Group #18017129 — subgroups

Subgroups — C₁₉×#11170 #18017129

Order 4864.

962 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 962 conjugacy classes of subgroups — too many to draw. Only its 222 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	C₂²×C₁₉×#100	C₂	1	2	yes	maximal
2	C₁₉×#2309	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₉×#344	C₂	1	4	yes	maximal, complemented
4	C₂³×D₄×C₁₉	C₂²	1	1	yes	characteristic
5	C₂²×M₁₆×C₁₉	C₂²	1	4	yes	—
6	C₂×C₁₉×#117	C₂²	1	2	yes	—
7	C₂×C₁₉×#100	—	2	4	no	—
8	C₂×C₁₉×#100	—	2	8	no	—
9	C₁₉×#390	—	2	4	no	—
10	C₁₉×#379	C₄	1	4	yes	—
11	C₁₉×#344	—	2	8	no	—
12	C₂⁵×C₁₉	C₂×C₄	1	2	yes	—
13	C₂²×D₄×C₁₉	—	2	4	no	—
14	C₂²×D₄×C₁₉	—	2	4	no	—
15	C₂²×D₄×C₁₉	—	2	4	no	—
16	C₂²×D₄×C₁₉	D₄	1	4	yes	—
17	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
18	C₂×M₁₆×C₁₉	—	2	4	no	—
19	C₂×M₁₆×C₁₉	—	2	4	no	—
20	C₂×M₁₆×C₁₉	—	2	4	no	—
21	C₂×M₁₆×C₁₉	—	2	4	no	—
22	C₂×M₁₆×C₁₉	—	2	4	no	—
23	C₂×M₁₆×C₁₉	D₄	1	8	yes	—
24	C₂²×C₈×C₁₉	—	2	4	no	—
25	C₁₉×#117	—	2	4	no	—
26	C₁₉×#117	—	2	4	no	—
27	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
28	C₂×C₂²⋊C₄×C₁₉	—	2	8	no	—
29	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₁₉	—	2	2	no	—
31	C₁₉×#100	—	2	16	no	—
32	C₂⁴×C₁₉	—	2	4	no	—
33	C₂⁴×C₁₉	—	2	8	no	—
34	C₂⁴×C₁₉	—	8	4	no	—
35	C₂⁴×C₁₉	C₂²⋊C₄	1	4	yes	—
36	C₂⁴×C₁₉	C₂²×C₄	1	1	yes	characteristic
37	C₂×D₄×C₁₉	—	2	4	no	—
38	C₂×D₄×C₁₉	—	2	8	no	—
39	C₂×D₄×C₁₉	—	2	8	no	—
40	C₂×D₄×C₁₉	—	2	4	no	—
41	C₂×D₄×C₁₉	—	4	4	no	—
42	C₂×D₄×C₁₉	—	4	4	no	—
43	C₂×D₄×C₁₉	—	4	4	no	—
44	C₂×D₄×C₁₉	—	4	4	no	—
45	C₂²×C₄×C₁₉	—	2	4	no	—
46	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
47	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₁₉	#41	1	4	yes	—
49	C₂²×C₄×C₁₉	#41	1	4	yes	—
50	M₁₆×C₁₉	—	2	8	no	—
51	M₁₆×C₁₉	—	2	8	no	—
52	M₁₆×C₁₉	—	4	4	no	—
53	M₁₆×C₁₉	—	4	4	no	—
54	C₂×C₈×C₁₉	—	2	8	no	—
55	C₂×C₈×C₁₉	—	4	4	no	—
56	C₂×C₈×C₁₉	—	4	4	no	—
57	C₄⋊C₄×C₁₉	—	4	2	no	—
58	C₄⋊C₄×C₁₉	—	4	2	no	—
59	C₂²⋊C₄×C₁₉	—	2	8	no	—
