Group #14569118 — subgroups

Subgroups — #14569118

Order 2432.

520 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 153 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	C₂×#10597321	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₃₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926471	C₂	1	2	yes	maximal, complemented
4	#11926457	C₂	1	2	yes	maximal, complemented
5	#11926449	C₂	1	2	yes	maximal, complemented
6	#11926445	C₂	1	2	yes	maximal
7	#11926435	C₂	1	4	yes	maximal, complemented
8	C₄×C₄⋊C₄×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₇₆	C₂²	1	1	yes	characteristic
10	C₂²×Dic₃₈	C₂²	1	1	yes	characteristic
11	C₂²×Dic₃₈	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
15	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	2	yes	—
16	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	2	yes	—
17	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	4	yes	—
18	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	4	yes	—
19	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
20	C₂×Dic₁₉⋊C₄	C₂²	1	2	yes	—
21	C₂×C₄×Dic₁₉	C₂²	1	2	yes	—
22	#10597321	—	2	4	no	—
23	C₄×Dic₃₈	—	2	4	no	—
24	C₁₉×#95	C₂²	1	2	yes	—
25	C₁₉⋊#95	C₂²	1	4	yes	—
26	C₁₉⋊#95	C₂²	1	4	yes	—
27	C₂³×D₁₉	C₂³	1	2	yes	—
28	C₂²×C₄×C₁₉	C₂³	1	4	yes	—
29	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
32	C₂²×Dic₁₉	C₂³	1	2	yes	—
33	C₂²×Dic₁₉	C₂³	1	2	yes	—
34	C₂²×Dic₁₉	C₂³	1	2	yes	—
35	C₂×D₇₆	—	2	2	no	—
36	C₂×D₇₆	—	2	4	no	—
37	C₂×Dic₃₈	—	2	2	no	—
38	C₂×Dic₃₈	—	2	4	no	—
39	C₂×Dic₃₈	—	2	2	no	—
40	C₂×Dic₃₈	—	2	4	no	—
41	C₄⋊C₄×C₁₉	—	2	4	no	—
42	C₄²×C₁₉	—	4	2	no	—
43	C₄²×C₁₉	D₄	1	4	yes	—
44	(C₂×D₁₉)⋊C₄	—	2	4	no	—
45	(C₂×D₁₉)⋊C₄	—	2	4	no	—
46	(C₂×D₁₉)⋊C₄	—	2	8	no	—
47	(C₂×D₁₉)⋊C₄	—	4	4	no	—
48	C₄⋊Dic₁₉	—	2	2	no	—
49	Dic₁₉⋊C₄	—	2	4	no	—
50	C₄×Dic₁₉	—	2	4	no	—
51	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
52	C₂²×D₁₉	—	2	2	no	—
53	C₂²×D₁₉	—	2	4	no	—
54	C₂²×D₁₉	—	2	4	no	—
55	C₂²×D₁₉	—	2	2	no	—
56	C₂²×D₁₉	—	2	2	no	—
57	C₂×C₄×C₁₉	—	2	2	no	—
58	C₂×C₄×C₁₉	—	2	1	no	—
59	C₂×C₄×C₁₉	—	2	1	no	—
60	C₂×C₄×C₁₉	—	2	4	no	—
61	C₂×C₄×C₁₉	—	2	4	no	—
62	C₂×C₄×C₁₉	—	2	4	no	—
63	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
64	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
66	C₂×C₄×C₁₉	#41	1	4	yes	—
67	C₂×Dic₁₉	—	2	2	no	—
68	C₂×Dic₁₉	—	2	2	no	—
69	C₂×Dic₁₉	—	2	2	no	—
70	C₂×Dic₁₉	—	2	2	no	—
71	C₂×Dic₁₉	—	2	4	no	—
72	C₂×Dic₁₉	—	2	2	no	—
73	C₂×Dic₁₉	—	2	2	no	—
74	C₂×Dic₁₉	#41	1	4	yes	—
75	D₇₆	—	4	4	no	—
76	Dic₃₈	—	2	8	no	—
77	Dic₃₈	—	4	4	no	—
78	#2592	—	19	1	no	maximal, Sylow
79	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
80	C₂²×C₁₉	C₂×#41	1	2	yes	—
81	C₂²×C₁₉	C₂×#41	1	2	yes	—
82	C₂²×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
83	C₂²×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
84	C₂×D₁₉	—	4	2	no	—
85	C₂×D₁₉	—	4	2	no	—
86	C₂×D₁₉	—	4	2	no	—
87	C₂×D₁₉	—	4	4	no	—
88	C₂×D₁₉	—	4	4	no	—
89	C₄×C₁₉	—	2	2	no	—
