Group #18085200 — subgroups

Subgroups — #18085200

Order 4864.

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

normal subgroups only

This group has 542 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	#14574760	C₂	1	2	yes	maximal
2	#14572013	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562321	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	#11926940	—	2	2	no	—
8	#11926890	—	2	2	no	—
9	#11926860	—	2	4	no	—
10	#11926834	—	2	4	no	—
11	#11926597	C₂²	1	2	yes	—
12	#11926450	—	2	4	no	—
13	#11926410	—	2	2	no	—
14	#11926289	—	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	—	2	2	no	—
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×C₄⋊Dic₁₉	—	2	2	no	—
31	C₂×C₄⋊Dic₁₉	D₄	1	4	yes	—
32	#10597315	—	2	2	no	—
33	#10597315	—	2	2	no	—
34	#10597315	—	2	2	no	—
35	#10597315	—	2	2	no	—
36	C₁₉×C₄⋊C₈	—	2	2	no	—
37	C₁₉×#104	—	4	4	no	—
38	#10597289	—	2	8	no	—
39	C₁₉⋊#95	—	2	8	no	—
40	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
41	#10597265	—	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	—	4	8	no	—
56	C₂×Dic₃₈	—	4	2	no	—
57	C₂×Dic₃₈	—	4	2	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	—
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	C₄⋊Dic₁₉	—	2	4	no	—
69	C₄⋊Dic₁₉	—	2	4	no	—
70	C₄⋊Dic₁₉	—	4	2	no	—
71	C₄⋊Dic₁₉	—	4	2	no	—
72	C₂×C₁₉⋊C₈	—	2	4	no	—
73	#10285	—	19	1	no	maximal, Sylow
74	C₂³×C₁₉	—	4	1	no	—
75	C₂³×C₁₉	#120	1	1	yes	characteristic
76	Q₈×C₁₉	—	4	2	no	—
77	D₄×C₁₉	—	4	4	no	—
78	D₄×C₁₉	—	4	2	no	—
79	D₄×C₁₉	—	8	2	no	—
80	C₂×C₄×C₁₉	—	2	4	no	—
81	C₂×C₄×C₁₉	—	2	4	no	—
82	C₂×C₄×C₁₉	—	4	4	no	—
83	C₂×C₄×C₁₉	—	4	1	no	—
84	C₂×C₄×C₁₉	—	8	1	no	—
85	C₂×C₄×C₁₉	—	8	1	no	—
86	C₂×C₄×C₁₉	#120	1	2	yes	—
87	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
88	C₂×C₄×C₁₉	#121	1	2	yes	—
89	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
90	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
91	C₂×C₄×C₁₉	#137	1	2	yes	—
92	C₂×Dic₁₉	—	4	8	no	—
93	C₂×Dic₁₉	—	4	4	no	—
94	Dic₃₈	—	8	4	no	—
95	Dic₃₈	—	8	4	no	—
96	C₈×C₁₉	—	8	1	no	—
97	C₁₉⋊C₈	—	4	4	no	—
98	#3227	—	19	2	no	—
99	#2862	—	19	1	no	—
100	#1962	D₁₉	1	1	yes	characteristic
101	#1549	—	19	2	no	—
102	#1276	—	19	1	no	—
103	C₂²×C₁₉	—	2	2	no	—
104	C₂²×C₁₉	—	8	2	no	—
105	C₂²×C₁₉	—	8	1	no	—
106	C₂²×C₁₉	#392	1	1	yes	characteristic
107	C₂²×C₁₉	#448	1	2	yes	—
108	C₄×C₁₉	—	2	2	no	—
109	C₄×C₁₉	—	4	4	no	—
110	C₄×C₁₉	—	8	1	no	—
111	C₄×C₁₉	#454	1	4	yes	—
112	C₄×C₁₉	#461	1	2	yes	—
113	C₄×C₁₉	#492	1	1	yes	characteristic
114	C₄×C₁₉	#497	1	1	yes	characteristic
115	Dic₁₉	—	8	4	no	—
116	#532	C₂×D₁₉	1	1	yes	characteristic
117	C₂×C₄⋊Q₈	—	19	2	no	—
118	#498	—	38	2	no	—
119	#486	—	38	2	no	—
120	#481	—	38	4	no	—
121	#476	—	38	4	no	—
122	#431	—	19	2	no	—
123	#423	C₂×D₁₉	1	1	yes	characteristic
124	C₂×#104	—	2	2	no	—
