Group #23776482 — subgroups

Subgroups — #23776482

Order 10112.

356 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 113 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	#19099282	C₂	1	2	yes	maximal, complemented
2	#19098954	C₂	1	2	yes	maximal, complemented
3	#19098782	C₂	1	2	yes	maximal, complemented
4	#19098776	C₂	1	2	yes	maximal
5	#19098771	C₂	1	2	yes	maximal, complemented
6	#19098693	C₂	1	1	yes	maximal, characteristic, complemented
7	#19098664	C₂	1	2	yes	maximal, complemented
8	C₄×Dic₃₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	#495×C₇₉	C₂	1	1	yes	maximal, characteristic
10	C₂×Dic₃₁₆	C₂²	1	1	yes	characteristic
11	C₂×#11929614	C₂²	1	2	yes	—
12	C₂×#11929612	C₂²	1	2	yes	—
13	#14579159	C₂²	1	2	yes	—
14	#14579154	—	2	2	no	—
15	#14579154	—	2	2	no	—
16	#14579148	C₂²	1	2	yes	—
17	#14579138	—	2	2	no	—
18	Q₈×Dic₇₉	C₂²	1	2	yes	—
19	C₄×Dic₁₅₈	C₂²	1	2	yes	—
20	C₄×Dic₁₅₈	C₂²	1	1	yes	characteristic
21	D₄×Dic₇₉	C₂²	1	2	yes	—
22	#14579117	—	2	2	no	—
23	#14579095	C₂²	1	1	yes	characteristic
24	C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
25	Dic₇₉⋊C₈	C₂²	1	2	yes	—
26	#14579086	C₂²	1	2	yes	—
27	#14579085	C₂²	1	2	yes	—
28	#14579082	C₂²	1	2	yes	—
29	#14579071	C₂²	1	2	yes	—
30	C₂×Q₁₆×C₇₉	C₂²	1	1	yes	characteristic
31	C₂×SD₁₆×C₇₉	C₂²	1	2	yes	—
32	C₂×D₈×C₇₉	C₂²	1	1	yes	characteristic
33	#124×C₇₉	C₂²	1	2	yes	—
34	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
35	C₂×Dic₁₅₈	D₄	1	1	yes	characteristic
36	C₂×Dic₁₅₈	C₂³	1	2	yes	—
37	C₂²×Dic₇₉	—	2	2	no	—
38	Dic₃₁₆	—	2	2	no	—
39	#11929614	—	4	2	no	—
40	#11929612	—	4	2	no	—
41	C₂×C₇₉⋊C₈	C₂³	1	2	yes	—
42	C₄⋊Dic₇₉	—	2	2	no	—
43	C₄⋊Dic₇₉	D₄	1	1	yes	characteristic
44	C₄⋊Dic₇₉	C₂³	1	2	yes	—
45	Dic₇₉⋊C₄	—	2	2	no	—
46	Dic₇₉⋊C₄	—	2	2	no	—
47	Dic₇₉⋊C₄	—	2	2	no	—
48	Dic₇₉⋊C₄	D₄	1	2	yes	—
49	C₂²⋊Dic₇₉	—	2	2	no	—
50	C₂²⋊Dic₇₉	—	2	2	no	—
51	C₂²⋊Dic₇₉	—	2	2	no	—
52	C₄×Dic₇₉	—	2	2	no	—
53	C₄×Dic₇₉	—	2	2	no	—
54	C₄×Dic₇₉	C₂³	1	2	yes	—
55	C₂×Q₈×C₇₉	C₂³	1	2	yes	—
56	C₂×D₄×C₇₉	C₂³	1	2	yes	—
57	Q₁₆×C₇₉	—	4	1	no	—
58	SD₁₆×C₇₉	—	2	4	no	—
59	D₈×C₇₉	—	4	1	no	—
60	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
61	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
62	C₂²⋊C₄×C₇₉	—	2	2	no	—
63	C₂²⋊C₄×C₇₉	—	2	2	no	—
64	C₄²×C₇₉	C₂³	1	1	yes	characteristic
65	C₂³×C₇₉	—	2	2	no	—
66	Q₈×C₇₉	—	2	2	no	—
67	Q₈×C₇₉	—	2	2	no	—
68	D₄×C₇₉	—	2	2	no	—
69	D₄×C₇₉	—	2	2	no	—
70	C₂×C₄×C₇₉	—	2	2	no	—
71	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
74	C₂×Dic₇₉	—	2	2	no	—
75	C₂×Dic₇₉	—	2	2	no	—
76	C₂×Dic₇₉	—	4	4	no	—
77	C₂×Dic₇₉	C₂×D₄	1	2	yes	—
78	C₂×Dic₇₉	C₂×D₄	1	2	yes	—
79	Dic₁₅₈	—	2	2	no	—
80	Dic₁₅₈	—	2	2	no	—
81	Dic₁₅₈	—	4	1	no	—
82	C₈×C₇₉	—	2	1	no	—
83	C₈×C₇₉	#41	1	2	yes	—
84	C₇₉⋊C₈	—	2	2	no	—
85	C₂²×C₇₉	—	4	2	no	—
86	C₂²×C₇₉	—	4	2	no	—
87	C₂²×C₇₉	—	4	2	no	—
88	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
89	C₄×C₇₉	—	2	1	no	—
90	C₄×C₇₉	—	2	1	no	—
91	C₄×C₇₉	—	4	2	no	—
92	C₄×C₇₉	C₂×#41	1	1	yes	characteristic
93	C₄×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
