Group #18069606 — subgroups

Subgroups — #18069606

Order 4864.

248 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 79 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	#14563685	C₂	1	1	yes	maximal, characteristic
2	#14563681	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#2113	C₂	1	1	yes	maximal, characteristic
4	C₂×#10597276	—	2	1	no	—
5	#11926582	—	2	1	no	—
6	#11926528	—	2	1	no	—
7	#11926518	—	2	1	no	—
8	C₁₉⋊#338	C₄	1	2	yes	complemented
9	#11926301	C₄	1	2	yes	—
10	C₂×C₁₉×#131	C₂²	1	1	yes	characteristic
11	C₂×C₁₉×M₃₂	C₄	1	1	yes	characteristic, complemented
12	C₂²×C₁₆×C₁₉	C₄	1	1	yes	characteristic, complemented
13	C₁₉×C₂²⋊C₁₆	—	2	1	no	—
14	C₁₉×C₂²⋊C₁₆	—	2	1	no	—
15	C₂×#41×C₁₉	D₄	1	1	yes	characteristic
16	C₁₉×#131	—	4	1	no	—
17	C₁₉×#131	—	4	1	no	—
18	C₂×M₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
19	C₂×M₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
20	C₂×M₁₆×C₁₉	Q₈	1	1	yes	characteristic
21	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
24	#10597372	—	2	2	no	—
25	C₂×#64520	—	2	2	no	—
26	C₁₉×M₃₂	—	4	1	no	—
27	C₂×C₁₆×C₁₉	—	2	1	no	—
28	C₂×C₁₆×C₁₉	—	2	1	no	—
29	C₂×C₁₆×C₁₉	—	4	1	no	—
30	#10597293	—	4	1	no	—
31	#10597291	—	4	1	no	—
32	#10597291	—	4	1	no	—
33	#10597290	—	4	1	no	—
34	#10597276	—	4	1	no	—
35	#10597276	—	4	1	no	—
36	#10597275	—	4	1	no	—
37	#10597274	—	4	1	no	—
38	#41×C₁₉	—	4	1	no	—
39	#41×C₁₉	—	4	1	no	—
40	C₂×Q₈×C₁₉	SD₁₆	1	1	yes	characteristic
41	C₂×D₄×C₁₉	D₈	1	1	yes	characteristic
42	C₂×D₄×C₁₉	SD₁₆	1	1	yes	characteristic
43	C₂×D₄×C₁₉	Q₁₆	1	1	yes	characteristic
44	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
47	M₁₆×C₁₉	—	4	1	no	—
48	M₁₆×C₁₉	—	4	1	no	—
49	M₁₆×C₁₉	—	4	1	no	—
50	C₂×C₈×C₁₉	—	2	1	no	—
51	C₂×C₈×C₁₉	—	2	1	no	—
52	C₂×C₈×C₁₉	—	2	1	no	—
53	C₂×C₈×C₁₉	—	2	1	no	—
54	C₂×C₈×C₁₉	—	2	1	no	—
55	C₂×C₈×C₁₉	—	2	1	no	—
56	C₂×C₈×C₁₉	C₄²	1	1	yes	characteristic
57	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
60	C₂²⋊Dic₁₉	—	4	2	no	—
61	C₄⋊Dic₁₉	—	4	2	no	—
62	C₄×Dic₁₉	—	4	2	no	—
63	#64520	—	4	2	no	—
64	#64520	—	4	2	no	—
65	C₂×C₁₉⋊C₈	—	4	2	no	—
66	C₁₆×C₁₉	—	4	1	no	—
67	C₁₆×C₁₉	—	4	1	no	—
