Group #18081166 — subgroups

Subgroups — #18081166

Order 4864.

294 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 131 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	#14562387	C₂	1	2	yes	maximal
2	C₂×#11926297	C₂	1	1	yes	maximal, characteristic
3	#14561927	C₂	1	1	yes	maximal, characteristic
4	#14561486	C₂	1	2	yes	maximal
5	C₄×C₁₉×#108	C₂	1	1	yes	maximal, characteristic
6	#11926597	C₂²	1	1	yes	characteristic
7	#11926597	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
9	#11926297	—	2	2	no	—
10	#11926297	—	2	2	no	—
11	#11926295	—	2	2	no	—
12	#11926266	—	2	2	no	—
13	C₂×C₁₉×#108	C₄	1	2	yes	—
14	C₄×M₁₆×C₁₉	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
16	C₁₉×#341	C₄	1	1	yes	characteristic
17	C₁₉×#341	C₄	1	1	yes	characteristic
18	C₂×M₁₆×C₁₉	C₂×C₄	1	2	yes	—
19	C₂×M₁₆×C₁₉	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
23	C₂²×C₁₉⋊C₈	—	2	2	no	—
24	C₁₉×#108	—	2	4	no	—
25	C₁₉×#97	—	2	2	no	—
26	C₄×C₈×C₁₉	—	2	2	no	—
27	C₄×C₈×C₁₉	D₄	1	2	yes	—
28	C₄×C₈×C₁₉	Q₈	1	2	yes	—
29	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
30	C₂²⋊(C₁₉⋊C₈)	Q₈	1	2	yes	—
31	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
32	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
33	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
34	C₄⋊(C₁₉⋊C₈)	D₄	1	1	yes	characteristic
35	C₄⋊(C₁₉⋊C₈)	D₄	1	1	yes	characteristic
36	C₄×C₁₉⋊C₈	—	2	1	no	—
37	C₄×C₁₉⋊C₈	—	2	1	no	—
38	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
41	M₁₆×C₁₉	—	2	2	no	—
42	M₁₆×C₁₉	—	2	2	no	—
43	M₁₆×C₁₉	—	2	2	no	—
44	M₁₆×C₁₉	M₁₆	1	4	yes	—
45	C₂×C₈×C₁₉	—	2	2	no	—
46	C₂×C₈×C₁₉	—	2	2	no	—
47	C₂×C₈×C₁₉	—	2	2	no	—
48	C₂×C₈×C₁₉	—	2	2	no	—
49	C₂×C₈×C₁₉	C₄⋊C₄	1	2	yes	—
50	C₂×C₈×C₁₉	C₄⋊C₄	1	2	yes	—
51	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
52	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
53	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
54	C₄²×C₁₉	#41	1	1	yes	characteristic
55	C₂×C₁₉⋊C₈	—	2	1	no	—
56	C₂×C₁₉⋊C₈	—	2	1	no	—
57	C₂×C₁₉⋊C₈	—	2	2	no	—
58	C₂×C₁₉⋊C₈	—	4	2	no	—
59	C₂×C₁₉⋊C₈	—	8	2	no	—
60	#9610	—	19	1	no	maximal, Sylow
61	C₂³×C₁₉	C₄×Q₈	1	1	yes	characteristic
62	C₂×C₄×C₁₉	—	2	1	no	—
63	C₂×C₄×C₁₉	—	2	1	no	—
64	C₂×C₄×C₁₉	—	4	1	no	—
65	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
68	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₁₉	#125	1	1	yes	characteristic
70	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
72	C₂×C₄×C₁₉	#131	1	2	yes	—
73	C₈×C₁₉	—	2	2	no	—
74	C₈×C₁₉	—	2	2	no	—
75	C₈×C₁₉	—	2	4	no	—
76	C₈×C₁₉	—	4	2	no	—
77	C₁₉⋊C₈	—	4	2	no	—
78	C₁₉⋊C₈	—	4	1	no	—
79	C₁₉⋊C₈	—	4	1	no	—
80	C₁₉⋊C₈	—	8	2	no	—
81	C₄×#108	D₁₉	1	1	yes	characteristic
82	#1565	—	19	2	no	—
83	C₂×#335	—	19	1	no	—
84	#1452	—	19	1	no	—
85	#1283	—	19	2	no	—
86	C₂²×C₁₉	—	2	1	no	—
87	C₂²×C₁₉	—	2	1	no	—
88	C₂²×C₁₉	#389	1	1	yes	characteristic
89	C₂²×C₁₉	#446	1	2	yes	—
90	C₄×C₁₉	—	2	1	no	—
91	C₄×C₁₉	—	2	1	no	—
92	C₄×C₁₉	—	2	1	no	—
93	C₄×C₁₉	#423	1	1	yes	characteristic
94	C₄×C₁₉	#424	1	1	yes	characteristic
