Group #18086672 — subgroups

Subgroups — #18086672

Order 4864.

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

normal subgroups only

This group has 382 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	#14574980	C₂	1	1	yes	maximal, characteristic
2	#14564612	C₂	1	2	yes	maximal
3	#14562323	C₂	1	2	yes	maximal
4	#14561510	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1831	C₂	1	1	yes	maximal, characteristic
6	#11927101	C₂²	1	2	yes	—
7	#11926840	—	2	2	no	—
8	#11926785	—	2	2	no	—
9	#11926755	—	2	2	no	—
10	#11926568	C₂²	1	2	yes	—
11	#11926304	—	2	2	no	—
12	#11926304	—	2	2	no	—
13	#11926292	—	2	2	no	—
14	#11926291	—	2	2	no	—
15	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₄³×C₁₉	C₂²	1	1	yes	characteristic
18	C₁₉×#336	—	2	2	no	—
19	C₂²×C₈×C₁₉	—	2	1	no	—
20	C₂²×C₈×C₁₉	D₄	1	2	yes	—
21	C₂×C₄²×C₁₉	—	2	1	no	—
22	C₂×C₄²×C₁₉	—	2	1	no	—
23	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
26	#10597372	—	2	2	no	—
27	#10597370	—	2	2	no	—
28	#10597370	—	2	2	no	—
29	C₂×#64520	—	2	2	no	—
30	C₂×Dic₇₆	—	2	2	no	—
31	#10597316	—	2	2	no	—
32	#10597315	D₄	1	2	yes	—
33	#10597315	D₄	1	2	yes	—
34	C₄×Dic₃₈	—	2	2	no	—
35	C₄×Dic₃₈	—	2	2	no	—
36	C₁₉×C₄⋊C₈	—	2	2	no	—
37	C₁₉×C₄⋊C₈	—	2	2	no	—
38	#10597275	—	2	2	no	—
39	#10597273	—	2	4	no	—
40	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
41	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
42	C₂²×C₄×C₁₉	—	2	1	no	—
43	C₂²×C₄×C₁₉	—	2	1	no	—
44	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
47	C₂×Dic₃₈	—	2	2	no	—
48	C₂×Dic₃₈	—	2	2	no	—
49	C₂×C₈×C₁₉	—	2	2	no	—
50	C₂×C₈×C₁₉	—	4	2	no	—
51	C₂×C₈×C₁₉	—	4	2	no	—
52	C₂×C₈×C₁₉	—	4	1	no	—
53	C₄²×C₁₉	—	2	1	no	—
54	C₄²×C₁₉	—	2	2	no	—
55	C₄²×C₁₉	—	4	1	no	—
56	C₄²×C₁₉	—	4	1	no	—
57	C₄²×C₁₉	—	4	2	no	—
58	C₄²×C₁₉	Q₁₆	1	2	yes	complemented
59	C₄²×C₁₉	C₂×D₄	1	2	yes	—
60	C₄²×C₁₉	#41	1	2	yes	—
61	C₂²⋊Dic₁₉	—	4	2	no	—
62	C₄⋊Dic₁₉	—	2	2	no	—
63	C₄⋊Dic₁₉	—	2	2	no	—
64	C₄⋊Dic₁₉	—	2	2	no	—
65	C₄⋊Dic₁₉	—	2	2	no	—
66	Dic₁₉⋊C₄	—	4	2	no	—
67	Dic₁₉⋊C₄	—	4	2	no	—
68	C₄×Dic₁₉	—	4	2	no	—
69	#64520	—	4	2	no	—
70	#64520	—	4	2	no	—
71	C₂×C₁₉⋊C₈	—	4	2	no	—
72	Dic₇₆	—	8	2	no	—
73	#10523	—	19	1	no	maximal, Sylow
74	C₂³×C₁₉	#121	1	1	yes	characteristic
75	C₂×C₄×C₁₉	—	2	1	no	—