68	#7227	—	19	1	no	maximal, Sylow
69	C₂³×C₁₉	#102	1	1	yes	characteristic
70	C₂³×C₁₉	#103	1	1	yes	characteristic
71	C₂³×C₁₉	#107	1	1	yes	characteristic
72	Q₈×C₁₉	—	4	1	no	—
73	D₄×C₁₉	—	4	1	no	—
74	D₄×C₁₉	—	4	1	no	—
75	D₄×C₁₉	—	4	1	no	—
76	C₂×C₄×C₁₉	—	2	1	no	—
77	C₂×C₄×C₁₉	—	2	1	no	—
78	C₂×C₄×C₁₉	—	2	1	no	—
79	C₂×C₄×C₁₉	—	2	1	no	—
80	C₂×C₄×C₁₉	—	2	1	no	—
81	C₂×C₄×C₁₉	—	2	1	no	—
82	C₂×C₄×C₁₉	#95	1	1	yes	characteristic
83	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
84	C₂×C₄×C₁₉	#103	1	1	yes	characteristic
85	C₂×C₄×C₁₉	#106	1	1	yes	characteristic
86	C₂×Dic₁₉	—	8	2	no	—
87	C₈×C₁₉	—	2	1	no	—
88	C₈×C₁₉	—	2	1	no	—
89	C₈×C₁₉	—	2	1	no	—
90	C₈×C₁₉	#99	1	1	yes	characteristic
91	C₈×C₁₉	#99	1	1	yes	characteristic
92	C₁₉⋊C₈	—	8	2	no	—
93	#2113	D₁₉	1	1	yes	characteristic
94	#1813	—	19	1	no	—
95	#1812	—	19	1	no	—
96	C₂²×C₁₉	—	2	1	no	—
97	C₂²×C₁₉	—	2	1	no	—
98	C₂²×C₁₉	—	2	1	no	—
99	C₂²×C₁₉	—	4	1	no	—
100	C₂²×C₁₉	—	4	1	no	—
101	C₂²×C₁₉	—	4	1	no	—
102	C₂²×C₁₉	#340	1	1	yes	characteristic
103	C₄×C₁₉	—	2	1	no	—
104	C₄×C₁₉	—	2	1	no	—
105	C₄×C₁₉	—	2	1	no	—
106	C₄×C₁₉	H(3,ℤ/4)	1	1	yes	characteristic
107	C₄×C₁₉	#342	1	1	yes	characteristic
108	Dic₁₉	—	16	2	no	—
109	C₂×#131	C₂×D₁₉	1	1	yes	characteristic
110	C₂×M₃₂	Dic₁₉	1	1	yes	characteristic, complemented
111	C₂²×C₁₆	Dic₁₉	1	1	yes	characteristic, complemented
112	C₂×#108	—	38	1	no	—
113	#427	—	38	1	no	—
114	#413	—	38	1	no	—
115	#410	—	38	1	no	—
116	C₂²⋊C₁₆	—	2	1	no	—
117	C₂²⋊C₁₆	—	2	1	no	—
118	#338	—	19	2	no	—
119	H(3,ℤ/4)	—	19	2	no	—
120	C₂×C₁₉	—	2	1	no	—
121	C₂×C₁₉	—	4	1	no	—
122	C₂×C₁₉	—	4	1	no	—
123	C₂×C₁₉	—	4	1	no	—
124	C₂×C₁₉	#1289	1	1	yes	characteristic
125	C₂×#41	D₇₆	1	1	yes	characteristic
126	#131	—	4	1	no	—
127	#131	—	4	1	no	—
128	C₂×M₁₆	—	38	2	no	—
129	C₂×M₁₆	Dic₃₈	1	1	yes	characteristic
130	C₂×M₁₆	C₄×D₁₉	1	1	yes	characteristic
131	C₂×M₁₆	C₄×D₁₉	1	1	yes	characteristic
132	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
133	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
134	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
135	#117	—	38	2	no	—
136	M₃₂	—	4	1	no	—
137	C₂×C₁₆	—	2	1	no	—
138	C₂×C₁₆	—	2	1	no	—
