Group #34208233 — subgroups

Subgroups — #34208233

Order 20224.

270 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 101 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×#19098386	C₂	1	2	yes	maximal
2	#23764143	C₂	1	1	yes	maximal, characteristic
3	#23763575	C₂	1	1	yes	maximal, characteristic
4	#23761509	C₂	1	2	yes	maximal
5	C₇₉×#1377	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579097	—	2	1	no	—
7	C₂×#14579097	C₂²	1	2	yes	—
8	C₂×#14579095	C₂²	1	1	yes	characteristic
9	C₂×#14579093	C₂²	1	1	yes	characteristic
10	#19098587	—	2	1	no	—
11	#19098461	—	2	1	no	—
12	#19098386	—	2	4	no	—
13	#19098326	—	2	1	no	—
14	C₂×M₃₂×C₇₉	C₂²	1	2	yes	—
15	C₂×#97×C₇₉	C₂²	1	1	yes	characteristic
16	C₂×#11929608	—	2	2	no	—
17	C₂×C₄⋊Dic₇₉	—	2	1	no	—
18	C₂×C₄⋊Dic₇₉	—	2	1	no	—
19	C₂×Dic₇₉⋊C₄	—	4	1	no	—
20	#14579097	—	2	2	no	—
21	#14579097	—	2	2	no	—
22	#14579097	—	4	2	no	—
23	#14579095	—	2	2	no	—
24	#14579093	—	2	1	no	—
25	#14579093	D₄	1	2	yes	—
26	C₄⋊(C₇₉⋊C₈)	—	4	2	no	—
27	C₇₉⋊#95	—	4	1	no	—
28	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
30	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
31	M₃₂×C₇₉	—	2	2	no	—
32	M₃₂×C₇₉	Q₈	1	4	yes	—
33	C₂×C₁₆×C₇₉	—	2	2	no	—
34	#97×C₇₉	—	2	2	no	—
35	C₂²×Dic₇₉	—	4	1	no	—
36	C₂²×Dic₇₉	—	4	1	no	—
37	#11929608	—	4	2	no	—
38	#11929608	—	4	2	no	—
39	C₂×C₇₉⋊C₈	—	4	2	no	—
40	C₄⋊Dic₇₉	—	2	2	no	—
41	C₄⋊Dic₇₉	—	4	1	no	—
42	C₄⋊Dic₇₉	—	4	1	no	—
43	C₄⋊Dic₇₉	—	4	1	no	—
44	Dic₇₉⋊C₄	—	8	2	no	—
45	C₂²×C₄×C₇₉	SD₁₆	1	1	yes	characteristic
46	C₂²×C₄×C₇₉	SD₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₇₉	—	2	1	no	—
49	C₂×C₈×C₇₉	—	2	1	no	—
50	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
51	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
53	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
54	C₂×C₈×C₇₉	#41	1	2	yes	—
55	C₄²×C₇₉	—	2	1	no	—
56	C₄²×C₇₉	—	2	1	no	—
57	C₁₆×C₇₉	—	2	4	no	—
58	C₂³×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
59	C₂×C₄×C₇₉	—	2	1	no	—
60	C₂×C₄×C₇₉	—	2	1	no	—
61	C₂×C₄×C₇₉	—	4	1	no	—
62	C₂×C₄×C₇₉	—	4	1	no	—
63	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
64	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
65	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
67	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
68	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
69	C₂×Dic₇₉	—	4	2	no	—
70	C₂×Dic₇₉	—	8	1	no	—
71	C₂×Dic₇₉	—	8	1	no	—
72	C₂×Dic₇₉	—	8	1	no	—
73	C₂×Dic₇₉	—	8	1	no	—
74	C₂×Dic₇₉	—	8	1	no	—
75	C₈×C₇₉	—	2	2	no	—
76	C₈×C₇₉	C₂²⋊Q₈	1	2	yes	—
77	C₈×C₇₉	#125	1	2	yes	—
78	C₇₉⋊C₈	—	8	2	no	—
79	C₂²×C₇₉	—	2	1	no	—
80	C₂²×C₇₉	—	2	1	no	—
81	C₂²×C₇₉	#461	1	1	yes	characteristic
82	C₂²×C₇₉	#481	1	1	yes	characteristic
83	C₂²×C₇₉	#498	1	1	yes	characteristic
84	C₄×C₇₉	—	4	1	no	—
85	C₄×C₇₉	—	4	1	no	—
86	C₄×C₇₉	#398	1	1	yes	characteristic
87	C₄×C₇₉	#460	1	1	yes	characteristic
88	C₄×C₇₉	#483	1	1	yes	characteristic
89	C₄×C₇₉	#501	1	1	yes	characteristic
90	Dic₇₉	—	8	2	no	—
91	Dic₇₉	—	16	1	no	—
92	#13510	—	79	1	no	maximal, Sylow
93	C₂×C₇₉	—	2	1	no	—
