Group #18075322 — subgroups

Subgroups — #18075322

Order 4864.

456 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 121 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	#125×Dic₁₉	C₂	1	1	yes	maximal, characteristic
2	#14563489	C₂	1	2	yes	maximal
3	C₂×#11926286	C₂	1	1	yes	maximal, characteristic
4	#14561517	C₂	1	2	yes	maximal
5	C₂×C₁₉×#330	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₉⋊C₈	—	2	2	no	—
7	C₂×#10597260	C₂²	1	2	yes	—
8	#11926443	C₄	1	2	yes	—
9	#11926430	—	2	2	no	—
10	#11926410	C₂²	1	1	yes	characteristic
11	C₄⋊C₄×Dic₁₉	—	2	2	no	—
12	C₄⋊C₄×Dic₁₉	C₄	1	2	yes	—
13	C₄²×Dic₁₉	C₂²	1	1	yes	characteristic
14	#11926298	—	2	2	no	—
15	#11926286	—	2	2	no	—
16	C₂×C₁₉×#125	C₂²	1	1	yes	characteristic
17	C₂×C₁₉×#97	C₂²	1	2	yes	—
18	C₁₉×#330	—	2	2	no	—
19	C₂²×C₈×C₁₉	—	2	2	no	—
20	C₁₉×#125	—	2	2	no	—
21	C₁₉×#125	D₄	1	2	yes	—
22	C₁₉×#125	Q₈	1	2	yes	—
23	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
24	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
26	C₂×C₄⋊Dic₁₉	—	2	2	no	—
27	C₂×C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
28	C₂×C₄×Dic₁₉	—	2	2	no	—
29	C₂×C₄×Dic₁₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×Dic₁₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₁₉	D₄	1	2	yes	—
32	C₂×C₄×Dic₁₉	D₄	1	2	yes	—
33	C₂²×C₁₉⋊C₈	—	2	2	no	—
34	C₁₉×#97	—	2	2	no	—
35	C₁₉×#97	—	2	2	no	—
36	C₁₉⋊#95	—	2	2	no	—
37	C₁₉⋊#95	—	2	2	no	—
38	C₁₉⋊#95	—	2	2	no	—
39	Dic₁₉⋊C₈	—	2	8	no	—
40	#10597260	—	2	2	no	—
41	#10597260	—	2	2	no	—
42	C₂²×C₄×C₁₉	—	2	2	no	—
43	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
45	C₂²×Dic₁₉	—	2	2	no	—
46	C₂²×Dic₁₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×Dic₁₉	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×Dic₁₉	C₂²⋊C₄	1	2	yes	—
49	C₂×C₈×C₁₉	—	2	4	no	—
50	C₂×C₈×C₁₉	—	4	2	no	—
51	C₂×C₈×C₁₉	—	4	2	no	—
52	C₄⋊C₄×C₁₉	—	2	2	no	—
53	C₄⋊C₄×C₁₉	—	2	4	no	—
54	C₄⋊C₄×C₁₉	—	2	4	no	—
55	C₄⋊C₄×C₁₉	C₄⋊C₄	1	4	yes	—
56	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
58	C₄²×C₁₉	#41	1	2	yes	—
59	C₄⋊Dic₁₉	—	4	4	no	—
60	C₄⋊Dic₁₉	—	4	2	no	—
61	C₄×Dic₁₉	—	2	2	no	—
62	C₄×Dic₁₉	—	2	4	no	—
63	C₄×Dic₁₉	—	4	1	no	—
64	C₄×Dic₁₉	—	4	2	no	—
65	C₄×Dic₁₉	—	4	4	no	—
66	C₂×C₁₉⋊C₈	—	2	4	no	—
67	C₂×C₁₉⋊C₈	—	4	2	no	—
68	C₂×C₁₉⋊C₈	—	4	2	no	—
69	#8584	—	19	1	no	maximal, Sylow
70	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₉	—	2	1	no	—
72	C₂×C₄×C₁₉	—	2	1	no	—
73	C₂×C₄×C₁₉	—	2	2	no	—
74	C₂×C₄×C₁₉	—	2	2	no	—
75	C₂×C₄×C₁₉	—	2	4	no	—
76	C₂×C₄×C₁₉	—	4	2	no	—
77	C₂×C₄×C₁₉	—	4	2	no	—
78	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
80	C₂×C₄×C₁₉	#121	1	2	yes	—
81	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
82	C₂×Dic₁₉	—	2	2	no	—
83	C₂×Dic₁₉	—	2	2	no	—
84	C₂×Dic₁₉	—	2	2	no	—
85	C₂×Dic₁₉	—	2	1	no	—
86	C₂×Dic₁₉	—	2	2	no	—
87	C₂×Dic₁₉	—	4	2	no	—
88	C₂×Dic₁₉	—	4	2	no	—
89	C₂×Dic₁₉	—	4	2	no	—
90	C₂×Dic₁₉	#101	1	2	yes	—
91	C₂×Dic₁₉	#104	1	4	yes	—
92	C₈×C₁₉	—	4	4	no	—
93	C₁₉⋊C₈	—	4	4	no	—
94	C₄×#125	—	19	1	no	—
95	#1767	—	19	2	no	—
96	C₂×#330	—	19	1	no	—
97	C₂×#330	D₁₉	1	1	yes	characteristic
98	#1295	—	19	2	no	—
99	C₂²×C₁₉	#380	1	1	yes	characteristic
100	C₂²×C₁₉	C₂×#101	1	1	yes	characteristic
101	C₂²×C₁₉	#413	1	1	yes	characteristic