139	C₂×C₁₆	—	4	1	no	—
140	#108	—	76	1	no	—
141	#108	—	76	1	no	—
142	#107	—	76	1	no	—
143	#106	—	76	1	no	—
144	#101	—	76	1	no	—
145	#100	—	76	1	no	—
146	#99	—	76	1	no	—
147	#99	—	76	1	no	—
148	C₁₉	#7227	1	1	yes	characteristic, complemented, Sylow
149	#41	—	4	1	no	—
150	#41	—	4	1	no	—
151	C₂×Q₈	#64516	1	1	yes	characteristic
152	C₂×D₄	#64516	1	1	yes	characteristic
153	C₂×D₄	D₁₅₂	1	1	yes	characteristic
154	C₂×D₄	Dic₇₆	1	1	yes	characteristic
155	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
156	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
157	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
158	M₁₆	—	4	1	no	—
159	M₁₆	—	4	1	no	—
160	M₁₆	—	4	1	no	—
161	M₁₆	—	76	2	no	—
162	M₁₆	—	76	2	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	—	76	2	no	—
170	C₂×C₈	C₄×Dic₁₉	1	1	yes	characteristic
171	C₂×C₈	Dic₁₉⋊C₄	1	1	yes	characteristic
172	C₂×C₈	Dic₁₉⋊C₄	1	1	yes	characteristic
173	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
174	C₄⋊C₄	—	76	2	no	—
175	C₂²⋊C₄	—	76	2	no	—
176	C₄²	—	76	2	no	—
177	C₁₆	—	4	1	no	—
178	C₁₆	—	4	1	no	—
179	C₂³	#10597273	1	1	yes	characteristic
180	C₂³	#10597275	1	1	yes	characteristic
181	C₂³	#10597278	1	1	yes	characteristic
182	Q₈	—	4	1	no	—
183	D₄	—	4	1	no	—
184	D₄	—	4	1	no	—
185	D₄	—	4	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	152	2	no	—
193	C₂×C₄	#10597273	1	1	yes	characteristic
194	C₂×C₄	#10597274	1	1	yes	characteristic
195	C₂×C₄	#10597278	1	1	yes	characteristic
196	C₂×C₄	C₁₉⋊#95	1	1	yes	characteristic
197	C₈	—	2	1	no	—
198	C₈	—	2	1	no	—
199	C₈	—	2	1	no	—
200	C₈	—	152	2	no	—
201	C₈	#10597291	1	1	yes	characteristic
202	C₈	#10597291	1	1	yes	characteristic
203	C₂²	—	2	1	no	—
204	C₂²	—	2	1	no	—
205	C₂²	—	2	1	no	—
206	C₂²	—	4	1	no	—
207	C₂²	—	4	1	no	—
208	C₂²	—	4	1	no	—
209	C₂²	#11926307	1	1	yes	characteristic
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	304	2	no	—
214	C₄	#11926301	1	1	yes	characteristic
215	C₄	#11926311	1	1	yes	characteristic
216	C₂	—	2	1	no	—
217	C₂	—	4	1	no	—
218	C₂	—	4	1	no	—
219	C₂	—	4	1	no	—
220	C₂	#14561503	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14563685