94	C₂×C₇₉	—	2	1	no	—
95	C₂×C₇₉	#2084	1	1	yes	characteristic
96	C₂×C₇₉	#2238	1	1	yes	characteristic
97	C₂×C₇₉	#2239	1	1	yes	characteristic
98	C₂×#365	—	79	2	no	—
99	#1944	—	79	1	no	—
100	#1849	—	79	1	no	—
101	#1387	—	79	2	no	—
102	#1377	D₇₉	1	1	yes	characteristic
103	C₇₉	#13510	1	1	yes	characteristic, complemented, Sylow
104	C₂×M₃₂	C₂×D₇₉	1	2	yes	—
105	C₂×#108	—	79	2	no	—
106	C₂×#108	—	158	1	no	—
107	C₂×#107	—	79	1	no	—
108	C₂×#106	—	79	1	no	—
109	#423	—	158	1	no	—
110	C₂×#97	C₂×D₇₉	1	1	yes	characteristic
111	#389	—	158	1	no	—
112	#365	—	158	4	no	—
113	#340	—	158	1	no	—
114	C₂×M₁₆	—	158	2	no	—
115	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
116	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
117	C₂×C₄⋊C₄	—	316	1	no	—
118	C₂×C₄⋊C₄	—	158	1	no	—
119	C₂×C₄⋊C₄	—	158	1	no	—
120	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
121	M₃₂	—	2	2	no	—
122	M₃₂	Dic₁₅₈	1	4	yes	—
123	C₂×C₁₆	—	2	2	no	—
124	#108	—	316	2	no	—
125	#108	—	158	2	no	—
126	#108	—	158	2	no	—
127	#107	—	158	2	no	—
128	#106	—	79	2	no	—
129	#106	—	158	1	no	—
130	C₄⋊C₈	—	316	2	no	—
131	#97	—	2	2	no	—
132	#95	—	316	1	no	—
133	C₂²×C₄	—	316	1	no	—
134	C₂²×C₄	—	316	1	no	—
135	C₂²×C₄	#11929613	1	1	yes	characteristic
136	C₂²×C₄	#11929613	1	1	yes	characteristic
137	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
138	M₁₆	—	316	2	no	—
139	M₁₆	—	316	2	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	316	2	no	—
143	C₂×C₈	#11929613	1	2	yes	—
144	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
145	C₂×C₈	C₂×Dic₁₅₈	1	2	yes	—
146	C₂×C₈	#11929623	1	2	yes	—
147	C₂×C₈	#11929625	1	1	yes	characteristic
148	C₄⋊C₄	—	316	1	no	—
149	C₄⋊C₄	—	316	1	no	—
150	C₄⋊C₄	—	316	1	no	—
151	C₄⋊C₄	—	632	2	no	—
152	C₄⋊C₄	—	158	2	no	—
153	C₄²	—	2	1	no	—
154	C₄²	—	2	1	no	—
155	C₁₆	—	2	4	no	—
156	C₂³	C₂×#11929613	1	1	yes	characteristic
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	4	1	no	—
160	C₂×C₄	—	4	1	no	—
161	C₂×C₄	—	316	2	no	—
162	C₂×C₄	—	632	1	no	—
163	C₂×C₄	—	632	1	no	—
164	C₂×C₄	—	632	1	no	—
165	C₂×C₄	—	632	1	no	—
166	C₂×C₄	—	632	1	no	—
167	C₂×C₄	#14579136	1	1	yes	characteristic
168	C₂×C₄	#14579145	1	1	yes	characteristic
169	C₂×C₄	#14579161	1	1	yes	characteristic
170	C₂×C₄	C₂×#11929613	1	1	yes	characteristic
171	C₂×C₄	#14579189	1	1	yes	characteristic
172	C₂×C₄	#14579194	1	1	yes	characteristic
173	C₈	—	2	2	no	—
174	C₈	—	632	2	no	—
175	C₈	#14579138	1	2	yes	—
176	C₈	#14579152	1	2	yes	—
177	C₂²	—	2	1	no	—
178	C₂²	—	2	1	no	—
179	C₂²	#19098769	1	1	yes	characteristic
180	C₂²	#19098882	1	1	yes	characteristic
181	C₂²	#19098961	1	1	yes	characteristic
182	C₄	—	4	1	no	—
183	C₄	—	4	1	no	—
184	C₄	—	632	2	no	—
185	C₄	—	1264	1	no	—
186	C₄	#19098495	1	1	yes	characteristic
187	C₄	#19098764	1	1	yes	characteristic
188	C₄	#19098894	1	1	yes	characteristic
189	C₄	#19098972	1	1	yes	characteristic
190	C₂	—	2	1	no	—
191	C₂	—	2	1	no	—
192	C₂	#23764878	1	1	yes	characteristic
193	C₂	#23765615	1	1	yes	characteristic
194	C₂	#23765620	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19098386

