊C₈	—	2	2	no	—
81	C₇₉⋊C₈	—	4	2	no	—
82	C₇₉⋊C₈	—	8	2	no	—
83	C₇₉⋊C₈	#101	1	2	yes	—
84	C₂²×C₇₉	—	2	1	no	—
85	C₂²×C₇₉	—	2	1	no	—
86	C₂²×C₇₉	#380	1	1	yes	characteristic
87	C₂²×C₇₉	C₂×#108	1	2	yes	—
88	C₄×C₇₉	—	2	1	no	—
89	C₄×C₇₉	—	2	4	no	—
90	C₄×C₇₉	—	2	2	no	—
91	C₄×C₇₉	—	2	2	no	—
92	C₄×C₇₉	—	4	2	no	—
93	C₄×C₇₉	C₂×#101	1	1	yes	characteristic
94	C₄×C₇₉	#413	1	1	yes	characteristic
95	#8623	—	79	1	no	maximal, Sylow
96	C₂×C₇₉	—	2	2	no	—
97	C₂×C₇₉	#1801	1	1	yes	characteristic
98	C₂×C₇₉	#1810	1	1	yes	characteristic
99	C₂×C₇₉	#1812	1	1	yes	characteristic
100	C₂×C₈×Q₈	—	79	1	no	—
101	#1568	—	79	2	no	—
102	#1467	—	79	1	no	—
103	#1467	D₇₉	1	1	yes	characteristic
104	#1290	—	79	2	no	—
105	C₇₉	#8623	1	1	yes	characteristic, complemented, Sylow
106	C₂×C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
107	C₈×Q₈	—	158	2	no	—
108	C₈×Q₈	—	158	2	no	—
109	#432	—	79	2	no	—
110	#432	C₂×D₇₉	1	2	yes	—
111	C₂×C₄⋊C₈	—	79	1	no	—
112	C₂×C₄⋊C₈	—	79	2	no	—
113	C₂×C₄×C₈	—	79	1	no	—
114	C₂×C₄×C₈	—	79	2	no	—
115	#335	—	158	2	no	—
116	#334	—	158	2	no	—
117	#324	—	2	2	no	—
118	#324	—	158	2	no	—
119	C₂²×Q₈	Dic₁₅₈	1	1	yes	characteristic
120	C₂²×C₈	—	79	1	no	—
121	C₂²×C₈	—	79	2	no	—
122	C₂²×C₈	—	79	1	no	—
123	C₄×Q₈	—	2	2	no	—
124	C₄×Q₈	—	2	2	no	—
125	C₂×C₄⋊C₄	Dic₁₅₈	1	1	yes	characteristic
126	C₂×C₄⋊C₄	C₂²⋊D₇₉	1	2	yes	—
127	C₂×C₄²	C₄×D₇₉	1	2	yes	—
128	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
129	C₄⋊C₈	—	2	2	no	—
130	C₄⋊C₈	—	158	4	no	—
131	C₄⋊C₈	—	158	2	no	—
132	C₄⋊C₈	—	158	2	no	—
133	C₂²⋊C₈	—	2	2	no	—
134	C₂²⋊C₈	—	158	2	no	—
135	#97	—	2	2	no	—
136	#97	—	158	2	no	—
137	C₄×C₈	—	79	2	no	—
138	C₄×C₈	—	79	2	no	—
139	C₄×C₈	—	158	2	no	—
140	C₄×C₈	—	158	2	no	—
141	C₂×Q₈	—	2	4	no	—
142	C₂×Q₈	—	2	2	no	—
143	C₂²×C₄	Dic₇₉⋊C₄	1	2	yes	—
144	C₂²×C₄	Dic₇₉⋊C₄	1	2	yes	—
145	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
147	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	—	316	2	no	—
150	C₂×C₈	—	316	2	no	—
151	C₂×C₈	—	316	1	no	—
152	C₂×C₈	—	79	1	no	—
153	C₂×C₈	—	79	1	no	—
154	C₂×C₈	—	79	2	no	—
155	C₂×C₈	—	158	2	no	—
156	C₂×C₈	—	158	2	no	—
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	2	2	no	—
159	C₄²	—	2	2	no	—
160	C₄²	—	2	2	no	—
161	C₄²	D₄×D₇₉	1	2	yes	—
162	C₄²	#11929623	1	2	yes	—
163	C₂³	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
164	Q₈	—	2	8	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	#14579096	1	4	yes	—
174	C₂×C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
175	C₂×C₄	#14579117	1	1	yes	characteristic
176	C₂×C₄	#14579133	1	2	yes	—
177	C₂×C₄	#14579138	1	2	yes	—
178	C₈	—	8	2	no	—
179	C₈	—	316	2	no	—
180	C₈	—	79	2	no	—
181	C₈	—	632	2	no	—
182	C₈	—	158	1	no	—
183	C₈	—	158	2	no	—
184	C₂²	—	2	1	no	—
185	C₂²	—	2	1	no	—
186	C₂²	#19098421	1	1	yes	characteristic
187	C₂²	C₂×#14579096	1	2	yes	—
188	C₄	—	2	1	no	—
189	C₄	—	2	4	no	—
190	C₄	—	2	2	no	—
191	C₄	—	2	2	no	—
192	C₄	—	4	2	no	—
193	C₄	#101×D₇₉	1	1	yes	characteristic
194	C₄	#19098546	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	#23763394	1	1	yes	characteristic
197	C₂	#23763436	1	1	yes	characteristic
198	C₂	#23763444	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×C₇₉⋊C₈

