Group #18069086 — subgroups

Subgroups — #18069086

Order 4864.

300 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 115 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	#14563778	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1745	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
5	C₂×C₄×C₁₉⋊C₈	C₄	1	2	yes	—
6	#11926299	—	2	1	no	—
7	#11926299	—	2	1	no	—
8	#11926299	—	2	1	no	—
9	#11926299	C₄	1	2	yes	—
10	C₂×C₁₉×#97	C₄	1	2	yes	—
11	C₄³×C₁₉	C₂²	1	1	yes	characteristic
12	C₁₉×#336	—	2	1	no	—
13	C₁₉×#336	—	2	1	no	—
14	C₂²×C₈×C₁₉	—	2	2	no	—
15	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
19	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
20	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₉	Q₈	1	1	yes	characteristic
22	C₂²×C₁₉⋊C₈	—	2	2	no	—
23	C₂²×C₁₉⋊C₈	—	2	2	no	—
24	C₁₉×#97	—	2	2	no	—
25	C₁₉×#97	C₈	1	4	yes	complemented
26	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
27	C₄×C₁₉⋊C₈	—	4	2	no	—
28	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
29	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₈×C₁₉	—	2	4	no	—
36	C₂×C₈×C₁₉	—	4	2	no	—
37	C₂×C₈×C₁₉	—	4	2	no	—
38	C₄²×C₁₉	—	2	1	no	—
39	C₄²×C₁₉	—	2	1	no	—
40	C₄²×C₁₉	—	2	2	no	—
41	C₄²×C₁₉	—	2	2	no	—
42	C₄²×C₁₉	—	4	1	no	—
43	C₄²×C₁₉	—	4	1	no	—
44	C₄²×C₁₉	—	4	1	no	—
45	C₄²×C₁₉	C₂×C₈	1	2	yes	—
46	C₄²×C₁₉	M₁₆	1	2	yes	—
47	C₂×C₁₉⋊C₈	—	2	4	no	—
48	C₂×C₁₉⋊C₈	—	4	2	no	—
49	C₂×C₁₉⋊C₈	—	4	2	no	—
50	C₂×C₁₉⋊C₈	—	4	2	no	—
51	C₂×C₁₉⋊C₈	—	4	2	no	—
52	C₂×C₁₉⋊C₈	—	4	2	no	—
53	#7023	—	19	1	no	maximal, Sylow
54	C₂³×C₁₉	#95	1	1	yes	characteristic
55	C₂×C₄×C₁₉	—	2	1	no	—
56	C₂×C₄×C₁₉	—	2	1	no	—
57	C₂×C₄×C₁₉	—	2	1	no	—
58	C₂×C₄×C₁₉	—	2	1	no	—
59	C₂×C₄×C₁₉	—	2	1	no	—
60	C₂×C₄×C₁₉	—	2	1	no	—
61	C₂×C₄×C₁₉	—	2	1	no	—
62	C₂×C₄×C₁₉	—	2	1	no	—
63	C₂×C₄×C₁₉	—	2	1	no	—
64	C₂×C₄×C₁₉	—	2	1	no	—
65	C₂×C₄×C₁₉	—	2	1	no	—
66	C₂×C₄×C₁₉	—	2	1	no	—
67	C₂×C₄×C₁₉	—	2	1	no	—
68	C₂×C₄×C₁₉	—	2	1	no	—
69	C₂×C₄×C₁₉	—	2	1	no	—
70	C₂×C₄×C₁₉	C₄×C₈	1	1	yes	characteristic
71	C₂×C₄×C₁₉	#97	1	1	yes	characteristic
72	C₂×C₄×C₁₉	C₂²⋊C₈	1	2	yes	—
73	C₂×C₄×C₁₉	#100	1	1	yes	characteristic
74	C₂×C₄×C₁₉	#101	1	1	yes	characteristic
75	C₂×C₄×C₁₉	#104	1	2	yes	—
76	C₂×C₄×C₁₉	C₄⋊C₈	1	2	yes	—
77	C₂×C₄×C₁₉	#108	1	2	yes	—
78	C₈×C₁₉	—	4	4	no	—
79	C₁₉⋊C₈	—	4	4	no	—
80	C₁₉⋊C₈	—	8	2	no	—
81	#1839	—	19	1	no	—
82	C₄×C₄⋊C₈	—	19	1	no	—
83	#1745	D₁₉	1	1	yes	characteristic
84	C₂²×C₁₉	#336	1	1	yes	characteristic
85	C₂²×C₁₉	H(3,ℤ/4)	1	1	yes	characteristic
86	C₂²×C₁₉	H(3,ℤ/4)	1	1	yes	characteristic
87	C₂²×C₁₉	#339	1	1	yes	characteristic
88	C₂²×C₁₉	#341	1	1	yes	characteristic
89	C₂²×C₁₉	#343	1	1	yes	characteristic
90	C₂²×C₁₉	#344	1	1	yes	characteristic
91	C₄×C₁₉	—	2	1	no	—
92	C₄×C₁₉	—	2	1	no	—
93	C₄×C₁₉	—	2	1	no	—
94	C₄×C₁₉	—	2	1	no	—
95	C₄×C₁₉	—	2	2	no	—
96	C₄×C₁₉	—	2	2	no	—
97	C₄×C₁₉	—	2	2	no	—
98	C₄×C₁₉	—	2	2	no	—
99	C₄×C₁₉	—	4	1	no	—
100	C₂×C₄⋊C₈	—	38	1	no	—
101	C₂×#97	Dic₁₉	1	2	yes	—
102	C₂×C₄×C₈	—	19	2	no	—
103	C₄³	C₂×D₁₉	1	1	yes	characteristic
