Group #18081998 — subgroups

Subgroups — #18081998

Order 4864.

414 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 155 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	1	yes	maximal, characteristic
2	#14563807	C₂	1	1	yes	maximal, characteristic
3	#14562319	C₂	1	2	yes	maximal
4	#14562315	C₂	1	2	yes	maximal
5	C₁₉×#1450	C₂	1	1	yes	maximal, characteristic
6	C₂×#10597315	C₂²	1	2	yes	—
7	#11926949	—	2	1	no	—
8	#11926949	—	2	1	no	—
9	#11926949	C₄	1	4	yes	—
10	C₂×#10597276	—	2	2	no	—
11	C₂×#10597275	—	2	2	no	—
12	#11926568	C₂²	1	2	yes	—
13	#11926292	—	2	4	no	—
14	#11926291	—	2	4	no	—
15	C₁₉×#431	C₂²	1	1	yes	characteristic
16	C₁₉×#431	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
18	C₁₉×#322	—	2	1	no	—
19	C₈²×C₁₉	—	2	1	no	—
20	C₂²×Dic₃₈	—	2	2	no	—
21	C₂²×C₈×C₁₉	D₄	1	2	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×#64520	—	2	2	no	—
26	#10597315	—	2	2	no	—
27	#10597315	—	2	2	no	—
28	#10597315	—	2	2	no	—
29	#10597315	C₂×C₄	1	4	yes	—
30	C₁₉×C₄⋊C₈	D₄	1	2	yes	—
31	C₁₉×C₄⋊C₈	D₄	1	2	yes	—
32	C₁₉×C₂²⋊C₈	—	2	1	no	—
33	C₁₉×C₂²⋊C₈	—	2	1	no	—
34	C₄×C₈×C₁₉	—	2	1	no	—
35	C₄×C₈×C₁₉	—	2	1	no	—
36	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
38	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
39	#10597276	—	4	4	no	—
40	#10597275	—	2	4	no	—
41	#10597275	—	2	4	no	—
42	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
43	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
44	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
46	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
47	C₂²×Dic₁₉	—	4	2	no	—
48	C₂×Dic₃₈	—	2	4	no	—
49	C₂×Dic₃₈	—	4	2	no	—
50	C₂×Dic₃₈	—	8	4	no	—
51	C₂×C₈×C₁₉	—	2	2	no	—
52	C₂×C₈×C₁₉	—	2	2	no	—
53	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
55	C₂×C₈×C₁₉	SD₁₆	1	4	yes	—
56	C₂×C₈×C₁₉	Q₁₆	1	4	yes	—
57	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
61	C₄⋊Dic₁₉	—	2	4	no	—
62	C₄⋊Dic₁₉	—	2	4	no	—
63	C₄⋊Dic₁₉	—	4	2	no	—
64	C₄⋊Dic₁₉	—	4	2	no	—
65	#64520	—	4	4	no	—
66	C₂×C₁₉⋊C₈	—	4	2	no	—
67	#9771	—	19	1	no	maximal, Sylow
68	C₂³×C₁₉	#124	1	1	yes	characteristic
69	C₂×C₄×C₁₉	—	2	2	no	—
70	C₂×C₄×C₁₉	—	2	2	no	—
71	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
73	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
74	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
75	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
76	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
77	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
78	C₂×C₄×C₁₉	C₂×Q₁₆	1	2	yes	—
79	C₂×Dic₁₉	—	4	4	no	—
80	C₂×Dic₁₉	—	8	4	no	—
81	Dic₃₈	—	8	4	no	—
82	Dic₃₈	—	8	4	no	—
83	C₈×C₁₉	—	4	2	no	—
84	C₈×C₁₉	—	4	2	no	—
85	C₈×C₁₉	#103	1	4	yes	—
86	C₈×C₁₉	#103	1	4	yes	—
87	C₁₉⋊C₈	—	8	2	no	—
88	C₂×#500	—	19	1	no	—
89	#1848	—	19	1	no	—
90	#1549	—	19	2	no	—
91	#1548	—	19	2	no	—
92	#1450	D₁₉	1	1	yes	characteristic
93	C₂²×C₁₉	—	2	1	no	—
94	C₂²×C₁₉	—	2	1	no	—
95	C₂²×C₁₉	#390	1	1	yes	characteristic
96	C₂²×C₁₉	#487	1	2	yes	—
97	C₄×C₁₉	—	2	1	no	—
98	C₄×C₁₉	—	2	1	no	—
99	C₄×C₁₉	C₂×#103	1	2	yes	—
100	C₄×C₁₉	#472	1	2	yes	—
101	C₄×C₁₉	#473	1	2	yes	—
102	C₄×C₁₉	#492	1	1	yes	characteristic
103	C₄×C₁₉	#494	1	1	yes	characteristic
