Group #18084786 — subgroups

Subgroups — #18084786

Order 4864.

542 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 105 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	#14574573	C₂	1	2	yes	maximal
2	#14564459	C₂	1	1	yes	maximal, characteristic
3	#14562199	C₂	1	2	yes	maximal, complemented
4	#14561480	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#3162	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10597320	C₂²	1	1	yes	characteristic
7	#11926918	—	2	2	no	—
8	#11926892	—	2	2	no	—
9	#11926744	—	2	4	no	—
10	#11926694	—	2	4	no	—
11	#11926597	C₂²	1	2	yes	—
12	C₂×#10597278	—	2	2	no	—
13	#11926410	C₂²	1	1	yes	characteristic
14	#11926281	—	2	4	no	—
15	C₁₉×#533	C₂²	1	2	yes	—
16	C₁₉×#501	—	2	2	no	—
17	C₁₉×#501	—	2	2	no	—
18	C₁₉×#427	—	2	2	no	—
19	C₂×C₁₉×#97	C₂²	1	1	yes	characteristic
20	C₂²×D₇₆	—	2	1	no	—
21	C₂²×D₇₆	—	2	2	no	—
22	C₂²×C₈×C₁₉	D₄	1	2	yes	—
23	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
24	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
25	C₁₉×#125	—	2	2	no	—
26	C₁₉×C₂²⋊Q₈	—	2	2	no	—
27	C₁₉×C₂²⋊Q₈	—	2	2	no	—
28	C₄×Q₈×C₁₉	—	2	2	no	—
29	C₄×Q₈×C₁₉	—	2	2	no	—
30	C₁₉×#117	—	2	2	no	—
31	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
32	C₂×#64527	—	2	8	no	—
33	C₂×C₄⋊Dic₁₉	—	2	1	no	—
34	#10597320	—	2	2	no	—
35	#10597320	—	2	1	no	—
36	#10597320	D₄	1	2	yes	—
37	C₁₉×#107	—	2	4	no	—
38	C₁₉×#106	—	2	2	no	—
39	C₁₉×#106	—	2	2	no	—
40	C₁₉×#97	—	2	2	no	—
41	C₁₉⋊#95	—	2	2	no	—
42	C₂²⋊(C₁₉⋊C₈)	D₄	1	4	yes	—
43	#10597278	—	4	4	no	—
44	#10597266	—	2	8	no	—
45	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
46	C₄×C₁₉⋊C₈	—	2	2	no	—
47	C₂³×D₁₉	—	4	1	no	—
48	C₂×Q₈×C₁₉	—	2	2	no	—
49	C₂×Q₈×C₁₉	—	2	2	no	—
50	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
51	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
52	C₂²×Dic₁₉	—	4	1	no	—
53	C₂×D₇₆	—	2	4	no	—
54	C₂×D₇₆	—	2	2	no	—
55	C₂×D₇₆	—	4	2	no	—
56	C₂×D₇₆	—	4	2	no	—
57	C₂×D₇₆	—	4	1	no	—
58	C₂×D₇₆	—	4	2	no	—
59	C₂×D₇₆	—	8	4	no	—
60	C₂×C₈×C₁₉	—	2	2	no	—
61	C₂×C₈×C₁₉	—	2	2	no	—
62	C₂×C₈×C₁₉	SD₁₆	1	4	yes	complemented
63	C₄⋊C₄×C₁₉	—	2	2	no	—
64	C₄⋊C₄×C₁₉	—	2	2	no	—
65	C₄⋊C₄×C₁₉	—	2	2	no	—
66	C₄⋊C₄×C₁₉	—	2	2	no	—
67	C₄⋊C₄×C₁₉	—	4	2	no	—
68	C₄⋊C₄×C₁₉	—	4	2	no	—
69	C₂²⋊C₄×C₁₉	—	4	2	no	—
