Group #18085337 — subgroups

Subgroups — #18085337

Order 4864.

618 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 169 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	C₂×#11926949	C₂	1	2	yes	maximal
2	C₂×#11926924	C₂	1	1	yes	maximal, characteristic
3	#14564596	C₂	1	2	yes	maximal
4	#14564437	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1273	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₇₆	—	2	2	no	—
7	C₂×#10597315	C₂²	1	1	yes	characteristic
8	C₂×#10597315	C₂²	1	2	yes	—
9	#11926949	—	2	4	no	—
10	#11926949	—	2	4	no	—
11	#11926924	—	2	4	no	—
12	C₂×#10597275	—	2	4	no	—
13	C₂×#10597273	—	2	2	no	—
14	#11926414	C₂²	1	1	yes	characteristic
15	#11926307	—	2	4	no	—
16	C₂×C₄×C₈×C₁₉	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
18	C₂²×Dic₃₈	—	2	1	no	—
19	C₂²×Dic₃₈	—	2	2	no	—
20	C₂²×C₈×C₁₉	D₄	1	2	yes	—
21	C₂²×C₈×C₁₉	D₄	1	4	yes	—
22	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
23	C₂×C₄⋊Dic₁₉	—	2	2	no	—
24	C₂×C₄⋊Dic₁₉	—	2	2	no	—
25	C₂×C₄⋊Dic₁₉	—	2	2	no	—
26	C₂×C₄⋊Dic₁₉	—	2	1	no	—
27	C₂×C₄⋊Dic₁₉	—	2	2	no	—
28	C₂×Dic₇₆	—	2	8	no	—
29	C₂×Dic₇₆	—	4	4	no	—
30	C₂×Dic₇₆	—	4	4	no	—
31	#10597315	—	2	2	no	—
32	#10597315	—	2	2	no	—
33	#10597315	—	2	2	no	—
34	#10597315	—	2	2	no	—
35	#10597315	—	2	1	no	—
36	#10597315	D₄	1	4	yes	—
37	#10597315	D₄	1	2	yes	—
38	C₄×C₈×C₁₉	—	2	2	no	—
39	C₄×C₈×C₁₉	—	2	2	no	—
40	C₄×C₈×C₁₉	—	2	2	no	—
41	#10597275	—	2	16	no	—
42	#10597273	—	4	4	no	—
43	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
45	C₂²×Dic₁₉	—	4	1	no	—
46	C₂²×Dic₁₉	—	4	2	no	—
47	C₂²×Dic₁₉	—	4	1	no	—
48	C₂×Dic₃₈	—	2	4	no	—
49	C₂×Dic₃₈	—	2	2	no	—
50	C₂×Dic₃₈	—	4	2	no	—
51	C₂×Dic₃₈	—	4	4	no	—
52	C₂×Dic₃₈	—	4	2	no	—
53	C₂×Dic₃₈	—	4	4	no	—
54	C₂×Dic₃₈	—	4	1	no	—
55	C₂×Dic₃₈	—	4	2	no	—
56	C₂×C₈×C₁₉	—	2	4	no	—
57	C₂×C₈×C₁₉	—	2	2	no	—
58	C₂×C₈×C₁₉	—	2	2	no	—
59	C₂×C₈×C₁₉	—	2	4	no	—
60	C₂×C₈×C₁₉	D₈	1	4	yes	—
61	C₂×C₈×C₁₉	Q₁₆	1	8	yes	complemented
62	C₄²×C₁₉	C₂×D₄	1	2	yes	—
63	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₁₉	#41	1	1	yes	characteristic
65	C₄⋊Dic₁₉	—	2	4	no	—
66	C₄⋊Dic₁₉	—	2	4	no	—
67	C₄⋊Dic₁₉	—	2	4	no	—
68	C₄⋊Dic₁₉	—	4	2	no	—
69	C₄⋊Dic₁₉	—	4	2	no	—
70	C₄⋊Dic₁₉	—	4	2	no	—
71	C₄⋊Dic₁₉	—	4	2	no	—
