Group #18069115 — subgroups

Subgroups — #18069115

Order 4864.

406 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 127 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	#14563615	C₂	1	2	yes	maximal
2	C₁₉×#1728	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
4	C₂×C₂²⋊(C₁₉⋊C₈)	C₄	1	4	yes	—
5	#11926299	—	2	2	no	—
6	C₂²×M₁₆×C₁₉	C₄	1	2	yes	—
7	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
8	C₁₉×#336	—	2	2	no	—
9	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
10	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
11	C₂×M₁₆×C₁₉	—	2	2	no	—
12	C₂×M₁₆×C₁₉	—	4	2	no	—
13	C₂×M₁₆×C₁₉	—	4	2	no	—
14	C₂×M₁₆×C₁₉	C₈	1	4	yes	complemented
15	C₂²×C₈×C₁₉	—	2	2	no	—
16	C₂×C₄²×C₁₉	—	4	2	no	—
17	C₂×C₄²×C₁₉	D₄	1	2	yes	—
18	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
19	C₂×C₄²×C₁₉	Q₈	1	1	yes	characteristic
20	C₂²×C₁₉⋊C₈	—	2	4	no	—
21	C₂²⋊(C₁₉⋊C₈)	—	4	4	no	—
22	C₄⋊(C₁₉⋊C₈)	—	4	4	no	—
23	C₂⁴×C₁₉	C₄²	1	1	yes	characteristic
24	C₂²×C₄×C₁₉	—	2	2	no	—
25	C₂²×C₄×C₁₉	—	2	1	no	—
26	C₂²×C₄×C₁₉	—	2	1	no	—
27	C₂²×C₄×C₁₉	—	2	2	no	—
28	C₂²×C₄×C₁₉	—	2	2	no	—
29	C₂²×C₄×C₁₉	—	4	2	no	—
30	C₂²×C₄×C₁₉	—	4	2	no	—
31	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
32	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
34	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₉	C₂×C₈	1	2	yes	—
36	C₂²×C₄×C₁₉	M₁₆	1	2	yes	—
37	M₁₆×C₁₉	—	4	4	no	—
38	M₁₆×C₁₉	—	4	4	no	—
39	C₂×C₈×C₁₉	—	2	4	no	—
40	C₂×C₈×C₁₉	—	4	2	no	—
41	C₂×C₈×C₁₉	—	4	2	no	—
42	C₄²×C₁₉	—	2	4	no	—
43	C₄²×C₁₉	—	4	1	no	—
44	C₄²×C₁₉	—	4	1	no	—
45	C₂×C₁₉⋊C₈	—	4	4	no	—
46	C₂×C₁₉⋊C₈	—	4	4	no	—
47	C₂×C₁₉⋊C₈	—	4	4	no	—
48	#7034	—	19	1	no	maximal, Sylow
49	C₂³×C₁₉	—	2	1	no	—
50	C₂³×C₁₉	—	2	2	no	—
51	C₂³×C₁₉	—	2	1	no	—
52	C₂³×C₁₉	—	2	2	no	—
53	C₂³×C₁₉	#95	1	1	yes	characteristic
54	C₂³×C₁₉	C₄×C₈	1	1	yes	characteristic
55	C₂³×C₁₉	#97	1	1	yes	characteristic
56	C₂×C₄×C₁₉	—	2	2	no	—
57	C₂×C₄×C₁₉	—	2	1	no	—
58	C₂×C₄×C₁₉	—	2	1	no	—
59	C₂×C₄×C₁₉	—	2	2	no	—
60	C₂×C₄×C₁₉	—	2	2	no	—
61	C₂×C₄×C₁₉	—	2	2	no	—
62	C₂×C₄×C₁₉	—	2	2	no	—
63	C₂×C₄×C₁₉	—	2	2	no	—
64	C₂×C₄×C₁₉	—	2	2	no	—
65	C₂×C₄×C₁₉	—	2	2	no	—
66	C₂×C₄×C₁₉	—	2	2	no	—
67	C₂×C₄×C₁₉	—	4	4	no	—
68	C₂×C₄×C₁₉	—	4	4	no	—
69	C₂×C₄×C₁₉	C₂²⋊C₈	1	2	yes	—
70	C₂×C₄×C₁₉	#99	1	2	yes	—
71	C₂×C₄×C₁₉	#100	1	2	yes	—
72	C₂×C₄×C₁₉	#100	1	1	yes	characteristic
73	C₂×C₄×C₁₉	#101	1	2	yes	—
74	C₂×C₄×C₁₉	#101	1	1	yes	characteristic
75	C₂×C₄×C₁₉	C₄⋊C₈	1	2	yes	—
76	C₈×C₁₉	—	4	4	no	—
77	C₁₉⋊C₈	—	8	4	no	—
78	#1797	—	19	2	no	—
79	#1728	D₁₉	1	1	yes	characteristic
80	C₂²×C₁₉	—	2	1	no	—
81	C₂²×C₁₉	—	2	1	no	—
82	C₂²×C₁₉	—	2	2	no	—
83	C₂²×C₁₉	—	2	2	no	—
84	C₂²×C₁₉	—	4	2	no	—
85	C₂²×C₁₉	—	4	1	no	—
86	C₂²×C₁₉	—	4	1	no	—
87	C₂²×C₁₉	#336	1	1	yes	characteristic
88	C₂²×C₁₉	#342	1	1	yes	characteristic
89	C₂²×C₁₉	#343	1	2	yes	—
90	C₂²×C₁₉	#343	1	1	yes	characteristic
91	C₂²×C₁₉	#344	1	2	yes	—
92	C₄×C₁₉	—	2	2	no	—
93	C₄×C₁₉	—	2	2	no	—
94	C₄×C₁₉	—	2	2	no	—
95	C₄×C₁₉	—	4	2	no	—
96	C₄×C₁₉	#323	1	2	yes	—