C₂

yes

maximal

#23764143

C₂

yes

maximal, characteristic

#23763575

C₂

yes

maximal, characteristic

#23761509

C₂

yes

maximal

C₇₉×#1377

C₂

yes

maximal, characteristic

C₂×#14579097

—

C₂×#14579097

C₂²

yes

—

C₂×#14579095

C₂²

yes

characteristic

C₂×#14579093

C₂²

yes

—

—

—

—

C₂×M₃₂×C₇₉

C₂²

yes

—

C₂×#97×C₇₉

C₂²

yes

characteristic

C₂×#11929608

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

C₂×Dic₇₉⋊C₄

—

—

—

—

—

—

D₄

yes

—

C₄⋊(C₇₉⋊C₈)

—

C₇₉⋊#95

—

C₂²×C₈×C₇₉

D₄

yes

characteristic

C₂²×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

characteristic

M₃₂×C₇₉

—

M₃₂×C₇₉

Q₈

yes

—

C₂×C₁₆×C₇₉

—

#97×C₇₉

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

C₂²×C₄×C₇₉

SD₁₆

yes

characteristic

C₂²×C₄×C₇₉

SD₁₆

yes

characteristic

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

SD₁₆

yes

—

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₂×Q₈

yes

—

C₂×C₈×C₇₉

#41

yes

characteristic

C₂×C₈×C₇₉

#41

yes

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₁₆×C₇₉

—

C₂³×C₇₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₉

C₂×SD₁₆

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₈×C₇₉

—

C₈×C₇₉

C₂²⋊Q₈

yes

—

C₈×C₇₉

#125

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

Dic₇₉

—

#13510

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

—

100

#1849

—

101

#1387

—

102

#1377

D₇₉

yes

characteristic

103

C₇₉

#13510

yes

characteristic, complemented, Sylow

104

C₂×M₃₂

C₂×D₇₉

yes

—

105

C₂×#108

—

106

C₂×#108

—

158

—

107

C₂×#107

—

108

C₂×#106

—

109

#423

—

158

—

110

C₂×#97

C₂×D₇₉

yes

characteristic

111

#389

—

158

—

112

#365

—

158

—

113

#340

—

158

—

114

C₂×M₁₆

—

158

—

115

C₂²×C₈

D₃₁₆

yes

characteristic

116

C₂²×C₈

C₂²×D₇₉

yes

characteristic

117

C₂×C₄⋊C₄

—

316

—

118

C₂×C₄⋊C₄

—

158

—

119

C₂×C₄⋊C₄

—

158

—

120

C₂×C₄²

D₃₁₆

yes

characteristic

121

M₃₂

—

122

M₃₂

Dic₁₅₈

yes

—

123

C₂×C₁₆

—

124

#108

—

316

—

125

#108

—

158

—

126

#108

—

158

—

127

#107

—

158

—

128

#106

—

129

#106

—

158

—

130

C₄⋊C₈

—

316

—

131

#97

—

132

#95

—

316

—

133

C₂²×C₄

—

316

—

134

C₂²×C₄

—

316

—

135

C₂²×C₄

#11929613

yes

characteristic

136

C₂²×C₄

#11929613

yes

characteristic

137

C₂²×C₄

C₂×D₃₁₆

yes

characteristic

138

M₁₆

—

316

—

139

M₁₆

—

316