72	C₄⋊Dic₁₉	—	4	4	no	—
73	Dic₇₆	—	4	16	no	—
74	#10309	—	19	1	no	maximal, Sylow
75	C₂³×C₁₉	#127	1	1	yes	characteristic
76	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
77	C₂×C₄×C₁₉	#120	1	2	yes	—
78	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
79	C₂×C₄×C₁₉	#121	1	2	yes	—
80	C₂×C₄×C₁₉	C₂×D₈	1	2	yes	—
81	C₂×C₄×C₁₉	C₂×Q₁₆	1	4	yes	—
82	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
83	C₂×C₄×C₁₉	#137	1	4	yes	—
84	C₂×Dic₁₉	—	4	2	no	—
85	C₂×Dic₁₉	—	4	4	no	—
86	C₂×Dic₁₉	—	8	1	no	—
87	C₂×Dic₁₉	—	8	2	no	—
88	C₂×Dic₁₉	—	8	1	no	—
89	C₂×Dic₁₉	—	8	2	no	—
90	C₂×Dic₁₉	—	8	1	no	—
91	C₂×Dic₁₉	—	8	2	no	—
92	Dic₃₈	—	4	8	no	—
93	Dic₃₈	—	4	4	no	—
94	Dic₃₈	—	8	2	no	—
95	Dic₃₈	—	8	4	no	—
96	C₈×C₁₉	—	2	4	no	—
97	C₈×C₁₉	—	2	8	no	—
98	C₂×#500	—	19	2	no	—
99	C₂×#494	—	19	1	no	—
100	#1990	—	19	2	no	—
101	#1956	—	19	1	no	—
102	#1273	D₁₉	1	1	yes	characteristic
103	C₂²×C₁₉	#392	1	1	yes	characteristic
104	C₂²×C₁₉	#493	1	1	yes	characteristic
105	C₂²×C₁₉	#494	1	2	yes	—
106	C₂²×C₁₉	#496	1	1	yes	characteristic
107	C₂²×C₁₉	#497	1	2	yes	—
108	C₄×C₁₉	#447	1	2	yes	—
109	C₄×C₁₉	#451	1	4	yes	—
110	C₄×C₁₉	#459	1	2	yes	—
111	C₄×C₁₉	#462	1	4	yes	—
112	Dic₁₉	—	8	4	no	—
113	Dic₁₉	—	8	2	no	—
114	Dic₁₉	—	16	1	no	—
115	C₂²×Q₁₆	—	38	2	no	—
116	C₂×C₄⋊Q₈	—	19	2	no	—
117	C₂×C₄⋊Q₈	—	19	1	no	—
118	#500	—	38	4	no	—
119	#500	—	38	4	no	—
120	#494	—	38	4	no	—
121	C₂×#107	—	38	4	no	—
122	C₂×#103	—	38	2	no	—
123	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
124	C₂×C₄×C₈	C₂×D₁₉	1	2	yes	—
125	#384	—	19	1	no	—
126	#340	—	38	4	no	—
127	C₂×C₁₉	#1713	1	1	yes	characteristic
128	C₂×C₁₉	#1722	1	1	yes	characteristic
129	C₂×C₁₉	#1723	1	2	yes	—
130	C₂×C₁₉	#2011	1	1	yes	characteristic
131	C₂×C₁₉	#2016	1	2	yes	—
132	C₂²×Q₈	—	38	2	no	—
133	C₂²×Q₈	—	38	1	no	—
134	C₂×Q₁₆	—	38	8	no	—
135	C₂×Q₁₆	—	76	4	no	—
136	C₂×Q₁₆	—	76	4	no	—
137	C₂²×C₈	D₇₆	1	2	yes	—
138	C₂²×C₈	D₇₆	1	4	yes	—
139	C₄⋊Q₈	—	19	4	no	—
140	C₄⋊Q₈	—	19	2	no	—
141	C₄⋊Q₈	—	38	2	no	—
142	C₄⋊Q₈	—	38	2	no	—
143	C₄⋊Q₈	—	38	2	no	—
144	C₄⋊Q₈	—	38	2	no	—
145	C₄⋊Q₈	—	38	1	no	—
146	C₂×C₄⋊C₄	—	38	2	no	—
147	C₂×C₄⋊C₄	—	38	2	no	—
148	C₂×C₄⋊C₄	—	38	2	no	—
149	C₂×C₄⋊C₄	—	38	2	no	—
150	C₂×C₄⋊C₄	—	38	1	no	—