97	C₄×C₁₉	#324	1	2	yes	—
98	C₂²×M₁₆	Dic₁₉	1	2	yes	—
99	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
100	C₂×C₄⋊C₈	—	38	2	no	—
101	C₂×C₂²⋊C₈	—	19	4	no	—
102	#336	—	2	2	no	—
103	#336	—	38	2	no	—
104	C₂×C₁₉	—	4	1	no	—
105	C₂×C₁₉	—	4	1	no	—
106	C₂×C₁₉	#1280	1	1	yes	characteristic
107	C₂×C₁₉	#1283	1	2	yes	—
108	C₂×C₁₉	#1305	1	2	yes	—
109	C₂×C₁₉	#1307	1	1	yes	characteristic
110	C₂×C₁₉	#1309	1	1	yes	characteristic
111	C₂³×C₄	C₄×D₁₉	1	2	yes	—
112	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
113	C₂×M₁₆	—	2	2	no	—
114	C₂×M₁₆	—	4	2	no	—
115	C₂×M₁₆	—	4	2	no	—
116	C₂×M₁₆	C₁₉⋊C₈	1	4	yes	complemented
117	C₂²×C₈	—	2	2	no	—
118	C₂²×C₈	—	38	4	no	—
119	C₂×C₄²	—	4	2	no	—
120	C₂×C₄²	Dic₃₈	1	1	yes	characteristic
121	C₂×C₄²	D₇₆	1	1	yes	characteristic
122	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
123	C₄⋊C₈	—	76	4	no	—
124	C₂²⋊C₈	—	76	4	no	—
125	C₁₉	#7034	1	1	yes	characteristic, complemented, Sylow
126	C₂⁴	C₄×Dic₁₉	1	1	yes	characteristic
127	C₂²×C₄	—	2	1	no	—
128	C₂²×C₄	—	2	2	no	—
129	C₂²×C₄	—	2	1	no	—
130	C₂²×C₄	—	2	2	no	—
131	C₂²×C₄	—	2	2	no	—
132	C₂²×C₄	—	4	2	no	—
133	C₂²×C₄	—	4	2	no	—
134	C₂²×C₄	C₂×C₁₉⋊C₈	1	2	yes	—
135	C₂²×C₄	#64520	1	2	yes	—
136	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
137	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
138	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
139	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
140	M₁₆	—	4	4	no	—
141	M₁₆	—	4	4	no	—
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	4	2	no	—
144	C₂×C₈	—	4	2	no	—
145	C₂×C₈	—	76	4	no	—
146	C₂×C₈	—	76	4	no	—
147	C₂×C₈	—	76	4	no	—
148	C₄²	—	2	4	no	—
149	C₄²	—	4	1	no	—
150	C₄²	—	4	1	no	—
151	C₂³	—	2	1	no	—
152	C₂³	—	2	2	no	—
153	C₂³	—	2	1	no	—
154	C₂³	—	2	2	no	—
155	C₂³	C₄×C₁₉⋊C₈	1	1	yes	characteristic
156	C₂³	#10597260	1	1	yes	characteristic
157	C₂³	C₁₉⋊#95	1	1	yes	characteristic
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	—	4	4	no	—
171	C₂×C₄	C₄⋊(C₁₉⋊C₈)	1	2	yes	—
172	C₂×C₄	#10597280	1	2	yes	—
173	C₂×C₄	#10597281	1	2	yes	—
174	C₂×C₄	C₂²⋊(C₁₉⋊C₈)	1	2	yes	—
175	C₂×C₄	#10597290	1	1	yes	characteristic
176	C₂×C₄	#10597291	1	2	yes	—
177	C₂×C₄	#10597293	1	1	yes	characteristic
178	C₈	—	4	4	no	—
179	C₈	—	152	4	no	—
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	—	2	2	no	—
183	C₂²	—	2	2	no	—
184	C₂²	—	4	2	no	—
185	C₂²	—	4	1	no	—
186	C₂²	—	4	1	no	—
187	C₂²	#11926299	1	1	yes	characteristic
188	C₂²	#11926311	1	1	yes	characteristic
189	C₂²	#11926312	1	2	yes	—
190	C₂²	#11926313	1	1	yes	characteristic
191	C₂²	#11926316	1	2	yes	—
192	C₄	—	2	2	no	—
193	C₄	—	2	2	no	—
194	C₄	—	2	2	no	—
195	C₄	—	4	2	no	—
196	C₄	#11926268	1	2	yes	—
197	C₄	#11926270	1	2	yes	—
198	C₂	—	4	1	no	—
199	C₂	—	4	1	no	—
200	C₂	#14561478	1	1	yes	characteristic
201	C₂	#14561486	1	2	yes	—
202	C₂	#14561540	1	2	yes	—
203	C₂	#14561546	1	1	yes	characteristic
204	C₂	C₁₉⋊#1309	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14563615