102	C₂²×C₁₉	C₂×#104	1	2	yes	—
103	C₂²×C₁₉	#421	1	2	yes	—
104	C₄×C₁₉	—	2	1	no	—
105	C₄×C₁₉	—	2	1	no	—
106	C₄×C₁₉	—	2	2	no	—
107	C₄×C₁₉	—	2	2	no	—
108	C₄×C₁₉	—	4	4	no	—
109	Dic₁₉	—	2	4	no	—
110	Dic₁₉	—	2	2	no	—
111	Dic₁₉	—	4	1	no	—
112	Dic₁₉	—	8	2	no	—
113	C₂×#125	C₂×D₁₉	1	1	yes	characteristic
114	C₂×C₄⋊C₈	—	38	2	no	—
115	C₂×#97	—	19	2	no	—
116	C₂×#97	C₂×D₁₉	1	2	yes	—
117	#389	—	19	2	no	—
118	#387	—	38	2	no	—
119	#382	—	19	1	no	—
120	C₄×C₄⋊C₄	—	19	2	no	—
121	C₄×C₄⋊C₄	—	38	2	no	—
122	C₄³	—	19	1	no	—
123	#336	—	38	2	no	—
124	#330	—	2	2	no	—
125	#330	—	38	2	no	—
126	C₂×C₁₉	#1526	1	1	yes	characteristic
127	C₂×C₁₉	#1532	1	2	yes	—
128	C₂×C₁₉	#1534	1	1	yes	characteristic
129	C₂×C₁₉	#1801	1	1	yes	characteristic
130	C₂×C₁₉	#1806	1	2	yes	—
131	C₂²×C₈	—	2	2	no	—
132	C₂²×C₈	—	38	2	no	—
133	#125	—	2	2	no	—
134	#125	Dic₃₈	1	2	yes	—
135	#125	C₂²⋊D₁₉	1	2	yes	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	19	2	no	—
138	C₂×C₄⋊C₄	—	38	2	no	—
139	C₂×C₄⋊C₄	C₄×D₁₉	1	2	yes	—
140	C₂×C₄²	—	19	1	no	—
141	C₂×C₄²	—	19	2	no	—
142	C₂×C₄²	—	19	1	no	—
143	C₂×C₄²	—	19	2	no	—
144	C₂×C₄²	—	38	2	no	—
145	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
146	C₄⋊C₈	—	38	8	no	—
147	#97	—	2	2	no	—
148	#97	—	2	2	no	—
149	#97	—	38	2	no	—
150	#97	—	38	2	no	—
151	#95	—	38	2	no	—
152	#95	—	38	2	no	—
153	#95	—	38	2	no	—
154	C₁₉	#8584	1	1	yes	characteristic, complemented, Sylow
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	19	2	no	—
157	C₂²×C₄	—	19	1	no	—
158	C₂²×C₄	—	19	1	no	—
159	C₂²×C₄	—	38	2	no	—
160	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
161	C₂²×C₄	D₄×D₁₉	1	2	yes	—
162	C₂×C₈	—	2	4	no	—
163	C₂×C₈	—	4	2	no	—
164	C₂×C₈	—	4	2	no	—
165	C₂×C₈	—	38	4	no	—
166	C₂×C₈	—	76	2	no	—
167	C₂×C₈	—	76	2	no	—
168	C₄⋊C₄	—	2	4	no	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	2	4	no	—
171	C₄⋊C₄	—	76	2	no	—
172	C₄⋊C₄	—	76	4	no	—
173	C₄⋊C₄	Dic₁₉⋊C₄	1	4	yes	—
174	C₄²	—	38	4	no	—
175	C₄²	—	38	2	no	—
176	C₄²	—	76	2	no	—
177	C₄²	—	76	1	no	—
178	C₄²	—	76	4	no	—
179	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
180	C₄²	#64543	1	2	yes	—
181	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
182	C₂³	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	19	4	no	—
191	C₂×C₄	—	19	2	no	—
192	C₂×C₄	—	38	2	no	—
193	C₂×C₄	—	38	2	no	—
194	C₂×C₄	—	38	2	no	—
195	C₂×C₄	—	38	2	no	—
196	C₂×C₄	—	38	1	no	—
197	C₂×C₄	—	76	2	no	—
198	C₂×C₄	—	76	2	no	—
199	C₂×C₄	—	76	2	no	—
200	C₂×C₄	#10597323	1	1	yes	characteristic
201	C₂×C₄	#10597324	1	2	yes	—
202	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
203	C₂×C₄	#10597385	1	2	yes	—
204	C₈	—	4	4	no	—
205	C₈	—	76	4	no	—
206	C₂²	#11926402	1	1	yes	characteristic
207	C₂²	#101×D₁₉	1	1	yes	characteristic
208	C₂²	#11926527	1	1	yes	characteristic
209	C₂²	#104×D₁₉	1	2	yes	—
210	C₂²	#11926559	1	2	yes	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	2	no	—
214	C₄	—	2	2	no	—
215	C₄	—	4	4	no	—
216	C₄	—	38	4	no	—
217	C₄	—	38	2	no	—
218	C₄	—	76	1	no	—
219	C₄	—	152	2	no	—
220	C₂	#14562228	1	1	yes	characteristic
221	C₂	#14562254	1	2	yes	—
222	C₂	#14562260	1	1	yes	characteristic
223	C₂	#14563629	1	1	yes	characteristic
224	C₂	#14563653	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#125×Dic₁₉