151	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
152	#107	—	38	16	no	—
153	#103	—	76	4	no	—
154	C₄×C₈	—	2	2	no	—
155	C₄×C₈	—	2	2	no	—
156	C₄×C₈	—	2	2	no	—
157	C₁₉	#10309	1	1	yes	characteristic, complemented, Sylow
158	C₂×Q₈	—	38	4	no	—
159	C₂×Q₈	—	38	2	no	—
160	C₂×Q₈	—	76	1	no	—
161	C₂×Q₈	—	76	2	no	—
162	C₂×Q₈	—	76	2	no	—
163	C₂×Q₈	—	76	4	no	—
164	C₂×Q₈	—	76	4	no	—
165	C₂×Q₈	—	76	2	no	—
166	C₂²×C₄	—	76	2	no	—
167	C₂²×C₄	—	76	1	no	—
168	C₂²×C₄	—	76	1	no	—
169	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
170	C₂²×C₄	C₂×D₇₆	1	2	yes	—
171	Q₁₆	—	76	16	no	—
172	C₂×C₈	—	2	4	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	2	4	no	—
176	C₂×C₈	D₁₅₂	1	4	yes	—
177	C₂×C₈	Dic₇₆	1	8	yes	complemented
178	C₄⋊C₄	—	38	4	no	—
179	C₄⋊C₄	—	38	4	no	—
180	C₄⋊C₄	—	38	4	no	—
181	C₄⋊C₄	—	76	2	no	—
182	C₄⋊C₄	—	76	2	no	—
183	C₄⋊C₄	—	76	2	no	—
184	C₄⋊C₄	—	76	2	no	—
185	C₄⋊C₄	—	76	4	no	—
186	C₄²	D₄×D₁₉	1	2	yes	—
187	C₄²	D₄×D₁₉	1	1	yes	characteristic
188	C₄²	#64545	1	1	yes	characteristic
189	C₂³	#10597320	1	1	yes	characteristic
190	Q₈	—	76	8	no	—
191	Q₈	—	76	4	no	—
192	Q₈	—	152	2	no	—
193	Q₈	—	152	4	no	—
194	C₂×C₄	—	76	2	no	—
195	C₂×C₄	—	76	4	no	—
196	C₂×C₄	—	152	1	no	—
197	C₂×C₄	—	152	2	no	—
198	C₂×C₄	—	152	1	no	—
199	C₂×C₄	—	152	2	no	—
200	C₂×C₄	—	152	2	no	—
201	C₂×C₄	—	152	1	no	—
202	C₂×C₄	#10597328	1	1	yes	characteristic
203	C₂×C₄	#10597328	1	2	yes	—
204	C₂×C₄	#10597341	1	1	yes	characteristic
205	C₂×C₄	#10597341	1	2	yes	—
206	C₂×C₄	C₂×D₁₅₂	1	2	yes	—
207	C₂×C₄	C₂×Dic₇₆	1	4	yes	—
208	C₂×C₄	#10597354	1	2	yes	—
209	C₂×C₄	#10597355	1	4	yes	—
210	C₈	—	2	4	no	—
211	C₈	—	2	8	no	—
212	C₂²	#11926454	1	1	yes	characteristic
213	C₂²	#11926921	1	1	yes	characteristic
214	C₂²	#11926924	1	2	yes	—
215	C₂²	#11926933	1	1	yes	characteristic
216	C₂²	#11926938	1	2	yes	—
217	C₄	—	304	1	no	—
218	C₄	—	152	4	no	—
219	C₄	—	152	2	no	—
220	C₄	#11926680	1	2	yes	—
221	C₄	#11926702	1	4	yes	—
222	C₄	#11926740	1	2	yes	—
223	C₄	#11926755	1	4	yes	—
224	C₂	#14563253	1	1	yes	characteristic
225	C₂	#14563306	1	1	yes	characteristic
226	C₂	#14563312	1	2	yes	—
227	C₂	#14564708	1	1	yes	characteristic
228	C₂	#14564737	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11926949

