Group #23773178 — subgroups

Subgroups — C₂×#19098871 #23773178

Order 10112.

384 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 157 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×#14579153	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₁₅₈	C₂	1	1	yes	maximal, characteristic
3	#19098871	C₂	1	8	yes	maximal, direct product factor
4	C₂×#14579095	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579093	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579086	C₂	1	1	yes	maximal, characteristic
7	C₂×#14579086	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊C₈×C₇₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₅₈	C₂²	1	1	yes	characteristic
10	#14579153	—	2	2	no	—
11	#14579153	C₂²	1	4	yes	—
12	C₄×Dic₁₅₈	—	2	2	no	—
13	C₄×Dic₁₅₈	C₂²	1	4	yes	—
14	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₇₉⋊C₄	—	2	1	no	—
18	C₂×Dic₇₉⋊C₄	—	2	1	no	—
19	C₂×Dic₇₉⋊C₄	—	2	1	no	—
20	C₂×C₄×Dic₇₉	—	2	1	no	—
21	#14579095	C₂²	1	4	yes	—
22	#14579093	C₂²	1	4	yes	—
23	#14579086	C₂²	1	4	yes	—
24	#14579086	C₂²	1	4	yes	—
25	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
26	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
28	C₄⋊C₈×C₇₉	C₂²	1	4	yes	—
29	C₂×Dic₁₅₈	—	2	1	no	—
30	C₂×Dic₁₅₈	—	2	2	no	—
31	C₂×Dic₁₅₈	Q₈	1	4	yes	—
32	C₂×Dic₁₅₈	C₂³	1	2	yes	—
33	C₂²×Dic₇₉	—	2	1	no	—
34	C₂²×Dic₇₉	—	2	1	no	—
35	C₂²×Dic₇₉	—	2	1	no	—
36	C₂²×Dic₇₉	—	2	1	no	—
37	C₄⋊Dic₇₉	—	2	1	no	—
38	C₄⋊Dic₇₉	—	2	1	no	—
39	C₄⋊Dic₇₉	—	2	1	no	—
40	C₄⋊Dic₇₉	C₂³	1	2	yes	—
41	C₄⋊Dic₇₉	C₂³	1	2	yes	—
42	C₄⋊Dic₇₉	C₂³	1	2	yes	—
43	Dic₇₉⋊C₄	—	2	4	no	—
44	Dic₇₉⋊C₄	—	2	4	no	—
45	Dic₇₉⋊C₄	—	2	4	no	—
46	C₄×Dic₇₉	—	2	4	no	—
47	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
48	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
49	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₇₉	—	2	2	no	—
51	C₂×C₈×C₇₉	—	2	2	no	—
52	C₂×C₈×C₇₉	C₂³	1	2	yes	—
53	C₂×C₈×C₇₉	C₂³	1	2	yes	—
54	C₄²×C₇₉	D₄	1	2	yes	—
55	C₄²×C₇₉	C₂³	1	2	yes	—
56	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₇₉	—	2	2	no	—
58	C₂×C₄×C₇₉	—	2	2	no	—
59	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₇₉	C₂×Q₈	1	2	yes	—
63	C₂×C₄×C₇₉	#41	1	2	yes	—
64	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
65	C₂×Dic₇₉	—	2	4	no	—
66	C₂×Dic₇₉	—	2	2	no	—
67	C₂×Dic₇₉	—	2	2	no	—
68	C₂×Dic₇₉	—	2	2	no	—
69	C₂×Dic₇₉	—	2	2	no	—
70	C₂×Dic₇₉	—	4	2	no	—
71	C₂×Dic₇₉	—	4	2	no	—
72	C₂×Dic₇₉	—	4	2	no	—
73	Dic₁₅₈	—	2	4	no	—
74	Dic₁₅₈	—	2	2	no	—
75	Dic₁₅₈	C₂×Q₈	1	4	yes	—
76	C₈×C₇₉	—	2	2	no	—
77	C₈×C₇₉	—	2	2	no	—
78	C₂²×C₇₉	C₂²⋊Q₈	1	2	yes	—
79	C₂²×C₇₉	#135	1	2	yes	—
80	C₂²×C₇₉	#137	1	2	yes	—
81	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
82	C₄×C₇₉	—	2	2	no	—
83	C₄×C₇₉	—	2	2	no	—
84	C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
85	C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
86	C₄×C₇₉	C₂×#41	1	1	yes	characteristic
87	Dic₇₉	—	2	4	no	—
88	Dic₇₉	—	4	2	no	—
89	Dic₇₉	—	4	2	no	—
