Group #18081236 — subgroups

Subgroups — #18081236

Order 4864.

298 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 123 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	#14564408	C₂	1	2	yes	maximal
2	C₂×#11926295	C₂	1	1	yes	maximal, characteristic
3	C₈×#64520	C₂	1	1	yes	maximal, characteristic
4	#14561530	C₂	1	2	yes	maximal
5	C₁₉×#1566	C₂	1	1	yes	maximal, characteristic
6	#11926597	C₂²	1	1	yes	characteristic
7	C₂×#10597279	—	2	2	no	—
8	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
9	C₄×#64520	C₂²	1	1	yes	characteristic
10	#11926308	—	2	2	no	—
11	#11926295	C₄	1	2	yes	—
12	#11926295	C₄	1	2	yes	—
13	#11926266	—	2	1	no	—
14	C₈×C₁₉⋊C₈	—	2	1	no	—
15	C₁₉×#423	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#334	—	2	1	no	—
18	C₁₉×#334	—	2	1	no	—
19	C₂×M₁₆×C₁₉	—	2	2	no	—
20	C₂²×C₈×C₁₉	Q₈	1	1	yes	characteristic
21	C₂²×C₈×C₁₉	Q₈	1	1	yes	characteristic
22	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
23	C₂×#64520	D₄	1	2	yes	—
24	C₂²×C₁₉⋊C₈	—	2	2	no	—
25	C₁₉×C₄⋊C₈	—	2	2	no	—
26	C₁₉×C₄⋊C₈	—	2	2	no	—
27	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
30	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
31	C₂²⋊(C₁₉⋊C₈)	Q₈	1	2	yes	—
32	#10597279	—	4	4	no	—
33	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
34	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
35	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	4	yes	—
36	#10597260	—	2	1	no	—
37	C₄×C₁₉⋊C₈	—	2	1	no	—
38	C₄×C₁₉⋊C₈	—	2	1	no	—
39	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
42	M₁₆×C₁₉	—	4	2	no	—
43	M₁₆×C₁₉	—	4	2	no	—
44	C₂×C₈×C₁₉	—	2	2	no	—
45	C₂×C₈×C₁₉	—	4	1	no	—
46	C₂×C₈×C₁₉	—	4	2	no	—
47	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
51	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
53	C₄²×C₁₉	#41	1	1	yes	characteristic
54	C₄²×C₁₉	#41	1	1	yes	characteristic
55	#64520	—	2	2	no	—
56	#64520	SD₁₆	1	4	yes	—
57	C₂×C₁₉⋊C₈	—	2	2	no	—
58	C₂×C₁₉⋊C₈	—	2	4	no	—
59	C₂×C₁₉⋊C₈	—	2	2	no	—
60	C₂×C₁₉⋊C₈	—	4	2	no	—
61	C₂×C₁₉⋊C₈	—	4	2	no	—
62	#9623	—	19	1	no	maximal, Sylow
63	C₂³×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₁₉	—	2	1	no	—
65	C₂×C₄×C₁₉	—	2	1	no	—
66	C₂×C₄×C₁₉	—	4	1	no	—
67	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
71	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₁₉	#125	1	1	yes	characteristic
73	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
74	C₂×C₄×C₁₉	#135	1	2	yes	—
75	C₈×C₁₉	—	2	1	no	—
76	C₈×C₁₉	—	2	1	no	—
77	C₈×C₁₉	—	2	1	no	—
78	C₈×C₁₉	—	8	2	no	—
79	C₈×C₁₉	#108	1	2	yes	complemented
80	C₁₉⋊C₈	—	2	4	no	—
81	C₁₉⋊C₈	—	4	2	no	—
82	C₁₉⋊C₈	—	4	4	no	—
83	#1950	—	19	2	no	—
84	#1566	D₁₉	1	1	yes	characteristic
85	C₂×#334	—	19	1	no	—
86	C₈×M₁₆	—	19	1	no	—
87	#1301	—	19	2	no	—
88	C₂²×C₁₉	—	2	1	no	—
89	C₂²×C₁₉	—	2	1	no	—
90	C₂²×C₁₉	#389	1	1	yes	characteristic
91	C₂²×C₁₉	#498	1	1	yes	characteristic
92	C₂²×C₁₉	#499	1	1	yes	characteristic
93	C₄×C₁₉	—	2	1	no	—
94	C₄×C₁₉	—	2	1	no	—
95	C₄×C₁₉	—	2	1	no	—
96	C₄×C₁₉	C₂×#108	1	1	yes	characteristic
97	C₄×C₁₉	#430	1	1	yes	characteristic
98	C₄×C₁₉	C₄×SD₁₆	1	2	yes	—
99	C₄×C₁₉	#465	1	2	yes	—
100	#431	—	19	1	no	—
101	C₂×#108	—	38	2	no	—
102	#423	C₂×D₁₉	1	2	yes	—
103	C₂×C₄⋊C₈	—	19	2	no	—
