Group #18085212 — subgroups

Subgroups — #18085212

Order 4864.

566 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 111 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	#14574557	C₂	1	2	yes	maximal
2	#14572013	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562317	C₂	1	2	yes	maximal, complemented
4	#14561468	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1962	C₂	1	1	yes	maximal, characteristic
6	C₂×#10597315	C₂²	1	2	yes	—
7	#11926916	—	2	2	no	—
8	#11926896	—	2	2	no	—
9	#11926749	—	2	4	no	—
10	#11926682	—	2	4	no	—
11	#11926597	C₂²	1	2	yes	—
12	#11926450	—	2	4	no	—
13	#11926410	—	2	2	no	—
14	#11926292	—	2	4	no	—
15	C₁₉×#532	C₂²	1	1	yes	characteristic
16	C₁₉×#423	C₂²	1	1	yes	characteristic
17	C₂×C₁₉×#104	—	2	2	no	—
18	C₄³×C₁₉	C₂²	1	1	yes	characteristic
19	C₂×#41×C₁₉	—	2	1	no	—
20	C₂²×Dic₃₈	D₄	1	4	yes	—
21	C₂×M₁₆×C₁₉	—	2	1	no	—
22	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
23	C₁₉×#127	D₄	1	1	yes	characteristic
24	C₁₉×#124	—	2	1	no	—
25	C₁₉×#121	—	2	2	no	—
26	C₄×D₄×C₁₉	—	2	2	no	—
27	C₂×C₄²×C₁₉	—	2	2	no	—
28	C₂×C₄²×C₁₉	D₄	1	2	yes	—
29	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
30	C₂×#64526	—	2	8	no	—
31	C₂×C₄⋊Dic₁₉	—	2	2	no	—
32	C₂×C₄⋊Dic₁₉	—	2	2	no	—
33	#10597315	—	2	2	no	—
34	#10597315	—	2	2	no	—
35	#10597315	—	2	2	no	—
36	#10597315	—	2	2	no	—
37	C₁₉×C₄⋊C₈	—	2	2	no	—
38	C₁₉×#104	—	4	4	no	—
39	C₁₉⋊#95	—	2	8	no	—
40	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
41	#10597267	—	2	8	no	—
42	C₄⋊(C₁₉⋊C₈)	D₄	1	4	yes	—
43	C₄×C₁₉⋊C₈	—	2	2	no	—
44	#41×C₁₉	—	4	2	no	—
45	#41×C₁₉	—	4	2	no	—
46	C₂×Q₈×C₁₉	—	2	1	no	—
47	C₂×D₄×C₁₉	—	2	2	no	—
48	C₂×D₄×C₁₉	—	2	1	no	—
49	C₂×D₄×C₁₉	—	4	1	no	—
50	C₂²×C₄×C₁₉	—	2	2	no	—
51	C₂²×C₄×C₁₉	—	4	1	no	—
52	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
54	C₂²×Dic₁₉	—	2	4	no	—
55	C₂×Dic₃₈	—	2	4	no	—
56	C₂×Dic₃₈	—	2	4	no	—
57	C₂×Dic₃₈	—	4	8	no	—
58	M₁₆×C₁₉	—	4	2	no	—
59	C₂×C₈×C₁₉	—	4	1	no	—
60	C₄⋊C₄×C₁₉	—	2	2	no	—
61	C₂²⋊C₄×C₁₉	—	4	2	no	—
62	C₄²×C₁₉	—	2	2	no	—
63	C₄²×C₁₉	—	4	4	no	—
64	C₄²×C₁₉	SD₁₆	1	4	yes	complemented
65	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₁₉	C₂×D₄	1	2	yes	—
67	C₄²×C₁₉	#41	1	1	yes	characteristic