95	C₄×C₁₉	#435	1	2	yes	—
96	C₄×C₁₉	#472	1	1	yes	characteristic
97	C₄×C₁₉	#473	1	1	yes	characteristic
98	#431	—	19	1	no	—
99	#431	—	19	1	no	—
100	C₂×#108	Dic₁₉	1	2	yes	—
101	C₂×C₄⋊C₈	—	19	2	no	—
102	C₄×M₁₆	C₂×D₁₉	1	2	yes	—
103	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
104	#341	Dic₁₉	1	1	yes	characteristic
105	#341	Dic₁₉	1	1	yes	characteristic
106	#335	—	38	2	no	—
107	#335	—	38	2	no	—
108	#334	—	38	2	no	—
109	#322	—	38	2	no	—
110	C₂×C₁₉	—	2	2	no	—
111	C₂×C₁₉	#1940	1	1	yes	characteristic
112	C₂×C₁₉	#1952	1	1	yes	characteristic
113	C₂×C₁₉	#1953	1	1	yes	characteristic
114	C₂×M₁₆	C₂×Dic₁₉	1	2	yes	—
115	C₂×M₁₆	C₂×Dic₁₉	1	2	yes	—
116	C₂²×C₈	—	38	2	no	—
117	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
118	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
119	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
120	#108	—	2	4	no	—
121	C₄⋊C₈	—	19	1	no	—
122	C₄⋊C₈	—	19	1	no	—
123	C₄⋊C₈	—	38	2	no	—
124	C₄⋊C₈	—	38	2	no	—
125	C₄⋊C₈	—	38	1	no	—
126	C₂²⋊C₈	—	19	2	no	—
127	C₂²⋊C₈	—	38	1	no	—
128	#97	—	2	2	no	—
129	C₄×C₈	—	2	2	no	—
130	C₄×C₈	—	38	1	no	—
131	C₄×C₈	—	38	1	no	—
132	C₄×C₈	Dic₃₈	1	2	yes	—
133	C₄×C₈	D₇₆	1	2	yes	—
134	C₁₉	#9610	1	1	yes	characteristic, complemented, Sylow
135	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
136	C₂²×C₄	#64545	1	1	yes	characteristic
137	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
138	M₁₆	—	2	2	no	—
139	M₁₆	—	2	2	no	—
140	M₁₆	—	2	2	no	—
141	M₁₆	#64520	1	4	yes	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	38	1	no	—
147	C₂×C₈	—	38	1	no	—
148	C₂×C₈	—	38	2	no	—
149	C₂×C₈	—	76	2	no	—
150	C₂×C₈	—	152	2	no	—
151	C₂×C₈	C₄⋊Dic₁₉	1	2	yes	—
152	C₂×C₈	C₄⋊Dic₁₉	1	2	yes	—
153	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
154	C₄²	C₂×D₇₆	1	1	yes	characteristic
155	C₄²	D₄×D₁₉	1	1	yes	characteristic
156	C₄²	#64543	1	1	yes	characteristic
157	C₂³	Q₈×Dic₁₉	1	1	yes	characteristic
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	4	1	no	—
161	C₂×C₄	#10597334	1	1	yes	characteristic
162	C₂×C₄	#10597336	1	1	yes	characteristic
163	C₂×C₄	#10597341	1	1	yes	characteristic
164	C₂×C₄	#10597343	1	1	yes	characteristic
165	C₂×C₄	C₂×#64520	1	2	yes	—
166	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
167	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
168	C₂×C₄	#10597394	1	2	yes	—
169	C₈	—	2	2	no	—
170	C₈	—	2	2	no	—
171	C₈	—	2	4	no	—
172	C₈	—	4	2	no	—
173	C₈	—	76	2	no	—
174	C₈	—	76	1	no	—
175	C₈	—	76	1	no	—
176	C₈	—	152	2	no	—
177	C₂²	—	2	1	no	—
178	C₂²	—	2	1	no	—
179	C₂²	#11926443	1	1	yes	characteristic
180	C₂²	#11926675	1	2	yes	—
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	#11926568	1	1	yes	characteristic
185	C₄	#11926571	1	1	yes	characteristic
186	C₄	#11926616	1	2	yes	—
187	C₄	#11926815	1	1	yes	characteristic
188	C₄	#11926820	1	1	yes	characteristic
189	C₂	—	2	2	no	—
190	C₂	#14564359	1	1	yes	characteristic
191	C₂	#14564421	1	1	yes	characteristic
192	C₂	#14564426	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14562387

