Group #18086737 — subgroups

Subgroups — #18086737

Order 4864.

342 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 85 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	#14563772	C₂	1	1	yes	maximal, characteristic
2	#14562373	C₂	1	2	yes	maximal
3	#14562090	C₂	1	2	yes	maximal
4	C₁₉⋊#1309	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#3255	C₂	1	1	yes	maximal, characteristic
6	#11926568	C₂²	1	2	yes	—
7	#11926568	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
9	#11926308	—	2	2	no	—
10	#11926290	—	2	2	no	—
11	#11926290	—	2	2	no	—
12	Q₈⋊(C₁₉⋊C₈)	—	2	4	no	—
13	C₁₉×#533	C₂²	1	2	yes	—
14	C₁₉×#475	—	2	2	no	—
15	C₁₉×#470	—	2	2	no	—
16	C₁₉×#462	—	2	2	no	—
17	C₁₉×#423	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₁₉	—	2	2	no	—
19	C₂×M₁₆×C₁₉	D₄	1	2	yes	—
20	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
21	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
22	C₁₉×#125	—	2	2	no	—
23	C₁₉×C₂²⋊Q₈	—	2	2	no	—
24	C₁₉×C₂²⋊Q₈	—	2	2	no	—
25	C₄×Q₈×C₁₉	—	2	2	no	—
26	C₄×Q₈×C₁₉	—	2	2	no	—
27	C₁₉×#117	—	2	2	no	—
28	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
29	C₂×#64520	—	2	1	no	—
30	C₂×#64520	—	2	2	no	—
31	C₂²×C₁₉⋊C₈	—	2	1	no	—
32	C₁₉×#106	—	2	2	no	—
33	C₁₉×C₄⋊C₈	—	2	2	no	—
34	C₁₉×#103	—	2	4	no	—
35	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
36	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
37	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
38	C₄×C₁₉⋊C₈	—	2	1	no	—
39	C₄×C₁₉⋊C₈	D₄	1	2	yes	—
40	C₂×Q₈×C₁₉	—	2	2	no	—
41	C₂×Q₈×C₁₉	—	2	2	no	—
42	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₉	#41	1	2	yes	—
44	Q₁₆×C₁₉	—	8	2	no	—
45	M₁₆×C₁₉	—	4	2	no	—
46	C₂×C₈×C₁₉	—	2	2	no	—
47	C₄⋊C₄×C₁₉	—	2	2	no	—
48	C₄⋊C₄×C₁₉	—	2	2	no	—
49	C₄⋊C₄×C₁₉	—	2	2	no	—
50	C₄⋊C₄×C₁₉	—	2	2	no	—
51	C₄⋊C₄×C₁₉	—	4	2	no	—
52	C₄⋊C₄×C₁₉	—	4	2	no	—
53	C₂²⋊C₄×C₁₉	—	4	2	no	—
54	C₄²×C₁₉	—	4	2	no	—
55	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₁₉	C₂×D₄	1	2	yes	—
57	C₄²×C₁₉	#41	1	1	yes	characteristic
58	#64520	—	4	2	no	—
59	#64520	—	8	2	no	—
60	C₂×C₁₉⋊C₈	—	2	2	no	—
61	C₂×C₁₉⋊C₈	—	4	1	no	—
62	C₂×C₁₉⋊C₈	—	4	1	no	—
63	C₂×C₁₉⋊C₈	—	4	1	no	—
64	C₂×C₁₉⋊C₈	—	4	2	no	—
65	#10535	—	19	1	no	maximal, Sylow
66	C₂³×C₁₉	#124	1	1	yes	characteristic
67	Q₈×C₁₉	—	4	2	no	—
68	Q₈×C₁₉	—	4	2	no	—
69	Q₈×C₁₉	—	4	2	no	—
70	Q₈×C₁₉	—	4	2	no	—
71	C₂×C₄×C₁₉	—	2	1	no	—
72	C₂×C₄×C₁₉	—	2	1	no	—
73	C₂×C₄×C₁₉	—	2	2	no	—
74	C₂×C₄×C₁₉	—	2	2	no	—
75	C₂×C₄×C₁₉	—	4	2	no	—
76	C₂×C₄×C₁₉	—	4	2	no	—
77	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
78	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
79	C₂×C₄×C₁₉	#121	1	2	yes	—
80	C₂×C₄×C₁₉	#123	1	2	yes	—
81	C₈×C₁₉	—	4	2	no	—
82	C₁₉⋊C₈	—	4	2	no	—
83	C₁₉⋊C₈	—	8	1	no	—
84	C₁₉⋊C₈	—	8	2	no	—
85	#3255	D₁₉	1	1	yes	characteristic
86	#1837	—	19	1	no	—
