Group #34188296 — subgroups

Subgroups — #34188296

Order 20224.

434 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 151 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	C₄×#14579153	C₂	1	1	yes	maximal, characteristic
2	#23763517	C₂	1	1	yes	maximal, characteristic
3	#23761240	C₂	1	4	yes	maximal
4	C₇₉×#1744	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579153	C₄	1	1	yes	characteristic
6	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
7	#19098461	C₄	1	2	yes	—
8	#19098447	—	2	2	no	—
9	#19098429	C₄	1	1	yes	characteristic
10	C₄×Dic₇₉⋊C₄	—	2	2	no	—
11	C₄×C₄⋊Dic₇₉	C₂²	1	2	yes	—
12	#19098318	—	2	2	no	—
13	C₂×C₄×C₈×C₇₉	C₂²	1	2	yes	—
14	C₄³×C₇₉	C₂²	1	1	yes	characteristic
15	#336×C₇₉	—	2	2	no	—
16	C₂²×C₇₉⋊C₈	—	2	2	no	—
17	#14579153	—	2	4	no	—
18	C₂×C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
19	C₂×C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
20	C₂×Dic₇₉⋊C₄	—	2	2	no	—
21	C₂×Dic₇₉⋊C₄	—	2	2	no	—
22	C₂×C₄×Dic₇₉	—	2	2	no	—
23	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
24	C₇₉⋊#95	—	2	2	no	—
25	C₇₉⋊#95	—	2	2	no	—
26	C₇₉⋊#95	—	2	2	no	—
27	C₂²×C₈×C₇₉	—	2	2	no	—
28	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₇₉	D₄	1	2	yes	—
31	C₂×C₄²×C₇₉	D₄	1	2	yes	—
32	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
33	C₄×C₈×C₇₉	—	2	2	no	—
34	C₄×C₈×C₇₉	Q₈	1	4	yes	—
35	C₂²×Dic₇₉	—	2	2	no	—
36	C₂²×Dic₇₉	—	2	2	no	—
37	C₂×C₇₉⋊C₈	—	4	2	no	—
38	C₂×C₇₉⋊C₈	—	4	4	no	—
39	C₄⋊Dic₇₉	—	2	2	no	—
40	C₄⋊Dic₇₉	—	2	4	no	—
41	C₄⋊Dic₇₉	M₁₆	1	4	yes	—
42	Dic₇₉⋊C₄	—	2	8	no	—
43	Dic₇₉⋊C₄	—	4	4	no	—
44	C₄×Dic₇₉	—	4	4	no	—
45	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₇₉	—	2	4	no	—
51	C₂×C₈×C₇₉	—	4	4	no	—
52	C₄²×C₇₉	—	2	2	no	—
53	C₄²×C₇₉	—	2	2	no	—
54	C₄²×C₇₉	—	2	4	no	—
55	C₄²×C₇₉	—	4	2	no	—
56	C₄²×C₇₉	C₂×Q₈	1	2	yes	—
57	C₄²×C₇₉	#41	1	2	yes	—
58	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₇₉	—	2	2	no	—
60	C₂×C₄×C₇₉	—	2	2	no	—
61	C₂×C₄×C₇₉	—	2	2	no	—
62	C₂×C₄×C₇₉	—	2	2	no	—
63	C₂×C₄×C₇₉	—	2	2	no	—
64	C₂×C₄×C₇₉	—	2	2	no	—
65	C₂×C₄×C₇₉	—	2	2	no	—
66	C₂×C₄×C₇₉	C₄×Q₈	1	2	yes	—
67	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
68	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
69	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
71	C₂×C₄×C₇₉	#131	1	2	yes	—
72	C₂×C₄×C₇₉	#135	1	4	yes	—
73	C₂×Dic₇₉	—	2	4	no	—
74	C₂×Dic₇₉	—	4	2	no	—
75	C₂×Dic₇₉	—	4	2	no	—
76	C₂×Dic₇₉	—	4	2	no	—
77	C₂×Dic₇₉	—	4	2	no	—
78	C₂×Dic₇₉	—	4	2	no	—
79	C₈×C₇₉	—	4	4	no	—
80	C₇₉⋊C₈	—	8	2	no	—
81	C₂²×C₇₉	#391	1	1	yes	characteristic
82	C₂²×C₇₉	#407	1	1	yes	characteristic
83	C₂²×C₇₉	#408	1	1	yes	characteristic
84	C₂²×C₇₉	#416	1	2	yes	—
85	C₂²×C₇₉	#421	1	2	yes	—
86	C₄×C₇₉	—	2	2	no	—
87	C₄×C₇₉	—	4	2	no	—
88	C₄×C₇₉	—	4	2	no	—
89	C₄×C₇₉	#446	1	4	yes	—
90	C₄×C₇₉	#488	1	2	yes	—
91	C₄×C₇₉	#499	1	2	yes	—
92	Dic₇₉	—	4	4	no	—
93	Dic₇₉	—	8	2	no	—
94	#9875	—	79	1	no	maximal, Sylow
95	C₂×C₇₉	#1482	1	2	yes	—