90	C₄×C₁₉	—	2	2	no	—
91	C₄×C₁₉	—	4	4	no	—
92	C₄×C₁₉	#124	1	4	yes	—
93	Dic₁₉	—	2	4	no	—
94	Dic₁₉	—	4	2	no	—
95	Dic₁₉	—	4	2	no	—
96	C₂×#124	—	19	1	no	—
97	C₂×C₄×Q₈	—	19	1	no	—
98	#397	—	19	2	no	—
99	#393	—	19	2	no	—
100	#391	—	19	2	no	—
101	#390	—	19	2	no	—
102	#388	—	19	4	no	—
103	C₄×C₄⋊C₄	D₁₉	1	1	yes	characteristic, complemented
104	C₂×C₁₉	#525	1	1	yes	characteristic
105	C₂×C₁₉	C₂×#124	1	1	yes	characteristic
106	C₂×C₁₉	#537	1	1	yes	characteristic
107	C₂×C₁₉	#556	1	2	yes	—
108	C₂×C₁₉	#559	1	2	yes	—
109	D₁₉	—	8	2	no	—
110	C₂²×Q₈	—	19	1	no	—
111	C₂²×Q₈	—	19	1	no	—
112	C₂²×D₄	—	19	1	no	—
113	#124	—	38	4	no	—
114	C₄×Q₈	—	38	4	no	—
115	C₂×C₄⋊C₄	—	19	2	no	—
116	C₂×C₄⋊C₄	—	19	1	no	—
117	C₂×C₄⋊C₄	C₂×D₁₉	1	2	yes	—
118	C₂×C₂²⋊C₄	—	19	2	no	—
119	C₂×C₂²⋊C₄	—	19	2	no	—
120	C₂×C₂²⋊C₄	—	19	4	no	—
121	C₂×C₂²⋊C₄	—	19	4	no	—
122	C₂×C₄²	—	19	2	no	—
123	C₂×C₄²	C₂×D₁₉	1	2	yes	—
124	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
125	#95	—	19	4	no	—
126	#95	—	19	4	no	—
127	#95	C₂×D₁₉	1	2	yes	—
128	C₁₉	#2592	1	1	yes	characteristic, complemented, Sylow
129	C₂⁴	—	19	2	no	—
130	C₂×Q₈	—	38	4	no	—
131	C₂×Q₈	—	38	2	no	—
132	C₂×Q₈	—	38	2	no	—
133	C₂×Q₈	—	38	4	no	—
134	C₂×D₄	—	38	2	no	—
135	C₂×D₄	—	38	4	no	—
136	C₂²×C₄	—	19	2	no	—
137	C₂²×C₄	—	19	2	no	—
138	C₂²×C₄	—	19	2	no	—
139	C₂²×C₄	C₂²×D₁₉	1	4	yes	—
140	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
141	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
142	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
143	C₄⋊C₄	—	2	4	no	—
144	C₄⋊C₄	—	38	4	no	—
145	C₄⋊C₄	—	38	2	no	—
146	C₂²⋊C₄	—	38	8	no	—
147	C₂²⋊C₄	—	38	4	no	—
148	C₂²⋊C₄	—	38	4	no	—
149	C₂²⋊C₄	—	76	4	no	—
150	C₄²	—	4	2	no	—
151	C₄²	—	38	4	no	—
152	C₄²	D₇₆	1	4	yes	—
153	C₂³	—	38	2	no	—
154	C₂³	—	38	4	no	—
155	C₂³	—	38	2	no	—
156	C₂³	—	38	2	no	—
157	C₂³	—	38	4	no	—
158	C₂³	C₂³×D₁₉	1	1	yes	characteristic
159	Q₈	—	38	8	no	—
160	Q₈	—	76	4	no	—
161	D₄	—	76	4	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	19	4	no	—
169	C₂×C₄	—	38	2	no	—
170	C₂×C₄	—	38	2	no	—
171	C₂×C₄	—	38	2	no	—
172	C₂×C₄	—	38	2	no	—
173	C₂×C₄	—	38	4	no	—
174	C₂×C₄	—	38	2	no	—
175	C₂×C₄	—	38	2	no	—
176	C₂×C₄	C₂×D₇₆	1	2	yes	—
177	C₂×C₄	C₂×D₇₆	1	2	yes	—
178	C₂×C₄	C₂×D₇₆	1	2	yes	—
179	C₂×C₄	#64541	1	4	yes	—
180	C₂²	—	76	2	no	—
181	C₂²	—	76	2	no	—
182	C₂²	—	76	2	no	—
183	C₂²	—	76	4	no	—
184	C₂²	—	76	4	no	—
185	C₂²	C₂²×D₇₆	1	1	yes	characteristic
186	C₂²	C₂×#64541	1	2	yes	—
187	C₂²	#10597430	1	1	yes	characteristic
188	C₂²	#10597433	1	1	yes	characteristic
189	C₂²	#41×D₁₉	1	2	yes	—
190	C₄	—	2	2	no	—
191	C₄	—	2	2	no	—
192	C₄	—	4	4	no	—
193	C₄	—	38	4	no	—
194	C₄	—	76	2	no	—
195	C₄	—	76	2	no	—
196	C₄	#10597321	1	4	yes	—
197	C₂	—	152	2	no	—
198	C₂	#11927056	1	1	yes	characteristic
199	C₂	C₂×#10597321	1	1	yes	characteristic
200	C₂	#11927125	1	1	yes	characteristic
201	C₂	#11927266	1	2	yes	—
202	C₂	#11927277	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10597321