68	#64526	—	8	8	no	—
69	C₄⋊Dic₁₉	—	4	2	no	—
70	C₄⋊Dic₁₉	—	4	2	no	—
71	C₄⋊Dic₁₉	—	4	2	no	—
72	C₄⋊Dic₁₉	—	4	2	no	—
73	C₂×C₁₉⋊C₈	—	2	4	no	—
74	#10287	—	19	1	no	maximal, Sylow
75	C₂³×C₁₉	—	4	1	no	—
76	C₂³×C₁₉	#120	1	1	yes	characteristic
77	Q₈×C₁₉	—	4	2	no	—
78	D₄×C₁₉	—	4	4	no	—
79	D₄×C₁₉	—	4	2	no	—
80	D₄×C₁₉	—	8	2	no	—
81	C₂×C₄×C₁₉	—	2	4	no	—
82	C₂×C₄×C₁₉	—	2	4	no	—
83	C₂×C₄×C₁₉	—	4	4	no	—
84	C₂×C₄×C₁₉	—	4	1	no	—
85	C₂×C₄×C₁₉	—	8	1	no	—
86	C₂×C₄×C₁₉	—	8	1	no	—
87	C₂×C₄×C₁₉	#120	1	2	yes	—
88	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
89	C₂×C₄×C₁₉	#121	1	2	yes	—
90	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
91	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
92	C₂×C₄×C₁₉	#137	1	2	yes	—
93	C₂×Dic₁₉	—	4	8	no	—
94	C₂×Dic₁₉	—	4	4	no	—
95	Dic₃₈	—	4	8	no	—
96	Dic₃₈	—	8	4	no	—
97	C₈×C₁₉	—	8	1	no	—
98	C₁₉⋊C₈	—	4	4	no	—
99	#3206	—	19	2	no	—
100	#2862	—	19	1	no	—
101	#1962	D₁₉	1	1	yes	characteristic
102	#1548	—	19	2	no	—
103	#1276	—	19	1	no	—
104	C₂²×C₁₉	—	2	2	no	—
105	C₂²×C₁₉	—	8	2	no	—
106	C₂²×C₁₉	—	8	1	no	—
107	C₂²×C₁₉	#392	1	1	yes	characteristic
108	C₂²×C₁₉	#448	1	2	yes	—
109	C₄×C₁₉	—	2	2	no	—
110	C₄×C₁₉	—	4	4	no	—
111	C₄×C₁₉	—	8	1	no	—
112	C₄×C₁₉	#455	1	4	yes	—
113	C₄×C₁₉	#461	1	2	yes	—
114	C₄×C₁₉	#492	1	1	yes	characteristic
115	C₄×C₁₉	#497	1	1	yes	characteristic
116	Dic₁₉	—	8	4	no	—
117	#532	C₂×D₁₉	1	1	yes	characteristic
118	C₂×C₄⋊Q₈	—	19	2	no	—
119	#492	—	38	2	no	—
120	#487	—	38	2	no	—
121	#461	—	38	4	no	—
122	#448	—	38	4	no	—
123	#431	—	19	2	no	—
124	#423	C₂×D₁₉	1	1	yes	characteristic
125	C₂×#104	—	2	2	no	—
126	#391	—	38	4	no	—
127	#382	—	38	2	no	—
128	C₄³	C₂×D₁₉	1	1	yes	characteristic
129	#333	—	38	4	no	—
130	C₂×C₁₉	—	2	2	no	—
131	C₂×C₁₉	—	16	1	no	—
132	C₂×C₁₉	#2004	1	2	yes	—
133	C₂×C₁₉	#2013	1	1	yes	characteristic
134	C₂×#41	—	2	1	no	—
135	C₂²×Q₈	—	19	4	no	—
136	C₂×SD₁₆	—	38	8	no	—
137	C₂×M₁₆	—	2	1	no	—
138	C₄⋊Q₈	—	38	2	no	—
139	C₄⋊Q₈	—	38	2	no	—
140	C₄⋊Q₈	—	38	2	no	—
141	C₄⋊Q₈	—	38	2	no	—
142	C₄⋊Q₈	D₇₆	1	1	yes	characteristic
143	#127	D₇₆	1	1	yes	characteristic
144	#124	—	2	1	no	—
145	#121	—	2	2	no	—
146	C₄×D₄	—	2	2	no	—
147	C₂×C₄⋊C₄	—	38	2	no	—
148	C₂×C₄⋊C₄	—	38	2	no	—
149	C₂×C₄²	—	2	2	no	—
