Group #34193320 — subgroups

Subgroups — #34193320

Order 20224.

442 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 153 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	#23764360	C₂	1	2	yes	maximal
2	#23764202	C₂	1	1	yes	maximal, characteristic
3	#23763559	C₂	1	2	yes	maximal
4	C₇₉⋊#1273	C₂	1	1	yes	maximal, characteristic
5	C₇₉×#1845	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579161	C₂²	1	1	yes	characteristic
7	C₂×#14579094	—	2	4	no	—
8	C₂×#14579086	—	2	2	no	—
9	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
10	#19098433	C₂²	1	1	yes	characteristic
11	#19098325	—	2	4	no	—
12	C₂×#107×C₇₉	—	2	2	no	—
13	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
14	#384×C₇₉	C₂²	1	2	yes	—
15	C₂²×Dic₁₅₈	—	2	1	no	—
16	C₂²×C₇₉⋊C₈	D₄	1	4	yes	—
17	#14579161	—	2	2	no	—
18	#14579161	—	2	1	no	—
19	#14579161	Q₈	1	2	yes	—
20	C₂×C₄⋊Dic₇₉	—	2	2	no	—
21	C₂×C₄⋊Dic₇₉	—	2	1	no	—
22	C₂×C₄⋊Dic₇₉	—	2	2	no	—
23	#14579094	—	4	8	no	—
24	#14579086	—	4	4	no	—
25	C₄×C₇₉⋊C₈	—	2	2	no	—
26	C₄×C₇₉⋊C₈	—	2	2	no	—
27	C₂²×C₈×C₇₉	D₄	1	2	yes	—
28	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
29	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
30	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
31	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
32	#107×C₇₉	—	4	4	no	—
33	C₄×C₈×C₇₉	—	2	2	no	—
34	C₂×Dic₁₅₈	—	2	2	no	—
35	C₂×Dic₁₅₈	—	4	1	no	—
36	C₂×Dic₁₅₈	—	4	2	no	—
37	C₂×Dic₁₅₈	—	4	2	no	—
38	C₂²×Dic₇₉	—	4	1	no	—
39	C₂²×Dic₇₉	—	4	1	no	—
40	C₂×C₇₉⋊C₈	—	2	4	no	—
41	C₂×C₇₉⋊C₈	—	2	4	no	—
42	C₂×C₇₉⋊C₈	Q₁₆	1	8	yes	—
43	C₄⋊Dic₇₉	—	2	4	no	—
44	C₄⋊Dic₇₉	—	4	2	no	—
45	C₄⋊Dic₇₉	—	4	2	no	—
46	C₄⋊Dic₇₉	—	4	4	no	—
47	C₂²×C₄×C₇₉	—	4	2	no	—
48	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
50	C₂×C₈×C₇₉	—	2	2	no	—
51	C₂×C₈×C₇₉	—	2	2	no	—
52	C₂×C₈×C₇₉	D₈	1	4	yes	—
53	C₄⋊C₄×C₇₉	—	4	4	no	—
54	C₄⋊C₄×C₇₉	—	4	4	no	—
55	C₄⋊C₄×C₇₉	—	4	4	no	—
56	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
57	C₄²×C₇₉	#41	1	2	yes	—
58	C₄²×C₇₉	#41	1	1	yes	characteristic
59	C₂³×C₇₉	#127	1	1	yes	characteristic
60	C₂×C₄×C₇₉	—	8	2	no	—
61	C₂×C₄×C₇₉	—	8	2	no	—
62	C₂×C₄×C₇₉	—	8	2	no	—
63	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
65	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
66	C₂×C₄×C₇₉	#123	1	2	yes	—
67	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
68	C₂×C₄×C₇₉	C₂×Q₁₆	1	4	yes	—
69	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	4	yes	—
70	C₂×C₄×C₇₉	#137	1	2	yes	—
71	C₂×Dic₇₉	—	4	2	no	—
72	C₂×Dic₇₉	—	8	1	no	—
73	C₂×Dic₇₉	—	8	1	no	—
74	C₂×Dic₇₉	—	8	1	no	—
75	C₂×Dic₇₉	—	8	2	no	—
76	Dic₁₅₈	—	4	4	no	—
77	Dic₁₅₈	—	8	2	no	—
78	C₈×C₇₉	—	2	4	no	—
79	C₇₉⋊C₈	—	2	8	no	—
80	C₂²×C₇₉	#399	1	1	yes	characteristic
81	C₂²×C₇₉	#493	1	1	yes	characteristic
82	C₂²×C₇₉	#494	1	2	yes	—
83	C₂²×C₇₉	#496	1	2	yes	—
84	C₂²×C₇₉	#497	1	1	yes	characteristic
85	C₄×C₇₉	—	16	2	no	—
86	C₄×C₇₉	#474	1	2	yes	—
87	C₄×C₇₉	#477	1	4	yes	—
88	C₄×C₇₉	#480	1	2	yes	—
89	C₄×C₇₉	#484	1	4	yes	—
