Group #34206844 — subgroups

Subgroups — #34206844

Order 20224.

398 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 159 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×#19098968	C₂	1	1	yes	maximal, characteristic
2	C₄×#14579095	C₂	1	1	yes	maximal, characteristic
3	#23761492	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579161	—	2	1	no	—
6	#19098968	—	2	2	no	—
7	#19098968	C₄	1	4	yes	complemented
8	C₂×#14579095	—	2	1	no	—
9	C₂×#14579095	C₂²	1	2	yes	—
10	C₂×#14579095	C₂²	1	2	yes	—
11	C₄×C₄⋊Dic₇₉	—	2	1	no	—
12	#19098326	—	2	1	no	—
13	C₂²×C₁₆×C₇₉	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
15	C₄⋊C₁₆×C₇₉	—	2	2	no	—
16	C₂²×Dic₁₅₈	—	4	1	no	—
17	#14579161	—	2	4	no	—
18	#14579161	—	4	2	no	—
19	C₂×C₄⋊Dic₇₉	—	2	2	no	—
20	C₂×C₄⋊Dic₇₉	—	2	2	no	—
21	C₂×C₄⋊Dic₇₉	—	4	1	no	—
22	C₂×C₄×Dic₇₉	—	4	1	no	—
23	#14579095	—	2	4	no	—
24	#14579095	—	2	2	no	—
25	#14579095	—	2	2	no	—
26	#14579095	C₂×C₄	1	4	yes	—
27	#14579095	Q₈	1	4	yes	complemented
28	C₇₉⋊#95	—	4	1	no	—
29	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
32	C₂×C₁₆×C₇₉	—	2	2	no	—
33	C₂×C₁₆×C₇₉	—	2	4	no	—
34	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₇₉	D₄	1	2	yes	—
36	C₂×Dic₁₅₈	—	4	4	no	—
37	C₂×Dic₁₅₈	—	8	4	no	—
38	C₂²×Dic₇₉	—	4	2	no	—
39	C₂²×Dic₇₉	—	4	2	no	—
40	C₄⋊Dic₇₉	—	2	4	no	—
41	C₄⋊Dic₇₉	—	2	4	no	—
42	C₄⋊Dic₇₉	—	4	4	no	—
43	C₄⋊Dic₇₉	—	4	2	no	—
44	C₄⋊Dic₇₉	—	4	2	no	—
45	C₄×Dic₇₉	—	8	2	no	—
46	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
47	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
48	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₇₉	—	2	2	no	—
50	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
51	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
54	C₂×C₈×C₇₉	#41	1	2	yes	—
55	C₄²×C₇₉	C₂²⋊C₄	1	2	yes	—
56	C₄²×C₇₉	SD₁₆	1	2	yes	—
57	C₁₆×C₇₉	—	4	2	no	—
58	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₇₉	—	2	2	no	—
60	C₂×C₄×C₇₉	#102	1	2	yes	—
61	C₂×C₄×C₇₉	#102	1	2	yes	—
62	C₂×C₄×C₇₉	SD₃₂	1	2	yes	—
63	C₂×C₄×C₇₉	Q₃₂	1	2	yes	—
64	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
66	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
67	C₂×C₄×C₇₉	#137	1	2	yes	—
68	C₂×Dic₇₉	—	4	4	no	—
69	C₂×Dic₇₉	—	4	4	no	—
70	C₂×Dic₇₉	—	8	2	no	—
71	C₂×Dic₇₉	—	8	2	no	—
72	C₂×Dic₇₉	—	8	4	no	—
73	Dic₁₅₈	—	8	8	no	—
74	C₈×C₇₉	—	2	2	no	—
75	C₈×C₇₉	C₄×Q₈	1	2	yes	—
76	C₈×C₇₉	#124	1	1	yes	characteristic
77	C₈×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
78	C₂²×C₇₉	C₂×#102	1	1	yes	characteristic
79	C₂²×C₇₉	#474	1	2	yes	—
80	C₂²×C₇₉	C₂×SD₃₂	1	1	yes	characteristic
81	C₂²×C₇₉	C₂×Q₃₂	1	1	yes	characteristic
82	C₂²×C₇₉	#509	1	2	yes	—
83	C₄×C₇₉	—	2	2	no	—
84	C₄×C₇₉	#358	1	4	yes	—
85	C₄×C₇₉	#391	1	1	yes	characteristic
86	C₄×C₇₉	#419	1	1	yes	characteristic
87	C₄×C₇₉	#475	1	2	yes	—
88	Dic₇₉	—	8	4	no	—
89	Dic₇₉	—	8	4	no	—
90	#13276	—	79	1	no	maximal, Sylow
91	C₂×C₇₉	#1988	1	1	yes	characteristic
92	C₂×C₇₉	C₂×#358	1	1	yes	characteristic
93	C₂×C₇₉	#2140	1	1	yes	characteristic
94	C₂×C₇₉	#2226	1	2	yes	—
