Group #14564069 — subgroups

Subgroups — #14564069

Order 2432.

360 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 113 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×#64520	C₂	1	1	yes	maximal, characteristic
2	C₂×#10597292	C₂	1	2	yes	maximal, complemented
3	#11926404	C₂	1	1	yes	maximal, characteristic
4	#11926306	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₉×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₁₉	C₂²	1	1	yes	characteristic
7	C₂³×C₄×C₁₉	C₂²	1	1	yes	characteristic
8	C₁₉×C₂²⋊Q₈	—	2	2	no	—
9	C₁₉×C₂²⋊Q₈	C₄	1	4	yes	complemented
10	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
11	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
13	C₂×C₂²⋊Dic₁₉	—	2	1	no	—
14	C₂×C₄⋊Dic₁₉	C₂²	1	2	yes	—
15	C₂×#64520	—	2	2	no	—
16	C₂×#64520	—	2	4	no	—
17	C₂²×C₁₉⋊C₈	C₂²	1	2	yes	—
18	#10597292	—	2	4	no	—
19	C₁₉⋊#95	—	2	1	no	—
20	C₂⁴×C₁₉	D₄	1	1	yes	characteristic
21	C₂×Q₈×C₁₉	—	2	1	no	—
22	C₂×Q₈×C₁₉	—	2	2	no	—
23	C₂×Q₈×C₁₉	—	2	2	no	—
24	C₂×Q₈×C₁₉	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₁₉	—	2	2	no	—
26	C₂²×C₄×C₁₉	—	2	2	no	—
27	C₂²×C₄×C₁₉	—	2	1	no	—
28	C₂²×C₄×C₁₉	—	2	1	no	—
29	C₂²×C₄×C₁₉	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₁₉	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₉	D₄	1	2	yes	—
32	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
33	C₂²×Dic₁₉	—	2	2	no	—
34	C₄⋊C₄×C₁₉	—	2	4	no	—
35	C₄⋊C₄×C₁₉	—	2	1	no	—
36	C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
37	C₂²⋊C₄×C₁₉	—	2	4	no	—
38	C₂²⋊Dic₁₉	—	4	2	no	—
39	C₄⋊Dic₁₉	—	2	2	no	—
40	C₄⋊Dic₁₉	—	2	2	no	—
41	#64520	—	2	8	no	—
42	C₂×C₁₉⋊C₈	—	2	2	no	—
43	C₂×C₁₉⋊C₈	—	2	2	no	—
44	C₂×C₁₉⋊C₈	D₄	1	4	yes	—
45	C₂³×C₁₉	—	2	2	no	—
46	C₂³×C₁₉	—	2	1	no	—
47	C₂³×C₁₉	—	2	2	no	—
48	C₂³×C₁₉	—	2	1	no	—
49	C₂³×C₁₉	C₂²⋊C₄	1	2	yes	—
50	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
51	Q₈×C₁₉	—	2	4	no	—
52	Q₈×C₁₉	—	4	1	no	—
53	Q₈×C₁₉	—	4	1	no	—
54	C₂×C₄×C₁₉	—	2	2	no	—
55	C₂×C₄×C₁₉	—	2	4	no	—
56	C₂×C₄×C₁₉	—	2	2	no	—
57	C₂×C₄×C₁₉	—	2	2	no	—
58	C₂×C₄×C₁₉	—	2	2	no	—
59	C₂×C₄×C₁₉	—	2	2	no	—
60	C₂×C₄×C₁₉	—	4	2	no	—
61	C₂×C₄×C₁₉	—	4	1	no	—
62	C₂×C₄×C₁₉	—	4	1	no	—
63	C₂×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
64	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
66	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₁₉	#41	1	2	yes	—
68	C₂×Dic₁₉	—	4	2	no	—
69	C₂×Dic₁₉	—	4	2	no	—
70	C₂×Dic₁₉	—	4	2	no	—
71	C₁₉⋊C₈	—	2	4	no	—
72	#1894	—	19	1	no	maximal, Sylow
73	C₂²×C₁₉	—	2	2	no	—
74	C₂²×C₁₉	—	2	2	no	—
75	C₂²×C₁₉	—	2	2	no	—
76	C₂²×C₁₉	—	4	2	no	—
77	C₂²×C₁₉	—	4	2	no	—
78	C₂²×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₁₉	#121	1	2	yes	—
80	C₂²×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₂²×C₁₉	#137	1	2	yes	—
82	C₂²×C₁₉	#137	1	1	yes	characteristic
83	C₄×C₁₉	—	2	2	no	—
84	C₄×C₁₉	—	4	2	no	—
85	C₄×C₁₉	—	4	2	no	—
86	C₄×C₁₉	C₄×D₄	1	2	yes	—
87	C₄×C₁₉	#120	1	1	yes	characteristic
