Group #18069691 — subgroups

Subgroups — #18069691

Order 4864.

320 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 99 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	C₂×C₂²⋊(C₁₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₂×#11926309	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1810	C₂	1	1	yes	maximal, characteristic
4	C₂²×#64520	—	2	1	no	—
5	C₂²×C₁₉⋊C₁₆	—	2	1	no	—
6	C₂²⋊(C₁₉⋊C₁₆)	—	2	1	no	—
7	C₂²⋊(C₁₉⋊C₁₆)	C₄	1	2	yes	—
8	#11926309	—	2	1	no	—
9	#11926309	C₄	1	2	yes	—
10	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
11	C₂×C₁₉×#108	C₄	1	2	yes	—
12	C₁₉×#407	—	2	1	no	—
13	C₁₉×#341	—	2	1	no	—
14	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
15	C₂×M₁₆×C₁₉	—	2	2	no	—
16	C₂²×C₈×C₁₉	—	4	2	no	—
17	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₉	Q₈	1	1	yes	characteristic
23	C₂×#64520	—	2	4	no	—
24	C₂×#64520	—	2	2	no	—
25	C₂×#64520	—	4	2	no	—
26	C₂×#64520	—	4	1	no	—
27	C₂²×C₁₉⋊C₈	—	4	1	no	—
28	C₁₉×#108	—	4	2	no	—
29	C₁₉×C₂²⋊C₈	—	4	1	no	—
30	C₁₉×C₂²⋊C₈	—	4	1	no	—
31	C₂×C₁₉⋊C₁₆	—	2	2	no	—
32	C₂×C₁₉⋊C₁₆	—	4	2	no	—
33	C₂⁴×C₁₉	Q₁₆	1	1	yes	characteristic
34	C₂²×C₄×C₁₉	—	2	2	no	—
35	C₂²×C₄×C₁₉	—	2	2	no	—
36	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
40	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
41	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
42	M₁₆×C₁₉	—	4	2	no	—
43	M₁₆×C₁₉	—	4	2	no	—
44	C₂×C₈×C₁₉	—	2	2	no	—
45	C₂×C₈×C₁₉	—	2	2	no	—
46	C₂×C₈×C₁₉	—	4	1	no	—
47	C₂×C₈×C₁₉	—	4	1	no	—
48	C₂×C₈×C₁₉	—	4	1	no	—
49	C₂×C₈×C₁₉	—	4	1	no	—
50	C₂×C₈×C₁₉	—	4	2	no	—
51	C₂×C₈×C₁₉	C₄²	1	1	yes	characteristic
52	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
55	#64520	—	2	4	no	—
56	#64520	—	4	4	no	—
57	#64520	—	4	2	no	—
58	C₂×C₁₉⋊C₈	—	4	4	no	—
59	C₂×C₁₉⋊C₈	—	4	2	no	—
60	C₁₉⋊C₁₆	—	4	2	no	—
61	#7261	—	19	1	no	maximal, Sylow
62	C₂³×C₁₉	—	2	2	no	—
63	C₂³×C₁₉	—	2	2	no	—
64	C₂³×C₁₉	—	4	1	no	—
65	C₂³×C₁₉	#103	1	1	yes	characteristic
66	C₂³×C₁₉	#103	1	1	yes	characteristic
67	C₂³×C₁₉	#107	1	1	yes	characteristic
68	C₂×C₄×C₁₉	—	2	2	no	—
69	C₂×C₄×C₁₉	—	2	2	no	—
70	C₂×C₄×C₁₉	—	2	2	no	—
71	C₂×C₄×C₁₉	—	2	1	no	—
72	C₂×C₄×C₁₉	—	2	1	no	—
73	C₂×C₄×C₁₉	—	2	1	no	—
74	C₂×C₄×C₁₉	—	2	1	no	—
75	C₂×C₄×C₁₉	#95	1	1	yes	characteristic
76	C₂×C₄×C₁₉	#99	1	2	yes	—
77	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
78	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
79	C₂×C₄×C₁₉	#104	1	2	yes	—
80	C₂×C₄×C₁₉	#106	1	1	yes	characteristic
81	C₈×C₁₉	—	2	1	no	—
82	C₈×C₁₉	—	2	1	no	—
83	C₈×C₁₉	—	2	1	no	—
84	C₈×C₁₉	—	2	1	no	—
85	C₈×C₁₉	—	8	2	no	—
86	C₁₉⋊C₈	—	4	4	no	—
87	C₂×C₂²⋊C₁₆	—	19	1	no	—
88	#1810	D₁₉	1	1	yes	characteristic
89	C₂×#341	—	19	1	no	—
90	C₂²×C₁₉	—	2	1	no	—
91	C₂²×C₁₉	—	2	1	no	—
92	C₂²×C₁₉	—	2	1	no	—
93	C₂²×C₁₉	—	2	1	no	—
94	C₂²×C₁₉	—	2	1	no	—
95	C₂²×C₁₉	—	2	1	no	—
96	C₂²×C₁₉	—	2	1	no	—
97	C₂²×C₁₉	—	4	4	no	—
98	C₂²×C₁₉	#328	1	2	yes	—
99	C₂²×C₁₉	#340	1	1	yes	characteristic
100	C₂²×C₁₉	#368	1	2	yes	—
101	C₄×C₁₉	—	2	1	no	—
102	C₄×C₁₉	—	2	1	no	—
103	C₄×C₁₉	#327	1	2	yes	—
104	C₄×C₁₉	#339	1	1	yes	characteristic
105	C₄×C₁₉	#342	1	1	yes	characteristic
106	C₂²×M₁₆	—	38	1	no	—
107	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