70	C₄²×C₁₉	—	4	2	no	—
71	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₁₉	C₂×D₄	1	2	yes	—
73	C₄²×C₁₉	#41	1	1	yes	characteristic
74	#64527	—	8	8	no	—
75	C₄⋊Dic₁₉	—	4	2	no	—
76	C₂×C₁₉⋊C₈	—	2	4	no	—
77	#10220	—	19	1	no	maximal, Sylow
78	C₂³×C₁₉	#127	1	1	yes	characteristic
79	C₂²×D₁₉	—	4	2	no	—
80	C₂²×D₁₉	—	8	4	no	—
81	C₂²×D₁₉	—	8	1	no	—
82	C₂²×D₁₉	—	8	1	no	—
83	Q₈×C₁₉	—	4	4	no	—
84	Q₈×C₁₉	—	8	2	no	—
85	C₂×C₄×C₁₉	—	4	2	no	—
86	C₂×C₄×C₁₉	—	4	2	no	—
87	C₂×C₄×C₁₉	—	4	2	no	—
88	C₂×C₄×C₁₉	#120	1	2	yes	—
89	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
90	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
91	C₂×C₄×C₁₉	#121	1	2	yes	—
92	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
93	C₂×C₄×C₁₉	#137	1	2	yes	—
94	C₂×Dic₁₉	—	8	2	no	—
95	C₂×Dic₁₉	—	8	1	no	—
96	D₇₆	—	4	8	no	—
97	D₇₆	—	4	4	no	—
98	D₇₆	—	8	4	no	—
99	D₇₆	—	8	2	no	—
100	C₈×C₁₉	—	2	4	no	—
101	C₁₉⋊C₈	—	4	4	no	—
102	#3208	—	19	2	no	—
103	#3162	D₁₉	1	1	yes	characteristic, complemented
104	#1961	—	19	1	no	—
105	#1518	—	19	2	no	—
106	#1281	—	19	1	no	—
107	C₂²×C₁₉	—	2	2	no	—
108	C₂²×C₁₉	#392	1	1	yes	characteristic
109	C₂²×C₁₉	#492	1	1	yes	characteristic
110	C₂²×C₁₉	#497	1	1	yes	characteristic
111	C₂×D₁₉	—	8	4	no	—
112	C₂×D₁₉	—	8	2	no	—
113	C₂×D₁₉	—	16	2	no	—
114	C₂×D₁₉	—	16	2	no	—
115	C₄×C₁₉	—	2	2	no	—
116	C₄×C₁₉	—	2	2	no	—
117	C₄×C₁₉	—	8	2	no	—
118	C₄×C₁₉	—	8	2	no	—
119	C₄×C₁₉	#448	1	2	yes	—
120	C₄×C₁₉	#461	1	2	yes	—
121	Dic₁₉	—	16	1	no	—
122	#533	C₂×D₁₉	1	2	yes	—
123	C₂×#127	—	19	1	no	—
124	#501	—	2	2	no	—
125	#501	—	2	2	no	—
126	#492	—	38	2	no	—
127	#486	—	38	2	no	—
128	#460	—	38	4	no	—
129	#450	—	38	4	no	—
130	#431	—	19	2	no	—
131	#427	—	2	2	no	—
132	C₂×#102	—	38	2	no	—
133	C₂×#97	C₂×D₁₉	1	1	yes	characteristic
134	#382	—	19	1	no	—
135	#328	—	38	4	no	—
136	C₂×C₁₉	—	2	1	no	—
137	C₂×C₁₉	—	2	1	no	—
138	C₂×C₁₉	#2008	1	2	yes	—
139	C₂×C₁₉	#2013	1	1	yes	characteristic
140	D₁₉	—	16	2	no	—
141	C₂²×D₄	—	38	2	no	—
142	C₂²×D₄	—	38	1	no	—
143	C₂×SD₁₆	—	38	8	no	—
144	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
145	C₄⋊Q₈	C₂²⋊D₁₉	1	2	yes	—
146	C₄⋊Q₈	C₂²⋊D₁₉	1	2	yes	—
147	#127	—	19	2	no	—
148	#127	—	38	1	no	—
149	#127	—	38	2	no	—
150	#125	—	2	2	no	—
151	C₂²⋊Q₈	—	2	2	no	—
152	C₂²⋊Q₈	—	2	2	no	—
153	C₄×Q₈	—	2	2	no	—
