Group #34188241 — subgroups

Subgroups — #34188241

Order 20224.

514 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 155 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	#23773828	C₂	1	1	yes	maximal, characteristic
2	#23763572	C₂	1	1	yes	maximal, characteristic, complemented
3	#23761852	C₂	1	4	yes	maximal, complemented
4	C₂×#322×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14579183	—	2	2	no	—
6	#19099113	C₂²	1	2	yes	—
7	#19098947	—	2	2	no	—
8	#19098943	C₄	1	1	yes	characteristic
9	#19098940	C₄	1	1	yes	characteristic
10	#19098936	C₄	1	2	yes	—
11	C₂×#14579097	—	2	2	no	—
12	#19098587	C₂²	1	2	yes	—
13	#19098292	—	2	8	no	—
14	C₂×C₄×C₈×C₇₉	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
16	#322×C₇₉	—	2	2	no	—
17	C₂×#11929624	—	2	2	no	—
18	#14579183	—	4	4	no	—
19	#14579183	—	4	4	no	—
20	C₂×Dic₃₁₆	—	2	2	no	—
21	C₂×#11929613	—	2	4	no	—
22	C₂×D₆₃₂	—	2	2	no	—
23	C₂×#11929608	—	2	2	no	—
24	#14579161	C₂×C₄	1	2	yes	—
25	#14579158	C₂×C₄	1	2	yes	—
26	#14579149	—	2	2	no	—
27	#14579135	—	2	4	no	—
28	C₄×D₃₁₆	—	2	4	no	—
29	#14579097	—	4	4	no	—
30	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
31	C₂²×C₈×C₇₉	—	2	2	no	—
32	C₂²×C₈×C₇₉	D₄	1	2	yes	—
33	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
34	C₄×C₈×C₇₉	—	2	2	no	—
35	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₇₉	D₄	1	2	yes	—
38	C₄×C₈×C₇₉	D₄	1	4	yes	—
39	#11929624	—	4	4	no	—
40	#11929624	—	4	4	no	—
41	C₂×Dic₁₅₈	—	2	2	no	—
42	C₂×D₃₁₆	—	2	2	no	—
43	C₂×D₃₁₆	—	2	4	no	—
44	C₂×C₂²⋊D₇₉	—	4	2	no	—
45	C₂×C₄×D₇₉	—	4	2	no	—
46	Dic₃₁₆	—	4	4	no	—
47	#11929613	—	4	8	no	—
48	D₆₃₂	—	4	4	no	—
49	#11929608	—	4	4	no	—
50	C₂×C₇₉⋊C₈	—	4	2	no	—
51	C₄⋊Dic₇₉	—	2	4	no	—
52	(C₂×D₇₉)⋊C₄	—	4	4	no	—
53	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₇₉	#41	1	2	yes	—
55	C₂×C₈×C₇₉	—	2	4	no	—
56	C₂×C₈×C₇₉	—	4	4	no	—
57	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
58	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
59	C₂×C₈×C₇₉	D₈	1	4	yes	—
60	C₂×C₈×C₇₉	SD₁₆	1	4	yes	—
61	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₇₉	C₂×D₄	1	2	yes	—
64	C₂³×C₇₉	#124	1	1	yes	characteristic
65	C₂²×D₇₉	—	4	2	no	—
66	C₂×C₄×C₇₉	—	2	4	no	—
67	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
69	C₂×C₄×C₇₉	#121	1	2	yes	—
70	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
71	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
72	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
73	C₂²⋊D₇₉	—	8	4	no	—
74	C₂×Dic₇₉	—	4	2	no	—
75	D₃₁₆	—	4	4	no	—
76	D₃₁₆	—	4	8	no	—
77	C₄×D₇₉	—	8	2	no	—
78	C₄×D₇₉	—	8	2	no	—
79	Dic₁₅₈	—	4	4	no	—
80	C₈×C₇₉	—	4	4	no	—
81	C₈×C₇₉	#102	1	4	yes	—
82	C₈×C₇₉	#102	1	4	yes	—
83	C₇₉⋊C₈	—	8	2	no	—
84	C₂²×C₇₉	—	2	2	no	—
85	C₂²×C₇₉	#390	1	1	yes	characteristic
86	C₂²×C₇₉	#486	1	2	yes	—
87	C₂×D₇₉	—	8	2	no	—
88	C₂×D₇₉	—	8	4	no	—
89	C₄×C₇₉	—	2	2	no	—
90	C₄×C₇₉	C₂×#102	1	2	yes	—
91	C₄×C₇₉	#444	1	4	yes	—
92	C₄×C₇₉	#492	1	1	yes	characteristic
93	C₄×C₇₉	#493	1	1	yes	characteristic
94	Dic₇₉	—	8	2	no	—
95	#9862	—	79	1	no	maximal, Sylow
96	C₂×C₇₉	—	2	2	no	—
97	C₂×C₇₉	#1957	1	1	yes	characteristic