104	Dic₁₉	—	8	4	no	—
105	C₂×C₄⋊Q₈	—	19	2	no	—
106	#500	—	19	4	no	—
107	#500	—	38	1	no	—
108	#500	—	38	1	no	—
109	#431	C₂×D₁₉	1	1	yes	characteristic
110	#431	C₂×D₁₉	1	1	yes	characteristic
111	C₂×#108	—	38	2	no	—
112	C₂×#107	—	38	2	no	—
113	#423	—	19	2	no	—
114	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
115	#333	—	38	4	no	—
116	#332	—	38	4	no	—
117	#322	—	2	1	no	—
118	C₈²	—	2	1	no	—
119	C₂×C₁₉	—	2	2	no	—
120	C₂×C₁₉	#1956	1	1	yes	characteristic
121	C₂×C₁₉	#1978	1	1	yes	characteristic
122	C₂×C₁₉	#1979	1	1	yes	characteristic
123	C₂²×Q₈	—	38	2	no	—
124	C₂×M₁₆	—	38	2	no	—
125	C₂²×C₈	D₇₆	1	2	yes	—
126	C₄⋊Q₈	—	19	4	no	—
127	C₄⋊Q₈	—	38	2	no	—
128	C₄⋊Q₈	—	38	2	no	—
129	C₄⋊Q₈	—	38	2	no	—
130	C₂×C₄⋊C₄	—	38	2	no	—
131	C₂×C₄⋊C₄	—	38	2	no	—
132	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
133	#108	—	76	4	no	—
134	#107	—	38	4	no	—
135	#107	—	38	4	no	—
136	C₄⋊C₈	—	38	2	no	—
137	C₄⋊C₈	—	38	2	no	—
138	C₄⋊C₈	D₇₆	1	2	yes	—
139	C₄⋊C₈	D₇₆	1	2	yes	—
140	C₂²⋊C₈	—	2	1	no	—
141	C₂²⋊C₈	—	2	1	no	—
142	C₄×C₈	—	2	1	no	—
143	C₄×C₈	—	2	1	no	—
144	C₄×C₈	C₄×D₁₉	1	2	yes	—
145	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
146	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
147	C₁₉	#9771	1	1	yes	characteristic, complemented, Sylow
148	C₂×Q₈	—	38	4	no	—
149	C₂×Q₈	—	76	2	no	—
150	C₂×Q₈	—	152	4	no	—
151	C₂²×C₄	—	76	2	no	—
152	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
153	C₂²×C₄	#64541	1	1	yes	characteristic
154	C₂²×C₄	#64541	1	1	yes	characteristic
155	M₁₆	—	76	4	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	76	2	no	—
159	C₂×C₈	#64516	1	4	yes	—
160	C₂×C₈	Dic₇₆	1	4	yes	—
161	C₂×C₈	(C₂×D₁₉)⋊C₄	1	2	yes	—
162	C₂×C₈	(C₂×D₁₉)⋊C₄	1	2	yes	—
163	C₄⋊C₄	—	38	4	no	—
164	C₄⋊C₄	—	38	4	no	—
165	C₄⋊C₄	—	76	2	no	—
166	C₄⋊C₄	—	76	2	no	—
167	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
168	C₄²	C₂×D₇₆	1	1	yes	characteristic
169	C₄²	C₂×D₇₆	1	1	yes	characteristic
170	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
171	C₂³	#10597321	1	1	yes	characteristic
172	Q₈	—	152	4	no	—
173	Q₈	—	152	4	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	76	4	no	—
177	C₂×C₄	—	152	4	no	—
178	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
179	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
180	C₂×C₄	#10597320	1	1	yes	characteristic
181	C₂×C₄	C₂×#64516	1	2	yes	—
182	C₂×C₄	C₂×Dic₇₆	1	2	yes	—
183	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
184	C₂×C₄	#10597376	1	1	yes	characteristic
185	C₂×C₄	#10597376	1	1	yes	characteristic
186	C₈	—	4	2	no	—
187	C₈	—	4	2	no	—
188	C₈	—	152	2	no	—
189	C₈	#10597273	1	4	yes	—
190	C₈	#10597273	1	4	yes	—
191	C₂²	—	2	1	no	—
192	C₂²	—	2	1	no	—
193	C₂²	#11926446	1	1	yes	characteristic
194	C₂²	#11926895	1	2	yes	—
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	152	4	no	—
198	C₄	C₂×#10597273	1	2	yes	—
199	C₄	#11926814	1	2	yes	—
200	C₄	#11926819	1	2	yes	—
201	C₄	#11926917	1	1	yes	characteristic
202	C₄	#11926924	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	#14564437	1	1	yes	characteristic
205	C₂	#14564545	1	1	yes	characteristic
206	C₂	#14564550	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11926949