104	#336	—	2	1	no	—
105	#336	—	2	1	no	—
106	#336	—	19	2	no	—
107	#336	—	38	1	no	—
108	#336	—	38	1	no	—
109	#336	—	38	1	no	—
110	C₂×C₁₉	#1274	1	1	yes	characteristic
111	C₂×C₁₉	#1278	1	1	yes	characteristic
112	C₂×C₁₉	#1283	1	1	yes	characteristic
113	C₂×C₁₉	#1285	1	1	yes	characteristic
114	C₂×C₁₉	#1287	1	1	yes	characteristic
115	C₂×C₁₉	#1288	1	1	yes	characteristic
116	C₂×C₁₉	#1290	1	1	yes	characteristic
117	C₂²×C₈	—	2	2	no	—
118	C₂²×C₈	—	38	2	no	—
119	C₂²×C₈	—	38	2	no	—
120	C₂×C₄²	Dic₃₈	1	1	yes	characteristic
121	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
122	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
123	C₂×C₄²	D₇₆	1	1	yes	characteristic
124	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
125	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
126	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
127	C₄⋊C₈	—	38	4	no	—
128	#97	—	2	2	no	—
129	#97	C₁₉⋊C₈	1	4	yes	complemented
130	C₄×C₈	—	76	2	no	—
131	C₁₉	#7023	1	1	yes	characteristic, complemented, Sylow
132	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
133	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
134	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
135	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
136	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
137	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
138	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	—	38	4	no	—
143	C₂×C₈	—	76	2	no	—
144	C₂×C₈	—	76	2	no	—
145	C₂×C₈	—	76	2	no	—
146	C₂×C₈	—	76	2	no	—
147	C₂×C₈	—	76	2	no	—
148	C₄²	—	2	2	no	—
149	C₄²	—	2	1	no	—
150	C₄²	—	2	1	no	—
151	C₄²	—	2	2	no	—
152	C₄²	—	4	1	no	—
153	C₄²	—	4	1	no	—
154	C₄²	—	4	1	no	—
155	C₄²	C₂×C₁₉⋊C₈	1	2	yes	—
156	C₄²	#64520	1	2	yes	—
157	C₂³	C₁₉⋊#95	1	1	yes	characteristic
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	C₄×C₁₉⋊C₈	1	1	yes	characteristic
174	C₂×C₄	#10597260	1	1	yes	characteristic
175	C₂×C₄	C₄⋊(C₁₉⋊C₈)	1	2	yes	—
176	C₂×C₄	#10597279	1	2	yes	—
177	C₂×C₄	#10597280	1	1	yes	characteristic
178	C₂×C₄	#10597281	1	1	yes	characteristic
179	C₂×C₄	#10597282	1	2	yes	—
180	C₂×C₄	C₂²⋊(C₁₉⋊C₈)	1	2	yes	—
181	C₈	—	4	4	no	—
182	C₈	—	76	4	no	—
183	C₈	—	152	2	no	—
184	C₂²	#11926299	1	1	yes	characteristic
185	C₂²	#11926300	1	1	yes	characteristic
186	C₂²	#11926300	1	1	yes	characteristic
187	C₂²	#11926305	1	1	yes	characteristic
188	C₂²	#11926308	1	1	yes	characteristic
189	C₂²	#11926312	1	1	yes	characteristic
190	C₂²	#11926316	1	1	yes	characteristic
191	C₄	—	2	2	no	—
192	C₄	—	2	2	no	—
193	C₄	—	2	1	no	—
194	C₄	—	2	1	no	—
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	2	2	no	—
198	C₄	—	2	2	no	—
199	C₄	—	4	1	no	—
200	C₂	#14561464	1	1	yes	characteristic
201	C₂	#14561473	1	1	yes	characteristic
202	C₂	#14561486	1	1	yes	characteristic
203	C₂	#14561490	1	1	yes	characteristic
204	C₂	#14561496	1	1	yes	characteristic
205	C₂	#14561501	1	1	yes	characteristic
206	C₂	#14561504	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14563778

