Group #14575036 — subgroups

Subgroups — #14575036

Order 2432.

354 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 117 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	#11927078	C₂	1	2	yes	maximal, complemented
2	#11926852	C₂	1	2	yes	maximal, complemented
3	#11926845	C₂	1	2	yes	maximal, complemented
4	#11926802	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926797	C₂	1	1	yes	maximal, characteristic
6	#11926790	C₂	1	2	yes	maximal, complemented
7	#11926767	C₂	1	2	yes	maximal, complemented
8	#11926760	C₂	1	2	yes	maximal, complemented
9	C₁₉×#423	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
11	C₂×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
13	#10597376	C₂²	1	2	yes	—
14	#10597375	—	2	2	no	—
15	C₄×C₂²⋊D₁₉	—	2	2	no	—
16	#10597370	C₂²	1	2	yes	—
17	C₂×Dic₇₆	C₂²	1	1	yes	characteristic
18	C₂×D₁₅₂	C₂²	1	1	yes	characteristic
19	C₂×#64516	C₂²	1	2	yes	—
20	#10597322	—	2	2	no	—
21	#10597321	C₂²	1	2	yes	—
22	C₄×D₇₆	C₂²	1	2	yes	—
23	#10597318	—	2	2	no	—
24	#10597316	C₂²	1	2	yes	—
25	C₄×Dic₃₈	C₂²	1	2	yes	—
26	C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
27	C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
28	#10597278	C₂²	1	2	yes	—
29	#10597278	C₂²	1	2	yes	—
30	#10597275	C₂²	1	2	yes	—
31	#10597274	C₂²	1	1	yes	characteristic
32	#10597274	C₂²	1	1	yes	characteristic
33	#10597273	C₂²	1	2	yes	—
34	#10597273	C₂²	1	2	yes	—
35	C₂²×C₄×C₁₉	D₄	1	2	yes	—
36	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
37	C₂×C₂²⋊D₁₉	—	2	2	no	—
38	C₂×D₇₆	C₂³	1	2	yes	—
39	C₂×C₄×D₁₉	—	2	2	no	—
40	C₂×Dic₃₈	C₂³	1	2	yes	—
41	M₁₆×C₁₉	—	4	1	no	—
42	M₁₆×C₁₉	—	4	1	no	—
43	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₁₉	C₂³	1	2	yes	—
45	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
46	C₄²×C₁₉	D₄	1	1	yes	characteristic
47	C₄²×C₁₉	D₄	1	1	yes	characteristic
48	C₄²×C₁₉	C₂³	1	2	yes	—
49	C₂²⋊Dic₁₉	—	2	2	no	—
50	(C₂×D₁₉)⋊C₄	—	2	2	no	—
51	(C₂×D₁₉)⋊C₄	—	2	2	no	—
52	(C₂×D₁₉)⋊C₄	—	2	2	no	—
53	(C₂×D₁₉)⋊C₄	—	2	2	no	—
54	C₄⋊Dic₁₉	C₂³	1	2	yes	—
55	C₄⋊Dic₁₉	C₂³	1	2	yes	—
56	Dic₁₉⋊C₄	—	2	2	no	—
57	Dic₁₉⋊C₄	—	2	2	no	—
58	Dic₁₉⋊C₄	—	2	2	no	—
59	Dic₁₉⋊C₄	—	2	2	no	—
60	C₄×Dic₁₉	—	2	2	no	—
61	Dic₇₆	—	4	1	no	—
62	D₁₅₂	—	4	1	no	—
63	#64516	—	4	2	no	—
64	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×D₁₉	—	2	2	no	—
66	C₂×C₄×C₁₉	—	2	1	no	—
67	C₂×C₄×C₁₉	—	2	1	no	—
68	C₂×C₄×C₁₉	—	4	2	no	—
69	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
70	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
72	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
73	C₂²⋊D₁₉	—	4	4	no	—
74	C₂×Dic₁₉	—	2	2	no	—
75	C₂×Dic₁₉	—	2	2	no	—
76	C₂×Dic₁₉	—	2	2	no	—
77	D₇₆	—	2	2	no	—
78	D₇₆	—	2	2	no	—
79	C₄×D₁₉	—	4	4	no	—
80	Dic₃₈	—	2	2	no	—
81	Dic₃₈	—	2	2	no	—
82	C₈×C₁₉	—	2	2	no	—
83	C₈×C₁₉	—	2	1	no	—
84	C₈×C₁₉	—	2	1	no	—
85	#3256	—	19	1	no	maximal, Sylow
86	C₂²×C₁₉	—	2	1	no	—
87	C₂²×C₁₉	—	2	2	no	—
88	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
89	C₂×D₁₉	—	4	2	no	—
90	C₂×D₁₉	—	4	2	no	—
91	C₂×D₁₉	—	4	2	no	—
92	C₄×C₁₉	—	2	1	no	—
93	C₄×C₁₉	—	2	2	no	—