C₂

yes

maximal

C₂×#11926924

C₂

yes

maximal, characteristic

#14564596

C₂

yes

maximal

#14564437

C₂

yes

maximal, characteristic

C₁₉×#1273

C₂

yes

maximal, characteristic

C₂²×Dic₇₆

—

C₂×#10597315

C₂²

yes

characteristic

C₂×#10597315

C₂²

yes

—

—

—

—

—

—

C₂²

yes

characteristic

#11926307

—

C₂×C₄×C₈×C₁₉

C₂²

yes

—

C₂×C₄×C₈×C₁₉

C₂²

yes

characteristic

C₂²×Dic₃₈

—

C₂²×Dic₃₈

—

C₂²×C₈×C₁₉

D₄

yes

—

C₂²×C₈×C₁₉

D₄

yes

—

C₂×C₄²×C₁₉

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₉

—

C₂×C₄⋊Dic₁₉

—

C₂×C₄⋊Dic₁₉

—

C₂×C₄⋊Dic₁₉

—

C₂×C₄⋊Dic₁₉

—

C₂×Dic₇₆

—

C₂×Dic₇₆

—

C₂×Dic₇₆

—

—

—

—

—

—

D₄

yes

—

#10597315

D₄

yes

—

C₄×C₈×C₁₉

—

C₄×C₈×C₁₉

—

C₄×C₈×C₁₉

—

#10597275

—

#10597273

—

C₂²×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₉

C₂×D₄

yes

—

C₂²×Dic₁₉

—

C₂²×Dic₁₉

—

C₂²×Dic₁₉

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

D₈

yes

—

C₂×C₈×C₁₉

Q₁₆

yes

complemented

C₄²×C₁₉

C₂×D₄

yes

—

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₄²×C₁₉

#41

yes

characteristic

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

Dic₇₆

—

—

C₂³×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

—

C₂×C₄×C₁₉

#121

yes

characteristic

C₂×C₄×C₁₉

#121

yes

—

C₂×C₄×C₁₉

C₂×D₈

yes

—

C₂×C₄×C₁₉

C₂×Q₁₆

yes

—

C₂×C₄×C₁₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₁₉

#137

yes

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

Dic₃₈

—

Dic₃₈

—

Dic₃₈

—

Dic₃₈

—

C₈×C₁₉

—

C₈×C₁₉

—

C₂×#500

—

C₂×#494

—

100

#1990

—

101

#1956

—

102

#1273

D₁₉

yes

characteristic

103

C₂²×C₁₉

#392

yes

characteristic

104

C₂²×C₁₉

#493

yes

characteristic

105

C₂²×C₁₉

#494

yes

—

106

C₂²×C₁₉

#496

yes

characteristic

107

C₂²×C₁₉

#497

yes

—

108

C₄×C₁₉

#447

yes

—

109

C₄×C₁₉

#451

yes

—

110

C₄×C₁₉

#459

yes

—

111

C₄×C₁₉

#462

yes

—

112

Dic₁₉

—

113

Dic₁₉

—

114

Dic₁₉

—

115

C₂²×Q₁₆

—

116

C₂×C₄⋊Q₈

—

117

C₂×C₄⋊Q₈

—

118

#500

—

119

#500

—

120

#494

—

121

C₂×#107

—

122

C₂×#103

—

123

C₂×C₄×C₈

C₂×D₁₉

yes

characteristic

124

C₂×C₄×C₈

C₂×D₁₉

yes

—

125

#384

—

126

#340

—

127

C₂×C₁₉

#1713

yes

characteristic

128

C₂×C₁₉

#1722

yes

characteristic

129

C₂×C₁₉

#1723

yes

—

130

C₂×C₁₉

#2011

yes

characteristic

131

C₂×C₁₉

#2016

yes

—

132

C₂²×Q₈

—

133

C₂²×Q₈

—

134

C₂×Q₁₆

—

135

C₂×Q₁₆

—

136

C₂×Q₁₆

—

137

C₂²×C₈

D₇₆

yes

—

138

C₂²×C₈

D₇₆

yes

—

139

C₄⋊Q₈

—

140

C₄⋊Q₈

—

141

C₄⋊Q₈

—

142

C₄⋊Q₈

—

143

C₄⋊Q₈

—

144

C₄⋊Q₈

—

145

C₄⋊Q₈

—

146

C₂×C₄⋊C₄

—

147

C₂×C₄⋊C₄

—

148

C₂×C₄⋊C₄

—

149

C₂×C₄⋊C₄

—

150

C₂×C₄⋊C₄

—

151

C₂×C₄²

C₂²×D₁₉

yes

characteristic

152

#107

—

153

#103

—