76	C₂×C₄×C₁₉	—	2	1	no	—
77	C₂×C₄×C₁₉	—	2	2	no	—
78	C₂×C₄×C₁₉	—	2	2	no	—
79	C₂×C₄×C₁₉	—	4	1	no	—
80	C₂×C₄×C₁₉	—	4	1	no	—
81	C₂×C₄×C₁₉	—	4	1	no	—
82	C₂×C₄×C₁₉	—	4	1	no	—
83	C₂×C₄×C₁₉	—	4	1	no	—
84	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
85	C₂×C₄×C₁₉	#121	1	2	yes	—
86	C₂×C₄×C₁₉	#123	1	2	yes	—
87	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
88	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
89	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
90	C₂×C₄×C₁₉	#137	1	2	yes	—
91	C₂×Dic₁₉	—	4	2	no	—
92	C₂×Dic₁₉	—	4	2	no	—
93	Dic₃₈	—	4	2	no	—
94	Dic₃₈	—	4	2	no	—
95	Dic₃₈	—	8	2	no	—
96	C₈×C₁₉	—	4	2	no	—
97	C₈×C₁₉	—	8	1	no	—
98	C₁₉⋊C₈	—	8	2	no	—
99	#3249	—	19	1	no	—
100	#1994	—	19	2	no	—
101	#1831	D₁₉	1	1	yes	characteristic
102	#1550	—	19	2	no	—
103	#1292	—	19	1	no	—
104	C₂²×C₁₉	—	2	1	no	—
105	C₂²×C₁₉	—	2	1	no	—
106	C₂²×C₁₉	#394	1	1	yes	characteristic
107	C₂²×C₁₉	#467	1	1	yes	characteristic
108	C₂²×C₁₉	#470	1	1	yes	characteristic
109	C₄×C₁₉	—	2	2	no	—
110	C₄×C₁₉	—	2	1	no	—
111	C₄×C₁₉	—	4	2	no	—
112	C₄×C₁₉	—	4	2	no	—
113	C₄×C₁₉	#454	1	1	yes	characteristic
114	C₄×C₁₉	#455	1	1	yes	characteristic
115	C₄×C₁₉	#462	1	2	yes	—
116	C₄×C₁₉	#464	1	2	yes	—
117	Dic₁₉	—	8	2	no	—
118	Dic₁₉	—	8	2	no	—
119	#533	—	19	2	no	—
120	#477	—	38	2	no	—
121	#467	—	38	2	no	—
122	#462	—	38	2	no	—
123	#423	—	19	2	no	—
124	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
125	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
126	C₄³	C₂×D₁₉	1	1	yes	characteristic
127	#339	—	38	2	no	—
128	#339	—	38	2	no	—
129	#336	—	2	2	no	—
130	#333	—	38	2	no	—
131	#332	—	38	2	no	—
132	C₂×C₁₉	—	2	1	no	—
133	C₂×C₁₉	—	2	1	no	—
134	C₂×C₁₉	#2040	1	1	yes	characteristic
135	C₂×C₁₉	#2050	1	1	yes	characteristic
136	C₂×C₁₉	#2056	1	1	yes	characteristic
137	C₂×Q₁₆	—	38	2	no	—
138	C₂×M₁₆	—	38	2	no	—
139	C₂²×C₈	—	2	1	no	—
140	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
141	C₄⋊Q₈	—	19	2	no	—
142	C₄⋊Q₈	—	19	2	no	—
143	#125	—	38	2	no	—
144	C₂²⋊Q₈	—	38	2	no	—
145	C₂²⋊Q₈	—	38	2	no	—
146	C₄×Q₈	—	38	2	no	—
147	C₄×Q₈	—	38	2	no	—
148	#117	—	38	2	no	—
149	C₂×C₄²	—	2	1	no	—
150	C₂×C₄²	—	2	1	no	—
151	C₂×C₄²	D₇₆	1	1	yes	characteristic
152	C₂×C₄²	D₇₆	1	1	yes	characteristic