94	C₄×C₁₉	—	2	2	no	—
95	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
96	C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
97	Dic₁₉	—	4	2	no	—
98	Dic₁₉	—	4	2	no	—
99	Dic₁₉	—	4	2	no	—
100	#529	—	19	2	no	—
101	#479	—	19	2	no	—
102	#478	—	19	2	no	—
103	#470	—	19	1	no	—
104	#469	—	19	1	no	—
105	#468	—	19	2	no	—
106	#464	—	19	2	no	—
107	#463	—	19	2	no	—
108	#423	D₁₉	1	1	yes	characteristic, complemented
109	C₂×C₁₉	—	4	1	no	—
110	C₂×C₁₉	#537	1	1	yes	characteristic
111	C₂×C₁₉	#575	1	2	yes	—
112	D₁₉	—	8	2	no	—
113	C₂×Q₁₆	—	19	1	no	—
114	C₂×SD₁₆	—	19	2	no	—
115	C₂×D₈	—	19	1	no	—
116	C₂×M₁₆	C₂×D₁₉	1	1	yes	characteristic
117	C₂×M₁₆	C₂×D₁₉	1	1	yes	characteristic
118	#126	—	38	2	no	—
119	#125	—	19	2	no	—
120	#124	—	19	2	no	—
121	#123	—	38	2	no	—
122	C₂²⋊Q₈	—	19	2	no	—
123	#121	—	19	2	no	—
124	C₄×Q₈	—	19	2	no	—
125	C₄×D₄	—	19	2	no	—
126	C₄×D₄	—	38	2	no	—
127	#117	—	38	2	no	—
128	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
129	#107	—	19	2	no	—
130	#106	—	19	1	no	—
131	#106	—	19	1	no	—
132	C₄⋊C₈	C₂×D₁₉	1	2	yes	—
133	C₄⋊C₈	C₂×D₁₉	1	2	yes	—
134	#103	—	19	2	no	—
135	#103	—	19	2	no	—
136	#102	—	19	2	no	—
137	#102	—	19	2	no	—
138	C₁₉	#3256	1	1	yes	characteristic, complemented, Sylow
139	C₂×Q₈	—	19	2	no	—
140	C₂×D₄	—	19	2	no	—
141	C₂×D₄	—	38	2	no	—
142	C₂²×C₄	—	38	2	no	—
143	C₂²×C₄	D₇₆	1	2	yes	—
144	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
145	Q₁₆	—	76	1	no	—
146	SD₁₆	—	76	2	no	—
147	D₈	—	76	1	no	—
148	M₁₆	—	4	1	no	—
149	M₁₆	—	4	1	no	—
150	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
151	C₂×C₈	C₂²×D₁₉	1	2	yes	—
152	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
153	C₄⋊C₄	—	19	2	no	—
154	C₄⋊C₄	—	19	2	no	—
155	C₄⋊C₄	—	38	2	no	—
156	C₄⋊C₄	—	38	2	no	—
157	C₄⋊C₄	—	38	2	no	—
158	C₄⋊C₄	—	38	2	no	—
159	C₂²⋊C₄	—	38	2	no	—
160	C₂²⋊C₄	—	38	2	no	—
161	C₂²⋊C₄	—	38	2	no	—
162	C₂²⋊C₄	—	38	2	no	—
163	C₂²⋊C₄	—	38	2	no	—
164	C₄²	—	38	2	no	—
165	C₄²	D₇₆	1	1	yes	characteristic
166	C₄²	D₇₆	1	1	yes	characteristic
167	C₄²	C₂²×D₁₉	1	2	yes	—
168	C₂³	—	38	2	no	—
169	C₂³	C₂×D₇₆	1	1	yes	characteristic
170	Q₈	—	38	2	no	—
171	Q₈	—	38	2	no	—
172	D₄	—	38	2	no	—
173	D₄	—	38	2	no	—
174	D₄	—	76	4	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	38	2	no	—
179	C₂×C₄	—	38	2	no	—
180	C₂×C₄	—	38	2	no	—
181	C₂×C₄	—	76	4	no	—
182	C₂×C₄	C₂×D₇₆	1	2	yes	—
183	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
184	C₂×C₄	C₂×D₇₆	1	2	yes	—
185	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
186	C₈	—	2	1	no	—
187	C₈	—	2	2	no	—
188	C₈	—	2	1	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	1	no	—
191	C₂²	—	76	2	no	—
192	C₂²	—	76	2	no	—
193	C₂²	—	76	2	no	—
194	C₂²	C₂²×D₇₆	1	1	yes	characteristic
195	C₄	—	2	2	no	—
196	C₄	—	2	2	no	—
197	C₄	—	2	1	no	—
198	C₄	—	76	2	no	—
199	C₄	—	76	2	no	—
200	C₄	—	76	2	no	—
201	C₄	#10597430	1	1	yes	characteristic
202	C₄	#10597433	1	1	yes	characteristic
203	C₂	—	4	1	no	—
204	C₂	—	152	2	no	—
205	C₂	#11927125	1	1	yes	characteristic
206	C₂	#11927344	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11927078

