Group #23771458 — subgroups

Subgroups — #23771458

Order 10112.

430 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 133 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×#14579161	C₂	1	1	yes	maximal, characteristic, complemented
2	#19098502	C₂	1	2	yes	maximal, complemented
3	#19098499	C₂	1	2	yes	maximal
4	#19098493	C₂	1	2	yes	maximal, complemented
5	#19098488	C₂	1	2	yes	maximal
6	#19098483	C₂	1	2	yes	maximal, complemented
7	#19098475	C₂	1	2	yes	maximal
8	#19098431	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#126×C₇₉	C₂	1	1	yes	maximal, characteristic
10	C₂²×Dic₁₅₈	C₂²	1	2	yes	—
11	#14579161	—	2	4	no	—
12	C₂×C₄⋊Dic₇₉	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₇₉	C₂²	1	2	yes	—
14	C₂×Dic₇₉⋊C₄	C₂²	1	4	yes	—
15	C₂×C₂²⋊Dic₇₉	C₂²	1	4	yes	—
16	C₇₉⋊#95	C₂²	1	2	yes	—
17	C₇₉⋊#95	C₂²	1	2	yes	—
18	C₇₉⋊#95	C₂²	1	2	yes	—
19	C₇₉⋊#95	C₂²	1	4	yes	—
20	C₇₉⋊#95	C₂²	1	4	yes	—
21	#126×C₇₉	—	2	4	no	—
22	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
28	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
29	C₂×Dic₁₅₈	—	2	4	no	—
30	C₂×Dic₁₅₈	—	4	4	no	—
31	C₂²×Dic₇₉	C₂³	1	4	yes	—
32	C₂²×Dic₇₉	C₂³	1	4	yes	—
33	C₄⋊Dic₇₉	—	2	4	no	—
34	C₄⋊Dic₇₉	—	2	4	no	—
35	Dic₇₉⋊C₄	—	4	4	no	—
36	C₂²⋊Dic₇₉	—	4	4	no	—
37	C₂⁴×C₇₉	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
44	C₄⋊C₄×C₇₉	—	2	2	no	—
45	C₄⋊C₄×C₇₉	—	2	2	no	—
46	C₄⋊C₄×C₇₉	—	2	2	no	—
47	C₂²⋊C₄×C₇₉	—	2	2	no	—
48	C₂²⋊C₄×C₇₉	—	2	2	no	—
49	C₂²⋊C₄×C₇₉	—	2	2	no	—
50	C₄²×C₇₉	D₄	1	4	yes	—
51	C₂³×C₇₉	—	2	1	no	—
52	C₂³×C₇₉	—	2	2	no	—
53	C₂³×C₇₉	—	2	2	no	—
54	C₂³×C₇₉	—	2	2	no	—
55	C₂³×C₇₉	C₂⁴	1	1	yes	characteristic
56	C₂×C₄×C₇₉	—	2	2	no	—
57	C₂×C₄×C₇₉	—	2	2	no	—
58	C₂×C₄×C₇₉	—	2	2	no	—
59	C₂×C₄×C₇₉	—	2	1	no	—
60	C₂×C₄×C₇₉	—	2	1	no	—
61	C₂×C₄×C₇₉	—	2	1	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	C₂×D₄	1	2	yes	—
66	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
67	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
68	C₂×Dic₇₉	—	2	4	no	—
69	C₂×Dic₇₉	—	2	4	no	—
70	C₂×Dic₇₉	—	2	4	no	—
71	C₂×Dic₇₉	—	2	4	no	—
72	C₂×Dic₇₉	—	2	4	no	—
73	C₂×Dic₇₉	—	2	4	no	—
74	Dic₁₅₈	—	4	8	no	—
75	C₂²×C₇₉	—	4	1	no	—
76	C₂²×C₇₉	—	4	1	no	—
77	C₂²×C₇₉	—	4	1	no	—
78	C₂²×C₇₉	—	4	4	no	—
79	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
80	C₂²×C₇₉	H(5,𝔽₂)	1	2	yes	—
81	C₂²×C₇₉	2₋¹⁺⁴	1	2	yes	—
82	C₂²×C₇₉	2₋¹⁺⁴	1	2	yes	—
83	C₄×C₇₉	—	2	2	no	—
84	C₄×C₇₉	—	2	2	no	—
85	C₄×C₇₉	—	2	2	no	—
86	C₄×C₇₉	—	4	1	no	—
87	C₄×C₇₉	—	4	1	no	—
88	C₄×C₇₉	—	4	1	no	—
89	Dic₇₉	—	4	4	no	—
90	Dic₇₉	—	4	4	no	—
91	C₂×C₇₉	—	8	1	no	—
92	C₂×C₇₉	#535	1	1	yes	characteristic
93	C₂×C₇₉	#536	1	1	yes	characteristic
94	C₂×C₇₉	#536	1	1	yes	characteristic
95	C₂×C₇₉	#563	1	4	yes	—
96	#2821	—	79	1	no	maximal, Sylow
97	C₇₉	#2821	1	1	yes	characteristic, complemented, Sylow
98	C₂×C₄⋊Q₈	—	79	1	no	—
99	C₂×#126	D₇₉	1	1	yes	characteristic
100	#400	—	79	2	no	—
101	#399	—	79	2	no	—
102	#398	—	79	2	no	—
103	#397	—	79	2	no	—
104	#395	—	79	2	no	—
105	#393	—	79	2	no	—
106	#383	—	79	1	no	—
107	C₂²×Q₈	—	79	2	no	—
108	C₄⋊Q₈	—	158	4	no	—
109	#126	—	2	4	no	—
110	C₂×C₄⋊C₄	—	79	4	no	—
111	C₂×C₄⋊C₄	—	79	2	no	—
112	C₂×C₄⋊C₄	—	79	2	no	—
113	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
114	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
115	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
116	C₂×C₂²⋊C₄	—	79	4	no	—
117	C₂×C₂²⋊C₄	C₂×D₇₉	1	1	yes	characteristic
118	C₂×C₂²⋊C₄	C₂×D₇₉	1	1	yes	characteristic
119	C₂×C₂²⋊C₄	C₂×D₇₉	1	1	yes	characteristic
120	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
121	#95	—	79	2	no	—
122	#95	—	79	2	no	—
123	#95	—	79	2	no	—
124	#95	—	79	4	no	—
125	#95	—	79	4	no	—
126	C₂⁴	C₂²×D₇₉	1	1	yes	characteristic
127	C₂×Q₈	—	316	4	no	—
128	C₂×Q₈	—	158	4	no	—
129	C₂²×C₄	—	79	4	no	—
130	C₂²×C₄	—	79	4	no	—
131	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
132	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
133	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
134	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
135	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
136	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
137	C₄⋊C₄	—	2	2	no	—
138	C₄⋊C₄	—	2	2	no	—
139	C₄⋊C₄	—	2	2	no	—
140	C₄⋊C₄	—	316	4	no	—
141	C₄⋊C₄	—	158	4	no	—
142	C₄⋊C₄	—	158	4	no	—
143	C₂²⋊C₄	—	2	2	no	—
144	C₂²⋊C₄	—	2	2	no	—
145	C₂²⋊C₄	—	2	2	no	—
146	C₂²⋊C₄	—	316	4	no	—
147	C₄²	C₂²⋊D₇₉	1	4	yes	—
148	C₂³	—	2	2	no	—
149	C₂³	—	2	2	no	—
150	C₂³	—	2	2	no	—
151	C₂³	—	2	1	no	—
152	C₂³	C₂³×D₇₉	1	1	yes	characteristic
153	Q₈	—	316	8	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	158	4	no	—
164	C₂×C₄	—	158	4	no	—
165	C₂×C₄	—	158	4	no	—
166	C₂×C₄	—	158	4	no	—
167	C₂×C₄	—	158	4	no	—
168	C₂×C₄	—	158	4	no	—
169	C₂×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
170	C₂×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
171	C₂×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
172	C₂²	—	4	4	no	—
173	C₂²	—	4	1	no	—
174	C₂²	—	4	1	no	—
175	C₂²	—	4	1	no	—
176	C₂²	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
177	C₂²	#14579207	1	2	yes	—
178	C₂²	#14579208	1	2	yes	—
179	C₂²	#14579208	1	2	yes	—
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	2	2	no	—
183	C₄	—	4	1	no	—
184	C₄	—	4	1	no	—
185	C₄	—	4	1	no	—
186	C₄	—	316	4	no	—
187	C₄	—	316	4	no	—
188	C₂	—	8	1	no	—
189	C₂	#19099127	1	1	yes	characteristic
190	C₂	#19099134	1	1	yes	characteristic
191	C₂	#19099134	1	1	yes	characteristic
192	C₂	#19099315	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579161

