Group #18069185 — subgroups

Subgroups — #18069185

Order 4864.

210 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 51 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	#14561559	C₂	1	1	yes	maximal, characteristic
2	#14561559	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1851	C₂	1	1	yes	maximal, characteristic
4	C₂²⋊(C₁₉⋊C₁₆)	—	2	1	no	—
5	C₂²⋊(C₁₉⋊C₁₆)	—	2	1	no	—
6	C₂²×M₁₆×C₁₉	C₄	1	1	yes	characteristic
7	C₂×C₁₉×C₄⋊C₈	—	2	1	no	—
8	C₂×C₁₉×C₂²⋊C₈	C₂²	1	1	yes	characteristic
9	C₄×C₂²⋊C₄×C₁₉	C₄	1	1	yes	characteristic
10	C₁₉×#336	—	2	1	no	—
11	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
12	C₂×M₁₆×C₁₉	—	4	2	no	—
13	C₂×M₁₆×C₁₉	—	4	1	no	—
14	C₂²×C₈×C₁₉	—	2	1	no	—
15	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
17	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
18	C₂×C₄²×C₁₉	—	2	1	no	—
19	C₁₉×C₄⋊C₈	—	4	2	no	—
20	C₁₉×C₂²⋊C₈	—	4	1	no	—
21	C₁₉×#95	C₈	1	1	yes	characteristic
22	C₁₉×#95	C₈	1	1	yes	characteristic
23	C₂×C₁₉⋊C₁₆	—	4	1	no	—
24	C₂×C₁₉⋊C₁₆	—	4	1	no	—
25	C₂⁴×C₁₉	M₁₆	1	1	yes	characteristic
26	C₂²×C₄×C₁₉	—	2	1	no	—
27	C₂²×C₄×C₁₉	—	2	1	no	—
28	C₂²×C₄×C₁₉	—	2	1	no	—
29	C₂²×C₄×C₁₉	—	2	1	no	—
30	C₂²×C₄×C₁₉	—	4	1	no	—
31	C₂²×C₄×C₁₉	—	4	1	no	—
32	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₉	C₂×C₈	1	1	yes	characteristic
34	M₁₆×C₁₉	—	4	2	no	—
35	M₁₆×C₁₉	—	4	2	no	—
36	C₂×C₈×C₁₉	—	2	2	no	—
37	C₂×C₈×C₁₉	—	2	1	no	—
38	C₂×C₈×C₁₉	—	2	1	no	—
39	C₂×C₈×C₁₉	—	4	1	no	—
40	C₂×C₈×C₁₉	—	4	1	no	—
41	C₂×C₈×C₁₉	—	4	1	no	—
42	C₂×C₈×C₁₉	—	4	1	no	—
43	C₂²⋊C₄×C₁₉	—	4	2	no	—
44	C₄²×C₁₉	—	4	1	no	—
45	C₄²×C₁₉	—	4	1	no	—
46	C₁₉⋊C₁₆	—	8	1	no	—
47	C₁₉⋊C₁₆	—	8	1	no	—
48	#7067	—	19	1	no	maximal, Sylow
49	C₂³×C₁₉	—	2	1	no	—
50	C₂³×C₁₉	—	2	1	no	—
51	C₂³×C₁₉	—	2	1	no	—
52	C₂³×C₁₉	—	4	1	no	—
53	C₂³×C₁₉	—	4	1	no	—
54	C₂³×C₁₉	C₂²⋊C₈	1	1	yes	characteristic
55	C₂×C₄×C₁₉	—	2	1	no	—
56	C₂×C₄×C₁₉	—	2	1	no	—
57	C₂×C₄×C₁₉	—	2	2	no	—
58	C₂×C₄×C₁₉	—	2	1	no	—
59	C₂×C₄×C₁₉	—	4	2	no	—
60	C₂×C₄×C₁₉	—	4	1	no	—
61	C₂×C₄×C₁₉	—	4	1	no	—
62	C₂×C₄×C₁₉	—	4	1	no	—
63	C₂×C₄×C₁₉	—	4	1	no	—
64	C₂×C₄×C₁₉	—	4	1	no	—
65	C₂×C₄×C₁₉	—	4	1	no	—
66	C₂×C₄×C₁₉	—	4	1	no	—
67	C₂×C₄×C₁₉	—	4	1	no	—
68	C₂×C₄×C₁₉	#99	1	1	yes	characteristic
69	C₂×C₄×C₁₉	#101	1	1	yes	characteristic
70	C₈×C₁₉	—	4	1	no	—
71	C₈×C₁₉	—	4	2	no	—
72	C₈×C₁₉	—	4	1	no	—
73	#1851	D₁₉	1	1	yes	characteristic
74	#1314	—	19	1	no	—
75	#1314	—	19	1	no	—
76	C₂²×C₁₉	—	2	1	no	—
77	C₂²×C₁₉	—	2	1	no	—
78	C₂²×C₁₉	—	4	1	no	—
79	C₂²×C₁₉	—	4	1	no	—
80	C₂²×C₁₉	—	4	1	no	—
81	C₂²×C₁₉	—	8	1	no	—
82	C₂²×C₁₉	—	8	1	no	—
83	C₂²×C₁₉	#324	1	1	yes	characteristic
84	C₂²×C₁₉	#350	1	1	yes	characteristic
85	C₂²×C₁₉	#350	1	1	yes	characteristic
86	C₄×C₁₉	—	2	1	no	—
87	C₄×C₁₉	—	4	2	no	—