88	C₄×C₁₉	#123	1	1	yes	characteristic
89	Dic₁₉	—	8	2	no	—
90	C₂²×M₁₆	—	19	1	no	—
91	C₂×C₂²⋊Q₈	D₁₉	1	1	yes	characteristic
92	C₂×#103	—	19	2	no	—
93	#380	—	19	1	no	—
94	#340	—	19	2	no	—
95	C₂×C₁₉	—	4	2	no	—
96	C₂×C₁₉	#386	1	1	yes	characteristic
97	C₂×C₁₉	#417	1	1	yes	characteristic
98	C₂×C₁₉	#419	1	1	yes	characteristic
99	C₂×C₁₉	#468	1	2	yes	—
100	C₂×C₁₉	#470	1	2	yes	—
101	C₂²×Q₈	C₂×D₁₉	1	1	yes	characteristic
102	C₂³×C₄	C₂×D₁₉	1	1	yes	characteristic
103	C₂×M₁₆	—	38	4	no	—
104	C₂×M₁₆	—	38	2	no	—
105	C₂²×C₈	—	19	2	no	—
106	C₂²⋊Q₈	—	2	2	no	—
107	C₂²⋊Q₈	Dic₁₉	1	4	yes	complemented
108	C₂×C₄⋊C₄	—	2	1	no	—
109	C₂×C₄⋊C₄	—	19	2	no	—
110	C₂×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
111	C₂×C₂²⋊C₄	—	2	1	no	—
112	C₂×C₂²⋊C₄	—	38	1	no	—
113	#103	—	38	4	no	—
114	#95	—	38	1	no	—
115	C₁₉	#1894	1	1	yes	characteristic, complemented, Sylow
116	C₂⁴	C₂²⋊D₁₉	1	1	yes	characteristic
117	C₂×Q₈	—	2	1	no	—
118	C₂×Q₈	—	2	2	no	—
119	C₂×Q₈	—	2	2	no	—
120	C₂×Q₈	C₂×Dic₁₉	1	2	yes	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	2	2	no	—
123	C₂²×C₄	—	2	1	no	—
124	C₂²×C₄	—	2	1	no	—
125	C₂²×C₄	—	38	2	no	—
126	C₂²×C₄	C₂×Dic₁₉	1	2	yes	—
127	C₂²×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
128	C₂²×C₄	C₂²⋊D₁₉	1	2	yes	—
129	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
130	M₁₆	—	38	8	no	—
131	C₂×C₈	—	19	4	no	—
132	C₂×C₈	—	38	2	no	—
133	C₂×C₈	—	38	2	no	—
134	C₄⋊C₄	—	2	4	no	—
135	C₄⋊C₄	—	2	1	no	—
136	C₄⋊C₄	—	38	2	no	—
137	C₄⋊C₄	—	38	2	no	—
138	C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
139	C₂²⋊C₄	—	2	4	no	—
140	C₂²⋊C₄	—	76	2	no	—
141	C₂³	—	2	2	no	—
142	C₂³	—	2	2	no	—
143	C₂³	—	2	1	no	—
144	C₂³	—	2	1	no	—
145	C₂³	C₂²⋊Dic₁₉	1	2	yes	—
146	C₂³	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
147	Q₈	—	2	4	no	—
148	Q₈	—	4	1	no	—
149	Q₈	—	4	1	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	—	4	1	no	—
158	C₂×C₄	—	4	1	no	—
159	C₂×C₄	—	76	2	no	—
160	C₂×C₄	—	76	2	no	—
161	C₂×C₄	—	76	2	no	—
162	C₂×C₄	C₂²⋊Dic₁₉	1	2	yes	—
163	C₂×C₄	D₄×D₁₉	1	2	yes	—
164	C₂×C₄	#64543	1	2	yes	—
165	C₂×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
166	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
167	C₈	—	38	4	no	—
168	C₂²	—	2	2	no	—
169	C₂²	—	2	2	no	—
170	C₂²	—	2	2	no	—
171	C₂²	—	4	2	no	—
172	C₂²	—	4	2	no	—
173	C₂²	#10597385	1	2	yes	—
174	C₂²	#10597388	1	2	yes	—
175	C₂²	#10597395	1	1	yes	characteristic
176	C₂²	#10597397	1	1	yes	characteristic
177	C₂²	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
178	C₄	—	2	2	no	—
179	C₄	—	4	2	no	—
180	C₄	—	4	2	no	—
181	C₄	—	152	2	no	—
182	C₄	D₄×Dic₁₉	1	2	yes	—
183	C₄	#10597381	1	1	yes	characteristic
184	C₄	#10597383	1	1	yes	characteristic
185	C₂	—	4	2	no	—
186	C₂	#11926424	1	1	yes	characteristic
187	C₂	#11926545	1	1	yes	characteristic
188	C₂	#11926553	1	1	yes	characteristic
189	C₂	#11926792	1	2	yes	—
190	C₂	#11926800	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#64520

