Group #14569448 — subgroups

Subgroups — #14569448

Order 2432.

508 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 165 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×#10597370	C₂	1	1	yes	maximal, characteristic
2	#11926483	C₂	1	2	yes	maximal, complemented
3	#11926477	C₂	1	4	yes	maximal, complemented
4	#11926464	C₂	1	2	yes	maximal, complemented
5	#11926417	C₂	1	2	yes	maximal
6	#11926407	C₂	1	2	yes	maximal
7	#11926404	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₄⋊C₄×C₁₉	C₂	1	1	yes	maximal, characteristic
9	C₂³×C₄×C₁₉	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₁₉	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₁₉	C₂²	1	1	yes	characteristic
12	C₂²×Dic₃₈	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
14	C₂×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
16	C₂×C₂²⋊Dic₁₉	C₂²	1	2	yes	—
17	C₂×C₂²⋊Dic₁₉	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₁₉	C₂²	1	2	yes	—
20	#10597370	—	2	4	no	—
21	C₂×Dic₁₉⋊C₄	C₂²	1	2	yes	—
22	C₂×Dic₁₉⋊C₄	C₂²	1	4	yes	—
23	C₁₉⋊#95	C₂²	1	2	yes	—
24	C₁₉⋊#95	C₂²	1	4	yes	—
25	C₁₉⋊#95	C₂²	1	4	yes	—
26	C₁₉⋊#95	C₂²	1	4	yes	—
27	C₂⁴×C₁₉	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₁₉	—	2	2	no	—
29	C₂²×C₄×C₁₉	—	2	2	no	—
30	C₂²×C₄×C₁₉	—	2	2	no	—
31	C₂²×C₄×C₁₉	—	4	2	no	—
32	C₂²×C₄×C₁₉	—	4	2	no	—
33	C₂²×C₄×C₁₉	D₄	1	4	yes	—
34	C₂²×C₄×C₁₉	Q₈	1	4	yes	complemented
35	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
36	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
39	C₂²×Dic₁₉	C₂³	1	2	yes	—
40	C₂²×Dic₁₉	C₂³	1	2	yes	—
41	C₂²×Dic₁₉	C₂³	1	4	yes	—
42	C₂×Dic₃₈	—	2	2	no	—
43	C₂×Dic₃₈	—	4	2	no	—
44	C₄⋊C₄×C₁₉	—	2	4	no	—
45	C₄⋊C₄×C₁₉	—	2	4	no	—
46	C₂²⋊Dic₁₉	—	2	4	no	—
47	C₂²⋊Dic₁₉	—	2	4	no	—
48	C₄⋊Dic₁₉	—	2	2	no	—
49	C₄⋊Dic₁₉	—	4	2	no	—
50	Dic₁₉⋊C₄	—	2	4	no	—
51	Dic₁₉⋊C₄	—	4	4	no	—
52	C₂³×C₁₉	—	2	2	no	—
53	C₂³×C₁₉	—	2	2	no	—
54	C₂³×C₁₉	C₂×D₄	1	2	yes	—
55	C₂³×C₁₉	C₂×Q₈	1	2	yes	—
56	C₂³×C₁₉	#41	1	2	yes	—
57	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
58	C₂×C₄×C₁₉	—	2	8	no	—
59	C₂×C₄×C₁₉	—	2	2	no	—
60	C₂×C₄×C₁₉	—	2	2	no	—
61	C₂×C₄×C₁₉	—	2	2	no	—
62	C₂×C₄×C₁₉	—	2	2	no	—
63	C₂×C₄×C₁₉	—	2	2	no	—
64	C₂×C₄×C₁₉	—	2	4	no	—
65	C₂×C₄×C₁₉	—	4	4	no	—
66	C₂×C₄×C₁₉	—	4	4	no	—
67	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
68	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₁₉	C₂×Q₈	1	4	yes	—
70	C₂×Dic₁₉	—	2	2	no	—
71	C₂×Dic₁₉	—	2	2	no	—
72	C₂×Dic₁₉	—	2	4	no	—
73	C₂×Dic₁₉	—	2	4	no	—
74	C₂×Dic₁₉	—	2	4	no	—
75	C₂×Dic₁₉	—	2	4	no	—
76	C₂×Dic₁₉	—	2	4	no	—
77	Dic₃₈	—	4	4	no	—
78	#2627	—	19	1	no	maximal, Sylow
79	C₂²×C₁₉	—	2	2	no	—
80	C₂²×C₁₉	—	2	2	no	—
81	C₂²×C₁₉	—	2	4	no	—
82	C₂²×C₁₉	—	2	4	no	—
83	C₂²×C₁₉	C₂²⋊Q₈	1	4	yes	—
84	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
85	C₂²×C₁₉	C₂²×Q₈	1	1	yes	characteristic
86	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
87	C₂²×C₁₉	H(5,𝔽₂)	1	2	yes	—
88	C₂²×C₁₉	2₋¹⁺⁴	1	2	yes	—
89	C₄×C₁₉	—	2	4	no	—
90	C₄×C₁₉	—	2	2	no	—
91	C₄×C₁₉	—	2	2	no	—
92	C₄×C₁₉	—	4	2	no	—
93	Dic₁₉	—	4	2	no	—