96	C₂×C₇₉	#1488	1	2	yes	—
97	C₂×C₇₉	#1985	1	1	yes	characteristic
98	C₂×C₇₉	#1991	1	1	yes	characteristic
99	C₂×C₇₉	#1995	1	1	yes	characteristic
100	C₄×#125	—	79	1	no	—
101	#1831	—	79	1	no	—
102	#1744	D₇₉	1	1	yes	characteristic
103	#1279	—	79	4	no	—
104	C₇₉	#9875	1	1	yes	characteristic, complemented, Sylow
105	C₂×#125	—	79	1	no	—
106	C₂×C₄⋊C₈	—	79	2	no	—
107	C₂×C₄×C₈	C₂×D₇₉	1	2	yes	—
108	#389	—	79	2	no	—
109	#387	—	158	2	no	—
110	#382	—	79	1	no	—
111	C₄×C₄⋊C₄	—	79	2	no	—
112	C₄×C₄⋊C₄	—	158	2	no	—
113	C₄³	C₂×D₇₉	1	1	yes	characteristic
114	#336	—	2	2	no	—
115	#336	—	158	2	no	—
116	C₂²×C₈	—	2	2	no	—
117	C₂²×C₈	—	158	2	no	—
118	#125	—	158	4	no	—
119	C₂×C₄⋊C₄	—	79	2	no	—
120	C₂×C₄⋊C₄	—	79	2	no	—
121	C₂×C₄⋊C₄	—	158	2	no	—
122	C₂×C₄⋊C₄	—	158	2	no	—
123	C₂×C₄²	—	158	2	no	—
124	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
125	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
126	C₂×C₄²	D₃₁₆	1	2	yes	—
127	C₂×C₄²	C₂²⋊D₇₉	1	2	yes	—
128	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
129	C₄⋊C₈	—	158	4	no	—
130	C₄×C₈	—	2	2	no	—
131	C₄×C₈	Dic₁₅₈	1	4	yes	—
132	#95	—	158	2	no	—
133	#95	—	158	2	no	—
134	#95	—	158	2	no	—
135	C₂²×C₄	—	158	2	no	—
136	C₂²×C₄	—	158	2	no	—
137	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	2	yes	—
138	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	2	yes	—
139	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
140	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
141	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	4	4	no	—
144	C₂×C₈	—	316	2	no	—
145	C₂×C₈	—	316	4	no	—
146	C₄⋊C₄	—	316	4	no	—
147	C₄⋊C₄	—	79	4	no	—
148	C₄⋊C₄	—	158	8	no	—
149	C₄⋊C₄	—	158	4	no	—
150	C₄⋊C₄	—	158	2	no	—
151	C₄²	—	2	2	no	—
152	C₄²	—	2	2	no	—
153	C₄²	—	2	4	no	—
154	C₄²	—	4	2	no	—
155	C₄²	—	316	4	no	—
156	C₄²	C₂×Dic₁₅₈	1	2	yes	—
157	C₄²	#11929624	1	2	yes	—
158	C₂³	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	316	2	no	—
167	C₂×C₄	—	316	2	no	—
168	C₂×C₄	—	316	2	no	—
169	C₂×C₄	—	316	2	no	—
170	C₂×C₄	—	316	2	no	—
171	C₂×C₄	—	158	4	no	—
172	C₂×C₄	C₄×Dic₁₅₈	1	2	yes	—
173	C₂×C₄	#14579140	1	2	yes	—
174	C₂×C₄	#14579149	1	1	yes	characteristic
175	C₂×C₄	#14579161	1	1	yes	characteristic
176	C₂×C₄	M₁₆×D₇₉	1	2	yes	—
177	C₂×C₄	#14579169	1	2	yes	—
178	C₂×C₄	#14579183	1	4	yes	—
179	C₈	—	4	4	no	—
180	C₈	—	632	2	no	—
181	C₂²	#19098469	1	1	yes	characteristic
182	C₂²	#19098526	1	1	yes	characteristic
183	C₂²	#19098531	1	1	yes	characteristic
184	C₂²	#19098560	1	2	yes	—
185	C₂²	#19098581	1	2	yes	—
186	C₄	—	2	2	no	—
187	C₄	—	4	2	no	—
188	C₄	—	4	2	no	—
189	C₄	—	316	4	no	—
190	C₄	—	632	2	no	—
191	C₄	#19098691	1	4	yes	—
192	C₄	#19098918	1	2	yes	—
193	C₄	#19098965	1	2	yes	—
194	C₂	#23761809	1	2	yes	—
195	C₂	#23761835	1	2	yes	—
196	C₂	#23764340	1	1	yes	characteristic
197	C₂	#23764365	1	1	yes	characteristic
198	C₂	#23764381	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14579153