90	Dic₇₉	—	4	2	no	—
91	C₂×C₇₉	#479	1	4	yes	complemented
92	C₂×C₇₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
93	C₂×C₇₉	C₂×#135	1	1	yes	characteristic
94	C₂×C₇₉	C₂×#137	1	1	yes	characteristic
95	C₂×#479	—	79	1	no	maximal, Sylow
96	C₇₉	C₂×#479	1	1	yes	characteristic, complemented, Sylow
97	C₂×#125	—	79	1	no	—
98	C₂×C₄×Q₈	—	79	1	no	—
99	#479	—	79	8	no	—
100	C₂×#107	—	79	1	no	—
101	C₂×#106	—	79	1	no	—
102	C₂×C₄⋊C₈	D₇₉	1	1	yes	characteristic
103	C₂×#103	—	79	1	no	—
104	C₂×#103	—	79	1	no	—
105	C₂²×Q₈	—	79	1	no	—
106	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
107	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
108	#125	—	79	4	no	—
109	#125	—	158	2	no	—
110	C₄×Q₈	—	79	4	no	—
111	C₄×Q₈	—	158	2	no	—
112	C₂×C₄⋊C₄	—	79	1	no	—
113	C₂×C₄⋊C₄	—	79	1	no	—
114	C₂×C₄⋊C₄	—	79	1	no	—
115	C₂×C₄⋊C₄	—	158	1	no	—
116	C₂×C₄⋊C₄	—	158	1	no	—
117	C₂×C₄⋊C₄	—	158	1	no	—
118	C₂×C₄²	—	158	1	no	—
119	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
120	#107	—	79	4	no	—
121	#106	—	79	4	no	—
122	C₄⋊C₈	C₂×D₇₉	1	4	yes	—
123	#103	—	79	4	no	—
124	#103	—	79	4	no	—
125	C₂×Q₈	—	79	4	no	—
126	C₂×Q₈	—	79	2	no	—
127	C₂×Q₈	—	158	1	no	—
128	C₂×Q₈	—	158	2	no	—
129	C₂²×C₄	—	158	1	no	—
130	C₂²×C₄	—	158	1	no	—
131	C₂²×C₄	—	158	1	no	—
132	C₂²×C₄	—	158	1	no	—
133	C₂²×C₄	D₃₁₆	1	1	yes	characteristic
134	C₂²×C₄	D₃₁₆	1	1	yes	characteristic
135	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	C₂²×D₇₉	1	2	yes	—
139	C₂×C₈	C₂²×D₇₉	1	2	yes	—
140	C₄⋊C₄	—	79	2	no	—
141	C₄⋊C₄	—	79	2	no	—
142	C₄⋊C₄	—	79	2	no	—
143	C₄⋊C₄	—	158	4	no	—
144	C₄⋊C₄	—	158	4	no	—
145	C₄⋊C₄	—	158	4	no	—
146	C₄⋊C₄	—	158	1	no	—
147	C₄⋊C₄	—	158	1	no	—
148	C₄⋊C₄	—	158	1	no	—
149	C₄²	—	158	4	no	—
150	C₄²	D₃₁₆	1	2	yes	—
151	C₄²	C₂²×D₇₉	1	2	yes	—
152	C₂³	C₂×D₃₁₆	1	1	yes	characteristic
153	Q₈	—	79	4	no	—
154	Q₈	—	158	2	no	—
155	Q₈	—	158	4	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	316	2	no	—
159	C₂×C₄	—	316	2	no	—
160	C₂×C₄	—	316	2	no	—
161	C₂×C₄	—	158	2	no	—
162	C₂×C₄	—	158	2	no	—
163	C₂×C₄	—	158	2	no	—
164	C₂×C₄	—	158	4	no	—
165	C₂×C₄	—	158	2	no	—
166	C₂×C₄	C₂×D₃₁₆	1	2	yes	—
167	C₂×C₄	C₂×D₃₁₆	1	2	yes	—
168	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
169	C₂×C₄	Q₈×D₇₉	1	2	yes	—
170	C₂×C₄	#11929623	1	2	yes	—
171	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
172	C₈	—	2	2	no	—
173	C₈	—	2	2	no	—
174	C₂²	#14579141	1	2	yes	—
175	C₂²	#14579183	1	2	yes	—
176	C₂²	#14579194	1	2	yes	—
177	C₂²	C₂²×D₃₁₆	1	1	yes	characteristic
178	C₄	—	2	2	no	—
179	C₄	—	2	2	no	—
180	C₄	—	316	2	no	—
181	C₄	—	316	2	no	—
182	C₄	—	316	2	no	—
183	C₄	—	158	4	no	—
184	C₄	#14579141	1	2	yes	—
185	C₄	C₂×Q₈×D₇₉	1	1	yes	characteristic
186	C₄	C₂×#11929623	1	1	yes	characteristic
187	C₂	#19098871	1	4	yes	direct product factor
188	C₂	C₂×#14579141	1	1	yes	characteristic
189	C₂	C₂×#14579183	1	1	yes	characteristic
190	C₂	C₂×#14579194	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579153

