Group #18071237 — subgroups

Subgroups — #18071237

Order 4864.

872 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 872 conjugacy classes of subgroups — too many to draw. Only its 95 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	#14564705	C₂	1	3	yes	maximal, complemented
2	#14561456	C₂	1	3	yes	maximal, complemented
3	C₁₉×#2679	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#64525	—	2	3	no	—
5	C₂×#10597320	C₂²	1	1	yes	characteristic, complemented
6	C₂×#10597266	—	2	3	no	—
7	#11926299	C₂²	1	3	yes	complemented
8	C₂×C₁₉×#120	—	2	3	no	—
9	C₁₉×#394	—	2	3	no	—
10	C₁₉×#390	C₂²	1	3	yes	—
11	C₂⁵×C₁₉	—	2	1	no	—
12	C₂²×D₄×C₁₉	—	2	3	no	—
13	C₂²×D₄×C₁₉	D₄	1	3	yes	—
14	C₂²×D₇₆	—	2	3	no	—
15	C₁₉×#120	—	4	6	no	—
16	C₁₉×#120	—	4	6	no	—
17	C₂×C₄⋊C₄×C₁₉	D₄	1	3	yes	—
18	C₂×C₂²⋊C₄×C₁₉	—	2	3	no	—
19	C₂×C₂²⋊C₄×C₁₉	—	2	6	no	—
20	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
21	C₂×#64525	—	4	6	no	—
22	C₂×#64525	—	4	6	no	—
23	C₂×#64525	—	4	6	no	—
24	C₂²×C₁₉⋊C₈	—	2	3	no	—
25	#10597320	—	4	2	no	—
26	C₁₉×#95	—	2	1	no	—
27	#10597266	—	4	6	no	—
28	C₂⁴×C₁₉	—	2	3	no	—
29	C₂⁴×C₁₉	—	4	6	no	—
30	C₂⁴×C₁₉	—	8	3	no	—
31	C₂⁴×C₁₉	—	8	1	no	—
32	C₂³×D₁₉	—	4	1	no	—
33	C₂×D₄×C₁₉	—	2	3	no	—
34	C₂×D₄×C₁₉	—	4	3	no	—
35	C₂×D₄×C₁₉	—	4	3	no	—
36	C₂×D₄×C₁₉	—	4	12	no	—
37	C₂×D₄×C₁₉	—	8	3	no	—
38	C₂×D₄×C₁₉	D₈	1	6	yes	complemented
39	C₂²×C₄×C₁₉	—	2	3	no	—
40	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
41	C₂×D₇₆	—	2	6	no	—
42	C₂×D₇₆	—	4	3	no	—
43	C₂×D₇₆	—	4	3	no	—
44	C₂×D₇₆	—	4	3	no	—
45	C₂×D₇₆	—	4	3	no	—
46	C₂×D₇₆	—	4	3	no	—
47	C₄⋊C₄×C₁₉	—	4	3	no	—
48	C₂²⋊C₄×C₁₉	—	4	6	no	—
49	C₂²⋊C₄×C₁₉	—	4	6	no	—
50	C₂²⋊C₄×C₁₉	—	4	6	no	—
51	C₄²×C₁₉	—	4	1	no	—
52	#64525	—	4	24	no	—
53	C₂×C₁₉⋊C₈	—	2	6	no	—
54	C₂×C₁₉⋊C₈	—	4	3	no	—
55	C₂×C₁₉⋊C₈	—	4	3	no	—
56	#7734	—	19	1	no	maximal, Sylow
57	C₂³×C₁₉	—	2	6	no	—
58	C₂³×C₁₉	—	4	6	no	—
59	C₂³×C₁₉	—	4	3	no	—
60	C₂³×C₁₉	—	4	6	no	—
61	C₂³×C₁₉	—	4	3	no	—
62	C₂³×C₁₉	—	8	12	no	—
63	C₂³×C₁₉	—	8	12	no	—
64	C₂³×C₁₉	—	8	4	no	—
65	C₂³×C₁₉	#120	1	1	yes	characteristic
66	C₂²×D₁₉	—	4	2	no	—
67	C₂²×D₁₉	—	8	3	no	—
68	C₂²×D₁₉	—	8	3	no	—
69	D₄×C₁₉	—	4	6	no	—
70	D₄×C₁₉	—	8	3	no	—
71	D₄×C₁₉	—	8	12	no	—
72	C₂×C₄×C₁₉	—	2	3	no	—
73	C₂×C₄×C₁₉	—	2	3	no	—
74	C₂×C₄×C₁₉	—	4	6	no	—
75	C₂×C₄×C₁₉	—	4	3	no	—
76	C₂×C₄×C₁₉	C₂×D₈	1	3	yes	—
77	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	3	yes	—
78	D₇₆	—	4	6	no	—
79	D₇₆	—	4	6	no	—
80	D₇₆	—	8	3	no	—
81	D₇₆	—	8	3	no	—
82	C₁₉⋊C₈	—	4	6	no	—
83	#2679	D₁₉	1	1	yes	characteristic, complemented
84	#2011	—	19	3	no	—
85	#1270	—	19	3	no	—
86	C₂²×C₁₉	—	4	6	no	—
87	C₂²×C₁₉	—	4	12	no	—
88	C₂²×C₁₉	—	8	16	no	—
89	C₂²×C₁₉	—	8	6	no	—
90	C₂²×C₁₉	—	8	3	no	—
91	C₂²×C₁₉	—	8	3	no	—
92	C₂²×C₁₉	#447	1	3	yes	—
