Group #14571694 — subgroups

Subgroups — #14571694

Order 2432.

430 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 133 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×#10597322	C₂	1	1	yes	maximal, characteristic
2	#11926482	C₂	1	2	yes	maximal, complemented
3	#11926479	C₂	1	2	yes	maximal
4	#11926475	C₂	1	2	yes	maximal, complemented
5	#11926470	C₂	1	2	yes	maximal
6	#11926463	C₂	1	2	yes	maximal, complemented
7	#11926455	C₂	1	2	yes	maximal
8	#11926412	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Q₈×C₁₉	C₂²	1	2	yes	—
11	C₁₉×C₄⋊Q₈	—	2	4	no	—
12	C₂×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
15	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	4	yes	—
19	C₂×Dic₁₉⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×Dic₁₉⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₉⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×Dic₁₉⋊C₄	C₂²	1	4	yes	—
23	#10597322	—	2	4	no	—
24	C₁₉⋊#95	C₂²	1	2	yes	—
25	C₁₉⋊#95	C₂²	1	2	yes	—
26	C₁₉⋊#95	C₂²	1	2	yes	—
27	C₁₉⋊#95	C₂²	1	4	yes	—
28	C₁₉⋊#95	C₂²	1	4	yes	—
29	C₂³×D₁₉	C₂³	1	1	yes	characteristic
30	C₂×Q₈×C₁₉	—	2	4	no	—
31	C₂×Q₈×C₁₉	—	4	4	no	—
32	C₂²×C₄×C₁₉	C₂³	1	4	yes	—
33	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
36	C₂²×Dic₁₉	C₂³	1	4	yes	—
37	C₂²×Dic₁₉	C₂³	1	1	yes	characteristic
38	C₂²×Dic₁₉	C₂³	1	1	yes	characteristic
39	C₂²×Dic₁₉	C₂³	1	1	yes	characteristic
40	C₄⋊C₄×C₁₉	—	2	4	no	—
41	C₄⋊C₄×C₁₉	—	2	4	no	—
42	C₄²×C₁₉	D₄	1	4	yes	—
43	(C₂×D₁₉)⋊C₄	—	2	2	no	—
44	(C₂×D₁₉)⋊C₄	—	2	2	no	—
45	(C₂×D₁₉)⋊C₄	—	2	2	no	—
46	(C₂×D₁₉)⋊C₄	—	4	4	no	—
47	Dic₁₉⋊C₄	—	2	2	no	—
48	Dic₁₉⋊C₄	—	2	2	no	—
49	Dic₁₉⋊C₄	—	2	2	no	—
50	Dic₁₉⋊C₄	—	4	4	no	—
51	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
52	C₂²×D₁₉	—	2	1	no	—
53	C₂²×D₁₉	—	2	2	no	—
54	C₂²×D₁₉	—	2	2	no	—
55	C₂²×D₁₉	—	2	2	no	—
56	Q₈×C₁₉	—	4	8	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	—	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×Dic₁₉	—	2	1	no	—
67	C₂×Dic₁₉	—	2	1	no	—
68	C₂×Dic₁₉	—	2	1	no	—
69	C₂×Dic₁₉	—	2	2	no	—
70	C₂×Dic₁₉	—	2	4	no	—
71	C₂×Dic₁₉	—	2	2	no	—
72	C₂×Dic₁₉	—	2	2	no	—
73	C₂×Dic₁₉	—	2	4	no	—
74	C₂×Dic₁₉	—	2	4	no	—
75	#2821	—	19	1	no	maximal, Sylow
76	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
77	C₂²×C₁₉	H(5,𝔽₂)	1	2	yes	—
78	C₂²×C₁₉	2₋¹⁺⁴	1	2	yes	—
79	C₂²×C₁₉	2₋¹⁺⁴	1	2	yes	—
80	C₂×D₁₉	—	4	1	no	—
81	C₂×D₁₉	—	4	1	no	—
82	C₂×D₁₉	—	4	1	no	—
83	C₂×D₁₉	—	4	4	no	—
84	C₄×C₁₉	—	2	2	no	—
85	C₄×C₁₉	—	2	2	no	—
86	C₄×C₁₉	—	2	2	no	—
87	C₄×C₁₉	—	4	4	no	—
88	Dic₁₉	—	4	1	no	—
89	Dic₁₉	—	4	1	no	—
90	Dic₁₉	—	4	1	no	—
91	Dic₁₉	—	4	4	no	—
92	C₂×C₄⋊Q₈	D₁₉	1	1	yes	characteristic, complemented
93	C₂×#126	—	19	1	no	—