154	C₄×Q₈	—	2	2	no	—
155	#117	—	2	2	no	—
156	C₂×C₄⋊C₄	—	38	1	no	—
157	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
158	#107	—	2	4	no	—
159	#106	—	2	2	no	—
160	#106	—	2	2	no	—
161	C₄⋊C₈	—	38	2	no	—
162	#102	—	38	8	no	—
163	#102	—	76	4	no	—
164	C₂²⋊C₈	—	19	4	no	—
165	#97	—	2	2	no	—
166	C₄×C₈	—	38	2	no	—
167	#95	—	38	2	no	—
168	C₁₉	#10220	1	1	yes	characteristic, complemented, Sylow
169	C₂⁴	—	76	1	no	—
170	C₂×Q₈	—	2	2	no	—
171	C₂×Q₈	—	2	2	no	—
172	C₂×D₄	—	38	4	no	—
173	C₂×D₄	—	38	2	no	—
174	C₂×D₄	—	76	2	no	—
175	C₂×D₄	—	76	2	no	—
176	C₂×D₄	—	76	2	no	—
177	C₂×D₄	—	76	1	no	—
178	C₂×D₄	—	152	4	no	—
179	C₂²×C₄	—	76	1	no	—
180	C₂²×C₄	D₄×D₁₉	1	2	yes	—
181	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
182	SD₁₆	—	152	8	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	38	4	no	—
186	C₂×C₈	#64527	1	4	yes	complemented
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	4	2	no	—
192	C₄⋊C₄	—	4	2	no	—
193	C₄⋊C₄	—	76	2	no	—
194	C₂²⋊C₄	—	4	2	no	—
195	C₄²	—	4	2	no	—
196	C₄²	D₄×D₁₉	1	1	yes	characteristic
197	C₄²	#64543	1	1	yes	characteristic
198	C₄²	C₂×C₂²⋊D₁₉	1	2	yes	—
199	C₂³	—	76	2	no	—
200	C₂³	—	152	4	no	—
201	C₂³	—	152	1	no	—
202	C₂³	—	152	1	no	—
203	C₂³	#10597386	1	1	yes	characteristic
204	Q₈	—	4	4	no	—
205	Q₈	—	8	2	no	—
206	D₄	—	76	8	no	—
207	D₄	—	76	4	no	—
208	D₄	—	152	4	no	—
209	D₄	—	152	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	152	2	no	—
214	C₂×C₄	—	152	1	no	—
215	C₂×C₄	#10597383	1	2	yes	—
216	C₂×C₄	#10597384	1	1	yes	characteristic
217	C₂×C₄	#10597385	1	2	yes	—
218	C₂×C₄	C₂×#64527	1	2	yes	—
219	C₂×C₄	#10597388	1	2	yes	—
220	C₂×C₄	#10597399	1	1	yes	characteristic
221	C₈	—	2	4	no	—
222	C₈	—	76	4	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	304	2	no	—
225	C₂²	—	304	2	no	—
226	C₂²	—	152	4	no	—
227	C₂²	—	152	2	no	—
228	C₂²	#11926452	1	1	yes	characteristic
229	C₂²	#11926918	1	1	yes	characteristic
230	C₂²	#11926936	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	2	2	no	—
233	C₄	—	8	2	no	—
234	C₄	—	8	2	no	—
235	C₄	—	304	1	no	—
236	C₄	#11926683	1	2	yes	—
237	C₄	#11926746	1	2	yes	—
238	C₂	—	2	1	no	—
239	C₂	—	2	1	no	—
240	C₂	—	304	2	no	—
241	C₂	#14564685	1	2	yes	—
242	C₂	#14564718	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14574573