150	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
151	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
152	C₄⋊C₈	—	2	2	no	—
153	C₄⋊C₈	—	19	4	no	—
154	#104	—	4	4	no	—
155	#103	—	38	8	no	—
156	C₂²⋊C₈	—	38	2	no	—
157	C₄×C₈	—	38	2	no	—
158	#95	—	38	8	no	—
159	C₁₉	#10287	1	1	yes	characteristic, complemented, Sylow
160	#41	—	4	2	no	—
161	#41	—	4	2	no	—
162	C₂×Q₈	—	2	1	no	—
163	C₂×Q₈	—	38	4	no	—
164	C₂×Q₈	—	38	4	no	—
165	C₂×Q₈	—	76	8	no	—
166	C₂×D₄	—	2	1	no	—
167	C₂×D₄	—	2	2	no	—
168	C₂×D₄	—	4	1	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	4	1	no	—
171	C₂²×C₄	—	38	4	no	—
172	C₂²×C₄	D₄×D₁₉	1	2	yes	—
173	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
174	SD₁₆	—	152	8	no	—
175	M₁₆	—	4	2	no	—
176	C₂×C₈	—	4	1	no	—
177	C₂×C₈	—	38	4	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	76	2	no	—
180	C₄⋊C₄	—	76	2	no	—
181	C₄⋊C₄	—	76	2	no	—
182	C₄⋊C₄	—	76	2	no	—
183	C₂²⋊C₄	—	4	2	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	4	4	no	—
186	C₄²	#64526	1	4	yes	complemented
187	C₄²	C₂×D₇₆	1	1	yes	characteristic
188	C₄²	#64541	1	1	yes	characteristic
189	C₄²	D₄×D₁₉	1	2	yes	—
190	C₂³	—	4	1	no	—
191	C₂³	#10597383	1	1	yes	characteristic
192	Q₈	—	4	2	no	—
193	Q₈	—	76	8	no	—
194	Q₈	—	152	4	no	—
195	D₄	—	4	4	no	—
196	D₄	—	4	2	no	—
197	D₄	—	8	2	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	4	1	no	—
202	C₂×C₄	—	8	1	no	—
203	C₂×C₄	—	8	1	no	—
204	C₂×C₄	—	76	8	no	—
205	C₂×C₄	—	76	4	no	—
206	C₂×C₄	#10597328	1	2	yes	—
207	C₂×C₄	#10597330	1	2	yes	—
208	C₂×C₄	#10597376	1	1	yes	characteristic
209	C₂×C₄	C₂×#64526	1	2	yes	—
210	C₂×C₄	#10597386	1	1	yes	characteristic
211	C₂×C₄	#10597397	1	2	yes	—
212	C₈	—	8	1	no	—
213	C₈	—	76	4	no	—
214	C₂²	—	2	2	no	—
215	C₂²	—	8	2	no	—
216	C₂²	—	8	1	no	—
217	C₂²	#11926453	1	1	yes	characteristic
218	C₂²	#11926682	1	2	yes	—
219	C₄	—	2	2	no	—
220	C₄	—	4	4	no	—
221	C₄	—	8	1	no	—
222	C₄	—	152	4	no	—
223	C₄	#11926723	1	4	yes	—
224	C₄	#11926749	1	2	yes	—
225	C₄	#11926916	1	1	yes	characteristic
226	C₄	#11926939	1	1	yes	characteristic
227	C₂	—	2	2	no	—
228	C₂	—	16	1	no	—
229	C₂	#14564663	1	2	yes	—
230	C₂	#14564717	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14574557