C₂

yes

maximal

C₂×#11926297

C₂

yes

maximal, characteristic

#14561927

C₂

yes

maximal, characteristic

#14561486

C₂

yes

maximal

C₄×C₁₉×#108

C₂

yes

maximal, characteristic

#11926597

C₂²

yes

characteristic

#11926597

C₂²

yes

characteristic

C₂×C₄⋊(C₁₉⋊C₈)

C₂²

yes

—

—

—

—

—

C₂×C₁₉×#108

C₄

yes

—

C₄×M₁₆×C₁₉

C₂²

yes

—

C₂×C₄×C₈×C₁₉

C₂²

yes

characteristic

C₁₉×#341

C₄

yes

characteristic

C₁₉×#341

C₄

yes

characteristic

C₂×M₁₆×C₁₉

C₂×C₄

yes

—

C₂×M₁₆×C₁₉

C₂×C₄

yes

—

C₂²×C₈×C₁₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₉

C₂³

yes

characteristic

C₂²×C₁₉⋊C₈

—

C₁₉×#108

—

C₁₉×#97

—

C₄×C₈×C₁₉

—

C₄×C₈×C₁₉

D₄

yes

—

C₄×C₈×C₁₉

Q₈

yes

—

C₂²⋊(C₁₉⋊C₈)

—

C₂²⋊(C₁₉⋊C₈)

Q₈

yes

—

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

D₄

yes

characteristic

C₄⋊(C₁₉⋊C₈)