C₂

yes

maximal, characteristic

#23762161

C₂

yes

maximal

#23761744

C₂

yes

maximal, characteristic

#23761269

C₂

yes

maximal

C₇₉×#1467

C₂

yes

maximal, characteristic

Q₈×C₇₉⋊C₈

—

Q₈×C₇₉⋊C₈

—

#19098620

C₂²

yes

—

C₂×C₄⋊(C₇₉⋊C₈)

C₄

yes

—

C₂×C₄⋊(C₇₉⋊C₈)

C₂²

yes

characteristic

C₂×C₄×C₇₉⋊C₈

C₄

yes

—

C₂×C₄×C₇₉⋊C₈

C₂²

yes

—

—

—

C₂×C₄×Q₈×C₇₉

C₂²

yes

characteristic

#432×C₇₉

C₂²

yes

—

#324×C₇₉

—

C₂²×C₇₉⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₇₉⋊C₈

C₂×C₄

yes

—

C₂²×C₇₉⋊C₈

C₂×C₄

yes

characteristic

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

C₂²⋊(C₇₉⋊C₈)

—

#14579072

—

C₄×C₇₉⋊C₈

—

C₄×C₇₉⋊C₈

—

C₄×C₇₉⋊C₈

D₄

yes

—

C₄×C₇₉⋊C₈

D₄

yes

—

C₂²×Q₈×C₇₉

Q₈

yes

characteristic

C₄×Q₈×C₇₉

—

C₄×Q₈×C₇₉

—

C₂×C₄⋊C₄×C₇₉

D₄

yes

—

C₂×C₄⋊C₄×C₇₉

Q₈

yes

characteristic

C₂×C₄²×C₇₉

C₂×C₄

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄⋊C₈×C₇₉

—

C₂²⋊C₈×C₇₉

—

#97×C₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

C₂²⋊C₄

yes

characteristic

C₂×C₇₉⋊C₈

C₂²⋊C₄

yes

—

C₂×C₇₉⋊C₈

C₂²⋊C₄

yes

characteristic

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂²×C₄×C₇₉

C₄⋊C₄

yes

—

C₂²×C₄×C₇₉

C₄⋊C₄

yes

—

C₂²×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂×D₄

yes

—

C₄²×C₇₉

#41

yes

—

C₂³×C₇₉

C₂×C₄⋊C₄

yes

characteristic

Q₈×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

#108

yes

—

C₂×C₄×C₇₉

C₂×C₂²⋊C₄

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

—

C₂×C₄×C₇₉

C₂²⋊Q₈

yes

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

#101

yes

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#380

yes

characteristic

C₂²×C₇₉

C₂×#108

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

100

C₂×C₈×Q₈

—

101

#1568

—

102

#1467

—

103

#1467

D₇₉

yes

characteristic

104

#1290

—

105

C₇₉

#8623

yes

characteristic, complemented, Sylow

106

C₂×C₄×Q₈

C₂×D₇₉

yes

characteristic

107

C₈×Q₈

—

158

—

108

C₈×Q₈

—

158

—

109

#432

—

110

#432

C₂×D₇₉

yes

—

111

C₂×C₄⋊C₈

—

112

C₂×C₄⋊C₈

—

113

C₂×C₄×C₈

—

114

C₂×C₄×C₈

—

115

#335

—

158

—

116

#334

—

158

—

117

#324

—

118

#324

—

158

—

119

C₂²×Q₈

Dic₁₅₈

yes

characteristic

120

C₂²×C₈

—

121

C₂²×C₈

—

122

C₂²×C₈

—

123

C₄×Q₈

—

124

C₄×Q₈

—

125

C₂×C₄⋊C₄

Dic₁₅₈

yes

characteristic

126

C₂×C₄⋊C₄

C₂²⋊D₇₉

yes

—

127

C₂×C₄²

C₄×D₇₉

yes

—

128

C₂×C₄²