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

316

—

143

C₂×C₈

#11929613

yes

—

144

C₂×C₈

D₄×D₇₉

yes

characteristic

145

C₂×C₈

C₂×Dic₁₅₈

yes

—

146

C₂×C₈

#11929623

yes

—

147

C₂×C₈

#11929625

yes

characteristic

148

C₄⋊C₄

—

316

—

149

C₄⋊C₄

—

316

—

150

C₄⋊C₄

—

316

—

151

C₄⋊C₄

—

632

—

152

C₄⋊C₄

—

158

—

153

C₄²

—

154

C₄²

—

155

C₁₆

—

156

C₂³

C₂×#11929613

yes

characteristic

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

316

—

162

C₂×C₄

—

632

—

163

C₂×C₄

—

632

—

164

C₂×C₄

—

632

—

165

C₂×C₄

—

632

—

166

C₂×C₄

—

632

—

167

C₂×C₄

#14579136

yes

characteristic

168

C₂×C₄

#14579145

yes

characteristic

169

C₂×C₄

#14579161

yes

characteristic

170

C₂×C₄

C₂×#11929613

yes

characteristic

171

C₂×C₄

#14579189

yes

characteristic

172

C₂×C₄

#14579194

yes

characteristic

173

C₈

—

174

C₈

—

632

—

175

C₈

#14579138

yes

—

176

C₈

#14579152

yes

—

177

C₂²

—

178

C₂²

—

179

C₂²

#19098769

yes

characteristic

180

C₂²

#19098882

yes

characteristic

181

C₂²

#19098961

yes

characteristic

182

C₄

—

183

C₄

—

184

C₄

—

632

—

185

C₄

—

1264

—

186

C₄

#19098495

yes

characteristic

187

C₄

#19098764

yes

characteristic

188

C₄

#19098894

yes

characteristic

189

C₄

#19098972

yes

characteristic

190

C₂

—

191

C₂

—

192

C₂

#23764878

yes

characteristic

193

C₂

#23765615

yes

characteristic

194

C₂

#23765620

yes

characteristic

	subgroup	G/H
center	C₂²	#19098882
commutator, cosocle	C₂²×C₈×C₇₉	C₂³
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#1377	C₂
socle	C₂²×C₇₉	#481
prime radical	C₂³×C₇₉	C₂×SD₁₆
hypercenter	#1377	D₇₉
hypocenter	C₇₉	#13510
LCS	G ▷ C₂²×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1377

subgroup

G/H

center

C₂²

#19098882

commutator, cosocle

C₂²×C₈×C₇₉

C₂³

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₇₉×#1377

C₂

socle

C₂²×C₇₉

#481

prime radical

C₂³×C₇₉

C₂×SD₁₆

D₇₉

C₇₉

G ▷ C₂²×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1377

N	H ≅ G/N	type
103. C₇₉	#13510	semidirect

H ≅ G/N

type

103. C₇₉

#13510

semidirect

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

subgroup	index
C₂×#19098386	2
#23764143	2
#23763575	2
#23761509	2
C₇₉×#1377	2
#13510	79

subgroup

index

C₇₉×#1377

Subgroups — #34208233

Conjugacy classes of subgroups

Complements

Maximal subgroups