87	#1561	—	19	2	no	—
88	#1493	—	19	2	no	—
89	#1309	—	19	1	no	—
90	C₂²×C₁₉	—	2	2	no	—
91	C₂²×C₁₉	—	2	1	no	—
92	C₂²×C₁₉	#394	1	1	yes	characteristic
93	C₄×C₁₉	—	2	2	no	—
94	C₄×C₁₉	—	2	1	no	—
95	C₄×C₁₉	—	8	2	no	—
96	C₄×C₁₉	—	8	2	no	—
97	C₄×C₁₉	#454	1	1	yes	characteristic
98	C₄×C₁₉	#455	1	1	yes	characteristic
99	C₄×C₁₉	#471	1	2	yes	—
100	C₄×C₁₉	#473	1	2	yes	—
101	#533	C₂×D₁₉	1	2	yes	—
102	#475	—	2	2	no	—
103	#470	—	2	2	no	—
104	#462	—	2	2	no	—
105	#423	—	19	1	no	—
106	#423	—	19	2	no	—
107	#423	C₂×D₁₉	1	1	yes	characteristic
108	C₂×C₄×C₈	—	19	1	no	—
109	#341	—	38	2	no	—
110	#332	—	38	2	no	—
111	#332	—	38	2	no	—
112	Q₈⋊C₈	—	38	4	no	—
113	C₂×C₁₉	—	4	1	no	—
114	C₂×C₁₉	#2040	1	1	yes	characteristic
115	C₂×C₁₉	#2052	1	2	yes	—
116	C₂×Q₁₆	—	2	2	no	—
117	C₂×M₁₆	—	38	1	no	—
118	C₂×M₁₆	—	38	2	no	—
119	C₂×M₁₆	C₂²⋊D₁₉	1	2	yes	—
120	C₂²×C₈	—	38	1	no	—
121	C₄⋊Q₈	C₂²⋊D₁₉	1	2	yes	—
122	C₄⋊Q₈	C₂²⋊D₁₉	1	2	yes	—
123	#125	—	2	2	no	—
124	C₂²⋊Q₈	—	2	2	no	—
125	C₂²⋊Q₈	—	2	2	no	—
126	C₄×Q₈	—	2	2	no	—
127	C₄×Q₈	—	2	2	no	—
128	#117	—	2	2	no	—
129	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
130	#106	—	2	2	no	—
131	C₄⋊C₈	—	2	2	no	—
132	C₄⋊C₈	—	38	2	no	—
133	C₄⋊C₈	—	38	2	no	—
134	C₄⋊C₈	—	38	2	no	—
135	#103	—	2	4	no	—
136	C₄×C₈	—	19	2	no	—
137	C₄×C₈	—	38	1	no	—
138	C₁₉	#10535	1	1	yes	characteristic, complemented, Sylow
139	C₂×Q₈	—	2	2	no	—
140	C₂×Q₈	—	2	2	no	—
141	C₂²×C₄	#64543	1	2	yes	—
142	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
143	Q₁₆	—	8	2	no	—
144	M₁₆	—	4	2	no	—
145	M₁₆	—	76	2	no	—
146	M₁₆	—	152	2	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	38	2	no	—
149	C₂×C₈	—	76	1	no	—
150	C₂×C₈	—	76	1	no	—
151	C₂×C₈	—	76	2	no	—
152	C₂×C₈	—	76	1	no	—
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	2	2	no	—
157	C₄⋊C₄	—	4	2	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₂²⋊C₄	—	4	2	no	—
160	C₄²	—	4	2	no	—
161	C₄²	D₄×D₁₉	1	1	yes	characteristic
162	C₄²	#64543	1	1	yes	characteristic
163	C₄²	C₂×C₂²⋊D₁₉	1	2	yes	—
164	C₂³	#10597382	1	1	yes	characteristic
165	Q₈	—	4	2	no	—
166	Q₈	—	4	2	no	—
167	Q₈	—	4	2	no	—
168	Q₈	—	4	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	#10597381	1	2	yes	—
176	C₂×C₄	#10597384	1	1	yes	characteristic
177	C₂×C₄	#10597385	1	2	yes	—
178	C₂×C₄	#10597399	1	1	yes	characteristic
179	C₈	—	4	2	no	—
180	C₈	—	76	2	no	—
181	C₈	—	152	2	no	—
182	C₈	—	152	1	no	—
183	C₂²	—	2	2	no	—
184	C₂²	—	2	1	no	—
185	C₂²	#11926458	1	1	yes	characteristic
186	C₄	—	2	2	no	—
187	C₄	—	2	1	no	—
188	C₄	—	8	2	no	—
189	C₄	—	8	2	no	—
190	C₄	#11926717	1	1	yes	characteristic
191	C₄	#11926722	1	1	yes	characteristic
192	C₄	#11926809	1	2	yes	—
193	C₄	#11926817	1	2	yes	—
194	C₂	—	4	1	no	—
195	C₂	#14564861	1	1	yes	characteristic
196	C₂	#14564931	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14563772