C₂

yes

maximal

#14564459

C₂

yes

maximal, characteristic

#14562199

C₂

yes

maximal, complemented

#14561480

C₂

yes

maximal, characteristic, complemented

C₁₉×#3162

C₂

yes

maximal, characteristic, complemented

C₂×#10597320

C₂²

yes

—

—

—

—

C₂²

yes

—

C₂×#10597278

—

#11926410

C₂²

yes

characteristic

#11926281

—

C₁₉×#533

C₂²

yes

—

C₁₉×#501

—

C₁₉×#501

—

C₁₉×#427

—

C₂×C₁₉×#97

C₂²

yes

characteristic

C₂²×D₇₆

—

C₂²×D₇₆

—

C₂²×C₈×C₁₉

D₄

yes

—

C₁₉×C₄⋊Q₈

D₄

yes

—

C₁₉×C₄⋊Q₈

D₄

yes

—

C₁₉×#125

—

C₁₉×C₂²⋊Q₈

—

C₁₉×C₂²⋊Q₈

—

C₄×Q₈×C₁₉

—

C₄×Q₈×C₁₉

—

C₁₉×#117

—

C₂×C₄²×C₁₉

C₂³

yes

characteristic

C₂×#64527

—

C₂×C₄⋊Dic₁₉

—

#10597320

—

#10597320

—

#10597320

D₄

yes

—

C₁₉×#107

—

C₁₉×#106

—

C₁₉×#106

—

C₁₉×#97

—

C₁₉⋊#95

—

C₂²⋊(C₁₉⋊C₈)

D₄

yes

—

#10597278

—

#10597266

—

C₄⋊(C₁₉⋊C₈)