94	#400	—	19	2	no	—
95	#399	—	19	2	no	—
96	#398	—	19	2	no	—
97	#397	—	19	2	no	—
98	#395	—	19	2	no	—
99	#393	—	19	2	no	—
100	#383	—	19	1	no	—
101	C₂×C₁₉	#535	1	1	yes	characteristic
102	C₂×C₁₉	#536	1	1	yes	characteristic
103	C₂×C₁₉	#536	1	1	yes	characteristic
104	C₂×C₁₉	#563	1	4	yes	—
105	D₁₉	—	8	1	no	—
106	C₂²×Q₈	C₂×D₁₉	1	2	yes	—
107	C₄⋊Q₈	—	2	4	no	—
108	#126	—	38	4	no	—
109	C₂×C₄⋊C₄	—	19	1	no	—
110	C₂×C₄⋊C₄	—	19	1	no	—
111	C₂×C₄⋊C₄	—	19	1	no	—
112	C₂×C₄⋊C₄	—	19	4	no	—
113	C₂×C₄⋊C₄	C₂×D₁₉	1	2	yes	—
114	C₂×C₄⋊C₄	C₂×D₁₉	1	2	yes	—
115	C₂×C₂²⋊C₄	—	19	1	no	—
116	C₂×C₂²⋊C₄	—	19	1	no	—
117	C₂×C₂²⋊C₄	—	19	1	no	—
118	C₂×C₂²⋊C₄	—	19	4	no	—
119	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
120	#95	—	19	2	no	—
121	#95	—	19	2	no	—
122	#95	—	19	2	no	—
123	#95	—	19	4	no	—
124	#95	—	19	4	no	—
125	C₁₉	#2821	1	1	yes	characteristic, complemented, Sylow
126	C₂⁴	—	19	1	no	—
127	C₂×Q₈	—	2	4	no	—
128	C₂×Q₈	—	4	4	no	—
129	C₂²×C₄	—	19	4	no	—
130	C₂²×C₄	—	19	1	no	—
131	C₂²×C₄	—	19	1	no	—
132	C₂²×C₄	—	19	1	no	—
133	C₂²×C₄	C₂²×D₁₉	1	4	yes	—
134	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
135	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
136	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
137	C₄⋊C₄	—	2	4	no	—
138	C₄⋊C₄	—	2	4	no	—
139	C₄⋊C₄	—	38	2	no	—
140	C₄⋊C₄	—	38	2	no	—
141	C₄⋊C₄	—	38	2	no	—
142	C₄⋊C₄	—	76	4	no	—
143	C₂²⋊C₄	—	38	2	no	—
144	C₂²⋊C₄	—	38	2	no	—
145	C₂²⋊C₄	—	38	2	no	—
146	C₂²⋊C₄	—	76	4	no	—
147	C₄²	C₂²⋊D₁₉	1	4	yes	—
148	C₂³	—	38	2	no	—
149	C₂³	—	38	2	no	—
150	C₂³	—	38	2	no	—
151	C₂³	—	38	1	no	—
152	C₂³	C₂³×D₁₉	1	1	yes	characteristic
153	Q₈	—	4	8	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	38	4	no	—
161	C₂×C₄	—	38	2	no	—
162	C₂×C₄	—	38	2	no	—
163	C₂×C₄	—	38	2	no	—
164	C₂×C₄	—	38	4	no	—
165	C₂×C₄	—	38	4	no	—
166	C₂×C₄	—	38	1	no	—
167	C₂×C₄	—	38	1	no	—
168	C₂×C₄	—	38	1	no	—
169	C₂×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
170	C₂×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
171	C₂×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
172	C₂²	—	76	1	no	—
173	C₂²	—	76	1	no	—
174	C₂²	—	76	1	no	—
175	C₂²	—	76	4	no	—
176	C₂²	#10597430	1	2	yes	—
177	C₂²	#10597433	1	2	yes	—
178	C₂²	#10597433	1	2	yes	—
179	C₂²	C₂²×C₂²⋊D₁₉	1	1	yes	characteristic
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	2	2	no	—
183	C₄	—	4	4	no	—
184	C₄	—	76	1	no	—
185	C₄	—	76	1	no	—
186	C₄	—	76	1	no	—
187	C₄	—	76	4	no	—
188	C₂	—	152	1	no	—
189	C₂	#11927111	1	1	yes	characteristic
190	C₂	#11927118	1	1	yes	characteristic
191	C₂	#11927118	1	1	yes	characteristic
192	C₂	#11927293	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10597322