C₂

yes

maximal, characteristic

C₄×C₄⋊(C₁₉⋊C₈)

C₂

yes

maximal, characteristic

C₁₉×#1745

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₁₉⋊C₈)

—

C₂×C₄×C₁₉⋊C₈

C₄

yes

—

—

—

—

C₄

yes

—

C₂×C₁₉×#97

C₄

yes

—

C₄³×C₁₉

C₂²

yes

characteristic

C₁₉×#336

—

C₁₉×#336

—

C₂²×C₈×C₁₉

—

C₂×C₄²×C₁₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₉

D₄

yes

characteristic

C₂×C₄²×C₁₉

D₄

yes

characteristic

C₂×C₄²×C₁₉

D₄

yes

characteristic

C₂×C₄²×C₁₉

Q₈

yes

characteristic

C₂²×C₁₉⋊C₈

—

C₂²×C₁₉⋊C₈

—

C₁₉×#97

—

C₁₉×#97

C₈

yes

complemented

C₄⋊(C₁₉⋊C₈)

—

C₄×C₁₉⋊C₈

—

C₂²×C₄×C₁₉

C₄²

yes

characteristic

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

C₂×C₈

yes

—

C₄²×C₁₉

M₁₆

yes

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

—

C₂³×C₁₉

yes

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

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₄×C₈

yes

characteristic

C₂×C₄×C₁₉

#97

yes

characteristic

C₂×C₄×C₁₉

C₂²⋊C₈

yes

—

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

—

C₂×C₄×C₁₉

C₄⋊C₈

yes

—

C₂×C₄×C₁₉

#108

yes

—

C₈×C₁₉

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

#1839

—

C₄×C₄⋊C₈

—

#1745

D₁₉

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

100

C₂×C₄⋊C₈

—

101

C₂×#97

Dic₁₉

yes

—

102

C₂×C₄×C₈

—

103

C₄³

C₂×D₁₉

yes

characteristic

104

#336

—

105

#336

—

106

#336

—

107

#336

—

108

#336

—

109

#336

—

110

C₂×C₁₉

#1274

yes

characteristic

111

C₂×C₁₉

#1278

yes

characteristic

112

C₂×C₁₉

#1283

yes

characteristic

113

C₂×C₁₉

#1285

yes

characteristic

114

C₂×C₁₉

#1287

yes

characteristic

115

C₂×C₁₉

#1288

yes

characteristic

116

C₂×C₁₉

#1290

yes

characteristic

117

C₂²×C₈

—

118

C₂²×C₈

—

119

C₂²×C₈

—

120

C₂×C₄²

Dic₃₈

yes

characteristic

121

C₂×C₄²

C₄×D₁₉

yes

characteristic

122

C₂×C₄²

C₄×D₁₉

yes

characteristic

123

C₂×C₄²

D₇₆

yes

characteristic

124

C₂×C₄²

C₂×Dic₁₉

yes

characteristic

125

C₂×C₄²

C₂²⋊D₁₉

yes

characteristic

126

C₂×C₄²

C₂²⋊D₁₉

yes

characteristic

127

C₄⋊C₈

—

128

#97

—

129

#97

C₁₉⋊C₈

yes

complemented

130

C₄×C₈

—

131

C₁₉

#7023

yes

characteristic, complemented, Sylow

132

C₂²×C₄

C₄×Dic₁₉

yes

characteristic

133

C₂²×C₄

Dic₁₉⋊C₄

yes

characteristic

134

C₂²×C₄

Dic₁₉⋊C₄

yes

characteristic

135

C₂²×C₄

C₄⋊Dic₁₉

yes

characteristic

