Group #18086741 — subgroups

Subgroups — #18086741

Order 4864.

342 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 81 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	#14575029	C₂	1	1	yes	maximal, characteristic
2	#14563772	C₂	1	1	yes	maximal, characteristic
3	#14562372	C₂	1	2	yes	maximal
4	#14562089	C₂	1	2	yes	maximal
5	C₁₉×#1309	C₂	1	1	yes	maximal, characteristic
6	#11927101	C₂²	1	2	yes	—
7	#11926830	—	2	2	no	—
8	#11926802	—	2	2	no	—
9	#11926755	—	2	2	no	—
10	#11926568	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
12	#11926308	—	2	2	no	—
13	#11926291	—	2	2	no	—
14	#11926289	—	2	2	no	—
15	#11926274	—	2	4	no	—
16	C₁₉×#423	C₂²	1	2	yes	—
17	C₁₉×#423	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₁₉	—	2	2	no	—
19	C₂×M₁₆×C₁₉	D₄	1	2	yes	—
20	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
21	#10597372	—	2	2	no	—
22	#10597370	—	2	2	no	—
23	#10597370	—	2	2	no	—
24	C₂×#64520	—	2	1	no	—
25	C₂²×C₁₉⋊C₈	—	2	1	no	—
26	C₂×Dic₇₆	—	2	2	no	—
27	#10597316	—	2	2	no	—
28	#10597315	D₄	1	2	yes	—
29	#10597315	D₄	1	2	yes	—
30	C₄×Dic₃₈	—	2	2	no	—
31	C₄×Dic₃₈	—	2	2	no	—
32	C₁₉×C₄⋊C₈	—	2	2	no	—
33	C₁₉×C₄⋊C₈	—	2	2	no	—
34	C₁₉×C₄⋊C₈	—	2	2	no	—
35	#10597274	—	2	2	no	—
36	#10597273	—	2	4	no	—
37	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
38	C₄×C₁₉⋊C₈	—	2	1	no	—
39	C₄×C₁₉⋊C₈	D₄	1	2	yes	—
40	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₉	#41	1	2	yes	—
42	C₂×Dic₃₈	—	2	2	no	—
43	C₂×Dic₃₈	—	2	2	no	—
44	M₁₆×C₁₉	—	4	2	no	—
45	M₁₆×C₁₉	—	8	2	no	—
46	C₂×C₈×C₁₉	—	2	2	no	—
47	C₂×C₈×C₁₉	—	4	2	no	—
48	C₄²×C₁₉	C₂×D₄	1	2	yes	—
49	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
50	C₄²×C₁₉	#41	1	1	yes	characteristic
51	C₂²⋊Dic₁₉	—	4	2	no	—
52	C₄⋊Dic₁₉	—	2	2	no	—
53	C₄⋊Dic₁₉	—	2	2	no	—
54	C₄⋊Dic₁₉	—	2	2	no	—
55	C₄⋊Dic₁₉	—	2	2	no	—
56	Dic₁₉⋊C₄	—	4	2	no	—
57	Dic₁₉⋊C₄	—	4	2	no	—
58	C₄×Dic₁₉	—	4	2	no	—
59	#64520	—	4	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	Dic₇₆	—	8	2	no	—
65	#10535	—	19	1	no	maximal, Sylow
66	C₂³×C₁₉	#124	1	1	yes	characteristic
67	C₂×C₄×C₁₉	—	2	1	no	—
68	C₂×C₄×C₁₉	—	2	1	no	—
69	C₂×C₄×C₁₉	—	2	2	no	—
70	C₂×C₄×C₁₉	—	2	2	no	—
71	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
72	C₂×C₄×C₁₉	#121	1	2	yes	—
73	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
74	C₂×C₄×C₁₉	#123	1	2	yes	—
75	C₂×Dic₁₉	—	4	2	no	—
76	C₂×Dic₁₉	—	4	2	no	—
77	Dic₃₈	—	4	2	no	—
78	Dic₃₈	—	4	2	no	—
79	Dic₃₈	—	4	2	no	—
80	Dic₃₈	—	4	2	no	—
81	C₈×C₁₉	—	4	2	no	—
82	C₈×C₁₉	—	8	2	no	—
83	C₁₉⋊C₈	—	4	2	no	—
84	C₁₉⋊C₈	—	8	1	no	—