C₂

yes

maximal, complemented

#11926852

C₂

yes

maximal, complemented

#11926845

C₂

yes

maximal, complemented

#11926802

C₂

yes

maximal, characteristic, complemented

#11926797

C₂

yes

maximal, characteristic

#11926790

C₂

yes

maximal, complemented

#11926767

C₂

yes

maximal, complemented

#11926760

C₂

yes

maximal, complemented

C₁₉×#423

C₂

yes

maximal, characteristic, complemented

C₂×M₁₆×C₁₉

C₂²

yes

characteristic

C₂×M₁₆×C₁₉

C₂²

yes

characteristic

C₂×C₄²×C₁₉

C₂²

yes

characteristic

#10597376

C₂²

yes

—

#10597375

—

C₄×C₂²⋊D₁₉

—

#10597370

C₂²

yes

—

C₂×Dic₇₆

C₂²

yes

characteristic

C₂×D₁₅₂

C₂²

yes

characteristic

C₂×#64516

C₂²

yes

—

#10597322

—

#10597321

C₂²

yes

—

C₄×D₇₆

C₂²

yes

—

#10597318

—

#10597316

C₂²

yes

—

C₄×Dic₃₈

C₂²

yes

—

C₁₉×C₄⋊C₈

C₂²

yes

—

C₁₉×C₄⋊C₈

C₂²

yes

—

#10597278

C₂²

yes

—

#10597278

C₂²

yes

—

#10597275

C₂²

yes

—

#10597274

C₂²

yes

characteristic

#10597274

C₂²

yes

characteristic

#10597273

C₂²

yes

—

#10597273

C₂²

yes

—

C₂²×C₄×C₁₉

D₄

yes

—

C₂²×C₄×C₁₉

C₂³

yes

characteristic

C₂×C₂²⋊D₁₉

—

C₂×D₇₆

C₂³

yes

—

C₂×C₄×D₁₉

—

C₂×Dic₃₈

C₂³

yes

—

M₁₆×C₁₉

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

C₂³

yes

characteristic

C₂×C₈×C₁₉

C₂³

yes

—

C₂×C₈×C₁₉

C₂³

yes

characteristic

C₄²×C₁₉

D₄

yes

characteristic

C₄²×C₁₉

D₄

yes

characteristic

C₄²×C₁₉

C₂³

yes

—

C₂²⋊Dic₁₉

—

(C₂×D₁₉)⋊C₄

—

(C₂×D₁₉)⋊C₄

—

(C₂×D₁₉)⋊C₄

—

(C₂×D₁₉)⋊C₄

—

C₄⋊Dic₁₉

C₂³

yes

—

C₄⋊Dic₁₉

C₂³

yes

—

Dic₁₉⋊C₄

—

Dic₁₉⋊C₄

—

Dic₁₉⋊C₄

—

Dic₁₉⋊C₄

—

C₄×Dic₁₉

—

Dic₇₆

—

D₁₅₂

—

#64516

—

C₂³×C₁₉

C₂×D₄

yes

characteristic

C₂²×D₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂×D₄

yes

—

C₂×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉

C₂×D₄

yes

—

C₂×C₄×C₁₉

C₂⁴

yes

characteristic

C₂²⋊D₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

D₇₆

—

D₇₆

—

C₄×D₁₉

—

Dic₃₈

—

Dic₃₈

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

#3256

—

maximal, Sylow

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

C₂²×D₄

yes

characteristic

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

H(5,𝔽₂)