C₂

yes

maximal, characteristic

#11926482

C₂

yes

maximal, complemented

#11926479

C₂

yes

maximal

#11926475

C₂

yes

maximal, complemented

#11926470

C₂

yes

maximal

#11926463

C₂

yes

maximal, complemented

#11926455

C₂

yes

maximal

#11926412

C₂

yes

maximal, characteristic, complemented

C₂×C₁₉×C₄⋊Q₈

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈×C₁₉

C₂²

yes

—

C₁₉×C₄⋊Q₈

—

C₂×C₄⋊C₄×C₁₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₉

C₂²

yes

—

C₂×C₄²×C₁₉

C₂²

yes

characteristic

C₂×(C₂×D₁₉)⋊C₄

C₂²

yes

characteristic

C₂×(C₂×D₁₉)⋊C₄

C₂²

yes

characteristic

C₂×(C₂×D₁₉)⋊C₄

C₂²

yes

characteristic

C₂×(C₂×D₁₉)⋊C₄

C₂²

yes

—

C₂×Dic₁₉⋊C₄

C₂²

yes

characteristic

C₂×Dic₁₉⋊C₄

C₂²

yes

characteristic

C₂×Dic₁₉⋊C₄

C₂²

yes

characteristic

C₂×Dic₁₉⋊C₄

C₂²

yes

—

#10597322

—

C₁₉⋊#95

C₂²

yes

—

C₁₉⋊#95

C₂²

yes

—

C₁₉⋊#95

C₂²

yes

—

C₁₉⋊#95

C₂²

yes

—

C₁₉⋊#95

C₂²

yes

—

C₂³×D₁₉

C₂³

yes

characteristic

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

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

characteristic

C₂²×Dic₁₉

C₂³

yes

characteristic

C₂²×Dic₁₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄²×C₁₉

D₄

yes

—

(C₂×D₁₉)⋊C₄

—

(C₂×D₁₉)⋊C₄

—

(C₂×D₁₉)⋊C₄

—

(C₂×D₁₉)⋊C₄

—

Dic₁₉⋊C₄

—

Dic₁₉⋊C₄

—

Dic₁₉⋊C₄

—

Dic₁₉⋊C₄

—

C₂³×C₁₉

C₂⁴

yes

characteristic

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂²×D₁₉

—

Q₈×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×D₄

yes

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

#2821

—

maximal, Sylow

C₂²×C₁₉

C₂²×D₄

yes

characteristic

C₂²×C₁₉

H(5,𝔽₂)