85	#3255	—	19	1	no	—
86	#1837	—	19	1	no	—
87	#1561	—	19	2	no	—
88	#1493	—	19	2	no	—
89	#1309	D₁₉	1	1	yes	characteristic
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	#454	1	1	yes	characteristic
96	C₄×C₁₉	#455	1	1	yes	characteristic
97	C₄×C₁₉	#471	1	2	yes	—
98	C₄×C₁₉	#473	1	2	yes	—
99	Dic₁₉	—	8	2	no	—
100	Dic₁₉	—	8	2	no	—
101	#533	—	19	2	no	—
102	#475	—	38	2	no	—
103	#470	—	38	2	no	—
104	#462	—	38	2	no	—
105	#423	—	19	1	no	—
106	#423	C₂×D₁₉	1	2	yes	—
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	—	38	2	no	—
117	C₂×M₁₆	—	2	2	no	—
118	C₂×M₁₆	—	38	1	no	—
119	C₂×M₁₆	D₇₆	1	2	yes	—
120	C₂²×C₈	—	38	1	no	—
121	C₄⋊Q₈	—	19	2	no	—
122	C₄⋊Q₈	—	19	2	no	—
123	#125	—	38	2	no	—
124	C₂²⋊Q₈	—	38	2	no	—
125	C₂²⋊Q₈	—	38	2	no	—
126	C₄×Q₈	—	38	2	no	—
127	C₄×Q₈	—	38	2	no	—
128	#117	—	38	2	no	—
129	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
130	#106	—	38	2	no	—
131	C₄⋊C₈	—	2	2	no	—
132	C₄⋊C₈	—	2	2	no	—
133	C₄⋊C₈	—	2	2	no	—
134	C₄⋊C₈	—	38	2	no	—
135	#103	—	38	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	—	38	2	no	—
140	C₂×Q₈	—	38	2	no	—
141	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
142	C₂²×C₄	#64541	1	2	yes	—
143	Q₁₆	—	152	2	no	—
144	M₁₆	—	4	2	no	—
145	M₁₆	—	8	2	no	—
146	M₁₆	—	76	2	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	—	38	2	no	—
150	C₂×C₈	—	76	1	no	—
151	C₂×C₈	—	76	1	no	—
152	C₂×C₈	—	76	1	no	—
153	C₄⋊C₄	—	38	2	no	—
154	C₄⋊C₄	—	38	2	no	—
155	C₄⋊C₄	—	38	2	no	—
156	C₄⋊C₄	—	38	2	no	—
157	C₄⋊C₄	—	76	2	no	—
158	C₄⋊C₄	—	76	2	no	—
159	C₂²⋊C₄	—	76	2	no	—
160	C₄²	—	76	2	no	—
161	C₄²	D₄×D₁₉	1	2	yes	—
162	C₄²	D₄×D₁₉	1	1	yes	characteristic
163	C₄²	#64545	1	1	yes	characteristic
164	C₂³	#10597321	1	1	yes	characteristic
165	Q₈	—	76	2	no	—
166	Q₈	—	76	2	no	—
167	Q₈	—	76	2	no	—
168	Q₈	—	76	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	—	76	2	no	—
174	C₂×C₄	—	76	2	no	—
175	C₂×C₄	#10597328	1	1	yes	characteristic
176	C₂×C₄	#10597330	1	2	yes	—
177	C₂×C₄	#10597340	1	2	yes	—
178	C₂×C₄	#10597341	1	1	yes	characteristic
179	C₈	—	4	2	no	—
180	C₈	—	8	2	no	—
181	C₈	—	76	2	no	—
182	C₈	—	152	1	no	—
183	C₂²	—	2	1	no	—
184	C₂²	—	2	2	no	—
185	C₂²	#11926462	1	1	yes	characteristic
186	C₄	—	2	1	no	—
187	C₄	—	2	2	no	—
188	C₄	—	152	2	no	—
189	C₄	—	152	2	no	—
190	C₄	#11926719	1	1	yes	characteristic
191	C₄	#11926724	1	1	yes	characteristic
192	C₄	#11926806	1	2	yes	—
193	C₄	#11926818	1	2	yes	—
194	C₂	—	4	1	no	—
195	C₂	#14564865	1	1	yes	characteristic
196	C₂	#14564937	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14575029