98	C₂×C₇₉	#1975	1	2	yes	—
99	D₇₉	—	16	2	no	—
100	#3147	—	79	1	no	—
101	#1848	—	79	1	no	—
102	#1492	—	79	4	no	—
103	C₂×#322	D₇₉	1	1	yes	characteristic, complemented
104	C₇₉	#9862	1	1	yes	characteristic, complemented, Sylow
105	C₂×#135	—	158	2	no	—
106	#532	—	79	2	no	—
107	#495	—	158	2	no	—
108	#494	—	79	1	no	—
109	#493	—	79	1	no	—
110	#492	—	79	2	no	—
111	C₂×#108	—	158	2	no	—
112	#423	—	79	2	no	—
113	C₂×C₄×C₈	C₂×D₇₉	1	2	yes	—
114	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
115	D₄⋊C₈	—	158	8	no	—
116	#322	—	2	2	no	—
117	C₂×#41	—	158	2	no	—
118	#135	—	316	4	no	—
119	#135	—	316	4	no	—
120	C₂×Q₁₆	—	158	2	no	—
121	C₂×SD₁₆	—	158	4	no	—
122	C₂×D₈	—	158	2	no	—
123	C₂×M₁₆	—	158	2	no	—
124	C₂²×C₈	—	2	2	no	—
125	C₂²×C₈	D₃₁₆	1	2	yes	—
126	C₄⋊Q₈	—	79	2	no	—
127	#127	—	79	2	no	—
128	#124	—	158	2	no	—
129	#121	—	158	4	no	—
130	C₄×D₄	—	158	4	no	—
131	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
132	#108	—	316	4	no	—
133	C₄⋊C₈	—	158	4	no	—
134	C₄×C₈	—	2	2	no	—
135	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
136	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
137	C₄×C₈	D₃₁₆	1	4	yes	—
138	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
139	#41	—	316	4	no	—
140	#41	—	316	4	no	—
141	C₂×Q₈	—	158	2	no	—
142	C₂×D₄	—	316	2	no	—
143	C₂×D₄	—	158	2	no	—
144	C₂×D₄	—	158	4	no	—
145	C₂²×C₄	—	316	2	no	—
146	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
147	C₂²×C₄	#11929624	1	2	yes	—
148	Q₁₆	—	316	4	no	—
149	SD₁₆	—	316	8	no	—
150	D₈	—	316	4	no	—
151	M₁₆	—	316	4	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	4	4	no	—
154	C₂×C₈	—	316	2	no	—
155	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
156	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
157	C₂×C₈	D₆₃₂	1	4	yes	—
158	C₂×C₈	#11929613	1	4	yes	—
159	C₄⋊C₄	—	158	4	no	—
160	C₂²⋊C₄	—	316	4	no	—
161	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
162	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
163	C₄²	C₂×D₃₁₆	1	2	yes	—
164	C₂³	—	316	2	no	—
165	C₂³	#14579149	1	1	yes	characteristic
166	Q₈	—	316	4	no	—
167	D₄	—	316	8	no	—
168	D₄	—	316	4	no	—
169	D₄	—	632	4	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	316	2	no	—
172	C₂×C₄	—	632	2	no	—
173	C₂×C₄	—	632	2	no	—
174	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
175	C₂×C₄	C₄×D₃₁₆	1	2	yes	—
176	C₂×C₄	#14579135	1	2	yes	—
177	C₂×C₄	#14579158	1	1	yes	characteristic
178	C₂×C₄	C₂×D₆₃₂	1	2	yes	—
179	C₂×C₄	C₂×#11929613	1	2	yes	—
180	C₈	—	4	4	no	—
181	C₈	—	632	2	no	—
182	C₈	#14579083	1	4	yes	—
183	C₈	#14579083	1	4	yes	—
184	C₂²	—	2	2	no	—
185	C₂²	—	632	4	no	—
186	C₂²	—	632	2	no	—
187	C₂²	#19098465	1	1	yes	characteristic
188	C₂²	#19098910	1	2	yes	—
189	C₄	—	2	2	no	—
190	C₄	—	632	2	no	—
191	C₄	C₂×#14579083	1	2	yes	—
192	C₄	#19098686	1	4	yes	—
193	C₄	#19098936	1	1	yes	characteristic
194	C₄	#19098940	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	—	1264	2	no	—
197	C₂	#23764206	1	1	yes	characteristic
198	C₂	#23764293	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23773828