136

C₂²×C₄

(C₂×D₁₉)⋊C₄

yes

characteristic

137

C₂²×C₄

(C₂×D₁₉)⋊C₄

yes

characteristic

138

C₂²×C₄

C₂²⋊Dic₁₉

yes

characteristic

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

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

C₄²

—

152

C₄²

—

153

C₄²

—

154

C₄²

—

155

C₄²

C₂×C₁₉⋊C₈

yes

—

156

C₄²

#64520

yes

—

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

—

172

C₂×C₄

—

173

C₂×C₄

C₄×C₁₉⋊C₈

yes

characteristic

174

C₂×C₄

#10597260

yes

characteristic

175

C₂×C₄

C₄⋊(C₁₉⋊C₈)

yes

—

176

C₂×C₄

#10597279

yes

—

177

C₂×C₄

#10597280

yes

characteristic

178

C₂×C₄

#10597281

yes

characteristic

179

C₂×C₄

#10597282

yes

—

180

C₂×C₄

C₂²⋊(C₁₉⋊C₈)

yes

—

181

C₈

—

182

C₈

—

183

C₈

—

152

—

184

C₂²

#11926299

yes

characteristic

185

C₂²

#11926300

yes

characteristic

186

C₂²

#11926300

yes

characteristic

187

C₂²

#11926305

yes

characteristic

188

C₂²

#11926308

yes

characteristic

189

C₂²

#11926312

yes

characteristic

190

C₂²

#11926316

yes

characteristic

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

—

199

C₄

—

200

C₂

#14561464

yes

characteristic

201

C₂

#14561473

yes

characteristic

202

C₂

#14561486

yes

characteristic

203

C₂

#14561490

yes

characteristic

204

C₂

#14561496

yes

characteristic

205

C₂

#14561501

yes

characteristic

206

C₂

#14561504

yes

characteristic

	subgroup	G/H
center	C₂³	C₁₉⋊#95
commutator	C₂×C₄×C₁₉	C₄×C₈
Frattini	C₄³	C₂×D₁₉
Fitting	C₁₉×#1745	C₂
socle, prime radical	C₂³×C₁₉	#95
cosocle	C₄³×C₁₉	C₂²
hypercenter	#1745	D₁₉
hypocenter	C₁₉	#7023
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1745

subgroup

G/H

center

C₂³

C₁₉⋊#95

commutator

C₂×C₄×C₁₉

C₄×C₈

Frattini

C₄³

C₂×D₁₉

Fitting

C₁₉×#1745

C₂

socle, prime radical

C₂³×C₁₉

#95

cosocle

C₄³×C₁₉

C₂²

D₁₉

C₁₉

G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1745

N	H ≅ G/N	type
25. C₁₉×#97	C₈	semidirect
129. #97	C₁₉⋊C₈	semidirect
131. C₁₉	#7023	semidirect

H ≅ G/N

type

25. C₁₉×#97

C₈

semidirect

129. #97

C₁₉⋊C₈

semidirect

131. C₁₉

#7023

semidirect

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

subgroup	index
#14563778	2
C₄×C₄⋊(C₁₉⋊C₈)	2
C₁₉×#1745	2
#7023	19

subgroup

index

#14563778

C₄×C₄⋊(C₁₉⋊C₈)

C₁₉×#1745

#7023

Subgroups — #18069086

Conjugacy classes of subgroups

Complements

Maximal subgroups