60	C₂²⋊C₄×C₁₉	—	8	8	no	—
61	C₄²×C₁₉	—	4	4	no	—
62	#11170	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂³×C₁₉	—	2	4	no	—
64	C₂³×C₁₉	—	2	8	no	—
65	C₂³×C₁₉	—	4	2	no	—
66	C₂³×C₁₉	—	4	8	no	—
67	C₂³×C₁₉	—	4	8	no	—
68	C₂³×C₁₉	—	8	8	no	—
69	C₂³×C₁₉	—	8	16	no	—
70	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂³×C₁₉	#117	1	2	yes	—
72	C₂³×C₁₉	C₄×D₄	1	4	yes	—
73	D₄×C₁₉	—	4	8	no	—
74	D₄×C₁₉	—	4	8	no	—
75	C₂×C₄×C₁₉	—	4	8	no	—
76	C₂×C₄×C₁₉	—	4	4	no	—
77	C₂×C₄×C₁₉	#120	1	2	yes	—
78	C₂×C₄×C₁₉	#121	1	4	yes	—
79	C₂×C₄×C₁₉	#121	1	4	yes	—
80	C₂×C₄×C₁₉	#123	1	4	yes	—
81	C₂×C₄×C₁₉	#123	1	4	yes	—
82	C₂×C₄×C₁₉	#123	1	2	yes	—
83	C₂×C₄×C₁₉	#124	1	4	yes	—
84	C₂×C₄×C₁₉	#126	1	4	yes	—
85	C₈×C₁₉	—	4	8	no	—
86	C₂²×#100	C₂×C₁₉	1	2	yes	—
87	#2309	C₂×C₁₉	1	1	yes	characteristic
88	C₂×#344	C₂×C₁₉	1	4	yes	complemented
89	C₂²×C₁₉	—	2	2	no	—
90	C₂²×C₁₉	—	2	4	no	—
91	C₂²×C₁₉	—	4	4	no	—
92	C₂²×C₁₉	—	4	8	no	—
93	C₂²×C₁₉	—	4	4	no	—
94	C₂²×C₁₉	—	8	8	no	—
95	C₂²×C₁₉	—	8	16	no	—
96	C₂²×C₁₉	#381	1	1	yes	characteristic
97	C₂²×C₁₉	#386	1	2	yes	—
98	C₂²×C₁₉	#388	1	4	yes	—
99	C₄×C₁₉	—	8	4	no	—
100	C₄×C₁₉	#394	1	4	yes	—
101	C₄×C₁₉	#397	1	4	yes	—
102	C₂³×D₄	C₂²×C₁₉	1	1	yes	characteristic
103	C₂²×M₁₆	C₂²×C₁₉	1	4	yes	—
104	C₂×#117	C₂²×C₁₉	1	2	yes	—
105	C₂×#100	—	2	4	no	—
106	C₂×#100	—	2	8	no	—
107	#390	—	2	4	no	—
108	#379	C₄×C₁₉	1	4	yes	—
109	#344	—	2	8	no	—
110	C₂×C₁₉	—	2	2	no	—
111	C₂×C₁₉	—	2	4	no	—
112	C₂×C₁₉	—	8	4	no	—
113	C₂×C₁₉	#1440	1	1	yes	characteristic
114	C₂×C₁₉	#1829	1	2	yes	—
115	C₂⁵	C₂×C₄×C₁₉	1	2	yes	—
116	C₂²×D₄	—	2	4	no	—
117	C₂²×D₄	—	2	4	no	—
118	C₂²×D₄	—	2	4	no	—
119	C₂²×D₄	D₄×C₁₉	1	4	yes	—
120	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
121	C₂×M₁₆	—	2	4	no	—
122	C₂×M₁₆	—	2	4	no	—
123	C₂×M₁₆	—	2	4	no	—
124	C₂×M₁₆	—	2	4	no	—
125	C₂×M₁₆	—	2	4	no	—
126	C₂×M₁₆	D₄×C₁₉	1	8	yes	—
127	C₂²×C₈	—	2	4	no	—
128	#117	—	2	4	no	—
129	#117	—	2	4	no	—
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₂²⋊C₄	—	2	8	no	—