C₂

yes

maximal, characteristic

C₂×#10597292

C₂

yes

maximal, complemented

#11926404

C₂

yes

maximal, characteristic

#11926306

C₂

yes

maximal, complemented

C₂×C₁₉×C₂²⋊Q₈

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₉

C₂²

yes

characteristic

C₂³×C₄×C₁₉

C₂²

yes

characteristic

C₁₉×C₂²⋊Q₈

—

C₁₉×C₂²⋊Q₈

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₁₉

—

C₂×C₂²⋊Dic₁₉

—

C₂×C₄⋊Dic₁₉

C₂²

yes

—

C₂×#64520

—

C₂×#64520

—

C₂²×C₁₉⋊C₈

C₂²

yes

—

#10597292

—

C₁₉⋊#95

—

C₂⁴×C₁₉

D₄

yes

characteristic

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

C₂×C₄

yes

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂×C₄

yes

—

C₂²×C₄×C₁₉

D₄

yes

characteristic

C₂²×C₄×C₁₉

D₄

yes

—

C₂²×C₄×C₁₉

C₂³

yes

characteristic

C₂²×Dic₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

C₂×C₄

yes

—

C₂²⋊C₄×C₁₉

—

C₂²⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

#64520

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

D₄

yes

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

C₂²⋊C₄

yes

—

C₂³×C₁₉

C₂×D₄

yes

characteristic

Q₈×C₁₉

—

Q₈×C₁₉

—

Q₈×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

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉

C₂×D₄

yes

—

C₂×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉

#41

yes

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₁₉⋊C₈

—

#1894

—

maximal, Sylow

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₉

#121

yes

—

C₂²×C₁₉

AGL(1,ℤ/8)