93	C₂²×C₁₉	#453	1	3	yes	—
94	C₂²×C₁₉	B(4,𝔽₂)	1	1	yes	characteristic
95	C₂×D₁₉	—	8	6	no	—
96	C₂×D₁₉	—	16	3	no	—
97	C₂×D₁₉	—	16	1	no	—
98	C₄×C₁₉	—	2	3	no	—
99	C₄×C₁₉	—	2	3	no	—
100	C₄×C₁₉	—	8	3	no	—
101	C₂²×D₈	—	38	3	no	—
102	C₂×#127	—	19	1	no	—
103	C₂×#120	—	2	3	no	—
104	C₂×#102	—	38	3	no	—
105	#394	—	2	3	no	—
106	#390	C₂×D₁₉	1	3	yes	—
107	#336	—	19	3	no	—
108	C₂×C₁₉	—	8	6	no	—
109	C₂×C₁₉	#1595	1	3	yes	—
110	C₂×C₁₉	#1619	1	3	yes	—
111	C₂×C₁₉	#1655	1	1	yes	characteristic
112	D₁₉	—	16	2	no	—
113	C₂⁵	—	2	1	no	—
114	C₂²×D₄	—	2	3	no	—
115	C₂²×D₄	—	38	3	no	—
116	C₂²×D₄	C₂²⋊D₁₉	1	3	yes	—
117	C₂×D₈	—	76	6	no	—
118	C₂×D₈	—	76	6	no	—
119	C₂×D₈	—	76	6	no	—
120	C₂²×C₈	—	38	3	no	—
121	#127	—	76	2	no	—
122	#120	—	4	6	no	—
123	#120	—	4	6	no	—
124	C₂×C₄⋊C₄	C₂²⋊D₁₉	1	3	yes	—
125	C₂×C₂²⋊C₄	—	2	3	no	—
126	C₂×C₂²⋊C₄	—	2	6	no	—
127	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
128	#102	—	76	6	no	—
129	#95	—	2	1	no	—
130	C₁₉	#7734	1	1	yes	characteristic, complemented, Sylow
131	C₂⁴	—	2	3	no	—
132	C₂⁴	—	4	6	no	—
133	C₂⁴	—	8	3	no	—
134	C₂⁴	—	8	1	no	—
135	C₂⁴	—	76	1	no	—
136	C₂×D₄	—	2	3	no	—
137	C₂×D₄	—	4	3	no	—
138	C₂×D₄	—	4	3	no	—
139	C₂×D₄	—	4	12	no	—
140	C₂×D₄	—	8	3	no	—
141	C₂×D₄	—	38	6	no	—
142	C₂×D₄	—	76	3	no	—
143	C₂×D₄	—	76	3	no	—
144	C₂×D₄	—	76	3	no	—
145	C₂×D₄	—	76	3	no	—
146	C₂×D₄	—	76	3	no	—
147	C₂×D₄	#64525	1	6	yes	complemented
148	C₂²×C₄	—	2	3	no	—
149	C₂²×C₄	C₂×C₂²⋊D₁₉	1	3	yes	—
150	D₈	—	76	24	no	—
151	C₂×C₈	—	38	6	no	—
152	C₂×C₈	—	76	3	no	—
153	C₂×C₈	—	76	3	no	—
154	C₄⋊C₄	—	4	3	no	—
155	C₂²⋊C₄	—	4	6	no	—
156	C₂²⋊C₄	—	4	6	no	—
157	C₂²⋊C₄	—	4	6	no	—
158	C₄²	—	4	1	no	—
159	C₂³	—	2	6	no	—
160	C₂³	—	4	6	no	—
161	C₂³	—	4	3	no	—
162	C₂³	—	4	6	no	—
163	C₂³	—	4	3	no	—
164	C₂³	—	8	12	no	—
165	C₂³	—	8	12	no	—
166	C₂³	—	8	4	no	—
167	C₂³	—	76	2	no	—
168	C₂³	—	152	3	no	—
169	C₂³	—	152	3	no	—
170	C₂³	#10597399	1	1	yes	characteristic
171	D₄	—	4	6	no	—
172	D₄	—	8	3	no	—
173	D₄	—	8	12	no	—
174	D₄	—	76	6	no	—
175	D₄	—	76	6	no	—
176	D₄	—	152	3	no	—
177	D₄	—	152	3	no	—
178	C₂×C₄	—	2	3	no	—
179	C₂×C₄	—	2	3	no	—
180	C₂×C₄	—	4	6	no	—
181	C₂×C₄	—	4	3	no	—
182	C₂×C₄	C₂×#64525	1	3	yes	—
183	C₂×C₄	#10597378	1	3	yes	—
184	C₈	—	76	6	no	—
185	C₂²	—	4	12	no	—
186	C₂²	—	4	6	no	—
187	C₂²	—	8	16	no	—
188	C₂²	—	8	3	no	—
189	C₂²	—	8	3	no	—
190	C₂²	—	8	6	no	—
191	C₂²	—	304	1	no	—
192	C₂²	—	304	3	no	—
193	C₂²	—	152	6	no	—
194	C₂²	#11926678	1	3	yes	—
195	C₂²	#11926712	1	3	yes	—
196	C₂²	#11926731	1	1	yes	characteristic
197	C₄	—	2	3	no	—
198	C₄	—	2	3	no	—
199	C₄	—	8	3	no	—
200	C₂	—	8	6	no	—
201	C₂	—	304	2	no	—
202	C₂	#14562553	1	3	yes	—
203	C₂	#14562689	1	3	yes	—
204	C₂	#14562924	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14564705