yes

—

C₂²×C₁₉

2₋¹⁺⁴

yes

—

C₂²×C₁₉

2₋¹⁺⁴

yes

—

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

Dic₁₉

—

Dic₁₉

—

Dic₁₉

—

Dic₁₉

—

C₂×C₄⋊Q₈

D₁₉

yes

characteristic, complemented

—

—

—

—

—

—

—

100

#383

—

101

C₂×C₁₉

#535

yes

characteristic

102

C₂×C₁₉

#536

yes

characteristic

103

C₂×C₁₉

#536

yes

characteristic

104

C₂×C₁₉

#563

yes

—

105

D₁₉

—

106

C₂²×Q₈

C₂×D₁₉

yes

—

107

C₄⋊Q₈

—

108

#126

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

C₂×D₁₉

yes

—

114

C₂×C₄⋊C₄

C₂×D₁₉

yes

—

115

C₂×C₂²⋊C₄

—

116

C₂×C₂²⋊C₄

—

117

C₂×C₂²⋊C₄

—

118

C₂×C₂²⋊C₄

—

119

C₂×C₄²

C₂×D₁₉

yes

characteristic

120

#95

—

121

#95

—

122

#95

—

123

#95

—

124

#95

—

125

C₁₉

#2821

yes

characteristic, complemented, Sylow

126

C₂⁴

—

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

C₂²×D₁₉

yes

—

134

C₂²×C₄

C₂²×D₁₉

yes

characteristic

135

C₂²×C₄

C₂²×D₁₉

yes

characteristic

136

C₂²×C₄

C₂²×D₁₉

yes

characteristic

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₂²⋊C₄

—

144

C₂²⋊C₄

—

145

C₂²⋊C₄

—

146

C₂²⋊C₄

—

147

C₄²

C₂²⋊D₁₉

yes

—

148

C₂³

—

149

C₂³

—

150

C₂³

—

151

C₂³

—

152

C₂³

C₂³×D₁₉

yes

characteristic

153

Q₈

—

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

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

C₂×C₂²⋊D₁₉

yes

—

170

C₂×C₄

C₂×C₂²⋊D₁₉

yes

—

171

C₂×C₄

C₂×C₂²⋊D₁₉

yes

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

175

C₂²

—

176

C₂²

#10597430

yes

—

177

C₂²

#10597433

yes

—

178

C₂²

#10597433

yes

—

179

C₂²

C₂²×C₂²⋊D₁₉

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

—

188

C₂

—

152

—

189

C₂

#11927111

yes

characteristic

190

C₂

#11927118

yes

characteristic

191

C₂

#11927118

yes

characteristic

192

C₂

#11927293

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₉
commutator, socle, cosocle	C₂³×C₁₉	C₂⁴
Fitting	C₂×C₁₉×C₄⋊Q₈	C₂
prime radical	C₂³×D₁₉	C₂³
hypercenter	C₂×C₄⋊Q₈	D₁₉
hypocenter	C₁₉	#2821
LCS	G ▷ C₂³×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₉

commutator, socle, cosocle

C₂³×C₁₉

C₂⁴

Fitting

C₂×C₁₉×C₄⋊Q₈

C₂

prime radical

C₂³×D₁₉

C₂³

hypercenter

C₂×C₄⋊Q₈

D₁₉

C₁₉

G ▷ C₂³×C₁₉ ▷ C₁₉

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #11926482	C₂	semidirect
4. #11926475	C₂	semidirect
6. #11926463	C₂	semidirect
8. #11926412	C₂	semidirect
9. C₂×C₁₉×C₄⋊Q₈	C₂	semidirect
92. C₂×C₄⋊Q₈	D₁₉	semidirect
125. C₁₉	#2821	semidirect

H ≅ G/N

type

2. #11926482

C₂

semidirect

4. #11926475

C₂

semidirect

6. #11926463

C₂

semidirect

8. #11926412

C₂

semidirect

9. C₂×C₁₉×C₄⋊Q₈

C₂

semidirect

92. C₂×C₄⋊Q₈

D₁₉

semidirect

125. C₁₉

#2821

semidirect

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

subgroup	index
C₂×#10597322	2
#11926482	2
#11926479	2
#11926475	2
#11926470	2
#11926463	2
#11926455	2
#11926412	2
C₂×C₁₉×C₄⋊Q₈	2
#2821	19

subgroup

index

C₂×C₁₉×C₄⋊Q₈

Subgroups — #14571694

Conjugacy classes of subgroups

Complements

Maximal subgroups