C₂

yes

maximal, characteristic

#23763572

C₂

yes

maximal, characteristic, complemented

#23761852

C₂

yes

maximal, complemented

C₂×#322×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×#14579183

—

#19099113

C₂²

yes

—

#19098947

—

#19098943

C₄

yes

characteristic

#19098940

C₄

yes

characteristic

#19098936

C₄

yes

—

C₂×#14579097

—

#19098587

C₂²

yes

—

#19098292

—

C₂×C₄×C₈×C₇₉

C₂²

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

#322×C₇₉

—

C₂×#11929624

—

#14579183

—

#14579183

—

C₂×Dic₃₁₆

—

C₂×#11929613

—

C₂×D₆₃₂

—

C₂×#11929608

—

#14579161

C₂×C₄

yes

—

#14579158

C₂×C₄

yes

—

#14579149

—

#14579135

—

C₄×D₃₁₆

—

#14579097

—

C₄⋊(C₇₉⋊C₈)

—

C₂²×C₈×C₇₉

—

C₂²×C₈×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄×C₈×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

—

C₄×C₈×C₇₉

D₄

yes

—

#11929624

—

#11929624

—

C₂×Dic₁₅₈

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×C₂²⋊D₇₉

—

C₂×C₄×D₇₉

—

Dic₃₁₆

—

#11929613

—

D₆₃₂

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

(C₂×D₇₉)⋊C₄

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

#41

yes

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₂×C₈×C₇₉

D₈

yes

—

C₂×C₈×C₇₉

SD₁₆

yes

—

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

—

C₂³×C₇₉

#124

yes

characteristic

C₂²×D₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄×D₄

yes

—

C₂×C₄×C₇₉

#121

yes

—

C₂×C₄×C₇₉

#127

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

—

C₂×C₄×C₇₉

C₂×SD₁₆

yes

—

C₂²⋊D₇₉

—

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

C₄×D₇₉

—

C₄×D₇₉

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

#102

yes

—

C₈×C₇₉

#102

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

#390

yes

characteristic

C₂²×C₇₉

#486

yes

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×#102

yes

—

C₄×C₇₉

#444

yes

—

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

#9862

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

#1957

yes

characteristic

C₂×C₇₉

#1975

yes

—

D₇₉

—

100

#3147

—

101

#1848

—

102

#1492

—

103

C₂×#322

D₇₉

yes

characteristic, complemented

104

C₇₉

#9862

yes

characteristic, complemented, Sylow

105

C₂×#135

—

158

—

106

#532

—

107

#495

—

158

—

108

#494

—

109

#493

—

110

#492

—

111

C₂×#108

—

158

—

112

#423

—

113

C₂×C₄×C₈

C₂×D₇₉

yes

—

114

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

115

D₄⋊C₈

—

158

—

116

#322

—

117

C₂×#41

—

158

—

118

#135

—

316

—

119

#135

—

316

—

120

C₂×Q₁₆

—

158

—

121

C₂×SD₁₆

—

158

—

122

C₂×D₈

—

158

—

123

C₂×M₁₆

—

158

—

124

C₂²×C₈

—

125

C₂²×C₈

D₃₁₆

yes

—

126

C₄⋊Q₈

—

127

#127

—

128

#124

—

158

—

129

#121

—

158

—

130

C₄×D₄

—

158

—

131

C₂×C₄²

C₂²×D₇₉