C₂

yes

maximal

C₁₉×#1728

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₁₉⋊C₈)

—

C₂×C₂²⋊(C₁₉⋊C₈)

C₄

yes

—

#11926299

—

C₂²×M₁₆×C₁₉

C₄

yes

—

C₂²×C₄²×C₁₉

C₂²

yes

characteristic

C₁₉×#336

—

C₂³×C₄×C₁₉

C₂×C₄

yes

characteristic

C₂³×C₄×C₁₉

C₂×C₄

yes

—

C₂×M₁₆×C₁₉

—

C₂×M₁₆×C₁₉

—

C₂×M₁₆×C₁₉

—

C₂×M₁₆×C₁₉

C₈

yes

complemented

C₂²×C₈×C₁₉

—

C₂×C₄²×C₁₉

—

C₂×C₄²×C₁₉

D₄

yes

—

C₂×C₄²×C₁₉

D₄

yes

characteristic

C₂×C₄²×C₁₉

Q₈

yes

characteristic

C₂²×C₁₉⋊C₈

—

C₂²⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

—

C₂⁴×C₁₉

C₄²

yes

characteristic

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

C₄⋊C₄

yes

—

C₂²×C₄×C₁₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

C₂×C₈

yes

—

C₂²×C₄×C₁₉

M₁₆

yes

—

M₁₆×C₁₉

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

#7034

—

maximal, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

#95

yes

characteristic

C₂³×C₁₉

C₄×C₈

yes

characteristic

C₂³×C₁₉

#97

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×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂²⋊C₈