88	C₄×C₁₉	—	4	1	no	—
89	C₄×C₁₉	—	8	1	no	—
90	C₄×C₁₉	AΣL(1,𝔽₁₆)	1	1	yes	characteristic
91	C₄×C₁₉	#352	1	1	yes	characteristic
92	C₂²×M₁₆	Dic₁₉	1	1	yes	characteristic
93	C₂×C₄⋊C₈	—	2	1	no	—
94	C₂×C₂²⋊C₈	C₂×D₁₉	1	1	yes	characteristic
95	C₄×C₂²⋊C₄	Dic₁₉	1	1	yes	characteristic
96	C₂²⋊C₁₆	—	38	1	no	—
97	C₂²⋊C₁₆	—	38	1	no	—
98	#336	—	2	1	no	—
99	C₂×C₁₉	—	2	1	no	—
100	C₂×C₁₉	—	2	1	no	—
101	C₂×C₁₉	—	8	1	no	—
102	C₂×C₁₉	#1323	1	1	yes	characteristic
103	C₂×C₁₉	#1326	1	1	yes	characteristic
104	C₂×C₁₉	#1327	1	1	yes	characteristic
105	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
106	C₂×M₁₆	—	4	2	no	—
107	C₂×M₁₆	—	4	1	no	—
108	C₂²×C₈	—	2	1	no	—
109	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
110	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
111	C₂×C₂²⋊C₄	—	2	1	no	—
112	C₂×C₄²	—	2	1	no	—
113	C₂×C₁₆	—	76	1	no	—
114	C₂×C₁₆	—	76	1	no	—
115	C₄⋊C₈	—	4	2	no	—
116	C₂²⋊C₈	—	4	1	no	—
117	#95	C₁₉⋊C₈	1	1	yes	characteristic
118	#95	C₁₉⋊C₈	1	1	yes	characteristic
119	C₁₉	#7067	1	1	yes	characteristic, complemented, Sylow
120	C₂⁴	#64520	1	1	yes	characteristic
121	C₂²×C₄	—	2	1	no	—
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	—	2	1	no	—
124	C₂²×C₄	—	2	1	no	—
125	C₂²×C₄	—	4	1	no	—
126	C₂²×C₄	—	4	1	no	—
127	C₂²×C₄	C₂×C₁₉⋊C₈	1	1	yes	characteristic
128	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
129	M₁₆	—	4	2	no	—
130	M₁₆	—	4	2	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	4	1	no	—
135	C₂×C₈	—	4	1	no	—
136	C₂×C₈	—	4	1	no	—
137	C₂×C₈	—	4	1	no	—
138	C₂²⋊C₄	—	4	2	no	—
139	C₄²	—	4	1	no	—
140	C₄²	—	4	1	no	—
141	C₁₆	—	152	1	no	—
142	C₁₆	—	152	1	no	—
143	C₂³	—	2	1	no	—
144	C₂³	—	2	1	no	—
145	C₂³	—	2	1	no	—
146	C₂³	—	4	1	no	—
147	C₂³	—	4	1	no	—
148	C₂³	C₂²⋊(C₁₉⋊C₈)	1	1	yes	characteristic
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	4	1	no	—
158	C₂×C₄	—	4	1	no	—
159	C₂×C₄	—	4	1	no	—
160	C₂×C₄	—	4	1	no	—
161	C₂×C₄	—	4	1	no	—
162	C₂×C₄	#10597291	1	1	yes	characteristic
163	C₂×C₄	#10597293	1	1	yes	characteristic
164	C₈	—	4	1	no	—
165	C₈	—	4	2	no	—
166	C₈	—	4	1	no	—
167	C₂²	—	2	1	no	—
168	C₂²	—	2	1	no	—
169	C₂²	—	4	1	no	—
170	C₂²	—	4	1	no	—
171	C₂²	—	4	1	no	—
172	C₂²	—	8	1	no	—
173	C₂²	—	8	1	no	—
174	C₂²	#11926270	1	1	yes	characteristic
175	C₂²	#11926328	1	1	yes	characteristic
176	C₂²	#11926328	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	4	2	no	—
179	C₄	—	4	1	no	—
180	C₄	—	8	1	no	—
181	C₄	#11926331	1	1	yes	characteristic
182	C₄	#11926334	1	1	yes	characteristic
183	C₂	—	2	1	no	—
184	C₂	—	2	1	no	—
185	C₂	—	8	1	no	—
186	C₂	#14561577	1	1	yes	characteristic
187	C₂	#14561583	1	1	yes	characteristic
188	C₂	#14561586	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14561559

