Group #18087664 — subgroups

Subgroups — #18087664

Order 4864.

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

normal subgroups only

This group has 264 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	#14563810	C₂	1	1	yes	maximal, characteristic
2	#14561538	C₂	1	2	yes	maximal
3	#14561532	C₂	1	2	yes	maximal
4	#14561466	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1940	C₂	1	1	yes	maximal, characteristic
6	#11926597	C₂²	1	2	yes	—
7	C₂×#10597276	—	2	1	no	—
8	C₂×#10597276	—	2	1	no	—
9	#11926568	C₂²	1	2	yes	—
10	C₂×C₁₉×C₄⋊C₈	—	2	1	no	—
11	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₁₉×#97	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
14	C₁₉×#336	—	2	1	no	—
15	C₂²×C₈×C₁₉	—	2	1	no	—
16	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
19	C₂×C₄²×C₁₉	—	2	1	no	—
20	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
21	C₂×#64520	—	2	2	no	—
22	C₁₉×C₄⋊C₈	—	2	2	no	—
23	C₁₉×C₄⋊C₈	—	4	2	no	—
24	C₁₉×#97	—	2	2	no	—
25	C₁₉×#95	Q₈	1	1	yes	characteristic
26	C₁₉×#95	Q₈	1	1	yes	characteristic
27	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
28	#10597276	—	4	2	no	—
29	#10597276	—	4	2	no	—
30	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
31	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
32	C₄×C₁₉⋊C₈	—	2	2	no	—
33	C₂²×C₄×C₁₉	—	2	1	no	—
34	C₂²×C₄×C₁₉	—	2	1	no	—
35	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
37	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
38	C₂×C₈×C₁₉	—	2	1	no	—
39	C₂×C₈×C₁₉	—	2	1	no	—
40	C₂×C₈×C₁₉	—	2	1	no	—
41	C₂×C₈×C₁₉	—	2	1	no	—
42	C₂×C₈×C₁₉	—	4	1	no	—
43	C₂×C₈×C₁₉	—	4	1	no	—
44	C₂×C₈×C₁₉	—	4	1	no	—
45	C₂×C₈×C₁₉	D₈	1	2	yes	—
46	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
47	C₄⋊C₄×C₁₉	—	4	2	no	—
48	C₄²×C₁₉	—	4	1	no	—
49	C₄²×C₁₉	—	4	1	no	—
50	C₄²×C₁₉	#41	1	2	yes	—
51	C₄²×C₁₉	#41	1	2	yes	—
52	#64520	—	4	2	no	—
53	#64520	—	4	2	no	—
54	C₂×C₁₉⋊C₈	—	4	2	no	—
55	C₂×C₁₉⋊C₈	—	4	2	no	—
56	#10687	—	19	1	no	maximal, Sylow
57	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
58	C₂×C₄×C₁₉	—	2	2	no	—
59	C₂×C₄×C₁₉	—	4	1	no	—
60	C₂×C₄×C₁₉	—	4	1	no	—
61	C₂×C₄×C₁₉	—	4	1	no	—
62	C₂×C₄×C₁₉	—	4	1	no	—
63	C₂×C₄×C₁₉	—	4	1	no	—
64	C₂×C₄×C₁₉	—	4	1	no	—
65	C₂×C₄×C₁₉	—	4	1	no	—
66	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
67	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
68	C₂×C₄×C₁₉	#125	1	2	yes	—
69	C₂×C₄×C₁₉	#126	1	2	yes	—
70	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
71	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
73	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
74	C₈×C₁₉	—	2	2	no	—
75	C₈×C₁₉	—	2	2	no	—
76	C₈×C₁₉	—	8	1	no	—
77	C₁₉⋊C₈	—	8	2	no	—
78	C₁₉⋊C₈	—	8	2	no	—
79	#1940	D₁₉	1	1	yes	characteristic
80	#1849	—	19	1	no	—
81	#1303	—	19	2	no	—
82	#1301	—	19	2	no	—
83	#1275	—	19	1	no	—
84	C₂²×C₁₉	—	2	1	no	—
85	C₂²×C₁₉	—	2	1	no	—
86	C₂²×C₁₉	#400	1	1	yes	characteristic
87	C₂²×C₁₉	#474	1	1	yes	characteristic
88	C₂²×C₁₉	#475	1	1	yes	characteristic
89	C₄×C₁₉	—	2	2	no	—
90	C₄×C₁₉	—	2	2	no	—