C₂

yes

maximal, complemented

#14561456

C₂

yes

maximal, complemented

C₁₉×#2679

C₂

yes

maximal, characteristic, complemented

C₂²×#64525

—

C₂×#10597320

C₂²

yes

characteristic, complemented

C₂×#10597266

—

#11926299

C₂²

yes

complemented

C₂×C₁₉×#120

—

C₁₉×#394

—

C₁₉×#390

C₂²

yes

—

C₂⁵×C₁₉

—

C₂²×D₄×C₁₉

—

C₂²×D₄×C₁₉

D₄

yes

—

C₂²×D₇₆

—

C₁₉×#120

—

C₁₉×#120

—

C₂×C₄⋊C₄×C₁₉

D₄

yes

—

C₂×C₂²⋊C₄×C₁₉

—

C₂×C₂²⋊C₄×C₁₉

—

C₂×C₄²×C₁₉

C₂³

yes

—

—

—

C₂²×C₁₉⋊C₈

—

#10597320

—

C₁₉×#95

—

#10597266

—

C₂⁴×C₁₉

—

C₂⁴×C₁₉

—

C₂⁴×C₁₉

—

C₂⁴×C₁₉

—

C₂³×D₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

D₈

yes

complemented

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂×D₄

yes

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

—

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

—

#64525

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

#7734

—

maximal, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

#120

yes

characteristic

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂²×D₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂×D₈

yes

—

C₂×C₄×C₁₉

AGL(1,ℤ/8)

yes

—

D₇₆

—

D₇₆

—

D₇₆

—

D₇₆

—

C₁₉⋊C₈

—

#2679

D₁₉

yes

characteristic, complemented

#2011

—

#1270

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#447

yes

—

C₂²×C₁₉

#453

yes

—

C₂²×C₁₉

B(4,𝔽₂)