108	C₂²×C₁₆	—	38	1	no	—
109	C₂×#108	Dic₁₉	1	2	yes	—
110	#407	—	2	1	no	—
111	C₂²⋊C₁₆	—	19	2	no	—
112	C₂²⋊C₁₆	—	38	1	no	—
113	#341	—	2	1	no	—
114	#341	—	19	2	no	—
115	#341	—	38	1	no	—
116	C₂×C₁₉	—	2	2	no	—
117	C₂×C₁₉	—	2	2	no	—
118	C₂×C₁₉	—	4	1	no	—
119	C₂×C₁₉	#1294	1	1	yes	characteristic
120	C₂×C₁₉	#1381	1	1	yes	characteristic
121	C₂×C₁₉	#1388	1	1	yes	characteristic
122	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
123	C₂×M₁₆	—	2	2	no	—
124	C₂×M₁₆	—	38	4	no	—
125	C₂×M₁₆	—	38	2	no	—
126	C₂×M₁₆	—	76	2	no	—
127	C₂×M₁₆	—	76	1	no	—
128	C₂²×C₈	—	4	2	no	—
129	C₂²×C₈	—	76	1	no	—
130	C₂²×C₈	Dic₃₈	1	1	yes	characteristic
131	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
132	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
133	C₂²×C₈	D₇₆	1	1	yes	characteristic
134	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
135	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
136	C₂×C₁₆	—	38	2	no	—
137	C₂×C₁₆	—	76	2	no	—
138	#108	—	4	2	no	—
139	C₂²⋊C₈	—	4	1	no	—
140	C₂²⋊C₈	—	4	1	no	—
141	C₁₉	#7261	1	1	yes	characteristic, complemented, Sylow
142	C₂⁴	#64528	1	1	yes	characteristic
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	—	2	2	no	—
145	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
146	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
147	C₂²×C₄	#64525	1	1	yes	characteristic
148	C₂²×C₄	#64526	1	1	yes	characteristic
149	C₂²×C₄	#64527	1	1	yes	characteristic
150	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
151	M₁₆	—	4	2	no	—
152	M₁₆	—	4	2	no	—
153	M₁₆	—	38	4	no	—
154	M₁₆	—	76	4	no	—
155	M₁₆	—	76	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	4	1	no	—
159	C₂×C₈	—	4	1	no	—
160	C₂×C₈	—	4	1	no	—
161	C₂×C₈	—	4	1	no	—
162	C₂×C₈	—	4	2	no	—
163	C₂×C₈	—	76	4	no	—
164	C₂×C₈	—	76	2	no	—
165	C₂×C₈	C₄×Dic₁₉	1	1	yes	characteristic
166	C₂×C₈	Dic₁₉⋊C₄	1	1	yes	characteristic
167	C₂×C₈	C₄⋊Dic₁₉	1	1	yes	characteristic
168	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
169	C₁₆	—	76	2	no	—
170	C₂³	—	2	2	no	—
171	C₂³	—	2	2	no	—
172	C₂³	—	4	1	no	—
173	C₂³	#10597264	1	1	yes	characteristic
174	C₂³	#10597267	1	1	yes	characteristic
175	C₂³	#10597292	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	#10597265	1	1	yes	characteristic
184	C₂×C₄	#10597266	1	1	yes	characteristic
185	C₂×C₄	C₁₉⋊#95	1	1	yes	characteristic
186	C₂×C₄	#10597289	1	1	yes	characteristic
187	C₂×C₄	#10597291	1	2	yes	—
188	C₂×C₄	#10597294	1	2	yes	—
189	C₈	—	2	1	no	—
190	C₈	—	2	1	no	—
191	C₈	—	2	1	no	—
192	C₈	—	2	1	no	—
193	C₈	—	8	2	no	—
194	C₈	—	76	4	no	—
195	C₂²	—	2	1	no	—
196	C₂²	—	2	1	no	—
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	—	2	1	no	—
201	C₂²	—	2	1	no	—
202	C₂²	—	4	4	no	—
203	C₂²	#11926281	1	2	yes	—
204	C₂²	#11926306	1	1	yes	characteristic
205	C₂²	#11926373	1	2	yes	—
206	C₄	—	2	1	no	—
207	C₄	—	2	1	no	—
208	C₄	#11926278	1	2	yes	—
209	C₄	#11926303	1	1	yes	characteristic
210	C₄	#11926311	1	1	yes	characteristic
211	C₂	—	2	2	no	—
212	C₂	—	2	2	no	—
213	C₂	—	4	1	no	—
214	C₂	#14561513	1	1	yes	characteristic
215	C₂	#14561728	1	1	yes	characteristic
216	C₂	#14561746	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂²⋊(C₁₉⋊C₁₆)