91	C₄×C₁₉	—	4	2	no	—
92	C₄×C₁₉	—	8	1	no	—
93	C₄×C₁₉	#480	1	1	yes	characteristic
94	C₄×C₁₉	#483	1	1	yes	characteristic
95	C₄×C₁₉	#486	1	1	yes	characteristic
96	C₄×C₁₉	#491	1	1	yes	characteristic
97	#431	—	19	2	no	—
98	C₂×#108	—	38	1	no	—
99	C₂×#108	—	38	1	no	—
100	#423	—	19	2	no	—
101	C₂×C₄⋊C₈	—	2	1	no	—
102	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
103	C₂×#97	C₂×D₁₉	1	1	yes	characteristic
104	C₄×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
105	#336	—	2	1	no	—
106	C₂×C₁₉	—	2	1	no	—
107	C₂×C₁₉	—	2	1	no	—
108	C₂×C₁₉	#2099	1	1	yes	characteristic
109	C₂×C₁₉	#2102	1	1	yes	characteristic
110	C₂×C₁₉	#2103	1	1	yes	characteristic
111	C₂×M₁₆	—	38	2	no	—
112	C₂²×C₈	—	2	1	no	—
113	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
114	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
115	C₂×C₄⋊C₄	—	2	1	no	—
116	C₂×C₄²	—	2	1	no	—
117	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
118	#108	—	76	2	no	—
119	#108	—	76	2	no	—
120	C₄⋊C₈	—	2	2	no	—
121	C₄⋊C₈	—	4	2	no	—
122	C₄⋊C₈	—	38	2	no	—
123	C₄⋊C₈	—	38	4	no	—
124	C₂²⋊C₈	—	38	2	no	—
125	#97	—	2	2	no	—
126	C₄×C₈	—	38	2	no	—
127	#95	Dic₃₈	1	1	yes	characteristic
128	#95	Dic₃₈	1	1	yes	characteristic
129	C₁₉	#10687	1	1	yes	characteristic, complemented, Sylow
130	C₂²×C₄	—	2	1	no	—
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
133	C₂²×C₄	#64541	1	1	yes	characteristic
134	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
135	M₁₆	—	76	2	no	—
136	M₁₆	—	76	2	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	4	1	no	—
142	C₂×C₈	—	4	1	no	—
143	C₂×C₈	—	4	1	no	—
144	C₂×C₈	—	76	2	no	—
145	C₂×C₈	—	76	2	no	—
146	C₂×C₈	#64525	1	2	yes	—
147	C₂×C₈	#64527	1	2	yes	—
148	C₄⋊C₄	—	4	2	no	—
149	C₄²	—	4	1	no	—
150	C₄²	—	4	1	no	—
151	C₄²	#64543	1	2	yes	—
152	C₄²	#64545	1	2	yes	—
153	C₂³	#10597370	1	1	yes	characteristic
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	4	1	no	—
158	C₂×C₄	—	4	1	no	—
159	C₂×C₄	—	4	1	no	—
160	C₂×C₄	—	4	1	no	—
161	C₂×C₄	—	4	1	no	—
162	C₂×C₄	#10597336	1	2	yes	—
163	C₂×C₄	#10597344	1	2	yes	—
164	C₂×C₄	C₂×#64525	1	1	yes	characteristic
165	C₂×C₄	#10597381	1	1	yes	characteristic
166	C₂×C₄	C₂×#64527	1	1	yes	characteristic
167	C₂×C₄	#10597393	1	1	yes	characteristic
168	C₂×C₄	#10597397	1	1	yes	characteristic
169	C₂×C₄	#10597397	1	1	yes	characteristic
170	C₈	—	2	2	no	—
171	C₈	—	2	2	no	—
172	C₈	—	8	1	no	—
173	C₈	—	152	2	no	—
174	C₈	—	152	2	no	—
175	C₂²	—	2	1	no	—
176	C₂²	—	2	1	no	—
177	C₂²	#11926483	1	1	yes	characteristic
178	C₂²	#11926824	1	1	yes	characteristic
179	C₂²	#11926829	1	1	yes	characteristic
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	4	2	no	—
183	C₄	—	8	1	no	—
184	C₄	#11926855	1	1	yes	characteristic
185	C₄	#11926872	1	1	yes	characteristic
186	C₄	#11926892	1	1	yes	characteristic
187	C₄	#11926914	1	1	yes	characteristic
188	C₂	—	2	1	no	—
189	C₂	—	2	1	no	—
190	C₂	#14565193	1	1	yes	characteristic
191	C₂	#14565210	1	1	yes	characteristic
192	C₂	#14565215	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14563810