C₂

yes

maximal, characteristic

C₂×C₄×Dic₃₈

C₂

yes

maximal, characteristic, complemented

#11926471

C₂

yes

maximal, complemented

#11926457

C₂

yes

maximal, complemented

#11926449

C₂

yes

maximal, complemented

#11926445

C₂

yes

maximal

#11926435

C₂

yes

maximal, complemented

C₄×C₄⋊C₄×C₁₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₇₆

C₂²

yes

characteristic

C₂²×Dic₃₈

C₂²

yes

characteristic

C₂²×Dic₃₈

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₉

C₂²

yes

—

C₂×C₄²×C₁₉

C₂²

yes

—

C₂×C₄²×C₁₉

C₂²

yes

characteristic

C₂×(C₂×D₁₉)⋊C₄

C₂²

yes

—

C₂×(C₂×D₁₉)⋊C₄

C₂²

yes

—

C₂×(C₂×D₁₉)⋊C₄

C₂²

yes

—

C₂×(C₂×D₁₉)⋊C₄

C₂²

yes

—

C₂×C₄⋊Dic₁₉

C₂²

yes

characteristic

C₂×Dic₁₉⋊C₄

C₂²

yes

—

C₂×C₄×Dic₁₉

C₂²

yes

—

#10597321

—

C₄×Dic₃₈

—

C₁₉×#95

C₂²

yes

—

C₁₉⋊#95

C₂²

yes

—

C₁₉⋊#95

C₂²

yes

—

C₂³×D₁₉

C₂³

yes

—

C₂²×C₄×C₁₉

C₂³

yes

—

C₂²×C₄×C₁₉

C₂³

yes

characteristic

C₂²×C₄×C₁₉

C₂³

yes

characteristic

C₂²×C₄×C₁₉

C₂³

yes

characteristic

C₂²×Dic₁₉

C₂³

yes

—

C₂²×Dic₁₉

C₂³

yes

—

C₂²×Dic₁₉

C₂³

yes

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₄⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

D₄

yes

—

(C₂×D₁₉)⋊C₄

—

(C₂×D₁₉)⋊C₄

—

(C₂×D₁₉)⋊C₄

—

(C₂×D₁₉)⋊C₄

—

C₄⋊Dic₁₉

—

Dic₁₉⋊C₄

—

C₄×Dic₁₉

—

C₂³×C₁₉

C₂⁴

yes

characteristic

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂×D₄

yes

—

C₂×C₄×C₁₉

C₂×D₄

yes

—

C₂×C₄×C₁₉

C₂×D₄

yes

—

C₂×C₄×C₁₉

#41

yes

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

#41

yes

—

D₇₆

—

Dic₃₈

—

Dic₃₈

—

#2592

—

maximal, Sylow

C₂²×C₁₉

C₂²×D₄

yes

characteristic

C₂²×C₁₉

C₂×#41

yes

—

C₂²×C₁₉

C₂×#41

yes

—

C₂²×C₁₉

H(5,𝔽₂)