C₂

yes

maximal, characteristic

#14563772

C₂

yes

maximal, characteristic

#14562372

C₂

yes

maximal

#14562089

C₂

yes

maximal

C₁₉×#1309

C₂

yes

maximal, characteristic

#11927101

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₂×C₄×C₁₉⋊C₈

C₂²

yes

—

—

—

—

C₁₉×#423

C₂²

yes

—

C₁₉×#423

C₂²

yes

characteristic

C₂×M₁₆×C₁₉

—

C₂×M₁₆×C₁₉

D₄

yes

—

C₂×C₄²×C₁₉

C₂³

yes

—

—

—

—

C₂²×C₁₉⋊C₈

—

C₂×Dic₇₆

—

#10597316

—

#10597315

D₄

yes

—

#10597315

D₄

yes

—

C₄×Dic₃₈

—

C₄×Dic₃₈

—

C₁₉×C₄⋊C₈

—

C₁₉×C₄⋊C₈

—

C₁₉×C₄⋊C₈

—

#10597274

—

#10597273

—

C₄⋊(C₁₉⋊C₈)

—

C₄×C₁₉⋊C₈

—

C₄×C₁₉⋊C₈

D₄

yes

—

C₂²×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₉

#41

yes

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

M₁₆×C₁₉

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₄²×C₁₉

C₂×D₄

yes

—

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₄²×C₁₉

#41

yes

characteristic

C₂²⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

Dic₁₉⋊C₄

—

Dic₁₉⋊C₄

—

C₄×Dic₁₉

—

#64520

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

Dic₇₆

—

—

C₂³×C₁₉

yes

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

#120

yes

characteristic

C₂×C₄×C₁₉

#121

yes

—

C₂×C₄×C₁₉

#121

yes

characteristic

C₂×C₄×C₁₉

#123

yes

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

Dic₃₈

—

Dic₃₈

—

Dic₃₈

—

Dic₃₈

—

C₈×C₁₉

—

C₈×C₁₉

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

—

—

—

—

D₁₉

yes

characteristic

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#394

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

yes

C₄×C₁₉

yes

C₄×C₁₉

yes

—

C₄×C₁₉

#473

yes

—

Dic₁₉

—

100

Dic₁₉

—

101

#533

—

102

#475

—

103

#470

—

104

#462

—

105

#423

—

106

#423

C₂×D₁₉

yes

—

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

D₇₆

yes

—

120

C₂²×C₈

—

121

C₄⋊Q₈

—

122

C₄⋊Q₈

—

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

C₂×D₇₆

yes

characteristic

142

C₂²×C₄

#64541

yes

—

143

Q₁₆

—

152

—

144

M₁₆

—

145

M₁₆

—

146

M₁₆

—

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

—

162

C₄²

D₄×D₁₉

yes

characteristic

163

C₄²

#64545

yes

characteristic

164

C₂³

#10597321

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

#10597328

yes

characteristic

176

C₂×C₄

#10597330

yes

—

177

C₂×C₄

#10597340

yes

—

178

C₂×C₄

#10597341

yes

characteristic

179

C₈

—

180

C₈

—

181

C₈

—

182

C₈

—

152

—

183

C₂²

—

184

C₂²

—

185

C₂²

#11926462

yes

characteristic

186

C₄

—

187

C₄

—

188

C₄

—

152

—

189

C₄

—

152

—

190

C₄

#11926719

yes

characteristic

191

C₄

#11926724

yes

characteristic

192

C₄

#11926806

yes

—

193

C₄

#11926818

yes

—

194

C₂

—

195

C₂

#14564865

yes

characteristic

196

C₂

#14564937

yes

—

	subgroup	G/H
center	C₂²	#11926462
commutator, cosocle	C₂×C₄²×C₁₉	C₂³
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1309	C₂
socle	C₂²×C₁₉	#394
prime radical	C₂³×C₁₉	#124
hypercenter	#1309	D₁₉
hypocenter	C₁₉	#10535
LCS	G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

subgroup

G/H

center

C₂²

#11926462

commutator, cosocle

C₂×C₄²×C₁₉

C₂³

Frattini

C₂×C₄²

C₂²×D₁₉

Fitting

C₁₉×#1309

C₂

C₂²×C₁₉

C₂³×C₁₉

D₁₉

C₁₉

G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

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
#14575029	2
#14563772	2
#14562372	2
#14562089	2
C₁₉×#1309	2
#10535	19

subgroup

index

C₁₉×#1309

Subgroups — #18086741

Conjugacy classes of subgroups

Complements

Maximal subgroups