95	C₂×C₇₉	#2227	1	2	yes	—
96	C₂×#500	—	79	1	no	—
97	C₂×C₄⋊C₁₆	D₇₉	1	1	yes	characteristic
98	C₄×#107	—	79	1	no	—
99	#1380	—	79	4	no	—
100	C₇₉	#13276	1	1	yes	characteristic, complemented, Sylow
101	C₂×C₄⋊Q₈	—	158	1	no	—
102	C₂²×C₁₆	C₂×D₇₉	1	2	yes	—
103	#500	—	79	4	no	—
104	#500	—	158	2	no	—
105	C₂×#107	—	79	2	no	—
106	C₂×#107	—	79	2	no	—
107	C₂×#107	—	158	1	no	—
108	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
109	C₄×C₄⋊C₄	—	158	1	no	—
110	C₄⋊C₁₆	—	2	2	no	—
111	#340	—	158	1	no	—
112	C₂²×Q₈	—	316	1	no	—
113	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
114	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
115	C₄⋊Q₈	—	316	2	no	—
116	C₄⋊Q₈	—	158	4	no	—
117	C₂×C₄⋊C₄	—	316	1	no	—
118	C₂×C₄⋊C₄	—	158	2	no	—
119	C₂×C₄⋊C₄	—	158	2	no	—
120	C₂×C₄²	—	316	1	no	—
121	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
122	C₂×C₁₆	—	2	2	no	—
123	C₂×C₁₆	—	2	4	no	—
124	#107	—	79	4	no	—
125	#107	—	79	4	no	—
126	#107	—	158	4	no	—
127	#107	—	158	2	no	—
128	#107	—	158	2	no	—
129	C₄×C₈	C₄×D₇₉	1	2	yes	—
130	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
131	#95	—	316	1	no	—
132	C₂×Q₈	—	316	4	no	—
133	C₂×Q₈	—	632	4	no	—
134	C₂²×C₄	—	316	2	no	—
135	C₂²×C₄	—	316	2	no	—
136	C₂²×C₄	D₆₃₂	1	1	yes	characteristic
137	C₂²×C₄	D₆₃₂	1	1	yes	characteristic
138	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
141	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
142	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
143	C₂×C₈	Q₈×D₇₉	1	2	yes	—
144	C₂×C₈	#11929623	1	2	yes	—
145	C₄⋊C₄	—	316	4	no	—
146	C₄⋊C₄	—	316	2	no	—
147	C₄⋊C₄	—	316	2	no	—
148	C₄⋊C₄	—	158	4	no	—
149	C₄⋊C₄	—	158	4	no	—
150	C₄²	—	632	2	no	—
151	C₄²	(C₂×D₇₉)⋊C₄	1	2	yes	—
152	C₄²	#11929613	1	2	yes	—
153	C₁₆	—	4	2	no	—
154	C₂³	C₂×D₆₃₂	1	1	yes	characteristic
155	Q₈	—	632	8	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	316	4	no	—
158	C₂×C₄	—	316	4	no	—
159	C₂×C₄	—	632	2	no	—
160	C₂×C₄	—	632	4	no	—
161	C₂×C₄	—	632	2	no	—
162	C₂×C₄	#14579083	1	2	yes	—
163	C₂×C₄	#14579083	1	2	yes	—
164	C₂×C₄	#14579106	1	2	yes	—
165	C₂×C₄	Dic₆₃₂	1	2	yes	—
166	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
167	C₂×C₄	#14579141	1	2	yes	—
168	C₂×C₄	C₂×#11929613	1	1	yes	characteristic
169	C₂×C₄	#14579194	1	2	yes	—
170	C₈	—	2	2	no	—
171	C₈	#14579126	1	2	yes	—
172	C₈	#14579148	1	1	yes	characteristic
173	C₈	#14579159	1	1	yes	characteristic
174	C₂²	C₂×#14579083	1	1	yes	characteristic
175	C₂²	#19098844	1	2	yes	—
176	C₂²	C₂×#14579106	1	1	yes	characteristic
177	C₂²	C₂×Dic₆₃₂	1	1	yes	characteristic
178	C₂²	#19099001	1	2	yes	—
179	C₄	—	2	2	no	—
180	C₄	—	632	4	no	—
181	C₄	—	632	4	no	—
182	C₄	#19098368	1	4	yes	—
183	C₄	#19098468	1	1	yes	characteristic
184	C₄	#19098573	1	1	yes	characteristic
185	C₄	#19098849	1	2	yes	—
186	C₂	#23764351	1	1	yes	characteristic
187	C₂	C₂×#19098368	1	1	yes	characteristic
188	C₂	#23765117	1	1	yes	characteristic
189	C₂	#23765548	1	2	yes	—
190	C₂	#23765553	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19098968