132	C₂×C₂²⋊C₄	C₂×C₄×C₁₉	1	4	yes	—
133	C₂×C₄²	—	2	2	no	—
134	#100	—	2	16	no	—
135	C₁₉	#11170	1	1	yes	characteristic, direct product factor, Sylow
136	C₂⁴	—	2	4	no	—
137	C₂⁴	—	2	8	no	—
138	C₂⁴	—	8	4	no	—
139	C₂⁴	C₂²⋊C₄×C₁₉	1	4	yes	—
140	C₂⁴	C₂²×C₄×C₁₉	1	1	yes	characteristic
141	C₂×D₄	—	2	4	no	—
142	C₂×D₄	—	2	8	no	—
143	C₂×D₄	—	2	8	no	—
144	C₂×D₄	—	2	4	no	—
145	C₂×D₄	—	4	4	no	—
146	C₂×D₄	—	4	4	no	—
147	C₂×D₄	—	4	4	no	—
148	C₂×D₄	—	4	4	no	—
149	C₂²×C₄	—	2	4	no	—
150	C₂²×C₄	C₂×D₄×C₁₉	1	4	yes	—
151	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
152	C₂²×C₄	#41×C₁₉	1	4	yes	—
153	C₂²×C₄	#41×C₁₉	1	4	yes	—
154	M₁₆	—	2	8	no	—
155	M₁₆	—	2	8	no	—
156	M₁₆	—	4	4	no	—
157	M₁₆	—	4	4	no	—
158	C₂×C₈	—	2	8	no	—
159	C₂×C₈	—	4	4	no	—
160	C₂×C₈	—	4	4	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₂²⋊C₄	—	2	8	no	—
164	C₂²⋊C₄	—	8	8	no	—
165	C₄²	—	4	4	no	—
166	C₂³	—	2	4	no	—
167	C₂³	—	2	8	no	—
168	C₂³	—	4	2	no	—
169	C₂³	—	4	8	no	—
170	C₂³	—	4	8	no	—
171	C₂³	—	8	8	no	—
172	C₂³	—	8	16	no	—
173	C₂³	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
174	C₂³	C₁₉×#117	1	2	yes	—
175	C₂³	C₄×D₄×C₁₉	1	4	yes	—
176	D₄	—	4	8	no	—
177	D₄	—	4	8	no	—
178	C₂×C₄	—	4	8	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	C₁₉×#120	1	2	yes	—
181	C₂×C₄	C₁₉×#121	1	4	yes	—
182	C₂×C₄	C₁₉×#121	1	4	yes	—
183	C₂×C₄	C₁₉×#123	1	4	yes	—
184	C₂×C₄	C₁₉×#123	1	4	yes	—
185	C₂×C₄	C₁₉×#123	1	2	yes	—
186	C₂×C₄	C₁₉×#124	1	4	yes	—
187	C₂×C₄	C₁₉×#126	1	4	yes	—
188	C₈	—	4	8	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	4	no	—
191	C₂²	—	4	4	no	—
192	C₂²	—	4	4	no	—
193	C₂²	—	4	8	no	—
194	C₂²	—	8	8	no	—
195	C₂²	—	8	16	no	—
196	C₂²	C₁₉×#381	1	1	yes	characteristic
197	C₂²	C₁₉×#386	1	2	yes	—
198	C₂²	C₁₉×#388	1	4	yes	—
199	C₄	—	8	4	no	—
200	C₄	C₁₉×#394	1	4	yes	—
201	C₄	C₁₉×#397	1	4	yes	—
202	C₂	—	2	2	no	—
203	C₂	—	2	4	no	—
204	C₂	—	8	4	no	—
205	C₂	C₁₉×#1440	1	1	yes	characteristic
206	C₂	C₁₉×#1829	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₁₉×#100