104	C₄×M₁₆	—	19	1	no	—
105	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
106	#341	—	38	2	no	—
107	#334	—	2	1	no	—
108	#334	—	2	1	no	—
109	#334	—	19	2	no	—
110	#334	—	19	2	no	—
111	#322	—	38	1	no	—
112	C₈²	—	38	1	no	—
113	C₂×C₁₉	—	2	1	no	—
114	C₂×C₁₉	—	2	1	no	—
115	C₂×C₁₉	#1941	1	1	yes	characteristic
116	C₂×C₁₉	#1950	1	1	yes	characteristic
117	C₂×C₁₉	#1953	1	1	yes	characteristic
118	C₂×M₁₆	—	2	2	no	—
119	C₂×M₁₆	—	19	2	no	—
120	C₂²×C₈	—	38	2	no	—
121	C₂²×C₈	Dic₃₈	1	1	yes	characteristic
122	C₂²×C₈	Dic₃₈	1	1	yes	characteristic
123	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
124	#108	—	76	4	no	—
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	2	2	no	—
127	C₄⋊C₈	—	19	4	no	—
128	C₄⋊C₈	—	38	2	no	—
129	C₄⋊C₈	—	38	1	no	—
130	C₂²⋊C₈	—	19	2	no	—
131	#97	—	38	1	no	—
132	C₄×C₈	—	38	1	no	—
133	C₄×C₈	—	38	1	no	—
134	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
135	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
136	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
137	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
138	C₁₉	#9623	1	1	yes	characteristic, complemented, Sylow
139	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
140	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
141	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
142	M₁₆	—	4	2	no	—
143	M₁₆	—	4	2	no	—
144	M₁₆	—	19	4	no	—
145	M₁₆	—	38	2	no	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	4	1	no	—
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	—	38	2	no	—
150	C₂×C₈	—	38	2	no	—
151	C₂×C₈	—	38	4	no	—
152	C₂×C₈	—	76	2	no	—
153	C₂×C₈	—	76	2	no	—
154	C₂×C₈	Dic₁₉⋊C₄	1	1	yes	characteristic
155	C₂×C₈	Dic₁₉⋊C₄	1	1	yes	characteristic
156	C₂×C₈	Dic₁₉⋊C₄	1	1	yes	characteristic
157	C₂×C₈	Dic₁₉⋊C₄	1	1	yes	characteristic
158	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
159	C₄²	#64543	1	1	yes	characteristic
160	C₄²	#64545	1	1	yes	characteristic
161	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
162	C₂³	#10597334	1	1	yes	characteristic
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	4	1	no	—
166	C₂×C₄	#10597333	1	1	yes	characteristic
167	C₂×C₄	#10597336	1	1	yes	characteristic
168	C₂×C₄	#10597339	1	1	yes	characteristic
169	C₂×C₄	SD₁₆×D₁₉	1	2	yes	—
170	C₂×C₄	#10597362	1	2	yes	—
171	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
172	C₂×C₄	#10597384	1	1	yes	characteristic
173	C₂×C₄	#10597392	1	1	yes	characteristic
174	C₈	—	2	1	no	—
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	8	2	no	—
178	C₈	—	38	4	no	—
179	C₈	—	76	2	no	—
180	C₈	—	76	4	no	—
181	C₈	#10597279	1	2	yes	complemented
182	C₂²	—	2	1	no	—
183	C₂²	—	2	1	no	—
184	C₂²	#11926440	1	1	yes	characteristic
185	C₂²	#11926941	1	1	yes	characteristic
186	C₂²	#11926944	1	1	yes	characteristic
187	C₄	—	2	1	no	—
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	C₂×#10597279	1	1	yes	characteristic
191	C₄	#11926592	1	1	yes	characteristic
192	C₄	#11926631	1	2	yes	—
193	C₄	#11926770	1	2	yes	—
194	C₂	—	2	1	no	—
195	C₂	—	2	1	no	—
196	C₂	#14564361	1	1	yes	characteristic
197	C₂	#14564408	1	1	yes	characteristic
198	C₂	#14564423	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14564408