C₂

yes

maximal, characteristic, complemented

#19098502

C₂

yes

maximal, complemented

#19098499

C₂

yes

maximal

#19098493

C₂

yes

maximal, complemented

#19098488

C₂

yes

maximal

#19098483

C₂

yes

maximal, complemented

#19098475

C₂

yes

maximal

#19098431

C₂

yes

maximal, characteristic, complemented

C₂×#126×C₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₅₈

C₂²

yes

—

#14579161

—

C₂×C₄⋊Dic₇₉

C₂²

yes

—

C₂×C₄⋊Dic₇₉

C₂²

yes

—

C₂×Dic₇₉⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₇₉

C₂²

yes

—

C₇₉⋊#95

C₂²

yes

—

C₇₉⋊#95

C₂²

yes

—

C₇₉⋊#95

C₂²

yes

—

C₇₉⋊#95

C₂²

yes

—

C₇₉⋊#95

C₂²

yes

—

#126×C₇₉

—

C₂×C₄⋊C₄×C₇₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₉

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₇₉

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₇₉

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₇₉

C₂²

yes

characteristic

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂²×Dic₇₉

C₂³

yes

—

C₂²×Dic₇₉

C₂³

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

—

C₂⁴×C₇₉

C₂³

yes

characteristic

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

D₄

yes

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

C₂⁴

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

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₂²×C₇₉

H(5,𝔽₂)