C₂

yes

maximal

C₁₉×#2309

C₂

yes

maximal, characteristic

C₂×C₁₉×#344

C₂

yes

maximal, complemented

C₂³×D₄×C₁₉

C₂²

yes

characteristic

C₂²×M₁₆×C₁₉

C₂²

yes

—

C₂×C₁₉×#117

C₂²

yes

—

C₂×C₁₉×#100

—

C₂×C₁₉×#100

—

C₁₉×#390

—

C₁₉×#379

C₄

yes

—

C₁₉×#344

—

C₂⁵×C₁₉

C₂×C₄

yes

—

C₂²×D₄×C₁₉

—

C₂²×D₄×C₁₉

—

C₂²×D₄×C₁₉

—

C₂²×D₄×C₁₉

D₄

yes

—

C₂³×C₄×C₁₉

C₂³

yes

characteristic

C₂×M₁₆×C₁₉

—

C₂×M₁₆×C₁₉

—

C₂×M₁₆×C₁₉

—

C₂×M₁₆×C₁₉

—

C₂×M₁₆×C₁₉

—

C₂×M₁₆×C₁₉

D₄

yes

—

C₂²×C₈×C₁₉

—

C₁₉×#117

—

C₁₉×#117

—

C₂×C₄⋊C₄×C₁₉

—

C₂×C₂²⋊C₄×C₁₉

—

C₂×C₂²⋊C₄×C₁₉

C₂×C₄

yes

—

C₂×C₄²×C₁₉

—

C₁₉×#100

—

C₂⁴×C₁₉

—

C₂⁴×C₁₉

—

C₂⁴×C₁₉

—

C₂⁴×C₁₉

C₂²⋊C₄

yes

—

C₂⁴×C₁₉

C₂²×C₄

yes

characteristic

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂×D₄

yes

—

C₂²×C₄×C₁₉

C₂×D₄

yes

—

C₂²×C₄×C₁₉

#41

yes

—

C₂²×C₄×C₁₉

#41

yes

—

M₁₆×C₁₉

—

M₁₆×C₁₉

—

M₁₆×C₁₉

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

—

#11170

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

C₂×C₂²⋊C₄

yes

characteristic

C₂³×C₁₉

#117

yes

—

C₂³×C₁₉

C₄×D₄

yes

—

D₄×C₁₉

—

D₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

#120

yes

—

C₂×C₄×C₁₉

#121

yes

—

C₂×C₄×C₁₉

#121

yes

—

C₂×C₄×C₁₉

#123

yes

—

C₂×C₄×C₁₉

#123

yes

—

C₂×C₄×C₁₉

#123

yes

—

C₂×C₄×C₁₉

#124

yes

—

C₂×C₄×C₁₉

#126

yes

—

C₈×C₁₉

—

C₂²×#100

C₂×C₁₉

yes

—

#2309

C₂×C₁₉

yes

characteristic

C₂×#344

C₂×C₁₉

yes

complemented

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#381

yes

characteristic

C₂²×C₁₉

#386

yes

—

C₂²×C₁₉

#388

yes

—

C₄×C₁₉

—

100

C₄×C₁₉

#394

yes

—

101

C₄×C₁₉

#397

yes

—

102

C₂³×D₄

C₂²×C₁₉

yes

characteristic

103

C₂²×M₁₆

C₂²×C₁₉

yes

—

104

C₂×#117

C₂²×C₁₉

yes

—

105

C₂×#100

—

106

C₂×#100

—

107

#390

—

108

#379

C₄×C₁₉

yes

—

109

#344

—

110

C₂×C₁₉

—

111

C₂×C₁₉

—

112

C₂×C₁₉

—

113

C₂×C₁₉

#1440

yes

characteristic

114

C₂×C₁₉

#1829

yes

—

115

C₂⁵

C₂×C₄×C₁₉

yes

—

116

C₂²×D₄

—

117

C₂²×D₄

—

118

C₂²×D₄

—

119

C₂²×D₄

D₄×C₁₉

yes

—

120

C₂³×C₄

C₂³×C₁₉

yes

characteristic

121

C₂×M₁₆

—

122

C₂×M₁₆

—

123

C₂×M₁₆

—

124

C₂×M₁₆

—

125

C₂×M₁₆

—

126

C₂×M₁₆

D₄×C₁₉

yes

—

127

C₂²×C₈

—

128

#117

—

129

#117

—

130

C₂×C₄⋊C₄

—

131

C₂×C₂²⋊C₄

—

132

C₂×C₂²⋊C₄

C₂×C₄×C₁₉

yes

—

133

C₂×C₄²

—

134

#100

—

135

C₁₉

#11170

yes

characteristic, direct product factor, Sylow

136

C₂⁴