94	Dic₁₉	—	4	2	no	—
95	Dic₁₉	—	4	4	no	—
96	C₂×C₂²⋊Q₈	—	19	1	no	—
97	C₂²×C₄⋊C₄	D₁₉	1	1	yes	characteristic
98	#400	—	19	2	no	—
99	#398	—	19	4	no	—
100	#395	—	19	2	no	—
101	#385	—	19	2	no	—
102	#381	—	19	2	no	—
103	#380	—	19	1	no	—
104	C₂×C₁₉	—	2	4	no	—
105	C₂×C₁₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
106	C₂×C₁₉	#534	1	1	yes	characteristic
107	C₂×C₁₉	#536	1	1	yes	characteristic
108	C₂×C₁₉	#539	1	2	yes	—
109	C₂×C₁₉	#544	1	2	yes	—
110	C₂²×Q₈	—	19	1	no	—
111	C₂³×C₄	C₂×D₁₉	1	1	yes	characteristic
112	C₂³×C₄	C₂×D₁₉	1	1	yes	characteristic
113	C₂³×C₄	C₂×D₁₉	1	1	yes	characteristic
114	C₂²⋊Q₈	—	38	4	no	—
115	C₂×C₄⋊C₄	—	2	4	no	—
116	C₂×C₄⋊C₄	—	19	2	no	—
117	C₂×C₄⋊C₄	—	19	1	no	—
118	C₂×C₄⋊C₄	—	19	2	no	—
119	C₂×C₄⋊C₄	—	19	4	no	—
120	C₂×C₄⋊C₄	C₂×D₁₉	1	2	yes	—
121	C₂×C₄⋊C₄	C₂×D₁₉	1	2	yes	—
122	C₂×C₂²⋊C₄	—	19	2	no	—
123	C₂×C₂²⋊C₄	—	19	2	no	—
124	#95	—	19	2	no	—
125	#95	—	19	4	no	—
126	#95	—	19	4	no	—
127	#95	—	19	4	no	—
128	C₁₉	#2627	1	1	yes	characteristic, complemented, Sylow
129	C₂⁴	C₂²×D₁₉	1	1	yes	characteristic
130	C₂×Q₈	—	38	2	no	—
131	C₂×Q₈	—	76	2	no	—
132	C₂²×C₄	—	2	2	no	—
133	C₂²×C₄	—	2	2	no	—
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	—	4	2	no	—
136	C₂²×C₄	—	4	2	no	—
137	C₂²×C₄	—	19	4	no	—
138	C₂²×C₄	—	19	2	no	—
139	C₂²×C₄	—	19	2	no	—
140	C₂²×C₄	Dic₃₈	1	4	yes	complemented
141	C₂²×C₄	C₂²⋊D₁₉	1	4	yes	—
142	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
143	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
144	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
145	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	—	2	4	no	—
148	C₄⋊C₄	—	38	4	no	—
149	C₄⋊C₄	—	38	2	no	—
150	C₄⋊C₄	—	76	4	no	—
151	C₄⋊C₄	—	76	2	no	—
152	C₂²⋊C₄	—	38	4	no	—
153	C₂²⋊C₄	—	38	4	no	—
154	C₂³	—	2	2	no	—
155	C₂³	—	2	2	no	—
156	C₂³	C₂×Dic₃₈	1	2	yes	—
157	C₂³	#64541	1	2	yes	—
158	C₂³	C₂×C₂²⋊D₁₉	1	2	yes	—
159	C₂³	C₂³×D₁₉	1	1	yes	characteristic
160	Q₈	—	76	4	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	8	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	—	38	4	no	—
171	C₂×C₄	—	38	4	no	—
172	C₂×C₄	—	38	4	no	—
173	C₂×C₄	—	38	4	no	—
174	C₂×C₄	—	38	4	no	—
175	C₂×C₄	—	38	2	no	—
176	C₂×C₄	—	38	2	no	—
177	C₂×C₄	C₂×Dic₃₈	1	4	yes	—
178	C₂×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
179	C₂×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
180	C₂²	—	2	2	no	—
181	C₂²	—	2	2	no	—
182	C₂²	—	2	4	no	—
183	C₂²	—	2	4	no	—
184	C₂²	#10597370	1	4	yes	—
185	C₂²	C₂²×Dic₃₈	1	1	yes	characteristic
186	C₂²	C₂×#64541	1	1	yes	characteristic
187	C₂²	#10597430	1	2	yes	—
188	C₂²	#10597433	1	2	yes	—
189	C₂²	C₂²×C₂²⋊D₁₉	1	1	yes	characteristic
190	C₄	—	2	4	no	—
191	C₄	—	2	2	no	—
192	C₄	—	2	2	no	—
193	C₄	—	4	2	no	—
194	C₄	—	76	2	no	—
195	C₄	—	76	2	no	—
196	C₄	—	76	4	no	—
197	C₂	—	2	4	no	—
198	C₂	C₂×#10597370	1	1	yes	characteristic
199	C₂	#11927103	1	1	yes	characteristic
200	C₂	#11927118	1	1	yes	characteristic
201	C₂	#11927141	1	2	yes	—
202	C₂	#11927175	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10597370