C₂

yes

maximal, characteristic

#14563681

C₂

yes

maximal, characteristic

C₁₉×#2113

C₂

yes

maximal, characteristic

—

—

—

—

C₁₉⋊#338

C₄

yes

complemented

#11926301

C₄

yes

—

C₂×C₁₉×#131

C₂²

yes

characteristic

C₂×C₁₉×M₃₂

C₄

yes

characteristic, complemented

C₂²×C₁₆×C₁₉

C₄

yes

characteristic, complemented

C₁₉×C₂²⋊C₁₆

—

C₁₉×C₂²⋊C₁₆

—

C₂×#41×C₁₉

D₄

yes

characteristic

C₁₉×#131

—

C₁₉×#131

—

C₂×M₁₆×C₁₉

C₂×C₄

yes

characteristic

C₂×M₁₆×C₁₉

C₂×C₄

yes

characteristic

C₂×M₁₆×C₁₉

Q₈

yes

characteristic

C₂²×C₈×C₁₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₂²×C₈×C₁₉

D₄

yes

characteristic

#10597372

—

C₂×#64520

—

C₁₉×M₃₂

—

C₂×C₁₆×C₁₉

—

C₂×C₁₆×C₁₉

—

C₂×C₁₆×C₁₉

—

—

—

—

—

—

—

—

—

#41×C₁₉

—

#41×C₁₉

—

C₂×Q₈×C₁₉

SD₁₆

yes

characteristic

C₂×D₄×C₁₉

D₈

yes

characteristic

C₂×D₄×C₁₉

SD₁₆

yes

characteristic

C₂×D₄×C₁₉

Q₁₆

yes

characteristic

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

C₄⋊C₄

yes

characteristic

M₁₆×C₁₉

—

M₁₆×C₁₉

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₄²

yes

characteristic

C₂×C₈×C₁₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₉

C₄⋊C₄

yes

characteristic

C₂²⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄×Dic₁₉

—

#64520

—

#64520

—

C₂×C₁₉⋊C₈

—

C₁₆×C₁₉

—

C₁₆×C₁₉

—

—

C₂³×C₁₉

yes

C₂³×C₁₉

yes

C₂³×C₁₉

yes

Q₈×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×Dic₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

yes

C₈×C₁₉

yes

C₁₉⋊C₈

—

#2113

D₁₉

yes

characteristic

#1813

—

#1812

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

100

C₂²×C₁₉

—

101

C₂²×C₁₉

—

102

C₂²×C₁₉

#340

yes

characteristic

103

C₄×C₁₉

—

104

C₄×C₁₉

—

105

C₄×C₁₉

—

106

C₄×C₁₉

H(3,ℤ/4)

yes

characteristic

107

C₄×C₁₉

#342

yes

characteristic

108

Dic₁₉

—

109

C₂×#131

C₂×D₁₉

yes

characteristic

110

C₂×M₃₂

Dic₁₉

yes

characteristic, complemented

111

C₂²×C₁₆

Dic₁₉

yes

characteristic, complemented

112

C₂×#108

—

113

#427

—

114

#413

—

115

#410

—

116

C₂²⋊C₁₆

—

117

C₂²⋊C₁₆

—

118

#338

—

119

H(3,ℤ/4)