—

137

C₂⁴

—

138

C₂⁴

—

139

C₂⁴

C₂²⋊C₄×C₁₉

yes

—

140

C₂⁴

C₂²×C₄×C₁₉

yes

characteristic

141

C₂×D₄

—

142

C₂×D₄

—

143

C₂×D₄

—

144

C₂×D₄

—

145

C₂×D₄

—

146

C₂×D₄

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂²×C₄

—

150

C₂²×C₄

C₂×D₄×C₁₉

yes

—

151

C₂²×C₄

C₂×D₄×C₁₉

yes

—

152

C₂²×C₄

#41×C₁₉

yes

—

153

C₂²×C₄

#41×C₁₉

yes

—

154

M₁₆

—

155

M₁₆

—

156

M₁₆

—

157

M₁₆

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₂²⋊C₄

—

164

C₂²⋊C₄

—

165

C₄²

—

166

C₂³

—

167

C₂³

—

168

C₂³

—

169

C₂³

—

170

C₂³

—

171

C₂³

—

172

C₂³

—

173

C₂³

C₂×C₂²⋊C₄×C₁₉

yes

characteristic

174

C₂³

C₁₉×#117

yes

—

175

C₂³

C₄×D₄×C₁₉

yes

—

176

D₄

—

177

D₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

C₁₉×#120

yes

—

181

C₂×C₄

C₁₉×#121

yes

—

182

C₂×C₄

C₁₉×#121

yes

—

183

C₂×C₄

C₁₉×#123

yes

—

184

C₂×C₄

C₁₉×#123

yes

—

185

C₂×C₄

C₁₉×#123

yes

—

186

C₂×C₄

C₁₉×#124

yes

—

187

C₂×C₄

C₁₉×#126

yes

—

188

C₈

—

189

C₂²

—

190

C₂²

—

191

C₂²

—

192

C₂²

—

193

C₂²

—

194

C₂²

—

195

C₂²

—

196

C₂²

C₁₉×#381

yes

characteristic

197

C₂²

C₁₉×#386

yes

—

198

C₂²

C₁₉×#388

yes

—

199

C₄

—

200

C₄

C₁₉×#394

yes

—

201

C₄

C₁₉×#397

yes

—

202

C₂

—

203

C₂

—

204

C₂

—

205

C₂

C₁₉×#1440

yes

characteristic

206

C₂

C₁₉×#1829

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₉	#381
commutator	C₂⁴	C₂²×C₄×C₁₉
Frattini, cosocle	C₂³×C₄	C₂³×C₁₉
prime radical	C₂³×D₄×C₁₉	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂³×C₄×C₁₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₉

#381

commutator

C₂⁴

C₂²×C₄×C₁₉

Frattini, cosocle

C₂³×C₄

C₂³×C₁₉

prime radical

C₂³×D₄×C₁₉

C₂²

LCS

G ▷ C₂⁴ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₉ ◁ C₂³×C₄×C₁₉ ◁ G

N	H ≅ G/N	type
3. C₂×C₁₉×#344	C₂	semidirect
62. #11170	C₁₉	direct (N × H)
88. C₂×#344	C₂×C₁₉	semidirect
135. C₁₉	#11170	direct (N × H)

H ≅ G/N

type

3. C₂×C₁₉×#344

C₂

semidirect

62. #11170

C₁₉

direct (N × H)

88. C₂×#344

C₂×C₁₉

semidirect

135. C₁₉

#11170

direct (N × H)

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

subgroup	index
C₂²×C₁₉×#100	2
C₁₉×#2309	2
C₂×C₁₉×#344	2
#11170	19

subgroup

index

C₂²×C₁₉×#100

C₁₉×#2309

C₂×C₁₉×#344

#11170

Subgroups — C₁₉×#11170 #18017129

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#11170 #18017129

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#11170 #18017129