C₂

yes

maximal, characteristic

C₄×#14579095

C₂

yes

maximal, characteristic

#23761492

C₂

yes

maximal

C₂×C₄⋊C₁₆×C₇₉

C₂

yes

maximal, characteristic

C₂×#14579161

—

#19098968

—

#19098968

C₄

yes

complemented

C₂×#14579095

—

C₂×#14579095

C₂²

yes

—

C₂×#14579095

C₂²

yes

—

C₄×C₄⋊Dic₇₉

—

#19098326

—

C₂²×C₁₆×C₇₉

C₂²

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₄⋊C₁₆×C₇₉

—

C₂²×Dic₁₅₈

—

#14579161

—

#14579161

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄×Dic₇₉

—

—

—

—

C₂×C₄

yes

—

#14579095

Q₈

yes

complemented

C₇₉⋊#95

—

C₂²×C₈×C₇₉

D₄

yes

characteristic

C₂²×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

characteristic

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

D₄

yes

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄×Dic₇₉

—

C₂²×C₄×C₇₉

D₈

yes

characteristic

C₂²×C₄×C₇₉

D₈

yes

characteristic

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₂×C₈×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₂×Q₈

yes

—

C₂×C₈×C₇₉

#41

yes

—

C₄²×C₇₉

C₂²⋊C₄

yes

—

C₄²×C₇₉

SD₁₆

yes

—

C₁₆×C₇₉

—

C₂³×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#102

yes

—

C₂×C₄×C₇₉

#102

yes

—

C₂×C₄×C₇₉

SD₃₂

yes

—

C₂×C₄×C₇₉

Q₃₂

yes

—

C₂×C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₉

#137

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

C₄×Q₈

yes

—

C₈×C₇₉

#124

yes

characteristic

C₈×C₇₉

C₄⋊Q₈

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

—

C₂²×C₇₉

C₂×SD₃₂

yes

characteristic

C₂²×C₇₉

C₂×Q₃₂

yes

characteristic

C₂²×C₇₉

#509

yes

—

C₄×C₇₉

—

C₄×C₇₉

#358

yes

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

Dic₇₉

—

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₂×C₇₉

#2227

yes

—

C₂×#500

—

C₂×C₄⋊C₁₆

D₇₉

yes

characteristic

C₄×#107

—

#1380

—

100

C₇₉

#13276

yes

characteristic, complemented, Sylow

101

C₂×C₄⋊Q₈

—

158

—

102

C₂²×C₁₆

C₂×D₇₉

yes

—

103

#500

—

104

#500

—

158

—

105

C₂×#107

—

106

C₂×#107

—

107

C₂×#107

—

158

—

108

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

109

C₄×C₄⋊C₄

—

158

—

110

C₄⋊C₁₆

—

111

#340

—

158

—

112

C₂²×Q₈

—

316

—

113

C₂²×C₈

D₃₁₆

yes

characteristic

114

C₂²×C₈

C₂²×D₇₉

yes

characteristic

115

C₄⋊Q₈

—

316

—

116

C₄⋊Q₈

—

158

—

117

C₂×C₄⋊C₄

—

316

—

118

C₂×C₄⋊C₄

—

158

—

119

C₂×C₄⋊C₄

—

158

—

120

C₂×C₄²

—

316

—

121

C₂×C₄²

D₃₁₆

yes

characteristic

122

C₂×C₁₆

—

123

C₂×C₁₆

—

124

#107

—

125

#107

—

126

#107

—

158

—

127

#107

—

158

—

128

#107

—

158

—

129

C₄×C₈

C₄×D₇₉

yes

—

130

C₄×C₈

C₂²⋊D₇₉

yes

—

131

#95

—

316

—

132

C₂×Q₈

—

316

—

133

C₂×Q₈

—

632

—

134

C₂²×C₄

—

316

—

135

C₂²×C₄

—

316