yes

characteristic

C₂²×C₁₉

2₋¹⁺⁴

yes

characteristic

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

#124

yes

—

Dic₁₉

—

Dic₁₉

—

Dic₁₉

—

C₂×#124

—

C₂×C₄×Q₈

—

#397

—

#393

—

100

#391

—

101

#390

—

102

#388

—

103

C₄×C₄⋊C₄

D₁₉

yes

characteristic, complemented

104

C₂×C₁₉

#525

yes

characteristic

105

C₂×C₁₉

C₂×#124

yes

characteristic

106

C₂×C₁₉

#537

yes

characteristic

107

C₂×C₁₉

#556

yes

—

108

C₂×C₁₉

#559

yes

—

109

D₁₉

—

110

C₂²×Q₈

—

111

C₂²×Q₈

—

112

C₂²×D₄

—

113

#124

—

114

C₄×Q₈

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄⋊C₄

C₂×D₁₉

yes

—

118

C₂×C₂²⋊C₄

—

119

C₂×C₂²⋊C₄

—

120

C₂×C₂²⋊C₄

—

121

C₂×C₂²⋊C₄

—

122

C₂×C₄²

—

123

C₂×C₄²

C₂×D₁₉

yes

—

124

C₂×C₄²

C₂×D₁₉

yes

characteristic

125

#95

—

126

#95

—

127

#95

C₂×D₁₉

yes

—

128

C₁₉

#2592

yes

characteristic, complemented, Sylow

129

C₂⁴

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×D₄

—

135

C₂×D₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

C₂²×D₁₉

yes

—

140

C₂²×C₄

C₂²×D₁₉

yes

characteristic

141

C₂²×C₄

C₂²×D₁₉

yes

characteristic

142

C₂²×C₄

C₂²×D₁₉

yes

characteristic

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₂²⋊C₄

—

147

C₂²⋊C₄

—

148

C₂²⋊C₄

—

149

C₂²⋊C₄

—

150

C₄²

—

151

C₄²

—

152

C₄²

D₇₆

yes

—

153

C₂³

—

154

C₂³

—

155

C₂³

—

156

C₂³

—

157

C₂³

—

158

C₂³

C₂³×D₁₉

yes

characteristic

159

Q₈

—

160

Q₈

—

161

D₄

—

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

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

C₂×D₇₆

yes

—

177

C₂×C₄

C₂×D₇₆

yes

—

178

C₂×C₄

C₂×D₇₆

yes

—

179

C₂×C₄

#64541

yes

—

180

C₂²

—

181

C₂²

—

182

C₂²

—

183

C₂²

—

184

C₂²

—

185

C₂²

C₂²×D₇₆

yes

characteristic

186

C₂²

C₂×#64541

yes

—

187

C₂²

#10597430

yes

characteristic

188

C₂²

#10597433

yes

characteristic

189

C₂²

#41×D₁₉

yes

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

#10597321

yes

—

197

C₂

—

152

—

198

C₂

#11927056

yes

characteristic

199

C₂

C₂×#10597321

yes

characteristic

200

C₂

#11927125

yes

characteristic

201

C₂

#11927266

yes

—

202

C₂

#11927277

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₉
commutator, socle, cosocle	C₂³×C₁₉	C₂⁴
Fitting	C₄×C₄⋊C₄×C₁₉	C₂
prime radical	C₂²×D₇₆	C₂²
hypercenter	C₄×C₄⋊C₄	D₁₉
hypocenter	C₁₉	#2592
LCS	G ▷ C₂³×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₉

commutator, socle, cosocle

C₂³×C₁₉

C₂⁴

Fitting

C₄×C₄⋊C₄×C₁₉

C₂

prime radical

C₂²×D₇₆

C₂²

hypercenter

C₄×C₄⋊C₄

D₁₉

C₁₉

G ▷ C₂³×C₁₉ ▷ C₁₉

C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
2. C₂×C₄×Dic₃₈	C₂	semidirect
3. #11926471	C₂	semidirect
4. #11926457	C₂	semidirect
5. #11926449	C₂	semidirect
7. #11926435	C₂	semidirect
8. C₄×C₄⋊C₄×C₁₉	C₂	semidirect
103. C₄×C₄⋊C₄	D₁₉	semidirect
128. C₁₉	#2592	semidirect

H ≅ G/N

type

2. C₂×C₄×Dic₃₈

C₂

semidirect

3. #11926471

C₂

semidirect

4. #11926457

C₂

semidirect

5. #11926449

C₂

semidirect

7. #11926435

C₂

semidirect

8. C₄×C₄⋊C₄×C₁₉

C₂

semidirect

103. C₄×C₄⋊C₄

D₁₉

semidirect

128. C₁₉

#2592

semidirect

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

subgroup	index
C₂×#10597321	2
C₂×C₄×Dic₃₈	2
#11926471	2
#11926457	2
#11926449	2
#11926445	2
#11926435	2
C₄×C₄⋊C₄×C₁₉	2
#2592	19

subgroup

index

C₂×C₄×Dic₃₈

C₄×C₄⋊C₄×C₁₉

Subgroups — #14569118

Conjugacy classes of subgroups

Complements

Maximal subgroups