C₂

yes

maximal, characteristic

#14561559

C₂

yes

maximal, characteristic

C₁₉×#1851

C₂

yes

maximal, characteristic

C₂²⋊(C₁₉⋊C₁₆)

—

C₂²⋊(C₁₉⋊C₁₆)

—

C₂²×M₁₆×C₁₉

C₄

yes

characteristic

C₂×C₁₉×C₄⋊C₈

—

C₂×C₁₉×C₂²⋊C₈

C₂²

yes

characteristic

C₄×C₂²⋊C₄×C₁₉

C₄

yes

characteristic

C₁₉×#336

—

C₂³×C₄×C₁₉

C₂×C₄

yes

characteristic

C₂×M₁₆×C₁₉

—

C₂×M₁₆×C₁₉

—

C₂²×C₈×C₁₉

—

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₂×C₂²⋊C₄×C₁₉

—

C₂×C₄²×C₁₉

—

C₁₉×C₄⋊C₈

—

C₁₉×C₂²⋊C₈

—

C₁₉×#95

C₈

yes

characteristic

C₁₉×#95

C₈

yes

characteristic

C₂×C₁₉⋊C₁₆

—

C₂×C₁₉⋊C₁₆

—

C₂⁴×C₁₉

M₁₆

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

yes

characteristic

C₂²×C₄×C₁₉

C₂×C₈

yes

characteristic

M₁₆×C₁₉

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₁₉⋊C₁₆

—

C₁₉⋊C₁₆

—

#7067

—

maximal, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

C₂²⋊C₈

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

yes

C₂×C₄×C₁₉

yes

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

#1851

D₁₉

yes

characteristic

#1314

—

#1314

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

AΣL(1,𝔽₁₆)