94	Dic₇₉	—	2	2	no	—
95	Dic₇₉	—	2	2	no	—
96	Dic₇₉	—	4	2	no	—
97	Dic₇₉	—	4	2	no	—
98	C₂×C₇₉	—	8	2	no	—
99	C₂×C₇₉	#550	1	1	yes	characteristic
100	C₂×C₇₉	#575	1	1	yes	characteristic
101	C₂×C₇₉	#578	1	1	yes	characteristic
102	#3402	—	79	1	no	maximal, Sylow
103	C₇₉	#3402	1	1	yes	characteristic, complemented, Sylow
104	#556	—	79	2	no	—
105	#497	—	79	2	no	—
106	#495	D₇₉	1	1	yes	characteristic
107	#464	—	79	2	no	—
108	#463	—	79	2	no	—
109	#462	—	79	2	no	—
110	#446	—	79	1	no	—
111	#440	—	79	2	no	—
112	C₄×Q₁₆	—	79	1	no	—
113	C₂×Q₁₆	—	79	2	no	—
114	C₂×Q₁₆	—	79	1	no	—
115	C₂×Q₁₆	C₂×D₇₉	1	1	yes	characteristic
116	C₂×SD₁₆	—	79	2	no	—
117	C₂×SD₁₆	C₂×D₇₉	1	2	yes	—
118	C₂×D₈	C₂×D₇₉	1	1	yes	characteristic
119	C₄⋊Q₈	—	79	2	no	—
120	#126	—	158	2	no	—
121	#126	—	158	2	no	—
122	#124	—	79	2	no	—
123	#124	C₂×D₇₉	1	2	yes	—
124	C₂²⋊Q₈	—	158	2	no	—
125	C₄×Q₈	—	79	1	no	—
126	C₄×Q₈	—	79	2	no	—
127	C₄×Q₈	—	79	2	no	—
128	C₄×D₄	—	79	2	no	—
129	#117	—	158	2	no	—
130	#107	—	79	1	no	—
131	C₄⋊C₈	—	79	2	no	—
132	C₄⋊C₈	—	79	1	no	—
133	#103	—	79	2	no	—
134	#103	—	79	2	no	—
135	#102	—	79	2	no	—
136	#97	—	79	2	no	—
137	C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
138	C₂×Q₈	—	79	1	no	—
139	C₂×Q₈	—	79	2	no	—
140	C₂×Q₈	C₂²×D₇₉	1	2	yes	—
141	C₂×D₄	C₂²×D₇₉	1	2	yes	—
142	C₂²×C₄	—	158	2	no	—
143	Q₁₆	—	4	1	no	—
144	Q₁₆	—	316	2	no	—
145	Q₁₆	—	158	2	no	—
146	SD₁₆	—	2	4	no	—
147	SD₁₆	—	316	2	no	—
148	D₈	—	4	1	no	—
149	C₂×C₈	—	79	2	no	—
150	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
151	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
152	C₄⋊C₄	—	79	2	no	—
153	C₄⋊C₄	—	79	1	no	—
154	C₄⋊C₄	—	79	2	no	—
155	C₄⋊C₄	—	158	2	no	—
156	C₄⋊C₄	—	158	2	no	—
157	C₄⋊C₄	—	158	2	no	—
158	C₄⋊C₄	—	158	2	no	—
159	C₂²⋊C₄	—	2	2	no	—
160	C₂²⋊C₄	—	2	2	no	—
161	C₂²⋊C₄	—	158	2	no	—
162	C₂²⋊C₄	—	158	2	no	—
163	C₂²⋊C₄	—	158	2	no	—
164	C₄²	—	79	2	no	—
165	C₄²	—	158	2	no	—
166	C₄²	—	158	2	no	—
167	C₄²	C₂²×D₇₉	1	1	yes	characteristic
168	C₂³	—	2	2	no	—
169	Q₈	—	2	2	no	—
170	Q₈	—	2	2	no	—
171	Q₈	—	316	1	no	—
172	Q₈	—	158	2	no	—
173	Q₈	—	158	2	no	—
174	D₄	—	2	2	no	—
175	D₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	316	4	no	—
178	C₂×C₄	—	79	2	no	—
179	C₂×C₄	—	79	2	no	—
180	C₂×C₄	—	158	2	no	—
181	C₂×C₄	—	158	2	no	—
182	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
183	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
184	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
185	C₈	—	2	1	no	—
186	C₈	—	158	2	no	—
187	C₈	#11929623	1	2	yes	—
188	C₂²	—	4	2	no	—
189	C₂²	—	4	2	no	—
190	C₂²	—	4	2	no	—
191	C₂²	C₂×D₄×D₇₉	1	1	yes	characteristic
192	C₄	—	2	1	no	—
193	C₄	—	2	1	no	—
194	C₄	—	4	2	no	—
195	C₄	—	316	2	no	—
196	C₄	—	316	2	no	—
197	C₄	—	158	2	no	—
198	C₄	—	158	2	no	—
199	C₄	C₂×#11929623	1	1	yes	characteristic
200	C₄	#14579206	1	1	yes	characteristic
201	C₂	—	8	2	no	—
202	C₂	#19099233	1	1	yes	characteristic
203	C₂	#19099360	1	1	yes	characteristic
204	C₂	#19099377	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19099282