90	Dic₇₉	—	8	2	no	—
91	Dic₇₉	—	16	1	no	—
92	#10674	—	79	1	no	maximal, Sylow
93	C₂×C₇₉	#1713	1	1	yes	characteristic
94	C₂×C₇₉	#1721	1	1	yes	characteristic
95	C₂×C₇₉	#1723	1	2	yes	—
96	C₂×C₇₉	#2093	1	1	yes	characteristic
97	C₂×C₇₉	#2095	1	2	yes	—
98	#1990	—	79	2	no	—
99	#1956	—	79	1	no	—
100	#1845	—	79	2	no	—
101	#1845	D₇₉	1	1	yes	characteristic
102	#1273	—	79	1	no	—
103	C₇₉	#10674	1	1	yes	characteristic, complemented, Sylow
104	C₂×C₄⋊Q₈	—	79	1	no	—
105	C₂×#107	—	2	2	no	—
106	C₂×#107	—	158	4	no	—
107	C₂×#103	—	158	2	no	—
108	C₂×C₄×C₈	—	79	2	no	—
109	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
110	#384	—	79	1	no	—
111	#384	C₂×D₇₉	1	2	yes	—
112	#340	—	158	4	no	—
113	C₂²×Q₈	—	158	1	no	—
114	C₂²×C₈	—	79	4	no	—
115	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
116	C₄⋊Q₈	—	79	2	no	—
117	C₄⋊Q₈	—	158	2	no	—
118	C₄⋊Q₈	—	158	1	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	—	2	2	no	—
122	C₂×C₄⋊C₄	—	158	2	no	—
123	C₂×C₄⋊C₄	—	158	1	no	—
124	C₂×C₄⋊C₄	—	158	2	no	—
125	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
126	#107	—	4	4	no	—
127	#107	—	316	8	no	—
128	#103	—	316	4	no	—
129	C₄×C₈	—	2	2	no	—
130	C₄×C₈	—	158	2	no	—
131	C₄×C₈	—	158	2	no	—
132	C₂×Q₈	—	316	1	no	—
133	C₂×Q₈	—	316	2	no	—
134	C₂×Q₈	—	316	2	no	—
135	C₂×Q₈	—	158	2	no	—
136	C₂²×C₄	—	4	2	no	—
137	C₂²×C₄	—	316	1	no	—
138	C₂²×C₄	—	316	1	no	—
139	C₂²×C₄	D₄×D₇₉	1	2	yes	—
140	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	79	8	no	—
144	C₂×C₈	—	158	4	no	—
145	C₂×C₈	—	158	4	no	—
146	C₂×C₈	#11929609	1	4	yes	—
147	C₄⋊C₄	—	4	4	no	—
148	C₄⋊C₄	—	4	4	no	—
149	C₄⋊C₄	—	4	4	no	—
150	C₄⋊C₄	—	316	2	no	—
151	C₄⋊C₄	—	316	2	no	—
152	C₄⋊C₄	—	316	4	no	—
153	C₄⋊C₄	—	158	4	no	—
154	C₄²	Q₈×D₇₉	1	1	yes	characteristic
155	C₄²	#11929624	1	2	yes	—
156	C₄²	#11929625	1	1	yes	characteristic
157	C₂³	#14579157	1	1	yes	characteristic
158	Q₈	—	316	4	no	—
159	Q₈	—	632	2	no	—
160	C₂×C₄	—	8	2	no	—
161	C₂×C₄	—	8	2	no	—
162	C₂×C₄	—	8	2	no	—
163	C₂×C₄	—	316	2	no	—
164	C₂×C₄	—	632	1	no	—
165	C₂×C₄	—	632	1	no	—
166	C₂×C₄	—	632	1	no	—
167	C₂×C₄	—	632	2	no	—
168	C₂×C₄	#14579137	1	1	yes	characteristic
169	C₂×C₄	#14579139	1	2	yes	—
170	C₂×C₄	#14579144	1	1	yes	characteristic
171	C₂×C₄	#14579146	1	2	yes	—
172	C₂×C₄	C₂×#11929609	1	2	yes	—
173	C₂×C₄	Q₁₆×D₇₉	1	4	yes	—
174	C₂×C₄	#14579185	1	4	yes	—
175	C₂×C₄	#14579192	1	2	yes	—
176	C₈	—	2	4	no	—
177	C₈	—	158	8	no	—
178	C₂²	#19098498	1	1	yes	characteristic
179	C₂²	#19098939	1	1	yes	characteristic
180	C₂²	#19098941	1	2	yes	—
181	C₂²	#19098951	1	2	yes	—
182	C₂²	#19098955	1	1	yes	characteristic
183	C₄	—	16	2	no	—
184	C₄	—	632	2	no	—
185	C₄	—	1264	1	no	—
186	C₄	#19098840	1	2	yes	—
187	C₄	#19098857	1	4	yes	—
188	C₄	#19098876	1	2	yes	—
189	C₄	#19098900	1	4	yes	—
190	C₂	#23763016	1	1	yes	characteristic
191	C₂	#23763067	1	1	yes	characteristic
192	C₂	#23763076	1	2	yes	—
193	C₂	#23764929	1	1	yes	characteristic
194	C₂	#23764935	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23764360