153	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
154	#107	—	38	2	no	—
155	C₄⋊C₈	—	2	2	no	—
156	C₄⋊C₈	—	2	2	no	—
157	C₄⋊C₈	—	38	2	no	—
158	C₄⋊C₈	—	38	2	no	—
159	#103	—	38	4	no	—
160	C₁₉	#10523	1	1	yes	characteristic, complemented, Sylow
161	C₂×Q₈	—	38	2	no	—
162	C₂×Q₈	—	38	2	no	—
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
166	C₂²×C₄	#64541	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
168	Q₁₆	—	152	2	no	—
169	M₁₆	—	76	2	no	—
170	M₁₆	—	76	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	4	2	no	—
173	C₂×C₈	—	4	2	no	—
174	C₂×C₈	—	4	1	no	—
175	C₂×C₈	—	76	2	no	—
176	C₄⋊C₄	—	38	2	no	—
177	C₄⋊C₄	—	38	2	no	—
178	C₄⋊C₄	—	38	2	no	—
179	C₄⋊C₄	—	38	2	no	—
180	C₄⋊C₄	—	76	2	no	—
181	C₄⋊C₄	—	76	2	no	—
182	C₂²⋊C₄	—	76	2	no	—
183	C₄²	—	2	1	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	4	1	no	—
186	C₄²	—	4	1	no	—
187	C₄²	—	4	2	no	—
188	C₄²	—	76	2	no	—
189	C₄²	Dic₇₆	1	2	yes	complemented
190	C₄²	D₄×D₁₉	1	2	yes	—
191	C₄²	#64545	1	2	yes	—
192	C₂³	#10597376	1	1	yes	characteristic
193	Q₈	—	76	2	no	—
194	Q₈	—	76	2	no	—
195	Q₈	—	152	2	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	—	4	1	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	—	4	1	no	—
204	C₂×C₄	—	4	1	no	—
205	C₂×C₄	—	76	2	no	—
206	C₂×C₄	—	76	2	no	—
207	C₂×C₄	#10597340	1	2	yes	—
208	C₂×C₄	#10597341	1	2	yes	—
209	C₂×C₄	#10597350	1	1	yes	characteristic
210	C₂×C₄	C₂×Dic₇₆	1	1	yes	characteristic
211	C₂×C₄	#10597355	1	2	yes	—
212	C₂×C₄	#10597383	1	1	yes	characteristic
213	C₂×C₄	#10597393	1	1	yes	characteristic
214	C₈	—	4	2	no	—
215	C₈	—	8	1	no	—
216	C₈	—	152	2	no	—
217	C₂²	—	2	1	no	—
218	C₂²	—	2	1	no	—
219	C₂²	#11926461	1	1	yes	characteristic
220	C₂²	#11926785	1	1	yes	characteristic
221	C₂²	#11926802	1	1	yes	characteristic
222	C₄	—	2	1	no	—
223	C₄	—	2	2	no	—
224	C₄	—	4	2	no	—
225	C₄	—	4	2	no	—
226	C₄	—	152	2	no	—
227	C₄	—	152	2	no	—
228	C₄	#11926716	1	1	yes	characteristic
229	C₄	#11926721	1	1	yes	characteristic
230	C₄	#11926755	1	2	yes	—
231	C₄	#11926767	1	2	yes	—
232	C₂	—	2	1	no	—
233	C₂	—	2	1	no	—
234	C₂	#14564864	1	1	yes	characteristic
235	C₂	#14564922	1	1	yes	characteristic
236	C₂	#14564958	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14574980