C₂

yes

maximal

C₂×#11926295

C₂

yes

maximal, characteristic

C₈×#64520

C₂

yes

maximal, characteristic

#14561530

C₂

yes

maximal

C₁₉×#1566

C₂

yes

maximal, characteristic

#11926597

C₂²

yes

characteristic

C₂×#10597279

—

C₂×C₄⋊(C₁₉⋊C₈)

C₂²

yes

—

C₄×#64520

C₂²

yes

characteristic

#11926308

—

#11926295

C₄

yes

—

#11926295

C₄

yes

—

#11926266

—

C₈×C₁₉⋊C₈

—

C₁₉×#423

C₂²

yes

—

C₂×C₄×C₈×C₁₉

C₂²

yes

characteristic

C₁₉×#334

—

C₁₉×#334

—

C₂×M₁₆×C₁₉

—

C₂²×C₈×C₁₉

Q₈

yes

characteristic

C₂²×C₈×C₁₉

Q₈

yes

characteristic

C₂×C₄²×C₁₉

C₂³

yes

characteristic

C₂×#64520

D₄

yes

—

C₂²×C₁₉⋊C₈

—

C₁₉×C₄⋊C₈

—

C₁₉×C₄⋊C₈

—

C₄×C₈×C₁₉

C₂×C₄

yes

characteristic

C₄×C₈×C₁₉

C₂×C₄

yes

characteristic

C₄×C₈×C₁₉

D₄

yes

characteristic

C₄×C₈×C₁₉

D₄

yes

characteristic

C₂²⋊(C₁₉⋊C₈)