C₂

yes

maximal

#14572013

C₂

yes

maximal, characteristic, complemented

#14562317

C₂

yes

maximal, complemented

#14561468

C₂

yes

maximal, characteristic, complemented

C₁₉×#1962

C₂

yes

maximal, characteristic

C₂×#10597315

C₂²

yes

—

—

—

—

—

C₂²

yes

—

#11926450

—

#11926410

—

#11926292

—

C₁₉×#532

C₂²

yes

characteristic

C₁₉×#423

C₂²

yes

characteristic

C₂×C₁₉×#104

—

C₄³×C₁₉

C₂²

yes

characteristic

C₂×#41×C₁₉

—

C₂²×Dic₃₈

D₄

yes

—

C₂×M₁₆×C₁₉

—

C₁₉×C₄⋊Q₈

D₄

yes

characteristic

C₁₉×#127

D₄

yes

characteristic

C₁₉×#124

—

C₁₉×#121

—

C₄×D₄×C₁₉

—

C₂×C₄²×C₁₉

—

C₂×C₄²×C₁₉

D₄

yes

—

C₂×C₄²×C₁₉

C₂³

yes

characteristic

C₂×#64526

—

C₂×C₄⋊Dic₁₉

—

C₂×C₄⋊Dic₁₉

—

—

—

—

—

C₁₉×C₄⋊C₈

—

C₁₉×#104

—

C₁₉⋊#95

—

C₂²⋊(C₁₉⋊C₈)

—

#10597267

—

C₄⋊(C₁₉⋊C₈)