C₂

yes

maximal, characteristic

#14564612

C₂

yes

maximal

#14562323

C₂

yes

maximal

#14561510

C₂

yes

maximal, characteristic

C₁₉×#1831

C₂

yes

maximal, characteristic

#11927101

C₂²

yes

—

—

—

—

C₂²

yes

—

—

—

—

—

C₂×C₁₉×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₁₉×C₄⋊C₈

C₂²

yes

characteristic

C₄³×C₁₉

C₂²

yes

characteristic

C₁₉×#336

—

C₂²×C₈×C₁₉

—

C₂²×C₈×C₁₉

D₄

yes

—

C₂×C₄²×C₁₉

—

C₂×C₄²×C₁₉

—

C₂×C₄²×C₁₉

D₄

yes

characteristic

C₂×C₄²×C₁₉

D₄

yes

characteristic

C₂×C₄²×C₁₉

C₂³

yes

—

—

—

—

C₂×Dic₇₆

—

#10597316

—

#10597315

D₄

yes

—

#10597315

D₄

yes

—

C₄×Dic₃₈

—

C₄×Dic₃₈

—

C₁₉×C₄⋊C₈

—

C₁₉×C₄⋊C₈

—

#10597275

—

#10597273

—

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₉

#41

yes

characteristic

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

Q₁₆

yes

complemented

C₄²×C₁₉

C₂×D₄

yes

—

C₄²×C₁₉

#41

yes

—

C₂²⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

Dic₁₉⋊C₄

—

Dic₁₉⋊C₄

—

C₄×Dic₁₉

—

#64520

—

#64520

—

C₂×C₁₉⋊C₈

—

Dic₇₆

—

—

C₂³×C₁₉

yes

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

#120

yes

characteristic

C₂×C₄×C₁₉

#121

yes

—

C₂×C₄×C₁₉

#123

yes

—

C₂×C₄×C₁₉

#124

yes

characteristic

C₂×C₄×C₁₉

C₂×Q₁₆

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

Dic₃₈

—

Dic₃₈

—

Dic₃₈

—

C₈×C₁₉

—

C₈×C₁₉

—

C₁₉⋊C₈

—

#3249

—

100

#1994

—

101

#1831

D₁₉

yes

characteristic

102

#1550

—

103

#1292

—

104

C₂²×C₁₉

—

105

C₂²×C₁₉

—

106

C₂²×C₁₉

#394

yes

characteristic

107

C₂²×C₁₉

#467

yes

characteristic

108

C₂²×C₁₉

#470

yes

characteristic

109

C₄×C₁₉

—

110

C₄×C₁₉

—

111

C₄×C₁₉

—

112

C₄×C₁₉

—

113

C₄×C₁₉

#454

yes

characteristic

114

C₄×C₁₉

#455

yes

characteristic

115

C₄×C₁₉

#462

yes

—

116

C₄×C₁₉

#464

yes

—

117

Dic₁₉

—

118

Dic₁₉

—

119

#533

—

120

#477

—

121

#467

—

122

#462

—

123

#423

—

124

C₂×C₄⋊C₈

C₂×D₁₉

yes

characteristic

125

C₂×C₄⋊C₈

C₂×D₁₉

yes

characteristic

126

C₄³

C₂×D₁₉

yes

characteristic

127

#339

—

128

#339

—

129

#336

—

130

#333

—

131

#332

—

132

C₂×C₁₉

—

133

C₂×C₁₉

—

134

C₂×C₁₉

#2040

yes

characteristic

135

C₂×C₁₉

#2050

yes

characteristic

136

C₂×C₁₉

#2056

yes

characteristic

137

C₂×Q₁₆

—

138

C₂×M₁₆

—

139

C₂²×C₈

—

140

C₂²×C₈

C₂²⋊D₁₉

yes

—

141

C₄⋊Q₈

—

142

C₄⋊Q₈

—

143

#125

—

144

C₂²⋊Q₈

—

145

C₂²⋊Q₈

—

146

C₄×Q₈

—

147

C₄×Q₈

—

148

#117

—

149

C₂×C₄²

—

150

C₂×C₄²

—

151

C₂×C₄²

D₇₆

yes

characteristic

152

C₂×C₄²

D₇₆

yes

characteristic

153

C₂×C₄²

C₂²×D₁₉

yes

characteristic

154

#107

—

155

C₄⋊C₈

—

156

C₄⋊C₈