C₂

yes

maximal, complemented

#19098954

C₂

yes

maximal, complemented

#19098782

C₂

yes

maximal, complemented

#19098776

C₂

yes

maximal

#19098771

C₂

yes

maximal, complemented

#19098693

C₂

yes

maximal, characteristic, complemented

#19098664

C₂

yes

maximal, complemented

C₄×Dic₃₁₆

C₂

yes

maximal, characteristic, complemented

#495×C₇₉

C₂

yes

maximal, characteristic

C₂×Dic₃₁₆

C₂²

yes

characteristic

C₂×#11929614

C₂²

yes

—

C₂×#11929612

C₂²

yes

—

#14579159

C₂²

yes

—

#14579154

—

#14579154

—

#14579148

C₂²

yes

—

#14579138

—

Q₈×Dic₇₉

C₂²

yes

—

C₄×Dic₁₅₈

C₂²

yes

—

C₄×Dic₁₅₈

C₂²

yes

characteristic

D₄×Dic₇₉

C₂²

yes

—

#14579117

—

#14579095

C₂²

yes

characteristic

C₄⋊(C₇₉⋊C₈)

C₂²

yes

characteristic

Dic₇₉⋊C₈

C₂²

yes

—

#14579086

C₂²

yes

—

#14579085

C₂²

yes

—

#14579082

C₂²

yes

—

#14579071

C₂²

yes

—

C₂×Q₁₆×C₇₉

C₂²

yes

characteristic

C₂×SD₁₆×C₇₉

C₂²

yes

—

C₂×D₈×C₇₉

C₂²

yes

characteristic

#124×C₇₉

C₂²

yes

—

C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

D₄

yes

characteristic

C₂×Dic₁₅₈

C₂³

yes

—

C₂²×Dic₇₉

—

Dic₃₁₆

—

#11929614

—

#11929612

—

C₂×C₇₉⋊C₈

C₂³

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

D₄

yes

characteristic

C₄⋊Dic₇₉

C₂³

yes

—

Dic₇₉⋊C₄

—

Dic₇₉⋊C₄

—

Dic₇₉⋊C₄

—

Dic₇₉⋊C₄

D₄

yes

—

C₂²⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₄×Dic₇₉

C₂³

yes

—

C₂×Q₈×C₇₉

C₂³

yes

—

C₂×D₄×C₇₉

C₂³

yes

—

Q₁₆×C₇₉

—

SD₁₆×C₇₉

—

D₈×C₇₉

—

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₂²⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