yes

—

C₂²×C₇₉

2₋¹⁺⁴

yes

—

C₂²×C₇₉

2₋¹⁺⁴

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

#2821

—

maximal, Sylow

C₇₉

#2821

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×#126

D₇₉

yes

characteristic

100

#400

—

101

#399

—

102

#398

—

103

#397

—

104

#395

—

105

#393

—

106

#383

—

107

C₂²×Q₈

—

108

C₄⋊Q₈

—

158

—

109

#126

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

C₂×D₇₉

yes

characteristic

114

C₂×C₄⋊C₄

C₂×D₇₉

yes

characteristic

115

C₂×C₄⋊C₄

C₂×D₇₉

yes

characteristic

116

C₂×C₂²⋊C₄

—

117

C₂×C₂²⋊C₄

C₂×D₇₉

yes

characteristic

118

C₂×C₂²⋊C₄

C₂×D₇₉

yes

characteristic

119

C₂×C₂²⋊C₄

C₂×D₇₉

yes

characteristic

120

C₂×C₄²

C₂×D₇₉

yes

characteristic

121

#95

—

122

#95

—

123

#95

—

124

#95

—

125

#95

—

126

C₂⁴

C₂²×D₇₉

yes

characteristic

127

C₂×Q₈

—

316

—

128

C₂×Q₈

—

158

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

C₂²×D₇₉

yes

characteristic

132

C₂²×C₄

C₂²×D₇₉

yes

characteristic

133

C₂²×C₄

C₂²×D₇₉

yes

characteristic

134

C₂²×C₄

C₂²×D₇₉

yes

characteristic

135

C₂²×C₄

C₂²×D₇₉

yes

characteristic

136

C₂²×C₄

C₂²×D₇₉

yes

characteristic

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

316

—

141

C₄⋊C₄

—

158

—

142

C₄⋊C₄

—

158

—

143

C₂²⋊C₄

—

144

C₂²⋊C₄

—

145

C₂²⋊C₄

—

146

C₂²⋊C₄

—

316

—

147

C₄²

C₂²⋊D₇₉

yes

—

148

C₂³

—

149

C₂³

—

150

C₂³

—

151

C₂³

—

152

C₂³

C₂³×D₇₉

yes

characteristic

153

Q₈

—

316

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

158

—

164

C₂×C₄

—

158

—

165

C₂×C₄

—

158

—

166

C₂×C₄

—

158

—

167

C₂×C₄

—

158

—

168

C₂×C₄

—

158

—

169

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

170

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

171

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

175

C₂²

—

176

C₂²

C₂²×C₂²⋊D₇₉

yes

characteristic

177

C₂²

#14579207

yes

—

178

C₂²

#14579208

yes

—

179

C₂²

#14579208

yes

—

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

316

—

187

C₄

—

316

—

188

C₂

—

189

C₂

#19099127

yes

characteristic

190

C₂

#19099134

yes

characteristic

191

C₂

#19099134

yes

characteristic

192

C₂

#19099315

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇₉
commutator, socle, cosocle	C₂³×C₇₉	C₂⁴
Fitting	C₂×#126×C₇₉	C₂
prime radical	C₂⁴×C₇₉	C₂³
hypercenter	C₂×#126	D₇₉
hypocenter	C₇₉	#2821
LCS	G ▷ C₂³×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×#126

subgroup

G/H

center, Frattini

C₂³

C₂³×D₇₉

commutator, socle, cosocle

C₂³×C₇₉

C₂⁴

Fitting

C₂×#126×C₇₉

C₂

prime radical

C₂⁴×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂³×C₇₉ ▷ C₇₉

C₁ ◁ C₂³ ◁ C₂×#126

N	H ≅ G/N	type
1. C₂×#14579161	C₂	semidirect
2. #19098502	C₂	semidirect
4. #19098493	C₂	semidirect
6. #19098483	C₂	semidirect
8. #19098431	C₂	semidirect
97. C₇₉	#2821	semidirect

H ≅ G/N

type

1. C₂×#14579161

C₂

semidirect

2. #19098502

C₂

semidirect

4. #19098493

C₂

semidirect

6. #19098483

C₂

semidirect

8. #19098431

C₂

semidirect

97. C₇₉

#2821

semidirect

Subgroups — #23771458

Conjugacy classes of subgroups

Complements