C₂

yes

maximal, characteristic

#23763517

C₂

yes

maximal, characteristic

#23761240

C₂

yes

maximal

C₇₉×#1744

C₂

yes

maximal, characteristic

C₂×#14579153

C₄

yes

characteristic

C₂×C₄⋊(C₇₉⋊C₈)

C₂²

yes

—

#19098461

C₄

yes

—

#19098447

—

#19098429

C₄

yes

characteristic

C₄×Dic₇₉⋊C₄

—

C₄×C₄⋊Dic₇₉

C₂²

yes

—

#19098318

—

C₂×C₄×C₈×C₇₉

C₂²

yes

—

C₄³×C₇₉

C₂²

yes

characteristic

#336×C₇₉

—

C₂²×C₇₉⋊C₈

—

#14579153

—

C₂×C₄⋊Dic₇₉

C₂×C₄

yes

—

C₂×C₄⋊Dic₇₉

C₂×C₄

yes

—

C₂×Dic₇₉⋊C₄

—

C₂×Dic₇₉⋊C₄

—

C₂×C₄×Dic₇₉

—

C₄⋊(C₇₉⋊C₈)

—

C₇₉⋊#95

—

C₇₉⋊#95

—

C₇₉⋊#95

—

C₂²×C₈×C₇₉

—

C₂×C₄²×C₇₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄×C₈×C₇₉

—

C₄×C₈×C₇₉

Q₈

yes

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

M₁₆

yes

—

Dic₇₉⋊C₄

—

Dic₇₉⋊C₄

—

C₄×Dic₇₉

—

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂×Q₈

yes

—

C₄²×C₇₉

#41

yes

—

C₂³×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₄×Q₈

yes

—

C₂×C₄×C₇₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₉

#124

yes

characteristic

C₂×C₄×C₇₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₉

C₂×M₁₆

yes

—

C₂×C₄×C₇₉

#131

yes

—

C₂×C₄×C₇₉

#135

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

—

C₂²×C₇₉

#421

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#446

yes

—

C₄×C₇₉

#488

yes

—

C₄×C₇₉

#499

yes

—

Dic₇₉

—

Dic₇₉

—

#9875

—

maximal, Sylow

C₂×C₇₉

#1482

yes

—

C₂×C₇₉

#1488

yes

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

100

—

101

#1831

—

102

#1744

D₇₉

yes

characteristic

103

#1279

—

104

C₇₉

#9875

yes

characteristic, complemented, Sylow

105

C₂×#125

—

106

C₂×C₄⋊C₈

—

107

C₂×C₄×C₈

C₂×D₇₉

yes

—

108

#389

—

109

#387

—

158

—

110

#382

—

111

C₄×C₄⋊C₄

—

112

C₄×C₄⋊C₄

—

158

—

113

C₄³

C₂×D₇₉

yes

characteristic

114

#336

—

115

#336

—

158

—

116

C₂²×C₈

—

117

C₂²×C₈

—

158

—

118

#125

—

158

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

158

—

122

C₂×C₄⋊C₄

—

158

—

123

C₂×C₄²

—

158

—

124

C₂×C₄²

C₄×D₇₉

yes

characteristic

125

C₂×C₄²

C₄×D₇₉

yes

characteristic

126

C₂×C₄²

D₃₁₆

yes

—

127

C₂×C₄²

C₂²⋊D₇₉

yes

—

128

C₂×C₄²

C₂²×D₇₉

yes

characteristic

129

C₄⋊C₈

—

158

—

130

C₄×C₈

—

131

C₄×C₈

Dic₁₅₈

yes

—

132

#95

—

158

—

133

#95

—

158

—

134

#95

—

158

—

135

C₂²×C₄

—

158

—