125	#391	—	38	4	no	—
126	#382	—	38	2	no	—
127	C₄³	C₂×D₁₉	1	1	yes	characteristic
128	#332	—	38	4	no	—
129	C₂×C₁₉	—	2	2	no	—
130	C₂×C₁₉	—	16	1	no	—
131	C₂×C₁₉	#2003	1	2	yes	—
132	C₂×C₁₉	#2013	1	1	yes	characteristic
133	C₂×#41	—	2	1	no	—
134	C₂²×Q₈	—	38	2	no	—
135	C₂×M₁₆	—	2	1	no	—
136	C₄⋊Q₈	—	38	2	no	—
137	C₄⋊Q₈	—	38	2	no	—
138	C₄⋊Q₈	—	38	2	no	—
139	C₄⋊Q₈	—	38	2	no	—
140	C₄⋊Q₈	D₇₆	1	1	yes	characteristic
141	#127	D₇₆	1	1	yes	characteristic
142	#124	—	2	1	no	—
143	#121	—	2	2	no	—
144	C₄×D₄	—	2	2	no	—
145	C₂×C₄⋊C₄	—	19	4	no	—
146	C₂×C₄⋊C₄	—	38	2	no	—
147	C₂×C₄²	—	2	2	no	—
148	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
149	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
150	#106	—	38	8	no	—
151	C₄⋊C₈	—	2	2	no	—
152	C₄⋊C₈	—	19	4	no	—
153	#104	—	4	4	no	—
154	#102	—	38	8	no	—
155	C₂²⋊C₈	—	38	2	no	—
156	C₄×C₈	—	38	2	no	—
157	#95	—	38	8	no	—
158	C₁₉	#10285	1	1	yes	characteristic, complemented, Sylow
159	#41	—	4	2	no	—
160	#41	—	4	2	no	—
161	C₂×Q₈	—	2	1	no	—
162	C₂×Q₈	—	76	8	no	—
163	C₂×Q₈	—	76	2	no	—
164	C₂×Q₈	—	76	2	no	—
165	C₂×D₄	—	2	1	no	—
166	C₂×D₄	—	2	2	no	—
167	C₂×D₄	—	4	1	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	4	1	no	—
170	C₂²×C₄	—	38	4	no	—
171	C₂²×C₄	D₄×D₁₉	1	2	yes	—
172	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
173	M₁₆	—	4	2	no	—
174	C₂×C₈	—	4	1	no	—
175	C₂×C₈	—	38	4	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	38	4	no	—
178	C₄⋊C₄	—	38	4	no	—
179	C₄⋊C₄	—	76	2	no	—
180	C₄⋊C₄	—	76	2	no	—
181	C₂²⋊C₄	—	4	2	no	—
182	C₄²	—	2	2	no	—
183	C₄²	—	4	4	no	—
184	C₄²	#64526	1	4	yes	—
185	C₄²	C₂×D₇₆	1	1	yes	characteristic
186	C₄²	#64541	1	1	yes	characteristic
187	C₄²	D₄×D₁₉	1	2	yes	—
188	C₂³	—	4	1	no	—
189	C₂³	#10597383	1	1	yes	characteristic
190	Q₈	—	4	2	no	—
191	Q₈	—	152	4	no	—
192	Q₈	—	152	4	no	—
193	D₄	—	4	4	no	—
194	D₄	—	4	2	no	—
195	D₄	—	8	2	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	—	4	1	no	—
200	C₂×C₄	—	8	1	no	—
201	C₂×C₄	—	8	1	no	—
202	C₂×C₄	—	76	8	no	—
203	C₂×C₄	—	76	4	no	—
204	C₂×C₄	#10597328	1	2	yes	—
205	C₂×C₄	#10597330	1	2	yes	—
206	C₂×C₄	#10597376	1	1	yes	characteristic
207	C₂×C₄	C₂×#64526	1	2	yes	—
208	C₂×C₄	#10597386	1	1	yes	characteristic
209	C₂×C₄	#10597397	1	2	yes	—
210	C₈	—	8	1	no	—
211	C₈	—	76	4	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	8	2	no	—
214	C₂²	—	8	1	no	—
215	C₂²	#11926453	1	1	yes	characteristic
216	C₂²	#11926682	1	2	yes	—
217	C₄	—	2	2	no	—
218	C₄	—	4	4	no	—
219	C₄	—	8	1	no	—
220	C₄	—	152	4	no	—
221	C₄	#11926718	1	4	yes	—
222	C₄	#11926749	1	2	yes	—
223	C₄	#11926916	1	1	yes	characteristic
224	C₄	#11926939	1	1	yes	characteristic
225	C₂	—	2	2	no	—
226	C₂	—	16	1	no	—
227	C₂	#14564656	1	2	yes	—
228	C₂	#14564717	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14574760