Q₈

yes

—

#10597279

—

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

C₂×C₄

yes

—

#10597260

—

C₄×C₁₉⋊C₈

—

C₄×C₁₉⋊C₈

—

C₂²×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₉

C₂×Q₈

yes

characteristic

C₂²×C₄×C₁₉

C₂×Q₈

yes

characteristic

M₁₆×C₁₉

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₉

C₄⋊C₄

yes

characteristic

C₄²×C₁₉

C₂²×C₄

yes

characteristic

C₄²×C₁₉

C₂×D₄

yes

C₄²×C₁₉

yes

C₄²×C₁₉

yes

—

SD₁₆

yes

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

#9623

—

maximal, Sylow

C₂³×C₁₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₉

C₄×D₄

yes

characteristic

C₂×C₄×C₁₉

C₄×Q₈

yes

characteristic

C₂×C₄×C₁₉

#121

yes

characteristic

C₂×C₄×C₁₉

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₁₉

#125

yes

characteristic

C₂×C₄×C₁₉

C₂×SD₁₆

yes

—

C₂×C₄×C₁₉

#135

yes

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

#108

yes

complemented

C₁₉⋊C₈

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

#1950

—

#1566

D₁₉

yes

characteristic

C₂×#334

—

C₈×M₁₆

—

#1301

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

yes

C₄×C₁₉

yes

C₄×C₁₉

C₄×SD₁₆

yes

—

C₄×C₁₉

#465

yes

—

100

#431

—

101

C₂×#108

—

102

#423

C₂×D₁₉

yes

—

103

C₂×C₄⋊C₈

—

104

C₄×M₁₆

—

105

C₂×C₄×C₈

C₂×D₁₉

yes

characteristic

106

#341

—

107

#334

—

108

#334

—

109

#334

—

110

#334

—

111

#322

—

112

C₈²

—

113

C₂×C₁₉

—

114

C₂×C₁₉

—

115

C₂×C₁₉

#1941

yes

characteristic

116

C₂×C₁₉

#1950

yes

characteristic

117

C₂×C₁₉

#1953

yes

characteristic

118

C₂×M₁₆

—

119

C₂×M₁₆

—

120

C₂²×C₈

—

121

C₂²×C₈

Dic₃₈

yes

characteristic

122

C₂²×C₈

Dic₃₈

yes

characteristic

123

C₂×C₄²

C₂²×D₁₉

yes

characteristic

124

#108

—

125

C₄⋊C₈

—

126

C₄⋊C₈

—

127

C₄⋊C₈

—

128

C₄⋊C₈

—

129

C₄⋊C₈

—

130

C₂²⋊C₈

—

131

#97

—

132

C₄×C₈

—

133

C₄×C₈

—

134

C₄×C₈

C₄×D₁₉

yes

characteristic

135

C₄×C₈

C₄×D₁₉

yes

characteristic

136

C₄×C₈

C₂²⋊D₁₉

yes

characteristic

137

C₄×C₈

C₂²⋊D₁₉

yes

characteristic

138

C₁₉

#9623

yes

characteristic, complemented, Sylow

139

C₂²×C₄

C₂×Dic₃₈

yes

characteristic

140

C₂²×C₄

D₄×D₁₉

yes

characteristic

141

C₂²×C₄

Q₈×D₁₉

yes

characteristic

142

M₁₆

—

143

M₁₆

—

144

M₁₆

—

145

M₁₆

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

Dic₁₉⋊C₄

yes

characteristic

155

C₂×C₈

Dic₁₉⋊C₄

yes

characteristic

156

C₂×C₈

Dic₁₉⋊C₄

yes

characteristic

157

C₂×C₈

Dic₁₉⋊C₄

yes

characteristic

158

C₄²

C₂×C₄×D₁₉

yes

characteristic

159

C₄²

#64543

yes

characteristic

160

C₄²

#64545

yes

characteristic

161

C₄²

C₂×C₂²⋊D₁₉

yes

characteristic

162

C₂³

#10597334

yes

characteristic

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

#10597333

yes

characteristic

167

C₂×C₄

#10597336

yes

characteristic

168

C₂×C₄

#10597339

yes

characteristic

169

C₂×C₄

SD₁₆×D₁₉

yes

—

170

C₂×C₄

#10597362

yes

—

171

C₂×C₄

C₂×Dic₁₉⋊C₄

yes

characteristic

172

C₂×C₄

#10597384

yes

characteristic

173

C₂×C₄

#10597392

yes

characteristic

174

C₈

—

175

C₈

—

176

C₈

—

177

C₈

—

178

C₈

—

179

C₈

—

180

C₈

—

181

C₈

#10597279

yes

complemented

182

C₂²

—

183

C₂²

—

184

C₂²

#11926440

yes

characteristic

185

C₂²

#11926941

yes

characteristic

186

C₂²

#11926944

yes

characteristic

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

C₂×#10597279

yes

characteristic

191

C₄

#11926592

yes

characteristic

192

C₄

#11926631

yes

—

193

C₄

#11926770

yes

—

194

C₂

—

195

C₂

—

196

C₂

#14564361

yes

characteristic

197

C₂

#14564408

yes

characteristic

198

C₂

#14564423

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#10597384
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1566	C₂
socle	C₂²×C₁₉	#389
prime radical	C₂³×C₁₉	C₄⋊Q₈
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1566	D₁₉
hypocenter	C₁₉	#9623
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

subgroup

G/H

center

C₂×C₄

#10597384

commutator

C₄²×C₁₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₉

Fitting

C₁₉×#1566

C₂

socle

C₂²×C₁₉

#389

prime radical

C₂³×C₁₉

C₄⋊Q₈

cosocle

C₂×C₄²×C₁₉

C₂³

D₁₉

C₁₉

G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

N	H ≅ G/N	type
79. C₈×C₁₉	#108	semidirect
138. C₁₉	#9623	semidirect
181. C₈	#10597279	semidirect

H ≅ G/N

type

79. C₈×C₁₉

#108

semidirect

138. C₁₉

#9623

semidirect

181. C₈

#10597279

semidirect

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

subgroup	index
#14564408	2
C₂×#11926295	2
C₈×#64520	2
#14561530	2
C₁₉×#1566	2
#9623	19

subgroup

index

C₁₉×#1566

Subgroups — #18081236

Conjugacy classes of subgroups

Complements

Maximal subgroups