Group #23763994 — subgroups

Subgroups — #23763994

Order 10112.

230 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 99 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	#19098462	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₄×Dic₇₉	C₂	1	1	yes	maximal, characteristic
3	#19098326	C₂	1	1	yes	maximal, characteristic
4	#19098325	C₂	1	1	yes	maximal, characteristic
5	#19098325	C₂	1	1	yes	maximal, characteristic
6	#19098317	C₂	1	1	yes	maximal, characteristic
7	C₂×#107×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₇₉	—	2	1	no	—
10	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₇₉	—	2	1	no	—
13	C₂×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
14	C₇₉⋊#95	—	2	1	no	—
15	C₇₉⋊#95	—	2	1	no	—
16	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
19	#107×C₇₉	C₄	1	4	yes	complemented
20	C₂²×Dic₇₉	—	2	1	no	—
21	C₂²×Dic₇₉	—	2	1	no	—
22	C₂²×Dic₇₉	D₄	1	1	yes	characteristic
23	C₂²×Dic₇₉	D₄	1	1	yes	characteristic
24	C₂×C₇₉⋊C₈	—	2	1	no	—
25	C₂×C₇₉⋊C₈	—	2	1	no	—
26	C₂×C₇₉⋊C₈	—	2	1	no	—
27	C₄⋊Dic₇₉	—	2	1	no	—
28	C₄⋊Dic₇₉	—	2	1	no	—
29	C₄⋊Dic₇₉	—	2	1	no	—
30	C₄⋊Dic₇₉	—	4	2	no	—
31	C₄⋊Dic₇₉	Q₈	1	2	yes	—
32	C₄×Dic₇₉	—	2	1	no	—
33	C₄×Dic₇₉	—	2	1	no	—
34	C₄×Dic₇₉	—	4	2	no	—
35	C₂²×C₄×C₇₉	—	2	1	no	—
36	C₂²×C₄×C₇₉	—	2	1	no	—
37	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
38	C₂×C₈×C₇₉	—	2	1	no	—
39	C₂×C₈×C₇₉	—	2	1	no	—
40	C₂×C₈×C₇₉	C₂×C₄	1	2	yes	—
41	C₄⋊C₄×C₇₉	—	2	1	no	—
42	C₄⋊C₄×C₇₉	—	2	1	no	—
43	C₄⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
44	C₄⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
45	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₇₉	—	2	2	no	—
47	C₂×C₄×C₇₉	—	2	2	no	—
48	C₂×C₄×C₇₉	—	4	1	no	—
49	C₂×C₄×C₇₉	—	4	1	no	—
50	C₂×C₄×C₇₉	—	4	1	no	—
51	C₂×C₄×C₇₉	—	4	1	no	—
52	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
55	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
56	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
57	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
58	C₂×Dic₇₉	—	2	1	no	—
59	C₂×Dic₇₉	—	2	1	no	—
60	C₂×Dic₇₉	—	2	2	no	—
61	C₂×Dic₇₉	—	2	1	no	—
62	C₂×Dic₇₉	—	2	1	no	—
63	C₂×Dic₇₉	—	2	1	no	—
64	C₂×Dic₇₉	—	4	1	no	—
65	C₂×Dic₇₉	—	4	1	no	—
66	C₂×Dic₇₉	—	4	1	no	—
67	C₂×Dic₇₉	—	4	1	no	—
68	C₂×Dic₇₉	—	4	1	no	—
69	C₂×Dic₇₉	Q₁₆	1	2	yes	—
70	C₈×C₇₉	—	2	2	no	—
71	C₇₉⋊C₈	—	4	1	no	—
72	C₂²×C₇₉	C₄×D₄	1	1	yes	characteristic
73	C₂²×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂²×C₇₉	#123	1	1	yes	characteristic
75	C₂²×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
76	C₂²×C₇₉	#135	1	1	yes	characteristic
77	C₂²×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₂²×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₄×C₇₉	—	4	2	no	—
80	C₄×C₇₉	—	4	2	no	—
81	C₄×C₇₉	#117	1	1	yes	characteristic
82	C₄×C₇₉	C₄×Q₈	1	1	yes	characteristic
83	C₄×C₇₉	#125	1	1	yes	characteristic
84	C₄×C₇₉	#126	1	1	yes	characteristic
85	Dic₇₉	—	2	2	no	—
86	Dic₇₉	—	4	2	no	—
87	Dic₇₉	—	4	1	no	—
88	Dic₇₉	—	8	1	no	—
89	C₂×C₇₉	#387	1	1	yes	characteristic
90	C₂×C₇₉	C₄×Q₁₆	1	1	yes	characteristic