D₄

yes

—

C₄×C₁₉⋊C₈

—

#41×C₁₉

—

#41×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂×D₄

yes

—

C₂²×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂²×Dic₁₉

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

SD₁₆

yes

complemented

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₄²×C₁₉

C₂×D₄

yes

—

C₄²×C₁₉

#41

yes

characteristic

#64526

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₂×C₁₉⋊C₈

—

#10287

—

maximal, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

#120

yes

characteristic

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

#120

yes

—

C₂×C₄×C₁₉

#121

yes

characteristic

C₂×C₄×C₁₉

#121

yes

—

C₂×C₄×C₁₉

#127

yes

characteristic

C₂×C₄×C₁₉

C₂×SD₁₆

yes

—

C₂×C₄×C₁₉

#137

yes

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

Dic₃₈

—

Dic₃₈

—

C₈×C₁₉

—

C₁₉⋊C₈

—

#3206

—

100

#2862

—

101

#1962

D₁₉

yes

characteristic

102

#1548

—

103

#1276

—

104

C₂²×C₁₉

—

105

C₂²×C₁₉

—

106

C₂²×C₁₉

—

107

C₂²×C₁₉

#392

yes

characteristic

108

C₂²×C₁₉

#448

yes

—

109

C₄×C₁₉

—

110

C₄×C₁₉

—

111

C₄×C₁₉

—

112

C₄×C₁₉

#455

yes

—

113

C₄×C₁₉

#461

yes

—

114

C₄×C₁₉

#492

yes

characteristic

115

C₄×C₁₉

#497

yes

characteristic

116

Dic₁₉

—

117

#532

C₂×D₁₉

yes

characteristic

118

C₂×C₄⋊Q₈

—

119

#492

—

120

#487

—

121

#461

—

122

#448

—

123

#431

—

124

#423

C₂×D₁₉

yes

characteristic

125

C₂×#104

—

126

#391

—

127

#382

—

128

C₄³

C₂×D₁₉

yes

characteristic

129

#333

—

130

C₂×C₁₉

—

131

C₂×C₁₉

—

132

C₂×C₁₉

#2004

yes

—

133

C₂×C₁₉

#2013

yes

characteristic

134

C₂×#41

—

135

C₂²×Q₈

—

136

C₂×SD₁₆

—

137

C₂×M₁₆

—

138

C₄⋊Q₈

—

139

C₄⋊Q₈

—

140

C₄⋊Q₈

—

141

C₄⋊Q₈

—

142

C₄⋊Q₈

D₇₆

yes

characteristic

143

#127

D₇₆

yes

characteristic

144

#124

—

145

#121

—

146

C₄×D₄

—

147

C₂×C₄⋊C₄

—

148

C₂×C₄⋊C₄

—

149

C₂×C₄²

—

150

C₂×C₄²

C₂²⋊D₁₉

yes

—

151

C₂×C₄²

C₂²×D₁₉

yes

characteristic

152

C₄⋊C₈

—

153

C₄⋊C₈

—

154

#104

—

155

#103

—

156

C₂²⋊C₈

—

157

C₄×C₈

—

158

#95

—

159

C₁₉

#10287

yes

characteristic, complemented, Sylow

160

#41

—

161

#41

—

162

C₂×Q₈

—

163

C₂×Q₈

—

164

C₂×Q₈

—

165

C₂×Q₈

—

166

C₂×D₄

—

167

C₂×D₄

—

168

C₂×D₄

—

169

C₂²×C₄

—

170

C₂²×C₄

—

171

C₂²×C₄

—

172

C₂²×C₄

D₄×D₁₉

yes

—

173

C₂²×C₄

C₂×C₂²⋊D₁₉

yes

characteristic

174

SD₁₆

—

152

—

175

M₁₆

—

176

C₂×C₈

—

177

C₂×C₈

—

178

C₄⋊C₄

—

179

C₄⋊C₄

—

180

C₄⋊C₄

—

181

C₄⋊C₄

—

182

C₄⋊C₄

—

183

C₂²⋊C₄

—

184

C₄²

—

185

C₄²

—

186

C₄²

#64526

yes

complemented

187

C₄²

C₂×D₇₆

yes

characteristic

188

C₄²

#64541

yes

characteristic

189

C₄²

D₄×D₁₉

yes

—

190

C₂³

—

191

C₂³

#10597383

yes

characteristic

192

Q₈

—

193

Q₈

—

194

Q₈

—

152

—

195

D₄

—

196

D₄

—

197

D₄

—

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

#10597328

yes

—

207

C₂×C₄

#10597330

yes

—

208

C₂×C₄

#10597376

yes

characteristic

209

C₂×C₄

C₂×#64526

yes

—

210

C₂×C₄

#10597386

yes

characteristic

211

C₂×C₄

#10597397

yes

—

212

C₈

—

213

C₈

—

214

C₂²

—

215

C₂²

—

216

C₂²

—

217

C₂²

#11926453

yes

characteristic

218

C₂²

#11926682

yes

—

219

C₄

—

220

C₄

—

221

C₄

—

222

C₄

—

152

—

223

C₄

#11926723

yes

—

224

C₄

#11926749

yes

—

225

C₄

#11926916

yes

characteristic

226

C₄

#11926939

yes

characteristic

227

C₂

—

228

C₂

—

229

C₂

#14564663

yes

—

230

C₂

#14564717

yes

characteristic

	subgroup	G/H
center	C₂²	#11926453
commutator, cosocle	C₂×C₄²×C₁₉	C₂³
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1962	C₂
socle	C₂²×C₁₉	#392
prime radical	C₁₉×#532	C₂²
hypercenter	#1962	D₁₉
hypocenter	C₁₉	#10287
LCS	G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1962

subgroup

G/H

center

C₂²

#11926453

commutator, cosocle

C₂×C₄²×C₁₉

C₂³

Frattini

C₂×C₄²

C₂²×D₁₉

Fitting

C₁₉×#1962

C₂

socle

C₂²×C₁₉

#392

prime radical

C₁₉×#532

C₂²

D₁₉

C₁₉

G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1962

N	H ≅ G/N	type
2. #14572013	C₂	semidirect
3. #14562317	C₂	semidirect
4. #14561468	C₂	semidirect
64. C₄²×C₁₉	SD₁₆	semidirect
159. C₁₉	#10287	semidirect
186. C₄²	#64526	semidirect

H ≅ G/N

type

2. #14572013

C₂

semidirect

3. #14562317

C₂

semidirect

4. #14561468

C₂

semidirect

64. C₄²×C₁₉

SD₁₆

semidirect

159. C₁₉

#10287

semidirect

186. C₄²

#64526

semidirect

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

subgroup	index
#14574557	2
#14572013	2
#14562317	2
#14561468	2
C₁₉×#1962	2
#10287	19

subgroup

index

C₁₉×#1962

Subgroups — #18085212

Conjugacy classes of subgroups

Complements

Maximal subgroups