C₂²×D₇₉

yes

characteristic

129

C₄⋊C₈

—

130

C₄⋊C₈

—

158

—

131

C₄⋊C₈

—

158

—

132

C₄⋊C₈

—

158

—

133

C₂²⋊C₈

—

134

C₂²⋊C₈

—

158

—

135

#97

—

136

#97

—

158

—

137

C₄×C₈

—

138

C₄×C₈

—

139

C₄×C₈

—

158

—

140

C₄×C₈

—

158

—

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂²×C₄

Dic₇₉⋊C₄

yes

—

144

C₂²×C₄

Dic₇₉⋊C₄

yes

—

145

C₂²×C₄

C₂×C₄×D₇₉

yes

characteristic

146

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

147

C₂²×C₄

C₂×Dic₁₅₈

yes

characteristic

148

C₂×C₈

—

149

C₂×C₈

—

316

—

150

C₂×C₈

—

316

—

151

C₂×C₈

—

316

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

158

—

156

C₂×C₈

—

158

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄²

—

160

C₄²

—

161

C₄²

D₄×D₇₉

yes

—

162

C₄²

#11929623

yes

—

163

C₂³

C₂×Dic₇₉⋊C₄

yes

characteristic

164

Q₈

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

#14579096

yes

—

174

C₂×C₄

C₂²⋊C₄×D₇₉

yes

characteristic

175

C₂×C₄

#14579117

yes

characteristic

176

C₂×C₄

#14579133

yes

—

177

C₂×C₄

#14579138

yes

—

178

C₈

—

179

C₈

—

316

—

180

C₈

—

181

C₈

—

632

—

182

C₈

—

158

—

183

C₈

—

158

—

184

C₂²

—

185

C₂²

—

186

C₂²

#19098421

yes

characteristic

187

C₂²

C₂×#14579096

yes

—

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

#101×D₇₉

yes

characteristic

194

C₄

#19098546

yes

characteristic

195

C₂

—

196

C₂

#23763394

yes

characteristic

197

C₂

#23763436

yes

characteristic

198

C₂

#23763444

yes

characteristic

	subgroup	G/H
center	C₂²	#19098421
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1467	C₂
socle	C₂²×C₇₉	#380
prime radical	C₂³×C₇₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1467	D₇₉
hypocenter	C₇₉	#8623
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1467

subgroup

G/H

center

C₂²

#19098421

commutator

C₂²×C₄×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#1467

C₂

socle

C₂²×C₇₉

#380

prime radical

C₂³×C₇₉

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1467

N	H ≅ G/N	type
105. C₇₉	#8623	semidirect

H ≅ G/N

type

105. C₇₉

#8623

semidirect

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

subgroup	index
C₂×Q₈×C₇₉⋊C₈	2
#23762161	2
#23761744	2
#23761269	2
C₇₉×#1467	2
#8623	79

subgroup

index

C₂×Q₈×C₇₉⋊C₈

C₇₉×#1467

Subgroups — #34181225

Conjugacy classes of subgroups

Complements

Maximal subgroups