91	C₂×C₇₉	#442	1	1	yes	characteristic
92	C₂×C₇₉	#477	1	1	yes	characteristic
93	C₂×C₇₉	#478	1	1	yes	characteristic
94	C₂×C₇₉	#482	1	1	yes	characteristic
95	C₂×C₇₉	#484	1	1	yes	characteristic
96	#1921	—	79	1	no	maximal, Sylow
97	C₇₉	#1921	1	1	yes	characteristic, complemented, Sylow
98	C₂×#107	D₇₉	1	1	yes	characteristic
99	#389	—	79	1	no	—
100	C₄×C₄⋊C₄	—	79	1	no	—
101	#340	—	79	1	no	—
102	#340	—	79	1	no	—
103	#340	—	79	1	no	—
104	#336	—	79	1	no	—
105	C₂²×C₈	—	79	1	no	—
106	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
107	C₂×C₄⋊C₄	—	79	1	no	—
108	C₂×C₄⋊C₄	—	79	1	no	—
109	C₂×C₄⋊C₄	—	158	1	no	—
110	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
111	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
112	C₂×C₄²	—	79	1	no	—
113	C₂×C₄²	—	158	1	no	—
114	#107	Dic₇₉	1	4	yes	complemented
115	#95	—	158	1	no	—
116	#95	—	158	1	no	—
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	—	79	1	no	—
120	C₂²×C₄	—	79	1	no	—
121	C₂²×C₄	—	158	1	no	—
122	C₂²×C₄	—	158	1	no	—
123	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	158	1	no	—
127	C₂×C₈	—	158	1	no	—
128	C₂×C₈	—	158	1	no	—
129	C₂×C₈	C₂×Dic₇₉	1	2	yes	—
130	C₄⋊C₄	—	2	1	no	—
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	—	316	2	no	—
133	C₄⋊C₄	—	79	2	no	—
134	C₄⋊C₄	—	158	1	no	—
135	C₄⋊C₄	—	158	1	no	—
136	C₄⋊C₄	—	158	1	no	—
137	C₄⋊C₄	C₂×Dic₇₉	1	2	yes	—
138	C₄⋊C₄	C₂×Dic₇₉	1	2	yes	—
139	C₄²	—	316	2	no	—
140	C₄²	—	158	1	no	—
141	C₄²	—	158	1	no	—
142	C₂³	D₄×D₇₉	1	1	yes	characteristic
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	316	1	no	—
150	C₂×C₄	—	316	1	no	—
151	C₂×C₄	—	316	1	no	—
152	C₂×C₄	—	316	1	no	—
153	C₂×C₄	—	316	1	no	—
154	C₂×C₄	—	79	2	no	—
155	C₂×C₄	—	158	2	no	—
156	C₂×C₄	—	158	1	no	—
157	C₂×C₄	—	158	1	no	—
158	C₂×C₄	—	158	1	no	—
159	C₂×C₄	—	158	1	no	—
160	C₂×C₄	—	158	1	no	—
161	C₂×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
162	C₂×C₄	Q₈×D₇₉	1	1	yes	characteristic
163	C₂×C₄	#11929623	1	1	yes	characteristic
164	C₂×C₄	#11929624	1	1	yes	characteristic
165	C₂×C₄	#11929624	1	1	yes	characteristic
166	C₂×C₄	#11929625	1	1	yes	characteristic
167	C₈	—	2	2	no	—
168	C₈	—	316	1	no	—
169	C₂²	D₄×Dic₇₉	1	1	yes	characteristic
170	C₂²	#14579139	1	1	yes	characteristic
171	C₂²	#14579146	1	1	yes	characteristic
172	C₂²	Q₁₆×D₇₉	1	1	yes	characteristic
173	C₂²	#14579184	1	1	yes	characteristic
174	C₂²	#14579185	1	1	yes	characteristic
175	C₂²	#14579185	1	1	yes	characteristic
176	C₄	—	4	2	no	—
177	C₄	—	4	2	no	—
178	C₄	—	316	1	no	—
179	C₄	—	316	2	no	—
180	C₄	—	632	1	no	—
181	C₄	—	158	2	no	—
182	C₄	#14579120	1	1	yes	characteristic
183	C₄	Q₈×Dic₇₉	1	1	yes	characteristic
184	C₄	#14579151	1	1	yes	characteristic
185	C₄	#14579156	1	1	yes	characteristic
186	C₂	#19098449	1	1	yes	characteristic
187	C₂	Q₁₆×Dic₇₉	1	1	yes	characteristic
188	C₂	#19098675	1	1	yes	characteristic
189	C₂	#19098857	1	1	yes	characteristic
190	C₂	#19098866	1	1	yes	characteristic
191	C₂	#19098890	1	1	yes	characteristic
192	C₂	#19098900	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19098462