C₂

yes

maximal, characteristic

#14562373

C₂

yes

maximal

#14562090

C₂

yes

maximal

C₁₉⋊#1309

C₂

yes

maximal, characteristic

C₁₉×#3255

C₂

yes

maximal, characteristic

#11926568

C₂²

yes

—

#11926568

C₂²

yes

characteristic

C₂×C₄×C₁₉⋊C₈

C₂²

yes

—

—

—

Q₈⋊(C₁₉⋊C₈)

—

C₁₉×#533

C₂²

yes

—

C₁₉×#475

—

C₁₉×#470

—

C₁₉×#462

—

C₁₉×#423

C₂²

yes

characteristic

C₂×Q₁₆×C₁₉

—

C₂×M₁₆×C₁₉

D₄

yes

—

C₁₉×C₄⋊Q₈

D₄

yes

—

C₁₉×C₄⋊Q₈

D₄

yes

—

C₁₉×#125

—

C₁₉×C₂²⋊Q₈

—

C₁₉×C₂²⋊Q₈

—

C₄×Q₈×C₁₉

—

C₄×Q₈×C₁₉

—

C₁₉×#117

—

C₂×C₄²×C₁₉

C₂³

yes

characteristic

C₂×#64520

—

C₂×#64520

—

C₂²×C₁₉⋊C₈

—

C₁₉×#106

—

C₁₉×C₄⋊C₈

—

C₁₉×#103

—

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

—

C₄×C₁₉⋊C₈

—

C₄×C₁₉⋊C₈

D₄

yes

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂²×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₉