—

157

C₄⋊C₈

—

158

C₄⋊C₈

—

159

#103

—

160

C₁₉

#10523

yes

characteristic, complemented, Sylow

161

C₂×Q₈

—

162

C₂×Q₈

—

163

C₂²×C₄

—

164

C₂²×C₄

—

165

C₂²×C₄

C₂×D₇₆

yes

characteristic

166

C₂²×C₄

#64541

yes

characteristic

167

C₂²×C₄

C₂×C₂²⋊D₁₉

yes

characteristic

168

Q₁₆

—

152

—

169

M₁₆

—

170

M₁₆

—

171

C₂×C₈

—

172

C₂×C₈

—

173

C₂×C₈

—

174

C₂×C₈

—

175

C₂×C₈

—

176

C₄⋊C₄

—

177

C₄⋊C₄

—

178

C₄⋊C₄

—

179

C₄⋊C₄

—

180

C₄⋊C₄

—

181

C₄⋊C₄

—

182

C₂²⋊C₄

—

183

C₄²

—

184

C₄²

—

185

C₄²

—

186

C₄²

—

187

C₄²

—

188

C₄²

—

189

C₄²

Dic₇₆

yes

complemented

190

C₄²

D₄×D₁₉

yes

—

191

C₄²

#64545

yes

—

192

C₂³

#10597376

yes

characteristic

193

Q₈

—

194

Q₈

—

195

Q₈

—

152

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

—

203

C₂×C₄

—

204

C₂×C₄

—

205

C₂×C₄

—

206

C₂×C₄

—

207

C₂×C₄

#10597340

yes

—

208

C₂×C₄

#10597341

yes

—

209

C₂×C₄

#10597350

yes

characteristic

210

C₂×C₄

C₂×Dic₇₆

yes

characteristic

211

C₂×C₄

#10597355

yes

—

212

C₂×C₄

#10597383

yes

characteristic

213

C₂×C₄

#10597393

yes

characteristic

214

C₈

—

215

C₈

—

216

C₈

—

152

—

217

C₂²

—

218

C₂²

—

219

C₂²

#11926461

yes

characteristic

220

C₂²

#11926785

yes

characteristic

221

C₂²

#11926802

yes

characteristic

222

C₄

—

223

C₄

—

224

C₄

—

225

C₄

—

226

C₄

—

152

—

227

C₄

—

152

—

228

C₄

#11926716

yes

characteristic

229

C₄

#11926721

yes

characteristic

230

C₄

#11926755

yes

—

231

C₄

#11926767

yes

—

232

C₂

—

233

C₂

—

234

C₂

#14564864

yes

characteristic

235

C₂

#14564922

yes

characteristic

236

C₂

#14564958

yes

characteristic

	subgroup	G/H
center	C₂²	#11926461
commutator, cosocle	C₂×C₄²×C₁₉	C₂³
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1831	C₂
socle	C₂²×C₁₉	#394
prime radical	C₂³×C₁₉	#121
hypercenter	#1831	D₁₉
hypocenter	C₁₉	#10523
LCS	G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1831

subgroup

G/H

center

C₂²

#11926461

commutator, cosocle

C₂×C₄²×C₁₉

C₂³

Frattini

C₂×C₄²

C₂²×D₁₉

Fitting

C₁₉×#1831

C₂

C₂²×C₁₉

C₂³×C₁₉

D₁₉

C₁₉

G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1831

N	H ≅ G/N	type
58. C₄²×C₁₉	Q₁₆	semidirect
160. C₁₉	#10523	semidirect
189. C₄²	Dic₇₆	semidirect

H ≅ G/N

type

58. C₄²×C₁₉

Q₁₆

semidirect

160. C₁₉

#10523

semidirect

189. C₄²

Dic₇₆

semidirect

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

subgroup	index
#14574980	2
#14564612	2
#14562323	2
#14561510	2
C₁₉×#1831	2
#10523	19

subgroup

index

C₁₉×#1831

Subgroups — #18086672

Conjugacy classes of subgroups

Complements

Maximal subgroups