C₂

yes

maximal, characteristic

C₄⋊C₄×Dic₇₉

C₂

yes

maximal, characteristic

#19098326

C₂

yes

maximal, characteristic

#19098325

C₂

yes

maximal, characteristic

#19098325

C₂

yes

maximal, characteristic

#19098317

C₂

yes

maximal, characteristic

C₂×#107×C₇₉

C₂

yes

maximal, characteristic

C₂²×C₇₉⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₉

C₂²

yes

characteristic

C₂×C₄×Dic₇₉

—

C₂×C₄×Dic₇₉

C₂²

yes

characteristic

C₇₉⋊#95

—

C₇₉⋊#95

—

C₂²×C₈×C₇₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₉

C₂²

yes

characteristic

#107×C₇₉

C₄

yes

complemented

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

D₄

yes

characteristic

C₂²×Dic₇₉

D₄

yes

characteristic

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

Q₈

yes

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂×C₄

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

C₂×C₄

yes

—

C₄⋊C₄×C₇₉

C₂×C₄

yes

—

C₂³×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₈

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

Q₁₆

yes

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

C₄×D₄

yes

characteristic

C₂²×C₇₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₇₉

#123

yes

characteristic

C₂²×C₇₉

C₂×Q₁₆

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#117

yes

characteristic

C₄×C₇₉

C₄×Q₈

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

#387

yes

characteristic

C₂×C₇₉

C₄×Q₁₆

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

C₂×#107

D₇₉

yes

characteristic

#389

—

100

C₄×C₄⋊C₄

—

101

#340

—

102

#340

—

103

#340

—

104

#336

—

105

C₂²×C₈

—

106

C₂²×C₈

C₂×D₇₉

yes

characteristic

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

158

—

110

C₂×C₄⋊C₄

C₂×D₇₉

yes

characteristic

111

C₂×C₄⋊C₄

C₂×D₇₉

yes

characteristic

112

C₂×C₄²

—

113

C₂×C₄²

—

158

—

114

#107

Dic₇₉

yes

complemented

115

#95

—

158

—

116

#95

—

158

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

158

—

122

C₂²×C₄

—

158

—

123

C₂²×C₄

C₂²×D₇₉

yes

characteristic

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

158

—

127

C₂×C₈

—

158

—

128

C₂×C₈

—

158

—

129

C₂×C₈

C₂×Dic₇₉

yes

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

316

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

158

—

135

C₄⋊C₄

—

158

—

136

C₄⋊C₄

—

158

—

137

C₄⋊C₄

C₂×Dic₇₉

yes

—

138

C₄⋊C₄

C₂×Dic₇₉

yes

—

139

C₄²

—

316

—

140

C₄²

—

158

—

141

C₄²

—

158

—

142

C₂³

D₄×D₇₉

yes

characteristic

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

316

—

150

C₂×C₄

—

316

—

151

C₂×C₄

—

316

—

152

C₂×C₄

—

316

—

153

C₂×C₄

—

316

—

154

C₂×C₄

—

155

C₂×C₄

—

158

—

156

C₂×C₄

—

158

—

157

C₂×C₄

—

158

—

158

C₂×C₄

—

158

—

159

C₂×C₄

—

158

—

160

C₂×C₄

—

158

—

161

C₂×C₄

C₂²×Dic₇₉

yes

characteristic

162

C₂×C₄

Q₈×D₇₉

yes

characteristic

163

C₂×C₄

#11929623

yes

characteristic

164

C₂×C₄

#11929624

yes

characteristic

165

C₂×C₄

#11929624

yes

characteristic

166

C₂×C₄

#11929625

yes

characteristic

167

C₈

—

168

C₈

—

316

—

169

C₂²

D₄×Dic₇₉

yes

characteristic

170

C₂²

#14579139

yes

characteristic

171

C₂²

#14579146

yes

characteristic

172

C₂²

Q₁₆×D₇₉

yes

characteristic

173

C₂²

#14579184

yes

characteristic

174

C₂²

#14579185

yes

characteristic

175

C₂²

#14579185

yes

characteristic

176

C₄

—

177

C₄

—

178

C₄

—

316

—

179

C₄

—

316

—

180

C₄

—

632

—

181

C₄

—

158

—

182

C₄

#14579120

yes

characteristic

183

C₄

Q₈×Dic₇₉

yes

characteristic

184

C₄

#14579151

yes

characteristic

185

C₄

#14579156

yes

characteristic

186

C₂

#19098449

yes

characteristic

187

C₂

Q₁₆×Dic₇₉

yes

characteristic

188

C₂

#19098675

yes

characteristic

189

C₂

#19098857

yes

characteristic

190

C₂

#19098866

yes

characteristic

191

C₂

#19098890

yes

characteristic

192

C₂

#19098900

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	C₂×#107×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×D₄
cosocle	C₂²×C₄×C₇₉	C₂³
hypercenter	C₂×#107	D₇₉
hypocenter	C₇₉	#1921
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₇₉

commutator

C₂×C₄×C₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

C₂×#107×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

C₂×D₄

cosocle

C₂²×C₄×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
19. #107×C₇₉	C₄	semidirect
97. C₇₉	#1921	semidirect
114. #107	Dic₇₉	semidirect

H ≅ G/N

type

19. #107×C₇₉

C₄

semidirect

97. C₇₉

#1921

semidirect

114. #107

Dic₇₉

semidirect

Subgroups — #23763994

Conjugacy classes of subgroups

Complements