yes

—

C₂×C₄×C₁₉

#99

yes

—

C₂×C₄×C₁₉

#100

yes

—

C₂×C₄×C₁₉

#100

yes

characteristic

C₂×C₄×C₁₉

#101

yes

—

C₂×C₄×C₁₉

#101

yes

characteristic

C₂×C₄×C₁₉

C₄⋊C₈

yes

—

C₈×C₁₉

—

C₁₉⋊C₈

—

#1797

—

#1728

D₁₉

yes

characteristic

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

—

C₂²×C₁₉

#343

yes

characteristic

C₂²×C₁₉

#344

yes

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

#323

yes

—

C₄×C₁₉

#324

yes

—

C₂²×M₁₆

Dic₁₉

yes

—

C₂²×C₄²

C₂×D₁₉

yes

characteristic

100

C₂×C₄⋊C₈

—

101

C₂×C₂²⋊C₈

—

102

#336

—

103

#336

—

104

C₂×C₁₉

—

105

C₂×C₁₉

—

106

C₂×C₁₉

#1280

yes

characteristic

107

C₂×C₁₉

#1283

yes

—

108

C₂×C₁₉

#1305

yes

—

109

C₂×C₁₉

#1307

yes

characteristic

110

C₂×C₁₉

#1309

yes

characteristic

111

C₂³×C₄

C₄×D₁₉

yes

—

112

C₂³×C₄

C₂×Dic₁₉

yes

characteristic

113

C₂×M₁₆

—

114

C₂×M₁₆

—

115

C₂×M₁₆

—

116

C₂×M₁₆

C₁₉⋊C₈

yes

complemented

117

C₂²×C₈

—

118

C₂²×C₈

—

119

C₂×C₄²

—

120

C₂×C₄²

Dic₃₈

yes

characteristic

121

C₂×C₄²

D₇₆

yes

characteristic

122

C₂×C₄²

C₂²⋊D₁₉

yes

—

123

C₄⋊C₈

—

124

C₂²⋊C₈

—

125

C₁₉

#7034

yes

characteristic, complemented, Sylow

126

C₂⁴

C₄×Dic₁₉

yes

characteristic

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

—

134

C₂²×C₄

C₂×C₁₉⋊C₈

yes

—

135

C₂²×C₄

#64520

yes

—

136

C₂²×C₄

Dic₁₉⋊C₄

yes

—

137

C₂²×C₄

C₄⋊Dic₁₉

yes

characteristic

138

C₂²×C₄

(C₂×D₁₉)⋊C₄

yes

—

139

C₂²×C₄

C₂²⋊Dic₁₉

yes

characteristic

140

M₁₆

—

141

M₁₆

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₄²

—

149

C₄²

—

150

C₄²

—

151

C₂³

—

152

C₂³

—

153

C₂³

—

154

C₂³

—

155

C₂³

C₄×C₁₉⋊C₈

yes

characteristic

156

C₂³

#10597260

yes

characteristic

157

C₂³

C₁₉⋊#95

yes

characteristic

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

C₄⋊(C₁₉⋊C₈)

yes

—

172

C₂×C₄

#10597280

yes

—

173

C₂×C₄

#10597281

yes

—

174

C₂×C₄

C₂²⋊(C₁₉⋊C₈)

yes

—

175

C₂×C₄

#10597290

yes

characteristic

176

C₂×C₄

#10597291

yes

—

177

C₂×C₄

#10597293

yes

characteristic

178

C₈

—

179

C₈

—

152

—

180

C₂²

—

181

C₂²

—

182

C₂²

—

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

#11926299

yes

characteristic

188

C₂²

#11926311

yes

characteristic

189

C₂²

#11926312

yes

—

190

C₂²

#11926313

yes

characteristic

191

C₂²

#11926316

yes

—

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

#11926268

yes

—

197

C₄

#11926270

yes

—

198

C₂

—

199

C₂

—

200

C₂

#14561478

yes

characteristic

201

C₂

#14561486

yes

—

202

C₂

#14561540

yes

—

203

C₂

#14561546

yes

characteristic

204

C₂

C₁₉⋊#1309

yes

characteristic

	subgroup	G/H
center	C₂³	C₁₉⋊#95
commutator	C₂³×C₁₉	C₄×C₈
Frattini	C₂²×C₄²	C₂×D₁₉
Fitting	C₁₉×#1728	C₂
socle	C₂³×C₁₉	#95
prime radical	C₂⁴×C₁₉	C₄²
cosocle	C₂²×C₄²×C₁₉	C₂²
hypercenter	#1728	D₁₉
hypocenter	C₁₉	#7034
LCS	G ▷ C₂³×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1728

subgroup

G/H

center

C₂³

C₁₉⋊#95

commutator

C₂³×C₁₉

C₄×C₈

Frattini

C₂²×C₄²

C₂×D₁₉

Fitting

C₁₉×#1728

C₂

socle

C₂³×C₁₉

#95

prime radical

C₂⁴×C₁₉

C₄²

cosocle

C₂²×C₄²×C₁₉

C₂²

D₁₉

C₁₉

G ▷ C₂³×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1728

N	H ≅ G/N	type
14. C₂×M₁₆×C₁₉	C₈	semidirect
116. C₂×M₁₆	C₁₉⋊C₈	semidirect
125. C₁₉	#7034	semidirect

H ≅ G/N

type

14. C₂×M₁₆×C₁₉

C₈

semidirect

116. C₂×M₁₆

C₁₉⋊C₈

semidirect

125. C₁₉

#7034

semidirect

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

subgroup	index
#14563615	2
C₁₉×#1728	2
#7034	19

subgroup

index

#14563615

C₁₉×#1728

#7034

Subgroups — #18069115

Conjugacy classes of subgroups

Complements

Maximal subgroups