—

C₄×C₁₉⋊C₈

—

C₂³×D₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂²×C₄×C₁₉

C₂×D₄

yes

—

C₂²×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂²×Dic₁₉

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

SD₁₆

yes

complemented

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₄²×C₁₉

C₂×D₄

yes

—

C₄²×C₁₉

#41

yes

characteristic

#64527

—

C₄⋊Dic₁₉

—

C₂×C₁₉⋊C₈

—

—

C₂³×C₁₉

yes

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂²×D₁₉

—

Q₈×C₁₉

—

Q₈×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

#120

yes

—

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

—

C₂×C₄×C₁₉

C₂×SD₁₆

yes

—

C₂×C₄×C₁₉

#137

yes

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

D₇₆

—

D₇₆

—

D₇₆

—

D₇₆

—

100

C₈×C₁₉

—

101

C₁₉⋊C₈

—

102

#3208

—

103

#3162

D₁₉

yes

characteristic, complemented

104

#1961

—

105

#1518

—

106

#1281

—

107

C₂²×C₁₉

—

108

C₂²×C₁₉

#392

yes

characteristic

109

C₂²×C₁₉

#492

yes

characteristic

110

C₂²×C₁₉

#497

yes

characteristic

111

C₂×D₁₉

—

112

C₂×D₁₉

—

113

C₂×D₁₉

—

114

C₂×D₁₉

—

115

C₄×C₁₉

—

116

C₄×C₁₉

—

117

C₄×C₁₉

—

118

C₄×C₁₉

—

119

C₄×C₁₉

#448

yes

—

120

C₄×C₁₉

#461

yes

—

121

Dic₁₉

—

122

#533

C₂×D₁₉

yes

—

123

C₂×#127

—

124

#501

—

125

#501

—

126

#492

—

127

#486

—

128

#460

—

129

#450

—

130

#431

—

131

#427

—

132

C₂×#102

—

133

C₂×#97

C₂×D₁₉

yes

characteristic

134

#382

—

135

#328

—

136

C₂×C₁₉

—

137

C₂×C₁₉

—

138

C₂×C₁₉

#2008

yes

—

139

C₂×C₁₉

#2013

yes

characteristic

140

D₁₉

—

141

C₂²×D₄

—

142

C₂²×D₄

—

143

C₂×SD₁₆

—

144

C₂²×C₈

C₂²⋊D₁₉

yes

—

145

C₄⋊Q₈

C₂²⋊D₁₉

yes

—

146

C₄⋊Q₈

C₂²⋊D₁₉

yes

—

147

#127

—

148

#127

—

149

#127

—

150

#125

—

151

C₂²⋊Q₈

—

152

C₂²⋊Q₈

—

153

C₄×Q₈

—

154

C₄×Q₈

—

155

#117

—

156

C₂×C₄⋊C₄

—

157

C₂×C₄²

C₂²×D₁₉

yes

characteristic

158

#107

—

159

#106

—

160

#106

—

161

C₄⋊C₈

—

162

#102

—

163

#102

—

164

C₂²⋊C₈

—

165

#97

—

166

C₄×C₈

—

167

#95

—

168

C₁₉

#10220

yes

characteristic, complemented, Sylow

169

C₂⁴

—

170

C₂×Q₈

—

171

C₂×Q₈

—

172

C₂×D₄

—

173

C₂×D₄

—

174

C₂×D₄

—

175

C₂×D₄

—

176

C₂×D₄

—

177

C₂×D₄

—

178

C₂×D₄

—

152

—

179

C₂²×C₄

—

180

C₂²×C₄

D₄×D₁₉

yes

—

181

C₂²×C₄

C₂×C₂²⋊D₁₉

yes

characteristic

182

SD₁₆

—

152

—

183

C₂×C₈

—

184

C₂×C₈

—

185

C₂×C₈

—

186

C₂×C₈

#64527

yes

complemented

187

C₄⋊C₄

—

188

C₄⋊C₄

—

189

C₄⋊C₄

—

190

C₄⋊C₄

—

191

C₄⋊C₄

—

192

C₄⋊C₄

—

193

C₄⋊C₄

—

194

C₂²⋊C₄

—

195

C₄²

—

196

C₄²

D₄×D₁₉

yes

characteristic

197

C₄²

#64543

yes

characteristic

198

C₄²

C₂×C₂²⋊D₁₉

yes

—

199

C₂³

—

200

C₂³

—

152

—

201

C₂³

—

152

—

202

C₂³

—

152

—

203

C₂³

#10597386

yes

characteristic

204

Q₈

—

205

Q₈

—

206

D₄

—

207

D₄

—

208

D₄

—

152

—

209

D₄

—

152

—

210

C₂×C₄

—

211

C₂×C₄

—

212

C₂×C₄

—

213

C₂×C₄

—

152

—

214

C₂×C₄

—

152

—

215

C₂×C₄

#10597383

yes

—

216

C₂×C₄

#10597384

yes

characteristic

217

C₂×C₄

#10597385

yes

—

218

C₂×C₄

C₂×#64527

yes

—

219

C₂×C₄

#10597388

yes

—

220

C₂×C₄

#10597399

yes

characteristic

221

C₈

—

222

C₈

—

223

C₂²

—

224

C₂²

—

304

—

225

C₂²

—

304

—

226

C₂²

—

152

—

227

C₂²

—

152

—

228

C₂²

#11926452

yes

characteristic

229

C₂²

#11926918

yes

characteristic

230

C₂²

#11926936

yes

characteristic

231

C₄

—

232

C₄

—

233

C₄

—

234

C₄

—

235

C₄

—

304

—

236

C₄

#11926683

yes

—

237

C₄

#11926746

yes

—

238

C₂

—

239

C₂

—

240

C₂

—

304

—

241

C₂

#14564685

yes

—

242

C₂

#14564718

yes

characteristic

	subgroup	G/H
center	C₂²	#11926452
commutator, cosocle	C₂×C₄²×C₁₉	C₂³
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#3162	C₂
socle	C₂²×C₁₉	#392
prime radical	C₂×#10597320	C₂²
hypercenter	#3162	D₁₉
hypocenter	C₁₉	#10220
LCS	G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3162

subgroup

G/H

center

C₂²

#11926452

commutator, cosocle

C₂×C₄²×C₁₉

C₂³

Frattini

C₂×C₄²

C₂²×D₁₉

Fitting

C₁₉×#3162

C₂

C₂²×C₁₉

C₂²

D₁₉

C₁₉

G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3162

N	H ≅ G/N	type
3. #14562199	C₂	semidirect
4. #14561480	C₂	semidirect
5. C₁₉×#3162	C₂	semidirect
62. C₂×C₈×C₁₉	SD₁₆	semidirect
103. #3162	D₁₉	semidirect
168. C₁₉	#10220	semidirect
186. C₂×C₈	#64527	semidirect

H ≅ G/N

type

3. #14562199

C₂

semidirect

4. #14561480

C₂

semidirect

5. C₁₉×#3162

C₂

semidirect

62. C₂×C₈×C₁₉

SD₁₆

semidirect

103. #3162

D₁₉

semidirect

168. C₁₉

#10220

semidirect

186. C₂×C₈

#64527

semidirect

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

subgroup	index
#14574573	2
#14564459	2
#14562199	2
#14561480	2
C₁₉×#3162	2
#10220	19

subgroup

index

C₁₉×#3162

Subgroups — #18084786

Conjugacy classes of subgroups

Complements

Maximal subgroups