C₂

yes

maximal, characteristic

#14563807

C₂

yes

maximal, characteristic

#14562319

C₂

yes

maximal

#14562315

C₂

yes

maximal

C₁₉×#1450

C₂

yes

maximal, characteristic

C₂×#10597315

C₂²

yes

—

#11926949

—

#11926949

—

#11926949

C₄

yes

—

C₂×#10597276

—

C₂×#10597275

—

#11926568

C₂²

yes

—

#11926292

—

#11926291

—

C₁₉×#431

C₂²

yes

characteristic

C₁₉×#431

C₂²

yes

characteristic

C₂×C₄×C₈×C₁₉

C₂²

yes

characteristic

C₁₉×#322

—

C₈²×C₁₉

—

C₂²×Dic₃₈

—

C₂²×C₈×C₁₉

D₄

yes

—

C₂×C₄²×C₁₉

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₉

—

C₂×C₄⋊Dic₁₉

—

—

—

—

—

C₂×C₄

yes

—

C₁₉×C₄⋊C₈

D₄

yes

—

C₁₉×C₄⋊C₈

D₄

yes

—

C₁₉×C₂²⋊C₈

—

C₁₉×C₂²⋊C₈

—

C₄×C₈×C₁₉

—

C₄×C₈×C₁₉

—

C₄×C₈×C₁₉

C₂×C₄

yes

—

C₄×C₈×C₁₉

D₄

yes

characteristic

C₄×C₈×C₁₉

D₄

yes

—

—

—

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

—

C₂²×C₄×C₁₉

C₂×D₄

yes

C₂²×C₄×C₁₉

yes

C₂²×C₄×C₁₉

yes

C₂²×Dic₁₉

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₂²⋊C₄

yes

—

C₂×C₈×C₁₉

C₂²⋊C₄

yes

—

C₂×C₈×C₁₉

SD₁₆

yes

—

C₂×C₈×C₁₉

Q₁₆

yes

—

C₄²×C₁₉

C₂²×C₄

yes

characteristic

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

#64520

—

C₂×C₁₉⋊C₈

—

—

C₂³×C₁₉

yes

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₉

C₄×D₄

yes

characteristic

C₂×C₄×C₁₉

C₄×D₄

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

C₂×SD₁₆

yes

—

C₂×C₄×C₁₉

C₂×Q₁₆

yes

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

Dic₃₈

—

Dic₃₈

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

#103

yes

—

C₈×C₁₉

#103

yes

—

C₁₉⋊C₈

—

—

—

—

—

D₁₉

yes

characteristic

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#390

yes

characteristic

C₂²×C₁₉

#487

yes

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

C₂×#103

yes

—

100

C₄×C₁₉

#472

yes

—

101

C₄×C₁₉

#473

yes

—

102

C₄×C₁₉

#492

yes

characteristic

103

C₄×C₁₉

#494

yes

characteristic

104

Dic₁₉

—

105

C₂×C₄⋊Q₈

—

106

#500

—

107

#500

—

108

#500

—

109

#431

C₂×D₁₉

yes

characteristic

110

#431

C₂×D₁₉

yes

characteristic

111

C₂×#108

—

112

C₂×#107

—

113

#423

—

114

C₂×C₄×C₈

C₂×D₁₉

yes

characteristic

115

#333

—

116

#332

—

117

#322

—

118

C₈²

—

119

C₂×C₁₉

—

120

C₂×C₁₉

#1956

yes

characteristic

121

C₂×C₁₉

#1978

yes

characteristic

122

C₂×C₁₉

#1979

yes

characteristic

123

C₂²×Q₈

—

124

C₂×M₁₆

—

125

C₂²×C₈

D₇₆

yes

—

126

C₄⋊Q₈

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

—

129

C₄⋊Q₈

—

130

C₂×C₄⋊C₄

—

131

C₂×C₄⋊C₄

—

132

C₂×C₄²

C₂²×D₁₉

yes

characteristic

133

#108

—

134

#107

—

135

#107

—

136

C₄⋊C₈

—

137

C₄⋊C₈

—

138

C₄⋊C₈

D₇₆

yes

—