C₂

yes

maximal, characteristic

#11926483

C₂

yes

maximal, complemented

#11926477

C₂

yes

maximal, complemented

#11926464

C₂

yes

maximal, complemented

#11926417

C₂

yes

maximal

#11926407

C₂

yes

maximal

#11926404

C₂

yes

maximal, characteristic

C₂²×C₄⋊C₄×C₁₉

C₂

yes

maximal, characteristic

C₂³×C₄×C₁₉

C₂²

yes

characteristic

C₂³×C₄×C₁₉

C₂²

yes

characteristic

C₂³×C₄×C₁₉

C₂²

yes

characteristic

C₂²×Dic₃₈

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₉

C₂²

yes

—

C₂×C₂²⋊Dic₁₉

C₂²

yes

—

C₂×C₂²⋊Dic₁₉

C₂²

yes

—

C₂×C₄⋊Dic₁₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₉

C₂²

yes

—

#10597370

—

C₂×Dic₁₉⋊C₄

C₂²

yes

—

C₂×Dic₁₉⋊C₄

C₂²

yes

—

C₁₉⋊#95

C₂²

yes

—

C₁₉⋊#95

C₂²

yes

—

C₁₉⋊#95

C₂²

yes

—

C₁₉⋊#95

C₂²

yes

—

C₂⁴×C₁₉

C₂³

yes

characteristic

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

D₄

yes

—

C₂²×C₄×C₁₉

Q₈

yes

complemented

C₂²×C₄×C₁₉

C₂³

yes

—

C₂²×C₄×C₁₉

C₂³

yes

characteristic

C₂²×C₄×C₁₉

C₂³

yes

characteristic

C₂²×C₄×C₁₉

C₂³

yes

—

C₂²×Dic₁₉

C₂³

yes

—

C₂²×Dic₁₉

C₂³

yes

—

C₂²×Dic₁₉

C₂³

yes

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₂²⋊Dic₁₉

—

C₂²⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

Dic₁₉⋊C₄

—

Dic₁₉⋊C₄

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

C₂×D₄

yes

—

C₂³×C₁₉

C₂×Q₈

yes

—

C₂³×C₁₉

#41

yes

—

C₂³×C₁₉

C₂⁴

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×D₄

yes

—

C₂×C₄×C₁₉

C₂×D₄

yes

—

C₂×C₄×C₁₉

C₂×Q₈

yes

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

Dic₃₈

—

#2627

—

maximal, Sylow

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

C₂²⋊Q₈

yes

—

C₂²×C₁₉

C₂²×D₄

yes

characteristic

C₂²×C₁₉

C₂²×Q₈

yes

characteristic

C₂²×C₁₉

C₂×#41

yes

characteristic

C₂²×C₁₉

H(5,𝔽₂)