C₂³

yes

characteristic

C₂³×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

C₂×D₄

yes

—

C₂×Dic₇₉

C₂×D₄

yes

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

#41

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×#41

yes

characteristic

C₄×C₇₉

H(5,𝔽₂)

yes

characteristic

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#550

yes

characteristic

100

C₂×C₇₉

#575

yes

characteristic

101

C₂×C₇₉

yes

102

—

103

C₇₉

#3402

yes

characteristic, complemented, Sylow

104

#556

—

105

#497

—

106

#495

D₇₉

yes

characteristic

107

#464

—

108

#463

—

109

#462

—

110

#446

—

111

#440

—

112

C₄×Q₁₆

—

113

C₂×Q₁₆

—

114

C₂×Q₁₆

—

115

C₂×Q₁₆

C₂×D₇₉

yes

characteristic

116

C₂×SD₁₆

—

117

C₂×SD₁₆

C₂×D₇₉

yes

—

118

C₂×D₈

C₂×D₇₉

yes

characteristic

119

C₄⋊Q₈

—

120

#126

—

158

—

121

#126

—

158

—

122

#124

—

123

#124

C₂×D₇₉

yes

—

124

C₂²⋊Q₈

—

158

—

125

C₄×Q₈

—

126

C₄×Q₈

—

127

C₄×Q₈

—

128

C₄×D₄

—

129

#117

—

158

—

130

#107

—

131

C₄⋊C₈

—

132

C₄⋊C₈

—

133

#103

—

134

#103

—

135

#102

—

136

#97

—

137

C₄×C₈

C₂×D₇₉

yes

characteristic

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂×Q₈

C₂²×D₇₉

yes

—

141

C₂×D₄

C₂²×D₇₉

yes

—

142

C₂²×C₄

—

158

—

143

Q₁₆

—

144

Q₁₆

—

316

—

145

Q₁₆

—

158

—

146

SD₁₆

—

147

SD₁₆

—

316

—

148

D₈

—

149

C₂×C₈

—

150

C₂×C₈

C₂²×D₇₉

yes

characteristic

151

C₂×C₈

C₂²×D₇₉

yes

characteristic

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

158

—

156

C₄⋊C₄

—

158

—

157

C₄⋊C₄

—

158

—

158

C₄⋊C₄

—

158

—

159

C₂²⋊C₄

—

160

C₂²⋊C₄

—

161

C₂²⋊C₄

—

158

—

162

C₂²⋊C₄

—

158

—

163

C₂²⋊C₄

—

158

—

164

C₄²

—

165

C₄²

—

158

—

166

C₄²

—

158

—

167

C₄²

C₂²×D₇₉

yes

characteristic

168

C₂³

—

169

Q₈

—

170

Q₈

—

171

Q₈

—

316

—

172

Q₈

—

158

—

173

Q₈

—

158

—

174

D₄

—

175

D₄

—

176

C₂×C₄

—

177

C₂×C₄

—

316

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

158

—

181

C₂×C₄

—

158

—

182

C₂×C₄

D₄×D₇₉

yes

characteristic

183

C₂×C₄

D₄×D₇₉

yes

characteristic

184

C₂×C₄

C₂³×D₇₉

yes

characteristic

185

C₈

—

186

C₈

—

158

—

187

C₈

#11929623

yes

—

188

C₂²

—

189

C₂²

—

190

C₂²

—

191

C₂²

C₂×D₄×D₇₉

yes

characteristic

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

316

—

196

C₄

—

316

—

197

C₄

—

158

—

198

C₄

—

158

—

199

C₄

C₂×#11929623

yes

characteristic

200

C₄

#14579206

yes

characteristic

201

C₂

—

202

C₂

#19099233

yes

characteristic

203

C₂

#19099360

yes

characteristic

204

C₂

#19099377

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₉
commutator, cosocle	C₂×C₄×C₇₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	#495×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₄
prime radical	C₂×D₈×C₇₉	C₂²
hypercenter	#495	D₇₉
hypocenter	C₇₉	#3402
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #495

subgroup

G/H

center

C₂²

C₂×D₄×D₇₉

commutator, cosocle

C₂×C₄×C₇₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

#495×C₇₉

C₂

socle

C₂²×C₇₉

C₂²×D₄

prime radical

C₂×D₈×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #495

N	H ≅ G/N	type
1. #19099282	C₂	semidirect
2. #19098954	C₂	semidirect
3. #19098782	C₂	semidirect
5. #19098771	C₂	semidirect
6. #19098693	C₂	semidirect
7. #19098664	C₂	semidirect
8. C₄×Dic₃₁₆	C₂	semidirect
103. C₇₉	#3402	semidirect

H ≅ G/N

type

1. #19099282

C₂

semidirect

2. #19098954

C₂

semidirect

3. #19098782

C₂

semidirect

5. #19098771

C₂

semidirect

6. #19098693

C₂

semidirect

7. #19098664

C₂

semidirect

8. C₄×Dic₃₁₆

C₂

semidirect

103. C₇₉

#3402

semidirect

Subgroups — #23776482

Conjugacy classes of subgroups

Complements