139

C₄⋊C₈

D₇₆

yes

—

140

C₂²⋊C₈

—

141

C₂²⋊C₈

—

142

C₄×C₈

—

143

C₄×C₈

—

144

C₄×C₈

C₄×D₁₉

yes

—

145

C₄×C₈

C₂²⋊D₁₉

yes

characteristic

146

C₄×C₈

C₂²⋊D₁₉

yes

characteristic

147

C₁₉

#9771

yes

characteristic, complemented, Sylow

148

C₂×Q₈

—

149

C₂×Q₈

—

150

C₂×Q₈

—

152

—

151

C₂²×C₄

—

152

C₂²×C₄

C₂×D₇₆

yes

characteristic

153

C₂²×C₄

#64541

yes

characteristic

154

C₂²×C₄

#64541

yes

characteristic

155

M₁₆

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

#64516

yes

—

160

C₂×C₈

Dic₇₆

yes

—

161

C₂×C₈

(C₂×D₁₉)⋊C₄

yes

—

162

C₂×C₈

(C₂×D₁₉)⋊C₄

yes

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

167

C₄²

C₂×C₄×D₁₉

yes

characteristic

168

C₄²

C₂×D₇₆

yes

characteristic

169

C₄²

C₂×D₇₆

yes

characteristic

170

C₄²

C₂×C₂²⋊D₁₉

yes

characteristic

171

C₂³

#10597321

yes

characteristic

172

Q₈

—

152

—

173

Q₈

—

152

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

152

—

178

C₂×C₄

C₄×D₇₆

yes

characteristic

179

C₂×C₄

C₄×D₇₆

yes

characteristic

180

C₂×C₄

#10597320

yes

characteristic

181

C₂×C₄

C₂×#64516

yes

—

182

C₂×C₄

C₂×Dic₇₆

yes

—

183

C₂×C₄

C₂×(C₂×D₁₉)⋊C₄

yes

characteristic

184

C₂×C₄

#10597376

yes

characteristic

185

C₂×C₄

#10597376

yes

characteristic

186

C₈

—

187

C₈

—

188

C₈

—

152

—

189

C₈

#10597273

yes

—

190

C₈

#10597273

yes

—

191

C₂²

—

192

C₂²

—

193

C₂²

#11926446

yes

characteristic

194

C₂²

#11926895

yes

—

195

C₄

—

196

C₄

—

197

C₄

—

152

—

198

C₄

C₂×#10597273

yes

—

199

C₄

#11926814

yes

—

200

C₄

#11926819

yes

—

201

C₄

#11926917

yes

characteristic

202

C₄

#11926924

yes

characteristic

203

C₂

—

204

C₂

#14564437

yes

characteristic

205

C₂

#14564545

yes

characteristic

206

C₂

#14564550

yes

characteristic

	subgroup	G/H
center	C₂²	#11926446
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1450	C₂
socle	C₂²×C₁₉	#390
prime radical	C₂³×C₁₉	#124
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1450	D₁₉
hypocenter	C₁₉	#9771
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

subgroup

G/H

center

C₂²

#11926446

commutator

C₄²×C₁₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₉

Fitting

C₁₉×#1450

C₂

C₂²×C₁₉

C₂³×C₁₉

C₂×C₄²×C₁₉

C₂³

D₁₉

C₁₉

G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

N	H ≅ G/N	type
147. C₁₉	#9771	semidirect

H ≅ G/N

type

147. C₁₉

#9771

semidirect

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

subgroup	index
C₂×#11926949	2
#14563807	2
#14562319	2
#14562315	2
C₁₉×#1450	2
#9771	19

subgroup

index

C₁₉×#1450

Subgroups — #18081998

Conjugacy classes of subgroups

Complements

Maximal subgroups