D₄

yes

characteristic

C₄×C₁₉⋊C₈

—

C₄×C₁₉⋊C₈

—

C₂²×C₄×C₁₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₉

C₂×Q₈

yes

characteristic

C₂²×C₄×C₁₉

#41

yes

characteristic

M₁₆×C₁₉

—

M₁₆×C₁₉

—

M₁₆×C₁₉

—

M₁₆×C₁₉

M₁₆

yes

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₄⋊C₄

yes

—

C₂×C₈×C₁₉

C₄⋊C₄

yes

—

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₄²×C₁₉

C₂×Q₈

yes

characteristic

C₄²×C₁₉

#41

yes

characteristic

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

#9610

—

maximal, Sylow

C₂³×C₁₉

C₄×Q₈

yes

characteristic

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₉

C₄×D₄

yes

characteristic

C₂×C₄×C₁₉

#121

yes

characteristic

C₂×C₄×C₁₉

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₁₉

#125

yes

characteristic

C₂×C₄×C₁₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₉

C₂×M₁₆

yes

—

C₂×C₄×C₁₉

#131

yes

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

C₄×#108

D₁₉

yes

—

—

—

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#389

yes

characteristic

C₂²×C₁₉

#446

yes

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

yes

C₄×C₁₉

yes

C₄×C₁₉

yes

—

C₄×C₁₉

yes

C₄×C₁₉

yes

—

—

100

C₂×#108

Dic₁₉

yes

—

101

C₂×C₄⋊C₈

—

102

C₄×M₁₆

C₂×D₁₉

yes

—

103

C₂×C₄×C₈

C₂×D₁₉

yes

characteristic

104

#341

Dic₁₉

yes

characteristic

105

#341

Dic₁₉

yes

characteristic

106

#335

—

107

#335

—

108

#334

—

109

#322

—

110

C₂×C₁₉

—

111

C₂×C₁₉

#1940

yes

characteristic

112

C₂×C₁₉

#1952

yes

characteristic

113

C₂×C₁₉

#1953

yes

characteristic

114

C₂×M₁₆

C₂×Dic₁₉

yes

—

115

C₂×M₁₆

C₂×Dic₁₉

yes

—

116

C₂²×C₈

—

117

C₂²×C₈

C₂×Dic₁₉

yes

characteristic

118

C₂²×C₈

C₂×Dic₁₉

yes

characteristic

119

C₂×C₄²

C₂²×D₁₉

yes

characteristic

120

#108

—

121

C₄⋊C₈

—

122

C₄⋊C₈

—

123

C₄⋊C₈

—

124

C₄⋊C₈

—

125

C₄⋊C₈

—

126

C₂²⋊C₈

—

127

C₂²⋊C₈

—

128

#97

—

129

C₄×C₈

—

130

C₄×C₈

—

131

C₄×C₈

—

132

C₄×C₈

Dic₃₈

yes

—

133

C₄×C₈

D₇₆

yes

—

134

C₁₉

#9610

yes

characteristic, complemented, Sylow

135

C₂²×C₄

Q₈×D₁₉

yes

characteristic

136

C₂²×C₄

#64545

yes

characteristic

137

C₂²×C₄

C₂²×Dic₁₉

yes

characteristic

138

M₁₆

—

139

M₁₆

—

140

M₁₆

—

141

M₁₆

#64520

yes

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

152

—

151

C₂×C₈

C₄⋊Dic₁₉

yes

—

152

C₂×C₈

C₄⋊Dic₁₉

yes

—

153

C₄²

C₂×Dic₃₈

yes

characteristic

154

C₄²

C₂×D₇₆

yes

characteristic

155

C₄²

D₄×D₁₉

yes

characteristic

156

C₄²

#64543

yes

characteristic

157

C₂³

Q₈×Dic₁₉

yes

characteristic

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

#10597334

yes

characteristic

162

C₂×C₄

#10597336

yes

characteristic

163

C₂×C₄

#10597341

yes

characteristic

164

C₂×C₄

#10597343

yes

characteristic

165

C₂×C₄

C₂×#64520

yes

—

166

C₂×C₄

C₂×C₄⋊Dic₁₉

yes

characteristic

167

C₂×C₄

D₄×Dic₁₉

yes

characteristic

168

C₂×C₄

#10597394

yes

—

169

C₈

—

170

C₈

—

171

C₈

—

172

C₈

—

173

C₈

—

174

C₈

—

175

C₈

—

176

C₈

—

152

—

177

C₂²

—

178

C₂²

—

179

C₂²

#11926443

yes

characteristic

180

C₂²

#11926675

yes

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

#11926568

yes

characteristic

185

C₄

#11926571

yes

characteristic

186

C₄

#11926616

yes

—

187

C₄

#11926815

yes

characteristic

188

C₄

#11926820

yes

characteristic

189

C₂

—

190

C₂

#14564359

yes

characteristic

191

C₂

#14564421

yes

characteristic

192

C₂

#14564426

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#10597341
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₄×C₁₉×#108	C₂
socle	C₂²×C₁₉	#389
prime radical	C₂³×C₁₉	C₄×Q₈
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₄×#108	D₁₉
hypocenter	C₁₉	#9610
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

subgroup

G/H

center

C₂×C₄

#10597341

commutator

C₂²×C₄×C₁₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₉

Fitting

C₄×C₁₉×#108

C₂

socle

C₂²×C₁₉

#389

prime radical

C₂³×C₁₉

C₄×Q₈

cosocle

C₂×C₄²×C₁₉

C₂³

D₁₉

C₁₉

G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

N	H ≅ G/N	type
134. C₁₉	#9610	semidirect

H ≅ G/N

type

134. C₁₉

#9610

semidirect

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

subgroup	index
#14562387	2
C₂×#11926297	2
#14561927	2
#14561486	2
C₄×C₁₉×#108	2
#9610	19

subgroup

index

C₄×C₁₉×#108

Subgroups — #18081166

Conjugacy classes of subgroups

Complements

Maximal subgroups