C₂

yes

maximal

#14572013

C₂

yes

maximal, characteristic, complemented

#14562321

C₂

yes

maximal, complemented

#14561468

C₂

yes

maximal, characteristic, complemented

C₁₉×#1962

C₂

yes

maximal, characteristic

C₂×#10597315

C₂²

yes

—

—

—

—

—

C₂²

yes

—

#11926450

—

#11926410

—

#11926289

—

C₁₉×#532

C₂²

yes

characteristic

C₁₉×#423

C₂²

yes

characteristic

C₂×C₁₉×#104

—

C₄³×C₁₉

C₂²

yes

characteristic

C₂×#41×C₁₉

—

C₂²×Dic₃₈

—

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×C₄⋊Dic₁₉

—

C₂×C₄⋊Dic₁₉

D₄

yes

—

—

—

—

—

C₁₉×C₄⋊C₈

—

C₁₉×#104

—

#10597289

—

C₁₉⋊#95

—

C₂²⋊(C₁₉⋊C₈)

—

#10597265

—

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

—

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₄²×C₁₉

C₂×D₄

yes

—

C₄²×C₁₉

#41

yes

characteristic

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₂×C₁₉⋊C₈

—

#10285

—

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

—

#3227

—

#2862

—

100

#1962

D₁₉

yes

characteristic

101

#1549

—

102

#1276

—

103

C₂²×C₁₉

—

104

C₂²×C₁₉

—

105

C₂²×C₁₉

—

106

C₂²×C₁₉

#392

yes

characteristic

107

C₂²×C₁₉

#448

yes

—

108

C₄×C₁₉

—

109

C₄×C₁₉

—

110

C₄×C₁₉

—

111

C₄×C₁₉

#454

yes

—

112

C₄×C₁₉

#461

yes

—

113

C₄×C₁₉

#492

yes

characteristic

114

C₄×C₁₉

#497

yes

characteristic

115

Dic₁₉

—

116

#532

C₂×D₁₉

yes

characteristic

117

C₂×C₄⋊Q₈

—

118

#498

—

119

#486

—

120

#481

—

121

#476

—

122

#431

—

123

#423

C₂×D₁₉

yes

characteristic

124

C₂×#104

—

125

#391

—

126

#382

—

127

C₄³

C₂×D₁₉

yes

characteristic

128

#332

—

129

C₂×C₁₉

—

130

C₂×C₁₉

—

131

C₂×C₁₉

#2003

yes

—

132

C₂×C₁₉

#2013

yes

characteristic

133

C₂×#41

—

134

C₂²×Q₈

—

135

C₂×M₁₆

—

136

C₄⋊Q₈

—

137

C₄⋊Q₈

—

138

C₄⋊Q₈

—

139

C₄⋊Q₈

—

140

C₄⋊Q₈

D₇₆

yes

characteristic

141

#127

D₇₆

yes

characteristic

142

#124

—

143

#121

—

144

C₄×D₄

—

145

C₂×C₄⋊C₄

—

146

C₂×C₄⋊C₄

—

147

C₂×C₄²

—

148

C₂×C₄²

C₂²⋊D₁₉

yes

—

149

C₂×C₄²

C₂²×D₁₉

yes

characteristic