yes

characteristic

C₄×C₁₉

#352

yes

characteristic

C₂²×M₁₆

Dic₁₉

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₂²⋊C₈

C₂×D₁₉

yes

characteristic

C₄×C₂²⋊C₄

Dic₁₉

yes

characteristic

C₂²⋊C₁₆

—

C₂²⋊C₁₆

—

#336

—

C₂×C₁₉

—

100

C₂×C₁₉

—

101

C₂×C₁₉

—

102

C₂×C₁₉

#1323

yes

characteristic

103

C₂×C₁₉

#1326

yes

characteristic

104

C₂×C₁₉

#1327

yes

characteristic

105

C₂³×C₄

C₂×Dic₁₉

yes

characteristic

106

C₂×M₁₆

—

107

C₂×M₁₆

—

108

C₂²×C₈

—

109

C₂²×C₈

C₂²⋊D₁₉

yes

characteristic

110

C₂²×C₈

C₂²⋊D₁₉

yes

characteristic

111

C₂×C₂²⋊C₄

—

112

C₂×C₄²

—

113

C₂×C₁₆

—

114

C₂×C₁₆

—

115

C₄⋊C₈

—

116

C₂²⋊C₈

—

117

#95

C₁₉⋊C₈

yes

characteristic

118

#95

C₁₉⋊C₈

yes

characteristic

119

C₁₉

#7067

yes

characteristic, complemented, Sylow

120

C₂⁴

#64520

yes

characteristic

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

C₂×C₁₉⋊C₈

yes

characteristic

128

C₂²×C₄

C₂²⋊Dic₁₉

yes

characteristic

129

M₁₆

—

130

M₁₆

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂²⋊C₄

—

139

C₄²

—

140

C₄²

—

141

C₁₆

—

152

—

142

C₁₆

—

152

—

143

C₂³

—

144

C₂³

—

145

C₂³

—

146

C₂³

—

147

C₂³

—

148

C₂³

C₂²⋊(C₁₉⋊C₈)

yes

characteristic

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

#10597291

yes

characteristic

163

C₂×C₄

#10597293

yes

characteristic

164

C₈

—

165

C₈

—

166

C₈

—

167

C₂²

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

#11926270

yes

characteristic

175

C₂²

#11926328

yes

characteristic

176

C₂²

#11926328

yes

characteristic

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

#11926331

yes

characteristic

182

C₄

#11926334

yes

characteristic

183

C₂

—

184

C₂

—

185

C₂

—

186

C₂

#14561577

yes

characteristic

187

C₂

#14561583

yes

characteristic

188

C₂

#14561586

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#10597291
commutator	C₂²×C₄×C₁₉	C₂×C₈
Frattini	C₂×C₂²⋊C₈	C₂×D₁₉
Fitting	C₁₉×#1851	C₂
socle	C₂²×C₁₉	#324
prime radical	C₂⁴×C₁₉	M₁₆
cosocle	C₂×C₁₉×C₂²⋊C₈	C₂²
hypercenter	#1851	D₁₉
hypocenter	C₁₉	#7067
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂³×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈ ◁ #1851

subgroup

G/H

center

C₂×C₄

#10597291

commutator

C₂²×C₄×C₁₉

C₂×C₈

Frattini

C₂×C₂²⋊C₈

C₂×D₁₉

Fitting

C₁₉×#1851

C₂

socle

C₂²×C₁₉

#324

prime radical

C₂⁴×C₁₉

M₁₆

cosocle

C₂×C₁₉×C₂²⋊C₈

C₂²

D₁₉

C₁₉

G ▷ C₂²×C₄×C₁₉ ▷ C₂³×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈ ◁ #1851

N	H ≅ G/N	type
119. C₁₉	#7067	semidirect

H ≅ G/N

type

119. C₁₉

#7067

semidirect

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

subgroup	index
#14561559	2
C₁₉×#1851	2
#7067	19

subgroup

index

#14561559

C₁₉×#1851

#7067

Subgroups — #18069185

Conjugacy classes of subgroups

Complements

Maximal subgroups