C₂

yes

maximal, characteristic

#14561538

C₂

yes

maximal

#14561532

C₂

yes

maximal

#14561466

C₂

yes

maximal, characteristic

C₁₉×#1940

C₂

yes

maximal, characteristic

#11926597

C₂²

yes

—

C₂×#10597276

—

C₂×#10597276

—

#11926568

C₂²

yes

—

C₂×C₁₉×C₄⋊C₈

—

C₂×C₁₉×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₁₉×#97

C₂²

yes

characteristic

C₄×C₄⋊C₄×C₁₉

C₂²

yes

characteristic

C₁₉×#336

—

C₂²×C₈×C₁₉

—

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄²×C₁₉

—

C₂×C₄²×C₁₉

C₂³

yes

characteristic

C₂×#64520

—

C₁₉×C₄⋊C₈

—

C₁₉×C₄⋊C₈

—

C₁₉×#97

—

C₁₉×#95

Q₈

yes

characteristic

C₁₉×#95

Q₈

yes

characteristic

C₂²⋊(C₁₉⋊C₈)

—

#10597276

—

#10597276

—

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

—

C₄×C₁₉⋊C₈

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₉

C₂×Q₈

yes

characteristic

C₂²×C₄×C₁₉

#41

yes

characteristic

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

D₈

yes

—

C₂×C₈×C₁₉

SD₁₆

yes

—

C₄⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

#41

yes

—

C₄²×C₁₉

#41

yes

—

#64520

—

#64520

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

#10687

—

maximal, Sylow

C₂³×C₁₉

C₂²⋊Q₈

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

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

—

C₂×C₄×C₁₉

#126

yes

—

C₂×C₄×C₁₉

C₂×D₈

yes

characteristic

C₂×C₄×C₁₉

C₂×SD₁₆

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

#1940

D₁₉

yes

—

—

—

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

yes

C₄×C₁₉

yes

C₄×C₁₉

yes

C₄×C₁₉

yes

—

—

—

100

#423

—

101

C₂×C₄⋊C₈

—

102

C₂×C₄⋊C₈

C₂×D₁₉

yes

characteristic

103

C₂×#97

C₂×D₁₉

yes

characteristic

104

C₄×C₄⋊C₄

C₂×D₁₉

yes

characteristic

105

#336

—

106

C₂×C₁₉

—

107

C₂×C₁₉

—

108

C₂×C₁₉

#2099

yes

characteristic

109

C₂×C₁₉

#2102

yes

characteristic

110

C₂×C₁₉

#2103

yes

characteristic

111

C₂×M₁₆

—

112

C₂²×C₈

—

113

C₂²×C₈

C₂²⋊D₁₉

yes

characteristic

114

C₂²×C₈

C₂²⋊D₁₉

yes

characteristic

115

C₂×C₄⋊C₄

—

116

C₂×C₄²

—

117

C₂×C₄²

C₂²×D₁₉

yes

characteristic

118

#108

—

119

#108

—

120

C₄⋊C₈

—

121

C₄⋊C₈

—

122

C₄⋊C₈

—

123

C₄⋊C₈

—

124

C₂²⋊C₈

—

125

#97

—

126

C₄×C₈

—

127

#95

Dic₃₈

yes

characteristic

128

#95

Dic₃₈

yes

characteristic

129

C₁₉

#10687

yes

characteristic, complemented, Sylow

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

C₂×Dic₃₈

yes

characteristic

133

C₂²×C₄

#64541

yes

characteristic

134

C₂²×C₄

C₂×C₂²⋊D₁₉

yes

characteristic

135

M₁₆

—

136

M₁₆

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

#64525

yes

—

147

C₂×C₈

#64527

yes

—

148

C₄⋊C₄

—

149

C₄²

—

150

C₄²

—

151

C₄²

#64543

yes

—

152

C₄²

#64545

yes

—

153

C₂³

#10597370

yes

characteristic

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

#10597336

yes

—

163

C₂×C₄

#10597344

yes

—

164

C₂×C₄

C₂×#64525

yes

characteristic

165

C₂×C₄

#10597381

yes

characteristic

166

C₂×C₄

C₂×#64527

yes

characteristic

167

C₂×C₄

#10597393

yes

characteristic

168

C₂×C₄

#10597397

yes

characteristic

169

C₂×C₄

#10597397

yes

characteristic

170

C₈

—

171

C₈

—

172

C₈

—

173

C₈

—

152

—

174

C₈

—

152

—

175

C₂²

—

176

C₂²

—

177

C₂²

#11926483

yes

characteristic

178

C₂²

#11926824

yes

characteristic

179

C₂²

#11926829

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

#11926855

yes

characteristic

185

C₄

#11926872

yes

characteristic

186

C₄

#11926892

yes

characteristic

187

C₄

#11926914

yes

characteristic

188

C₂

—

189

C₂

—

190

C₂

#14565193

yes

characteristic

191

C₂

#14565210

yes

characteristic

192

C₂

#14565215

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#10597393
commutator, cosocle	C₂×C₄²×C₁₉	C₂³
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1940	C₂
socle	C₂²×C₁₉	#400
prime radical	C₂³×C₁₉	C₂²⋊Q₈
hypercenter	#1940	D₁₉
hypocenter	C₁₉	#10687
LCS	G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1940

subgroup

G/H

center

C₂×C₄

#10597393

commutator, cosocle

C₂×C₄²×C₁₉

C₂³

Frattini

C₂×C₄²

C₂²×D₁₉

Fitting

C₁₉×#1940

C₂

socle

C₂²×C₁₉

#400

prime radical

C₂³×C₁₉

C₂²⋊Q₈

D₁₉

C₁₉

G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1940

N	H ≅ G/N	type
129. C₁₉	#10687	semidirect

H ≅ G/N

type

129. C₁₉

#10687

semidirect

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

subgroup	index
#14563810	2
#14561538	2
#14561532	2
#14561466	2
C₁₉×#1940	2
#10687	19

subgroup

index

C₁₉×#1940

Subgroups — #18087664

Conjugacy classes of subgroups

Complements

Maximal subgroups