150

#106

—

151

C₄⋊C₈

—

152

C₄⋊C₈

—

153

#104

—

154

#102

—

155

C₂²⋊C₈

—

156

C₄×C₈

—

157

#95

—

158

C₁₉

#10285

yes

characteristic, complemented, Sylow

159

#41

—

160

#41

—

161

C₂×Q₈

—

162

C₂×Q₈

—

163

C₂×Q₈

—

164

C₂×Q₈

—

165

C₂×D₄

—

166

C₂×D₄

—

167

C₂×D₄

—

168

C₂²×C₄

—

169

C₂²×C₄

—

170

C₂²×C₄

—

171

C₂²×C₄

D₄×D₁₉

yes

—

172

C₂²×C₄

C₂×C₂²⋊D₁₉

yes

characteristic

173

M₁₆

—

174

C₂×C₈

—

175

C₂×C₈

—

176

C₄⋊C₄

—

177

C₄⋊C₄

—

178

C₄⋊C₄

—

179

C₄⋊C₄

—

180

C₄⋊C₄

—

181

C₂²⋊C₄

—

182

C₄²

—

183

C₄²

—

184

C₄²

#64526

yes

—

185

C₄²

C₂×D₇₆

yes

characteristic

186

C₄²

#64541

yes

characteristic

187

C₄²

D₄×D₁₉

yes

—

188

C₂³

—

189

C₂³

#10597383

yes

characteristic

190

Q₈

—

191

Q₈

—

152

—

192

Q₈

—

152

—

193

D₄

—

194

D₄

—

195

D₄

—

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

#10597328

yes

—

205

C₂×C₄

#10597330

yes

—

206

C₂×C₄

#10597376

yes

characteristic

207

C₂×C₄

C₂×#64526

yes

—

208

C₂×C₄

#10597386

yes

characteristic

209

C₂×C₄

#10597397

yes

—

210

C₈

—

211

C₈

—

212

C₂²

—

213

C₂²

—

214

C₂²

—

215

C₂²

#11926453

yes

characteristic

216

C₂²

#11926682

yes

—

217

C₄

—

218

C₄

—

219

C₄

—

220

C₄

—

152

—

221

C₄

#11926718

yes

—

222

C₄

#11926749

yes

—

223

C₄

#11926916

yes

characteristic

224

C₄

#11926939

yes

characteristic

225

C₂

—

226

C₂

—

227

C₂

#14564656

yes

—

228

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	#10285
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. #14562321	C₂	semidirect
4. #14561468	C₂	semidirect
158. C₁₉	#10285	semidirect

H ≅ G/N

type

2. #14572013

C₂

semidirect

3. #14562321

C₂

semidirect

4. #14561468

C₂

semidirect

158. C₁₉

#10285

semidirect

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

subgroup	index
#14574760	2
#14572013	2
#14562321	2
#14561468	2
C₁₉×#1962	2
#10285	19

subgroup

index

C₁₉×#1962

Subgroups — #18085200

Conjugacy classes of subgroups

Complements

Maximal subgroups