—

120

C₂×C₁₉

—

121

C₂×C₁₉

—

122

C₂×C₁₉

—

123

C₂×C₁₉

—

124

C₂×C₁₉

#1289

yes

characteristic

125

C₂×#41

D₇₆

yes

characteristic

126

#131

—

127

#131

—

128

C₂×M₁₆

—

129

C₂×M₁₆

Dic₃₈

yes

characteristic

130

C₂×M₁₆

C₄×D₁₉

yes

characteristic

131

C₂×M₁₆

C₄×D₁₉

yes

characteristic

132

C₂²×C₈

C₂×Dic₁₉

yes

characteristic

133

C₂²×C₈

C₂²⋊D₁₉

yes

characteristic

134

C₂²×C₈

C₂²⋊D₁₉

yes

characteristic

135

#117

—

136

M₃₂

—

137

C₂×C₁₆

—

138

C₂×C₁₆

—

139

C₂×C₁₆

—

140

#108

—

141

#108

—

142

#107

—

143

#106

—

144

#101

—

145

#100

—

146

#99

—

147

#99

—

148

C₁₉

#7227

yes

characteristic, complemented, Sylow

149

#41

—

150

#41

—

151

C₂×Q₈

#64516

yes

characteristic

152

C₂×D₄

#64516

yes

characteristic

153

C₂×D₄

D₁₅₂

yes

characteristic

154

C₂×D₄

Dic₇₆

yes

characteristic

155

C₂²×C₄

C₄⋊Dic₁₉

yes

characteristic

156

C₂²×C₄

(C₂×D₁₉)⋊C₄

yes

characteristic

157

C₂²×C₄

(C₂×D₁₉)⋊C₄

yes

characteristic

158

M₁₆

—

159

M₁₆

—

160

M₁₆

—

161

M₁₆

—

162

M₁₆

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

C₄×Dic₁₉

yes

characteristic

171

C₂×C₈

Dic₁₉⋊C₄

yes

characteristic

172

C₂×C₈

Dic₁₉⋊C₄

yes

characteristic

173

C₂×C₈

C₂²⋊Dic₁₉

yes

characteristic

174

C₄⋊C₄

—

175

C₂²⋊C₄

—

176

C₄²

—

177

C₁₆

—

178

C₁₆

—

179

C₂³

#10597273

yes

characteristic

180

C₂³

#10597275

yes

characteristic

181

C₂³

#10597278

yes

characteristic

182

Q₈

—

183

D₄

—

184

D₄

—

185

D₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

152

—

193

C₂×C₄

#10597273

yes

characteristic

194

C₂×C₄

#10597274

yes

characteristic

195

C₂×C₄

#10597278

yes

characteristic

196

C₂×C₄

C₁₉⋊#95

yes

characteristic

197

C₈

—

198

C₈

—

199

C₈

—

200

C₈

—

152

—

201

C₈

#10597291

yes

characteristic

202

C₈

#10597291

yes

characteristic

203

C₂²

—

204

C₂²

—

205

C₂²

—

206

C₂²

—

207

C₂²

—

208

C₂²

—

209

C₂²

#11926307

yes

characteristic

210

C₄

—

211

C₄

—

212

C₄

—

213

C₄

—

304

—

214

C₄

#11926301

yes

characteristic

215

C₄

#11926311

yes

characteristic

216

C₂

—

217

C₂

—

218

C₂

—

219

C₂

—

220

C₂

#14561503

yes

characteristic

	subgroup	G/H
center	C₄	#11926301
commutator	C₂×C₈×C₁₉	C₄²
Frattini	C₂×#131	C₂×D₁₉
Fitting	C₁₉×#2113	C₂
socle	C₂×C₁₉	#1289
prime radical	C₂×#41×C₁₉	D₄
cosocle	C₂×C₁₉×#131	C₂²
hypercenter	#2113	D₁₉
hypocenter	C₁₉	#7227
LCS	G ▷ C₂×C₈×C₁₉ ▷ C₄×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #2113

subgroup

G/H

center

C₄

#11926301

commutator

C₂×C₈×C₁₉

C₄²

Frattini

C₂×#131

C₂×D₁₉

Fitting

C₁₉×#2113

C₂

socle

C₂×C₁₉

#1289

prime radical

C₂×#41×C₁₉

D₄

cosocle

C₂×C₁₉×#131

C₂²

D₁₉

C₁₉

G ▷ C₂×C₈×C₁₉ ▷ C₄×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ #2113

N	H ≅ G/N	type
8. C₁₉⋊#338	C₄	semidirect
11. C₂×C₁₉×M₃₂	C₄	semidirect
12. C₂²×C₁₆×C₁₉	C₄	semidirect
110. C₂×M₃₂	Dic₁₉	semidirect
111. C₂²×C₁₆	Dic₁₉	semidirect
148. C₁₉	#7227	semidirect

H ≅ G/N

type

8. C₁₉⋊#338

C₄

semidirect

11. C₂×C₁₉×M₃₂

C₄

semidirect

12. C₂²×C₁₆×C₁₉

C₄

semidirect

110. C₂×M₃₂

Dic₁₉

semidirect

111. C₂²×C₁₆

Dic₁₉

semidirect

148. C₁₉

#7227

semidirect

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

subgroup	index
#14563685	2
#14563681	2
C₁₉×#2113	2
#7227	19

subgroup

index

#14563685

#14563681

C₁₉×#2113

#7227

Subgroups — #18069606

Conjugacy classes of subgroups

Complements

Maximal subgroups