yes

characteristic

C₄×C₁₉

2₋¹⁺⁴

yes

characteristic

Dic₁₉

—

Dic₁₉

—

Dic₁₉

—

100

#529

—

101

#479

—

102

#478

—

103

#470

—

104

#469

—

105

#468

—

106

#464

—

107

#463

—

108

#423

D₁₉

yes

characteristic, complemented

109

C₂×C₁₉

—

110

C₂×C₁₉

#537

yes

characteristic

111

C₂×C₁₉

#575

yes

—

112

D₁₉

—

113

C₂×Q₁₆

—

114

C₂×SD₁₆

—

115

C₂×D₈

—

116

C₂×M₁₆

C₂×D₁₉

yes

characteristic

117

C₂×M₁₆

C₂×D₁₉

yes

characteristic

118

#126

—

119

#125

—

120

#124

—

121

#123

—

122

C₂²⋊Q₈

—

123

#121

—

124

C₄×Q₈

—

125

C₄×D₄

—

126

C₄×D₄

—

127

#117

—

128

C₂×C₄²

C₂×D₁₉

yes

characteristic

129

#107

—

130

#106

—

131

#106

—

132

C₄⋊C₈

C₂×D₁₉

yes

—

133

C₄⋊C₈

C₂×D₁₉

yes

—

134

#103

—

135

#103

—

136

#102

—

137

#102

—

138

C₁₉

#3256

yes

characteristic, complemented, Sylow

139

C₂×Q₈

—

140

C₂×D₄

—

141

C₂×D₄

—

142

C₂²×C₄

—

143

C₂²×C₄

D₇₆

yes

—

144

C₂²×C₄

C₂²×D₁₉

yes

characteristic

145

Q₁₆

—

146

SD₁₆

—

147

D₈

—

148

M₁₆

—

149

M₁₆

—

150

C₂×C₈

C₂²×D₁₉

yes

characteristic

151

C₂×C₈

C₂²×D₁₉

yes

—

152

C₂×C₈

C₂²×D₁₉

yes

characteristic

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₂²⋊C₄

—

160

C₂²⋊C₄

—

161

C₂²⋊C₄

—

162

C₂²⋊C₄

—

163

C₂²⋊C₄

—

164

C₄²

—

165

C₄²

D₇₆

yes

characteristic

166

C₄²

D₇₆

yes

characteristic

167

C₄²

C₂²×D₁₉

yes

—

168

C₂³

—

169

C₂³

C₂×D₇₆

yes

characteristic

170

Q₈

—

171

Q₈

—

172

D₄

—

173

D₄

—

174

D₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

C₂×D₇₆

yes

—

183

C₂×C₄

C₂×D₇₆

yes

characteristic

184

C₂×C₄

C₂×D₇₆

yes

—

185

C₂×C₄

C₂³×D₁₉

yes

characteristic

186

C₈

—

187

C₈

—

188

C₈

—

189

C₂²

—

190

C₂²

—

191

C₂²

—

192

C₂²

—

193

C₂²

—

194

C₂²

C₂²×D₇₆

yes

characteristic

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

—

201

C₄

#10597430

yes

characteristic

202

C₄

#10597433

yes

characteristic

203

C₂

—

204

C₂

—

152

—

205

C₂

#11927125

yes

characteristic

206

C₂

#11927344

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₇₆
commutator, cosocle	C₂×C₄×C₁₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉
Fitting	C₁₉×#423	C₂
socle	C₂²×C₁₉	C₂²×D₄
prime radical	#11926797	C₂
hypercenter	#423	D₁₉
hypocenter	C₁₉	#3256
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂²×D₇₆

commutator, cosocle

C₂×C₄×C₁₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₉

Fitting

C₁₉×#423

C₂

socle

C₂²×C₁₉

C₂²×D₄

C₂

D₁₉

C₁₉

G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
1. #11927078	C₂	semidirect
2. #11926852	C₂	semidirect
3. #11926845	C₂	semidirect
4. #11926802	C₂	semidirect
6. #11926790	C₂	semidirect
7. #11926767	C₂	semidirect
8. #11926760	C₂	semidirect
9. C₁₉×#423	C₂	semidirect
108. #423	D₁₉	semidirect
138. C₁₉	#3256	semidirect

H ≅ G/N

type

1. #11927078

C₂

semidirect

2. #11926852

C₂

semidirect

3. #11926845

C₂

semidirect

4. #11926802

C₂

semidirect

6. #11926790

C₂

semidirect

7. #11926767

C₂

semidirect

8. #11926760

C₂

semidirect

9. C₁₉×#423

C₂

semidirect

108. #423

D₁₉

semidirect

138. C₁₉

#3256

semidirect

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

subgroup	index
#11927078	2
#11926852	2
#11926845	2
#11926802	2
#11926797	2
#11926790	2
#11926767	2
#11926760	2
C₁₉×#423	2
#3256	19

subgroup

index

C₁₉×#423

Subgroups — #14575036

Conjugacy classes of subgroups

Complements

Maximal subgroups