yes

characteristic

C₂²×C₁₉

#137

yes

—

C₂²×C₁₉

#137

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

C₄×D₄

yes

—

C₄×C₁₉

yes

C₄×C₁₉

yes

Dic₁₉

—

C₂²×M₁₆

—

C₂×C₂²⋊Q₈

D₁₉

yes

—

—

—

C₂×C₁₉

—

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

—

100

C₂×C₁₉

#470

yes

—

101

C₂²×Q₈

C₂×D₁₉

yes

characteristic

102

C₂³×C₄

C₂×D₁₉

yes

characteristic

103

C₂×M₁₆

—

104

C₂×M₁₆

—

105

C₂²×C₈

—

106

C₂²⋊Q₈

—

107

C₂²⋊Q₈

Dic₁₉

yes

complemented

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

C₂×D₁₉

yes

characteristic

111

C₂×C₂²⋊C₄

—

112

C₂×C₂²⋊C₄

—

113

#103

—

114

#95

—

115

C₁₉

#1894

yes

characteristic, complemented, Sylow

116

C₂⁴

C₂²⋊D₁₉

yes

characteristic

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂×Q₈

C₂×Dic₁₉

yes

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

C₂×Dic₁₉

yes

—

127

C₂²×C₄

C₂²⋊D₁₉

yes

characteristic

128

C₂²×C₄

C₂²⋊D₁₉

yes

—

129

C₂²×C₄

C₂²×D₁₉

yes

characteristic

130

M₁₆

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

C₂×Dic₁₉

yes

—

139

C₂²⋊C₄

—

140

C₂²⋊C₄

—

141

C₂³

—

142

C₂³

—

143

C₂³

—

144

C₂³

—

145

C₂³

C₂²⋊Dic₁₉

yes

—

146

C₂³

C₂×C₂²⋊D₁₉

yes

characteristic

147

Q₈

—

148

Q₈

—

149

Q₈

—

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

C₂²⋊Dic₁₉

yes

—

163

C₂×C₄

D₄×D₁₉

yes

—

164

C₂×C₄

#64543

yes

—

165

C₂×C₄

C₂²×Dic₁₉

yes

characteristic

166

C₂×C₄

C₂×C₂²⋊D₁₉

yes

characteristic

167

C₈

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

—

173

C₂²

#10597385

yes

—

174

C₂²

#10597388

yes

—

175

C₂²

#10597395

yes

characteristic

176

C₂²

#10597397

yes

characteristic

177

C₂²

C₂×C₂²⋊Dic₁₉

yes

characteristic

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

—

152

—

182

C₄

D₄×Dic₁₉

yes

—

183

C₄

#10597381

yes

characteristic

184

C₄

#10597383

yes

characteristic

185

C₂

—

186

C₂

#11926424

yes

characteristic

187

C₂

#11926545

yes

characteristic

188

C₂

#11926553

yes

characteristic

189

C₂

#11926792

yes

—

190

C₂

#11926800

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₉
commutator	C₂×C₄×C₁₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉
Fitting	C₂×C₁₉×C₂²⋊Q₈	C₂
socle	C₂³×C₁₉	C₂×D₄
prime radical	C₂⁴×C₁₉	D₄
cosocle	C₂²×C₄×C₁₉	C₂³
hypercenter	C₂×C₂²⋊Q₈	D₁₉
hypocenter	C₁₉	#1894
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₂²⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₉

commutator

C₂×C₄×C₁₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉

Fitting

C₂×C₁₉×C₂²⋊Q₈

C₂

socle

C₂³×C₁₉

C₂×D₄

prime radical

C₂⁴×C₁₉

D₄

cosocle

C₂²×C₄×C₁₉

C₂³

hypercenter

C₂×C₂²⋊Q₈

D₁₉

hypocenter

C₁₉

#1894

LCS

G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₂²⋊Q₈

N	H ≅ G/N	type
2. C₂×#10597292	C₂	semidirect
4. #11926306	C₂	semidirect
9. C₁₉×C₂²⋊Q₈	C₄	semidirect
107. C₂²⋊Q₈	Dic₁₉	semidirect
115. C₁₉	#1894	semidirect

H ≅ G/N

type

2. C₂×#10597292

C₂

semidirect

4. #11926306

C₂

semidirect

9. C₁₉×C₂²⋊Q₈

C₄

semidirect

107. C₂²⋊Q₈

Dic₁₉

semidirect

115. C₁₉

#1894

semidirect

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

subgroup	index
C₂²×#64520	2
C₂×#10597292	2
#11926404	2
#11926306	2
C₂×C₁₉×C₂²⋊Q₈	2
#1894	19

subgroup

index

C₂²×#64520

C₂×C₁₉×C₂²⋊Q₈

Subgroups — #14564069

Conjugacy classes of subgroups

Complements

Maximal subgroups