—

136

C₂²×C₄

D₆₃₂

yes

characteristic

137

C₂²×C₄

D₆₃₂

yes

characteristic

138

C₂²×C₄

C₂×D₃₁₆

yes

characteristic

139

C₂×C₈

—

140

C₂×C₈

(C₂×D₇₉)⋊C₄

yes

—

141

C₂×C₈

C₂×C₄×D₇₉

yes

characteristic

142

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

143

C₂×C₈

Q₈×D₇₉

yes

—

144

C₂×C₈

#11929623

yes

—

145

C₄⋊C₄

—

316

—

146

C₄⋊C₄

—

316

—

147

C₄⋊C₄

—

316

—

148

C₄⋊C₄

—

158

—

149

C₄⋊C₄

—

158

—

150

C₄²

—

632

—

151

C₄²

(C₂×D₇₉)⋊C₄

yes

—

152

C₄²

#11929613

yes

—

153

C₁₆

—

154

C₂³

C₂×D₆₃₂

yes

characteristic

155

Q₈

—

632

—

156

C₂×C₄

—

157

C₂×C₄

—

316

—

158

C₂×C₄

—

316

—

159

C₂×C₄

—

632

—

160

C₂×C₄

—

632

—

161

C₂×C₄

—

632

—

162

C₂×C₄

#14579083

yes

—

163

C₂×C₄

#14579083

yes

—

164

C₂×C₄

#14579106

yes

—

165

C₂×C₄

Dic₆₃₂

yes

—

166

C₂×C₄

C₂×(C₂×D₇₉)⋊C₄

yes

characteristic

167

C₂×C₄

#14579141

yes

—

168

C₂×C₄

C₂×#11929613

yes

characteristic

169

C₂×C₄

#14579194

yes

—

170

C₈

—

171

C₈

#14579126

yes

—

172

C₈

#14579148

yes

characteristic

173

C₈

#14579159

yes

characteristic

174

C₂²

C₂×#14579083

yes

characteristic

175

C₂²

#19098844

yes

—

176

C₂²

C₂×#14579106

yes

characteristic

177

C₂²

C₂×Dic₆₃₂

yes

characteristic

178

C₂²

#19099001

yes

—

179

C₄

—

180

C₄

—

632

—

181

C₄

—

632

—

182

C₄

#19098368

yes

—

183

C₄

#19098468

yes

characteristic

184

C₄

#19098573

yes

characteristic

185

C₄

#19098849

yes

—

186

C₂

#23764351

yes

characteristic

187

C₂

C₂×#19098368

yes

characteristic

188

C₂

#23765117

yes

characteristic

189

C₂

#23765548

yes

—

190

C₂

#23765553

yes

—

	subgroup	G/H
center	C₂³	C₂×D₆₃₂
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×C₄⋊C₁₆×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×D₈
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₇₉
hypocenter	C₇₉	#13276
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

subgroup

G/H

center

C₂³

C₂×D₆₃₂

commutator

C₂×C₈×C₇₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₂×C₄⋊C₁₆×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

C₂×D₈

cosocle

C₂²×C₈×C₇₉

C₂³

hypercenter

C₂×C₄⋊C₁₆

D₇₉

hypocenter

C₇₉

#13276

LCS

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

N	H ≅ G/N	type
7. #19098968	C₄	semidirect
27. #14579095	Q₈	semidirect
100. C₇₉	#13276	semidirect

H ≅ G/N

type

7. #19098968

C₄

semidirect

27. #14579095

Q₈

semidirect

100. C₇₉

#13276

semidirect

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

subgroup	index
C₂×#19098968	2
C₄×#14579095	2
#23761492	2
C₂×C₄⋊C₁₆×C₇₉	2
#13276	79

subgroup

index

C₂×C₄⋊C₁₆×C₇₉

Subgroups — #34206844

Conjugacy classes of subgroups

Complements

Maximal subgroups