yes

characteristic

132

#108

—

316

—

133

C₄⋊C₈

—

158

—

134

C₄×C₈

—

135

C₄×C₈

C₄×D₇₉

yes

characteristic

136

C₄×C₈

C₄×D₇₉

yes

characteristic

137

C₄×C₈

D₃₁₆

yes

—

138

C₄×C₈

C₂²⋊D₇₉

yes

—

139

#41

—

316

—

140

#41

—

316

—

141

C₂×Q₈

—

158

—

142

C₂×D₄

—

316

—

143

C₂×D₄

—

158

—

144

C₂×D₄

—

158

—

145

C₂²×C₄

—

316

—

146

C₂²×C₄

C₂×D₃₁₆

yes

characteristic

147

C₂²×C₄

#11929624

yes

—

148

Q₁₆

—

316

—

149

SD₁₆

—

316

—

150

D₈

—

316

—

151

M₁₆

—

316

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

316

—

155

C₂×C₈

(C₂×D₇₉)⋊C₄

yes

—

156

C₂×C₈

(C₂×D₇₉)⋊C₄

yes

—

157

C₂×C₈

D₆₃₂

yes

—

158

C₂×C₈

#11929613

yes

—

159

C₄⋊C₄

—

158

—

160

C₂²⋊C₄

—

316

—

161

C₄²

C₂×C₄×D₇₉

yes

characteristic

162

C₄²

C₂×C₂²⋊D₇₉

yes

characteristic

163

C₄²

C₂×D₃₁₆

yes

—

164

C₂³

—

316

—

165

C₂³

#14579149

yes

characteristic

166

Q₈

—

316

—

167

D₄

—

316

—

168

D₄

—

316

—

169

D₄

—

632

—

170

C₂×C₄

—

171

C₂×C₄

—

316

—

172

C₂×C₄

—

632

—

173

C₂×C₄

—

632

—

174

C₂×C₄

C₂×(C₂×D₇₉)⋊C₄

yes

characteristic

175

C₂×C₄

C₄×D₃₁₆

yes

—

176

C₂×C₄

#14579135

yes

—

177

C₂×C₄

#14579158

yes

characteristic

178

C₂×C₄

C₂×D₆₃₂

yes

—

179

C₂×C₄

C₂×#11929613

yes

—

180

C₈

—

181

C₈

—

632

—

182

C₈

#14579083

yes

—

183

C₈

#14579083

yes

—

184

C₂²

—

185

C₂²

—

632

—

186

C₂²

—

632

—

187

C₂²

#19098465

yes

characteristic

188

C₂²

#19098910

yes

—

189

C₄

—

190

C₄

—

632

—

191

C₄

C₂×#14579083

yes

—

192

C₄

#19098686

yes

—

193

C₄

#19098936

yes

characteristic

194

C₄

#19098940

yes

characteristic

195

C₂

—

196

C₂

—

1264

—

197

C₂

#23764206

yes

characteristic

198

C₂

#23764293

yes

—

	subgroup	G/H
center	C₂×C₄	#14579158
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₂×#322×C₇₉	C₂
socle	C₂²×C₇₉	#390
prime radical	#23773828	C₂
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₂×#322	D₇₉
hypocenter	C₇₉	#9862
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

subgroup

G/H

center

C₂×C₄

#14579158

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₂×#322×C₇₉

C₂

C₂²×C₇₉

C₂

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

N	H ≅ G/N	type
2. #23763572	C₂	semidirect
3. #23761852	C₂	semidirect
4. C₂×#322×C₇₉	C₂	semidirect
103. C₂×#322	D₇₉	semidirect
104. C₇₉	#9862	semidirect

H ≅ G/N

type

2. #23763572

C₂

semidirect

3. #23761852

C₂

semidirect

4. C₂×#322×C₇₉

C₂

semidirect

103. C₂×#322

D₇₉

semidirect

104. C₇₉

#9862

semidirect

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

subgroup	index
#23773828	2
#23763572	2
#23761852	2
C₂×#322×C₇₉	2
#9862	79

subgroup

index

C₂×#322×C₇₉

Subgroups — #34188241

Conjugacy classes of subgroups

Complements

Maximal subgroups