136

C₂²×C₄

—

158

—

137

C₂²×C₄

(C₂×D₇₉)⋊C₄

yes

—

138

C₂²×C₄

(C₂×D₇₉)⋊C₄

yes

—

139

C₂²×C₄

C₂×C₄×D₇₉

yes

characteristic

140

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

141

C₂²×C₄

C₂×D₃₁₆

yes

characteristic

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

316

—

145

C₂×C₈

—

316

—

146

C₄⋊C₄

—

316

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

158

—

149

C₄⋊C₄

—

158

—

150

C₄⋊C₄

—

158

—

151

C₄²

—

152

C₄²

—

153

C₄²

—

154

C₄²

—

155

C₄²

—

316

—

156

C₄²

C₂×Dic₁₅₈

yes

—

157

C₄²

#11929624

yes

—

158

C₂³

C₂×(C₂×D₇₉)⋊C₄

yes

characteristic

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

316

—

167

C₂×C₄

—

316

—

168

C₂×C₄

—

316

—

169

C₂×C₄

—

316

—

170

C₂×C₄

—

316

—

171

C₂×C₄

—

158

—

172

C₂×C₄

C₄×Dic₁₅₈

yes

—

173

C₂×C₄

#14579140

yes

—

174

C₂×C₄

#14579149

yes

characteristic

175

C₂×C₄

#14579161

yes

characteristic

176

C₂×C₄

M₁₆×D₇₉

yes

—

177

C₂×C₄

#14579169

yes

—

178

C₂×C₄

#14579183

yes

—

179

C₈

—

180

C₈

—

632

—

181

C₂²

#19098469

yes

characteristic

182

C₂²

#19098526

yes

characteristic

183

C₂²

#19098531

yes

characteristic

184

C₂²

#19098560

yes

—

185

C₂²

#19098581

yes

—

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

316

—

190

C₄

—

632

—

191

C₄

#19098691

yes

—

192

C₄

#19098918

yes

—

193

C₄

#19098965

yes

—

194

C₂

#23761809

yes

—

195

C₂

#23761835

yes

—

196

C₂

#23764340

yes

characteristic

197

C₂

#23764365

yes

characteristic

198

C₂

#23764381

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₁₆
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1744	C₂
socle, prime radical	C₂³×C₇₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1744	D₇₉
hypocenter	C₇₉	#9875
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

subgroup

G/H

center

C₂²×C₄

C₂×D₃₁₆

commutator

C₂²×C₄×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#1744

C₂

socle, prime radical

C₂³×C₇₉

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

N	H ≅ G/N	type
104. C₇₉	#9875	semidirect

H ≅ G/N

type

104. C₇₉

#9875

semidirect

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

subgroup	index
C₄×#14579153	2
#23763517	2
#23761240	2
C₇₉×#1744	2
#9875	79

subgroup

index

C₇₉×#1744

Subgroups — #34188296

Conjugacy classes of subgroups

Complements

Maximal subgroups