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₅₈

C₂

yes

maximal, characteristic

#19098871

C₂

yes

maximal, direct product factor

C₂×#14579095

C₂

yes

maximal, characteristic

C₂×#14579093

C₂

yes

maximal, characteristic

C₂×#14579086

C₂

yes

maximal, characteristic

C₂×#14579086

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₅₈

C₂²

yes

characteristic

#14579153

—

#14579153

C₂²

yes

—

C₄×Dic₁₅₈

—

C₄×Dic₁₅₈

C₂²

yes

—

C₂×C₄⋊Dic₇₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₉

C₂²

yes

characteristic

C₂×Dic₇₉⋊C₄

—

C₂×Dic₇₉⋊C₄

—

C₂×Dic₇₉⋊C₄

—

C₂×C₄×Dic₇₉

—

#14579095

C₂²

yes

—

#14579093

C₂²

yes

—

#14579086

C₂²

yes

—

#14579086

C₂²

yes

—

C₂²×C₈×C₇₉

C₂²

yes

characteristic

C₂²×C₈×C₇₉

C₂²

yes

characteristic

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₇₉

C₂²

yes

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

Q₈

yes

—

C₂×Dic₁₅₈

C₂³

yes

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

C₂³

yes

—

C₄⋊Dic₇₉

C₂³

yes

—

C₄⋊Dic₇₉

C₂³

yes

—

Dic₇₉⋊C₄

—

Dic₇₉⋊C₄

—

Dic₇₉⋊C₄

—

C₄×Dic₇₉

—

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂³

yes

—

C₂×C₈×C₇₉

C₂³

yes

—

C₄²×C₇₉

D₄

yes

—

C₄²×C₇₉

C₂³

yes

—

C₂³×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₈

yes

—

C₂×C₄×C₇₉

#41

yes

—

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

C₂×Q₈

yes

—

C₈×C₇₉

—

C₈×C₇₉

—

C₂²×C₇₉

C₂²⋊Q₈

yes

—

C₂²×C₇₉

#135

yes

—

C₂²×C₇₉

#137

yes

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂²⋊Q₈

yes

—

C₄×C₇₉

C₂²×Q₈

yes

characteristic

C₄×C₇₉

C₂×#41

yes

characteristic

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

#479

yes

complemented

C₂×C₇₉

C₂×C₂²⋊Q₈

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

C₂×#125

—

C₂×C₄×Q₈

—

#479

—

100

C₂×#107

—

101

C₂×#106

—

102

C₂×C₄⋊C₈

D₇₉

yes

characteristic

103

C₂×#103

—

104

C₂×#103

—

105

C₂²×Q₈

—

106

C₂²×C₈

C₂×D₇₉

yes

characteristic

107

C₂²×C₈

C₂×D₇₉

yes

characteristic

108

#125

—

109

#125

—

158

—

110

C₄×Q₈

—

111

C₄×Q₈

—

158

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

158

—

116

C₂×C₄⋊C₄

—

158

—

117

C₂×C₄⋊C₄

—

158

—

118

C₂×C₄²

—

158

—

119

C₂×C₄²

C₂×D₇₉

yes

characteristic

120

#107

—

121

#106

—

122

C₄⋊C₈

C₂×D₇₉

yes

—

123

#103

—

124

#103

—

125

C₂×Q₈

—

126

C₂×Q₈

—

127

C₂×Q₈

—

158

—

128

C₂×Q₈

—

158

—

129

C₂²×C₄

—

158

—

130

C₂²×C₄

—

158

—

131

C₂²×C₄

—

158