C₂

yes

maximal

#23764202

C₂

yes

maximal, characteristic

#23763559

C₂

yes

maximal

C₇₉⋊#1273

C₂

yes

maximal, characteristic

C₇₉×#1845

C₂

yes

maximal, characteristic

C₂×#14579161

C₂²

yes

characteristic

C₂×#14579094

—

C₂×#14579086

—

C₂×C₄×C₇₉⋊C₈

C₂²

yes

—

#19098433

C₂²

yes

characteristic

#19098325

—

C₂×#107×C₇₉

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

#384×C₇₉

C₂²

yes

—

C₂²×Dic₁₅₈

—

C₂²×C₇₉⋊C₈

D₄

yes

—

#14579161

—

#14579161

—

#14579161

Q₈

yes

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

#14579094

—

#14579086

—

C₄×C₇₉⋊C₈

—

C₄×C₇₉⋊C₈

—

C₂²×C₈×C₇₉

D₄

yes

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

#107×C₇₉

—

C₄×C₈×C₇₉

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

Q₁₆

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

D₈

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂×Q₈

yes

characteristic

C₄²×C₇₉

#41

yes

—

C₄²×C₇₉

yes

C₂³×C₇₉

yes

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₉

#123

yes

characteristic

C₂×C₄×C₇₉

#123

yes

—

C₂×C₄×C₇₉

C₂×D₈

yes

—

C₂×C₄×C₇₉

C₂×Q₁₆

yes

—

C₂×C₄×C₇₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₇₉

#137

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

—

C₂²×C₇₉

#496

yes

—

C₂²×C₇₉

#497

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

#474

yes

—

C₄×C₇₉

#477

yes

—

C₄×C₇₉

#480

yes

—

C₄×C₇₉

#484

yes

—

Dic₇₉

—

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₂×C₇₉

#2093

yes

characteristic

C₂×C₇₉

#2095

yes

—

#1990

—

#1956

—

100

#1845

—

101

#1845

D₇₉

yes

characteristic

102

#1273

—

103

C₇₉

#10674

yes

characteristic, complemented, Sylow

104

C₂×C₄⋊Q₈

—

105

C₂×#107

—

106

C₂×#107

—

158

—

107

C₂×#103

—

158

—

108

C₂×C₄×C₈

—

109

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

110

#384

—

111

#384

C₂×D₇₉

yes

—

112

#340

—

158

—

113

C₂²×Q₈

—

158

—

114

C₂²×C₈

—

115

C₂²×C₈

C₂²⋊D₇₉

yes

—

116

C₄⋊Q₈

—

117

C₄⋊Q₈

—

158

—

118

C₄⋊Q₈

—

158

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

—

158

—

123

C₂×C₄⋊C₄

—

158

—

124

C₂×C₄⋊C₄

—

158

—

125

C₂×C₄²

C₂²×D₇₉

yes

characteristic

126

#107

—

127

#107

—

316

—

128

#103

—

316

—

129

C₄×C₈

—

130

C₄×C₈

—

158

—

131

C₄×C₈

—

158

—

132

C₂×Q₈

—

316

—

133

C₂×Q₈

—

316

—

134

C₂×Q₈

—

316

—

135

C₂×Q₈

—