yes

characteristic

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

100

C₄×C₁₉

—

101

C₂²×D₈

—

102

C₂×#127

—

103

C₂×#120

—

104

C₂×#102

—

105

#394

—

106

#390

C₂×D₁₉

yes

—

107

#336

—

108

C₂×C₁₉

—

109

C₂×C₁₉

#1595

yes

—

110

C₂×C₁₉

#1619

yes

—

111

C₂×C₁₉

#1655

yes

characteristic

112

D₁₉

—

113

C₂⁵

—

114

C₂²×D₄

—

115

C₂²×D₄

—

116

C₂²×D₄

C₂²⋊D₁₉

yes

—

117

C₂×D₈

—

118

C₂×D₈

—

119

C₂×D₈

—

120

C₂²×C₈

—

121

#127

—

122

#120

—

123

#120

—

124

C₂×C₄⋊C₄

C₂²⋊D₁₉

yes

—

125

C₂×C₂²⋊C₄

—

126

C₂×C₂²⋊C₄

—

127

C₂×C₄²

C₂²×D₁₉

yes

characteristic

128

#102

—

129

#95

—

130

C₁₉

#7734

yes

characteristic, complemented, Sylow

131

C₂⁴

—

132

C₂⁴

—

133

C₂⁴

—

134

C₂⁴

—

135

C₂⁴

—

136

C₂×D₄

—

137

C₂×D₄

—

138

C₂×D₄

—

139

C₂×D₄

—

140

C₂×D₄

—

141

C₂×D₄

—

142

C₂×D₄

—

143

C₂×D₄

—

144

C₂×D₄

—

145

C₂×D₄

—

146

C₂×D₄

—

147

C₂×D₄

#64525

yes

complemented

148

C₂²×C₄

—

149

C₂²×C₄

C₂×C₂²⋊D₁₉

yes

—

150

D₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₄⋊C₄

—

155

C₂²⋊C₄

—

156

C₂²⋊C₄

—

157

C₂²⋊C₄

—

158

C₄²

—

159

C₂³

—

160

C₂³

—

161

C₂³

—

162

C₂³

—

163

C₂³

—

164

C₂³

—

165

C₂³

—

166

C₂³

—

167

C₂³

—

168

C₂³

—

152

—

169

C₂³

—

152

—

170

C₂³

#10597399

yes

characteristic

171

D₄

—

172

D₄

—

173

D₄

—

174

D₄

—

175

D₄

—

176

D₄

—

152

—

177

D₄

—

152

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

C₂×#64525

yes

—

183

C₂×C₄

#10597378

yes

—

184

C₈

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

189

C₂²

—

190

C₂²

—

191

C₂²

—

304

—

192

C₂²

—

304

—

193

C₂²

—

152

—

194

C₂²

#11926678

yes

—

195

C₂²

#11926712

yes

—

196

C₂²

#11926731

yes

characteristic

197

C₄

—

198

C₄

—

199

C₄

—

200

C₂

—

201

C₂

—

304

—

202

C₂

#14562553

yes

—

203

C₂

#14562689

yes

—

204

C₂

#14562924

yes

characteristic

	subgroup	G/H
center	C₂³	#10597399
commutator, cosocle	C₂×C₄²×C₁₉	C₂³
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#2679	C₂
socle	C₂³×C₁₉	#120
hypercenter	#2679	D₁₉
hypocenter	C₁₉	#7734
LCS	G ▷ C₂×C₄²×C₁₉ ▷ C₂³×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2679

subgroup

G/H

center

C₂³

#10597399

commutator, cosocle

C₂×C₄²×C₁₉

C₂³

Frattini

C₂×C₄²

C₂²×D₁₉

Fitting

C₁₉×#2679

C₂

C₂³×C₁₉

D₁₉

C₁₉

G ▷ C₂×C₄²×C₁₉ ▷ C₂³×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2679

N	H ≅ G/N	type
1. #14564705	C₂	semidirect
2. #14561456	C₂	semidirect
3. C₁₉×#2679	C₂	semidirect
5. C₂×#10597320	C₂²	semidirect
7. #11926299	C₂²	semidirect
38. C₂×D₄×C₁₉	D₈	semidirect
83. #2679	D₁₉	semidirect
130. C₁₉	#7734	semidirect
147. C₂×D₄	#64525	semidirect

H ≅ G/N

type

1. #14564705

C₂

semidirect

2. #14561456

C₂

semidirect

3. C₁₉×#2679

C₂

semidirect

5. C₂×#10597320

C₂²

semidirect

7. #11926299

C₂²

semidirect

38. C₂×D₄×C₁₉

D₈

semidirect

83. #2679

D₁₉

semidirect

130. C₁₉

#7734

semidirect

147. C₂×D₄

#64525

semidirect

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

subgroup	index
#14564705	2
#14561456	2
C₁₉×#2679	2
#7734	19

subgroup

index

#14564705

#14561456

C₁₉×#2679

#7734

Subgroups — #18071237

Conjugacy classes of subgroups

Complements

Maximal subgroups