C₂

yes

maximal, characteristic

#14563489

C₂

yes

maximal

C₂×#11926286

C₂

yes

maximal, characteristic

#14561517

C₂

yes

maximal

C₂×C₁₉×#330

C₂

yes

maximal, characteristic

C₂×Dic₁₉⋊C₈

—

C₂×#10597260

C₂²

yes

—

#11926443

C₄

yes

—

#11926430

—

#11926410

C₂²

yes

characteristic

C₄⋊C₄×Dic₁₉

—

C₄⋊C₄×Dic₁₉

C₄

yes

—

C₄²×Dic₁₉

C₂²

yes

characteristic

#11926298

—

#11926286

—

C₂×C₁₉×#125

C₂²

yes

characteristic

C₂×C₁₉×#97

C₂²

yes

—

C₁₉×#330

—

C₂²×C₈×C₁₉

—

C₁₉×#125

—

C₁₉×#125

D₄

yes

—

C₁₉×#125

Q₈

yes

—

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

C₂×C₄

yes

—

C₂×C₄²×C₁₉

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₉

—

C₂×C₄⋊Dic₁₉

C₂×C₄

yes

—

C₂×C₄×Dic₁₉

—

C₂×C₄×Dic₁₉

C₂×C₄

yes

characteristic

C₂×C₄×Dic₁₉

C₂×C₄

yes

characteristic

C₂×C₄×Dic₁₉

D₄

yes

—

C₂×C₄×Dic₁₉

D₄

yes

—

C₂²×C₁₉⋊C₈

—

C₁₉×#97

—

C₁₉×#97

—

C₁₉⋊#95

—

C₁₉⋊#95

—

C₁₉⋊#95

—

Dic₁₉⋊C₈

—

#10597260

—

#10597260

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₉

C₂×D₄

yes

—

C₂²×Dic₁₉

—

C₂²×Dic₁₉

C₂²⋊C₄

yes

characteristic

C₂²×Dic₁₉

C₂²⋊C₄

yes

characteristic

C₂²×Dic₁₉

C₂²⋊C₄

yes

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

C₄⋊C₄

yes

—

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₄²×C₁₉

C₂×Q₈

yes

characteristic

C₄²×C₁₉

#41

yes

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄×Dic₁₉

—

C₄×Dic₁₉

—

C₄×Dic₁₉

—

C₄×Dic₁₉

—

C₄×Dic₁₉

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

#8584

—

maximal, Sylow

C₂³×C₁₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂×C₄⋊C₄

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

—

C₂×C₄×C₁₉

C₂²⋊Q₈

yes

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

#101

yes

—

C₂×Dic₁₉

#104

yes

—

C₈×C₁₉

—

C₁₉⋊C₈

—

—

—

—

D₁₉

yes

characteristic

#1295

—

C₂²×C₁₉

#380

yes

characteristic

100

C₂²×C₁₉

C₂×#101

yes

characteristic

101

C₂²×C₁₉

#413

yes

characteristic

102

C₂²×C₁₉

C₂×#104

yes

—

103

C₂²×C₁₉

#421

yes

—

104

C₄×C₁₉

—

105

C₄×C₁₉

—

106

C₄×C₁₉

—

107

C₄×C₁₉

—

108

C₄×C₁₉

—

109

Dic₁₉

—

110

Dic₁₉

—

111

Dic₁₉

—

112

Dic₁₉

—

113

C₂×#125

C₂×D₁₉

yes

characteristic

114

C₂×C₄⋊C₈

—

115

C₂×#97

—

116

C₂×#97

C₂×D₁₉

yes

—

117

#389

—

118

#387

—

119

#382

—

120

C₄×C₄⋊C₄

—

121

C₄×C₄⋊C₄

—

122

C₄³

—

123

#336

—

124

#330

—

125

#330

—

126

C₂×C₁₉

#1526

yes

characteristic

127

C₂×C₁₉

#1532

yes

—

128

C₂×C₁₉

#1534

yes

characteristic

129

C₂×C₁₉

#1801

yes

characteristic

130