158

—

136

C₂²×C₄

—

137

C₂²×C₄

—

316

—

138

C₂²×C₄

—

316

—

139

C₂²×C₄

D₄×D₇₉

yes

—

140

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

158

—

145

C₂×C₈

—

158

—

146

C₂×C₈

#11929609

yes

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

316

—

151

C₄⋊C₄

—

316

—

152

C₄⋊C₄

—

316

—

153

C₄⋊C₄

—

158

—

154

C₄²

Q₈×D₇₉

yes

characteristic

155

C₄²

#11929624

yes

—

156

C₄²

#11929625

yes

characteristic

157

C₂³

#14579157

yes

characteristic

158

Q₈

—

316

—

159

Q₈

—

632

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

316

—

164

C₂×C₄

—

632

—

165

C₂×C₄

—

632

—

166

C₂×C₄

—

632

—

167

C₂×C₄

—

632

—

168

C₂×C₄

#14579137

yes

characteristic

169

C₂×C₄

#14579139

yes

—

170

C₂×C₄

#14579144

yes

characteristic

171

C₂×C₄

#14579146

yes

—

172

C₂×C₄

C₂×#11929609

yes

—

173

C₂×C₄

Q₁₆×D₇₉

yes

—

174

C₂×C₄

#14579185

yes

—

175

C₂×C₄

#14579192

yes

—

176

C₈

—

177

C₈

—

158

—

178

C₂²

#19098498

yes

characteristic

179

C₂²

#19098939

yes

characteristic

180

C₂²

#19098941

yes

—

181

C₂²

#19098951

yes

—

182

C₂²

#19098955

yes

characteristic

183

C₄

—

184

C₄

—

632

—

185

C₄

—

1264

—

186

C₄

#19098840

yes

—

187

C₄

#19098857

yes

—

188

C₄

#19098876

yes

—

189

C₄

#19098900

yes

—

190

C₂

#23763016

yes

characteristic

191

C₂

#23763067

yes

characteristic

192

C₂

#23763076

yes

—

193

C₂

#23764929

yes

characteristic

194

C₂

#23764935

yes

—

	subgroup	G/H
center	C₂³	#14579157
commutator, cosocle	C₂×C₄²×C₇₉	C₂³
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1845	C₂
socle, prime radical	C₂³×C₇₉	#127
hypercenter	#1845	D₇₉
hypocenter	C₇₉	#10674
LCS	G ▷ C₂×C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

subgroup

G/H

center

C₂³

#14579157

commutator, cosocle

C₂×C₄²×C₇₉

C₂³

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#1845

C₂

C₂³×C₇₉

D₇₉

C₇₉

G ▷ C₂×C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

N	H ≅ G/N	type
103. C₇₉	#10674	semidirect

H ≅ G/N

type

103. C₇₉

#10674

semidirect

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

subgroup	index
#23764360	2
#23764202	2
#23763559	2
C₇₉⋊#1273	2
C₇₉×#1845	2
#10674	79

subgroup

index

#23764360

#23764202

#23763559

C₇₉⋊#1273

C₇₉×#1845

#10674

Subgroups — #34193320

Conjugacy classes of subgroups

Complements

Maximal subgroups