—

132

C₂²×C₄

—

158

—

133

C₂²×C₄

D₃₁₆

yes

characteristic

134

C₂²×C₄

D₃₁₆

yes

characteristic

135

C₂²×C₄

C₂²×D₇₉

yes

characteristic

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

C₂²×D₇₉

yes

—

139

C₂×C₈

C₂²×D₇₉

yes

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

158

—

144

C₄⋊C₄

—

158

—

145

C₄⋊C₄

—

158

—

146

C₄⋊C₄

—

158

—

147

C₄⋊C₄

—

158

—

148

C₄⋊C₄

—

158

—

149

C₄²

—

158

—

150

C₄²

D₃₁₆

yes

—

151

C₄²

C₂²×D₇₉

yes

—

152

C₂³

C₂×D₃₁₆

yes

characteristic

153

Q₈

—

154

Q₈

—

158

—

155

Q₈

—

158

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

316

—

159

C₂×C₄

—

316

—

160

C₂×C₄

—

316

—

161

C₂×C₄

—

158

—

162

C₂×C₄

—

158

—

163

C₂×C₄

—

158

—

164

C₂×C₄

—

158

—

165

C₂×C₄

—

158

—

166

C₂×C₄

C₂×D₃₁₆

yes

—

167

C₂×C₄

C₂×D₃₁₆

yes

—

168

C₂×C₄

C₂×D₃₁₆

yes

characteristic

169

C₂×C₄

Q₈×D₇₉

yes

—

170

C₂×C₄

#11929623

yes

—

171

C₂×C₄

C₂³×D₇₉

yes

characteristic

172

C₈

—

173

C₈

—

174

C₂²

#14579141

yes

—

175

C₂²

#14579183

yes

—

176

C₂²

#14579194

yes

—

177

C₂²

C₂²×D₃₁₆

yes

characteristic

178

C₄

—

179

C₄

—

180

C₄

—

316

—

181

C₄

—

316

—

182

C₄

—

316

—

183

C₄

—

158

—

184

C₄

#14579141

yes

—

185

C₄

C₂×Q₈×D₇₉

yes

characteristic

186

C₄

C₂×#11929623

yes

characteristic

187

C₂

#19098871

yes

direct product factor

188

C₂

C₂×#14579141

yes

characteristic

189

C₂

C₂×#14579183

yes

characteristic

190

C₂

C₂×#14579194

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₃₁₆
commutator, cosocle	C₂×C₄×C₇₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	C₂×C₄⋊C₈×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₇₉
hypocenter	C₇₉	C₂×#479
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₃₁₆

commutator, cosocle

C₂×C₄×C₇₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

C₂×C₄⋊C₈×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₇₉

hypocenter

C₇₉

C₂×#479

LCS

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
3. #19098871	C₂	direct (N × H)
91. C₂×C₇₉	#479	semidirect
96. C₇₉	C₂×#479	semidirect
187. C₂	#19098871	direct (N × H)

H ≅ G/N

type

3. #19098871

C₂

direct (N × H)

91. C₂×C₇₉

#479

semidirect

96. C₇₉

C₂×#479

semidirect

187. C₂

#19098871

direct (N × H)

Subgroups — C₂×#19098871 #23773178

Conjugacy classes of subgroups

Complements

Subgroups — C2×#19098871 #23773178

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#19098871 #23773178