C₂×C₁₉

#1806

yes

—

131

C₂²×C₈

—

132

C₂²×C₈

—

133

#125

—

134

#125

Dic₃₈

yes

—

135

#125

C₂²⋊D₁₉

yes

—

136

C₂×C₄⋊C₄

—

137

C₂×C₄⋊C₄

—

138

C₂×C₄⋊C₄

—

139

C₂×C₄⋊C₄

C₄×D₁₉

yes

—

140

C₂×C₄²

—

141

C₂×C₄²

—

142

C₂×C₄²

—

143

C₂×C₄²

—

144

C₂×C₄²

—

145

C₂×C₄²

C₂²×D₁₉

yes

characteristic

146

C₄⋊C₈

—

147

#97

—

148

#97

—

149

#97

—

150

#97

—

151

#95

—

152

#95

—

153

#95

—

154

C₁₉

#8584

yes

characteristic, complemented, Sylow

155

C₂²×C₄

—

156

C₂²×C₄

—

157

C₂²×C₄

—

158

C₂²×C₄

—

159

C₂²×C₄

—

160

C₂²×C₄

C₂×C₄×D₁₉

yes

characteristic

161

C₂²×C₄

D₄×D₁₉

yes

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

171

C₄⋊C₄

—

172

C₄⋊C₄

—

173

C₄⋊C₄

Dic₁₉⋊C₄

yes

—

174

C₄²

—

175

C₄²

—

176

C₄²

—

177

C₄²

—

178

C₄²

—

179

C₄²

C₂×Dic₃₈

yes

characteristic

180

C₄²

#64543

yes

—

181

C₄²

C₂×C₂²⋊D₁₉

yes

characteristic

182

C₂³

C₂²⋊C₄×D₁₉

yes

characteristic

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

#10597323

yes

characteristic

201

C₂×C₄

#10597324

yes

—

202

C₂×C₄

C₂×Dic₁₉⋊C₄

yes

characteristic

203

C₂×C₄

#10597385

yes

—

204

C₈

—

205

C₈

—

206

C₂²

#11926402

yes

characteristic

207

C₂²

#101×D₁₉

yes

characteristic

208

C₂²

#11926527

yes

characteristic

209

C₂²

#104×D₁₉

yes

—

210

C₂²

#11926559

yes

—

211

C₄

—

212

C₄

—

213

C₄

—

214

C₄

—

215

C₄

—

216

C₄

—

217

C₄

—

218

C₄

—

219

C₄

—

152

—

220

C₂

#14562228

yes

characteristic

221

C₂

#14562254

yes

—

222

C₂

#14562260

yes

characteristic

223

C₂

#14563629

yes

characteristic

224

C₂

#14563653

yes

—

	subgroup	G/H
center	C₂³	C₂²⋊C₄×D₁₉
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#330	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#330	D₁₉
hypocenter	C₁₉	#8584
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#330

subgroup

G/H

center

C₂³

C₂²⋊C₄×D₁₉

commutator

C₂²×C₄×C₁₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₉

Fitting

C₂×C₁₉×#330

C₂

socle, prime radical

C₂³×C₁₉

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₁₉

C₂³

D₁₉

C₁₉

G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#330

N	H ≅ G/N	type
154. C₁₉	#8584	semidirect

H ≅ G/N

type

154. C₁₉

#8584

semidirect

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

subgroup	index
#125×Dic₁₉	2
#14563489	2
C₂×#11926286	2
#14561517	2
C₂×C₁₉×#330	2
#8584	19

subgroup

index

#125×Dic₁₉

C₂×C₁₉×#330

Subgroups — #18075322

Conjugacy classes of subgroups

Complements

Maximal subgroups