154

C₄×C₈

—

155

C₄×C₈

—

156

C₄×C₈

—

157

C₁₉

#10309

yes

characteristic, complemented, Sylow

158

C₂×Q₈

—

159

C₂×Q₈

—

160

C₂×Q₈

—

161

C₂×Q₈

—

162

C₂×Q₈

—

163

C₂×Q₈

—

164

C₂×Q₈

—

165

C₂×Q₈

—

166

C₂²×C₄

—

167

C₂²×C₄

—

168

C₂²×C₄

—

169

C₂²×C₄

C₂×D₇₆

yes

characteristic

170

C₂²×C₄

C₂×D₇₆

yes

—

171

Q₁₆

—

172

C₂×C₈

—

173

C₂×C₈

—

174

C₂×C₈

—

175

C₂×C₈

—

176

C₂×C₈

D₁₅₂

yes

—

177

C₂×C₈

Dic₇₆

yes

complemented

178

C₄⋊C₄

—

179

C₄⋊C₄

—

180

C₄⋊C₄

—

181

C₄⋊C₄

—

182

C₄⋊C₄

—

183

C₄⋊C₄

—

184

C₄⋊C₄

—

185

C₄⋊C₄

—

186

C₄²

D₄×D₁₉

yes

—

187

C₄²

D₄×D₁₉

yes

characteristic

188

C₄²

#64545

yes

characteristic

189

C₂³

#10597320

yes

characteristic

190

Q₈

—

191

Q₈

—

192

Q₈

—

152

—

193

Q₈

—

152

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

152

—

197

C₂×C₄

—

152

—

198

C₂×C₄

—

152

—

199

C₂×C₄

—

152

—

200

C₂×C₄

—

152

—

201

C₂×C₄

—

152

—

202

C₂×C₄

#10597328

yes

characteristic

203

C₂×C₄

#10597328

yes

—

204

C₂×C₄

#10597341

yes

characteristic

205

C₂×C₄

#10597341

yes

—

206

C₂×C₄

C₂×D₁₅₂

yes

—

207

C₂×C₄

C₂×Dic₇₆

yes

—

208

C₂×C₄

#10597354

yes

—

209

C₂×C₄

#10597355

yes

—

210

C₈

—

211

C₈

—

212

C₂²

#11926454

yes

characteristic

213

C₂²

#11926921

yes

characteristic

214

C₂²

#11926924

yes

—

215

C₂²

#11926933

yes

characteristic

216

C₂²

#11926938

yes

—

217

C₄

—

304

—

218

C₄

—

152

—

219

C₄

—

152

—

220

C₄

#11926680

yes

—

221

C₄

#11926702

yes

—

222

C₄

#11926740

yes

—

223

C₄

#11926755

yes

—

224

C₂

#14563253

yes

characteristic

225

C₂

#14563306

yes

characteristic

226

C₂

#14563312

yes

—

227

C₂

#14564708

yes

characteristic

228

C₂

#14564737

yes

—

	subgroup	G/H
center	C₂³	#10597320
commutator, cosocle	C₂×C₄²×C₁₉	C₂³
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1273	C₂
socle, prime radical	C₂³×C₁₉	#127
hypercenter	#1273	D₁₉
hypocenter	C₁₉	#10309
LCS	G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

subgroup

G/H

center

C₂³

#10597320

commutator, cosocle

C₂×C₄²×C₁₉

C₂³

Frattini

C₂×C₄²

C₂²×D₁₉

Fitting

C₁₉×#1273

C₂

C₂³×C₁₉

D₁₉

C₁₉

G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

N	H ≅ G/N	type
61. C₂×C₈×C₁₉	Q₁₆	semidirect
157. C₁₉	#10309	semidirect
177. C₂×C₈	Dic₇₆	semidirect

H ≅ G/N

type

61. C₂×C₈×C₁₉

Q₁₆

semidirect

157. C₁₉

#10309

semidirect

177. C₂×C₈

Dic₇₆

semidirect

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

subgroup	index
C₂×#11926949	2
C₂×#11926924	2
#14564596	2
#14564437	2
C₁₉×#1273	2
#10309	19

subgroup

index

C₁₉×#1273

Subgroups — #18085337

Conjugacy classes of subgroups

Complements

Maximal subgroups