#41

yes

—

Q₁₆×C₁₉

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₄²×C₁₉

C₂×D₄

yes

—

C₄²×C₁₉

yes

—

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

—

C₂³×C₁₉

yes

Q₈×C₁₉

—

Q₈×C₁₉

—

Q₈×C₁₉

—

Q₈×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

#123

yes

—

C₈×C₁₉

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

#3255

D₁₉

yes

—

—

—

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#394

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

yes

C₄×C₁₉

yes

C₄×C₁₉

yes

—

100

C₄×C₁₉

#473

yes

—

101

#533

C₂×D₁₉

yes

—

102

#475

—

103

#470

—

104

#462

—

105

#423

—

106

#423

—

107

#423

C₂×D₁₉

yes

characteristic

108

C₂×C₄×C₈

—

109

#341

—

110

#332

—

111

#332

—

112

Q₈⋊C₈

—

113

C₂×C₁₉

—

114

C₂×C₁₉

#2040

yes

characteristic

115

C₂×C₁₉

#2052

yes

—

116

C₂×Q₁₆

—

117

C₂×M₁₆

—

118

C₂×M₁₆

—

119

C₂×M₁₆

C₂²⋊D₁₉

yes

—

120

C₂²×C₈

—

121

C₄⋊Q₈

C₂²⋊D₁₉

yes

—

122

C₄⋊Q₈

C₂²⋊D₁₉

yes

—

123

#125

—

124

C₂²⋊Q₈

—

125

C₂²⋊Q₈

—

126

C₄×Q₈

—

127

C₄×Q₈

—

128

#117

—

129

C₂×C₄²

C₂²×D₁₉

yes

characteristic

130

#106

—

131

C₄⋊C₈

—

132

C₄⋊C₈

—

133

C₄⋊C₈

—

134

C₄⋊C₈

—

135

#103

—

136

C₄×C₈

—

137

C₄×C₈

—

138

C₁₉

#10535

yes

characteristic, complemented, Sylow

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂²×C₄

#64543

yes

—

142

C₂²×C₄

C₂×C₂²⋊D₁₉

yes

characteristic

143

Q₁₆

—

144

M₁₆

—

145

M₁₆

—

146

M₁₆

—

152

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₂²⋊C₄

—

160

C₄²

—

161

C₄²

D₄×D₁₉

yes

characteristic

162

C₄²

#64543

yes

characteristic

163

C₄²

C₂×C₂²⋊D₁₉

yes

—

164

C₂³

#10597382

yes

characteristic

165

Q₈

—

166

Q₈

—

167

Q₈

—

168

Q₈

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

#10597381

yes

—

176

C₂×C₄

#10597384

yes

characteristic

177

C₂×C₄

#10597385

yes

—

178

C₂×C₄

#10597399

yes

characteristic

179

C₈

—

180

C₈

—

181

C₈

—

152

—

182

C₈

—

152

—

183

C₂²

—

184

C₂²

—

185

C₂²

#11926458

yes

characteristic

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

#11926717

yes

characteristic

191

C₄

#11926722

yes

characteristic

192

C₄

#11926809

yes

—

193

C₄

#11926817

yes

—

194

C₂

—

195

C₂

#14564861

yes

characteristic

196

C₂

#14564931

yes

—

	subgroup	G/H
center	C₂²	#11926458
commutator, cosocle	C₂×C₄²×C₁₉	C₂³
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#3255	C₂
socle	C₂²×C₁₉	#394
prime radical	C₂³×C₁₉	#124
hypercenter	#3255	D₁₉
hypocenter	C₁₉	#10535
LCS	G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3255

subgroup

G/H

center

C₂²

#11926458

commutator, cosocle

C₂×C₄²×C₁₉

C₂³

Frattini

C₂×C₄²

C₂²×D₁₉

Fitting

C₁₉×#3255

C₂

C₂²×C₁₉

C₂³×C₁₉

D₁₉

C₁₉

G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3255

N	H ≅ G/N	type
138. C₁₉	#10535	semidirect

H ≅ G/N

type

138. C₁₉

#10535

semidirect

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

subgroup	index
#14563772	2
#14562373	2
#14562090	2
C₁₉⋊#1309	2
C₁₉×#3255	2
#10535	19

subgroup

index

#14563772

#14562373

#14562090

C₁₉⋊#1309

C₁₉×#3255

#10535

Subgroups — #18086737

Conjugacy classes of subgroups

Complements

Maximal subgroups