yes

—

C₂²×C₁₉

2₋¹⁺⁴

yes

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

Dic₁₉

—

Dic₁₉

—

Dic₁₉

—

C₂×C₂²⋊Q₈

—

C₂²×C₄⋊C₄

D₁₉

yes

characteristic

#400

—

#398

—

100

#395

—

101

#385

—

102

#381

—

103

#380

—

104

C₂×C₁₉

—

105

C₂×C₁₉

C₂×C₂²⋊Q₈

yes

characteristic

106

C₂×C₁₉

#534

yes

characteristic

107

C₂×C₁₉

#536

yes

characteristic

108

C₂×C₁₉

#539

yes

—

109

C₂×C₁₉

#544

yes

—

110

C₂²×Q₈

—

111

C₂³×C₄

C₂×D₁₉

yes

characteristic

112

C₂³×C₄

C₂×D₁₉

yes

characteristic

113

C₂³×C₄

C₂×D₁₉

yes

characteristic

114

C₂²⋊Q₈

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

C₂×D₁₉

yes

—

121

C₂×C₄⋊C₄

C₂×D₁₉

yes

—

122

C₂×C₂²⋊C₄

—

123

C₂×C₂²⋊C₄

—

124

#95

—

125

#95

—

126

#95

—

127

#95

—

128

C₁₉

#2627

yes

characteristic, complemented, Sylow

129

C₂⁴

C₂²×D₁₉

yes

characteristic

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂²×C₄

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

—

140

C₂²×C₄

Dic₃₈

yes

complemented

141

C₂²×C₄

C₂²⋊D₁₉

yes

—

142

C₂²×C₄

C₂²×D₁₉

yes

—

143

C₂²×C₄

C₂²×D₁₉

yes

—

144

C₂²×C₄

C₂²×D₁₉

yes

characteristic

145

C₂²×C₄

C₂²×D₁₉

yes

characteristic

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₂²⋊C₄

—

153

C₂²⋊C₄

—

154

C₂³

—

155

C₂³

—

156

C₂³

C₂×Dic₃₈

yes

—

157

C₂³

#64541

yes

—

158

C₂³

C₂×C₂²⋊D₁₉

yes

—

159

C₂³

C₂³×D₁₉

yes

characteristic

160

Q₈

—

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

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

C₂×Dic₃₈

yes

—

178

C₂×C₄

C₂×C₂²⋊D₁₉

yes

—

179

C₂×C₄

C₂×C₂²⋊D₁₉

yes

—

180

C₂²

—

181

C₂²

—

182

C₂²

—

183

C₂²

—

184

C₂²

#10597370

yes

—

185

C₂²

C₂²×Dic₃₈

yes

characteristic

186

C₂²

C₂×#64541

yes

characteristic

187

C₂²

#10597430

yes

—

188

C₂²

#10597433

yes

—

189

C₂²

C₂²×C₂²⋊D₁₉

yes

characteristic

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

—

197

C₂

—

198

C₂

C₂×#10597370

yes

characteristic

199

C₂

#11927103

yes

characteristic

200

C₂

#11927118

yes

characteristic

201

C₂

#11927141

yes

—

202

C₂

#11927175

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₉
commutator, socle, cosocle	C₂³×C₁₉	C₂⁴
Fitting	C₂²×C₄⋊C₄×C₁₉	C₂
prime radical	C₂⁴×C₁₉	C₂³
hypercenter	C₂²×C₄⋊C₄	D₁₉
hypocenter	C₁₉	#2627
LCS	G ▷ C₂³×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₉

commutator, socle, cosocle

C₂³×C₁₉

C₂⁴

Fitting

C₂²×C₄⋊C₄×C₁₉

C₂

prime radical

C₂⁴×C₁₉

C₂³

hypercenter

C₂²×C₄⋊C₄

D₁₉

C₁₉

G ▷ C₂³×C₁₉ ▷ C₁₉

C₁ ◁ C₂³ ◁ C₂²×C₄⋊C₄

N	H ≅ G/N	type
2. #11926483	C₂	semidirect
3. #11926477	C₂	semidirect
4. #11926464	C₂	semidirect
34. C₂²×C₄×C₁₉	Q₈	semidirect
128. C₁₉	#2627	semidirect
140. C₂²×C₄	Dic₃₈	semidirect

H ≅ G/N

type

2. #11926483

C₂

semidirect

3. #11926477

C₂

semidirect

4. #11926464

C₂

semidirect

34. C₂²×C₄×C₁₉

Q₈

semidirect

128. C₁₉

#2627

semidirect

140. C₂²×C₄

Dic₃₈

semidirect

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

subgroup	index
C₂×#10597370	2
#11926483	2
#11926477	2
#11926464	2
#11926417	2
#11926407	2
#11926404	2
C₂²×C₄⋊C₄×C₁₉	2
#2627	19

subgroup

index

C₂²×C₄⋊C₄×C₁₉

Subgroups — #14569448

Conjugacy classes of subgroups

Complements

Maximal subgroups