C₂

yes

maximal, characteristic

C₂×#11926309

C₂

yes

maximal, characteristic

C₁₉×#1810

C₂

yes

maximal, characteristic

C₂²×#64520

—

C₂²×C₁₉⋊C₁₆

—

C₂²⋊(C₁₉⋊C₁₆)

—

C₂²⋊(C₁₉⋊C₁₆)

C₄

yes

—

#11926309

—

#11926309

C₄

yes

—

C₂³×C₈×C₁₉

C₂²

yes

characteristic

C₂×C₁₉×#108

C₄

yes

—

C₁₉×#407

—

C₁₉×#341

—

C₂³×C₄×C₁₉

D₄

yes

characteristic

C₂×M₁₆×C₁₉

—

C₂²×C₈×C₁₉

—

C₂²×C₈×C₁₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₂²×C₈×C₁₉

Q₈

yes

—

—

—

—

C₂²×C₁₉⋊C₈

—

C₁₉×#108

—

C₁₉×C₂²⋊C₈

—

C₁₉×C₂²⋊C₈

—

C₂×C₁₉⋊C₁₆

—

C₂×C₁₉⋊C₁₆

—

C₂⁴×C₁₉

Q₁₆

yes

characteristic

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

D₈

yes

characteristic

C₂²×C₄×C₁₉

SD₁₆

yes

characteristic

C₂²×C₄×C₁₉

SD₁₆

yes

characteristic

M₁₆×C₁₉

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₄²

yes

characteristic

C₂×C₈×C₁₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₉

C₄⋊C₄

yes

—

—

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₁₉⋊C₁₆

—

#7261

—

maximal, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

yes

C₂³×C₁₉

yes

C₂³×C₁₉

yes

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

#95

yes

characteristic

C₂×C₄×C₁₉

#99

yes

—

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

—

C₂×C₄×C₁₉

#106

yes

characteristic

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₁₉⋊C₈

—

C₂×C₂²⋊C₁₆

—

#1810

D₁₉

yes

characteristic

C₂×#341

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#328

yes

—

C₂²×C₁₉

#340

yes

characteristic

100

C₂²×C₁₉

#368

yes

—

101

C₄×C₁₉

—

102

C₄×C₁₉

—

103

C₄×C₁₉

#327

yes

—

104

C₄×C₁₉

#339

yes

characteristic

105

C₄×C₁₉

#342

yes

characteristic

106

C₂²×M₁₆

—

107

C₂³×C₈

C₂×D₁₉

yes

characteristic

108

C₂²×C₁₆

—

109

C₂×#108

Dic₁₉

yes

—

110

#407

—

111

C₂²⋊C₁₆

—

112

C₂²⋊C₁₆

—

113

#341

—

114

#341

—

115

#341

—

116

C₂×C₁₉

—

117

C₂×C₁₉

—

118

C₂×C₁₉

—

119

C₂×C₁₉

#1294

yes

characteristic

120

C₂×C₁₉

#1381

yes

characteristic

121

C₂×C₁₉

#1388

yes

characteristic

122

C₂³×C₄

C₂²⋊D₁₉

yes

characteristic

123

C₂×M₁₆

—

124

C₂×M₁₆

—

125

C₂×M₁₆

—

126

C₂×M₁₆

—

127

C₂×M₁₆

—

128

C₂²×C₈

—

129

C₂²×C₈

—

130

C₂²×C₈

Dic₃₈

yes

characteristic

131

C₂²×C₈

C₄×D₁₉

yes

characteristic

132

C₂²×C₈

C₄×D₁₉

yes

characteristic

133

C₂²×C₈

D₇₆

yes

characteristic

134

C₂²×C₈

C₂×Dic₁₉

yes

characteristic

135

C₂²×C₈

C₂²⋊D₁₉

yes

characteristic

136

C₂×C₁₆

—

137

C₂×C₁₆

—

138

#108

—

139

C₂²⋊C₈

—

140

C₂²⋊C₈

—

141

C₁₉

#7261

yes

characteristic, complemented, Sylow

142

C₂⁴

#64528

yes

characteristic

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

Dic₁₉⋊C₄

yes

characteristic

146

C₂²×C₄

(C₂×D₁₉)⋊C₄

yes

characteristic

147

C₂²×C₄

#64525

yes

characteristic

148

C₂²×C₄

#64526

yes

characteristic

149

C₂²×C₄

#64527

yes

characteristic

150

C₂²×C₄

C₂²⋊Dic₁₉

yes

characteristic

151

M₁₆

—

152

M₁₆

—

153

M₁₆

—

154

M₁₆

—

155

M₁₆

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

C₄×Dic₁₉

yes

characteristic

166

C₂×C₈

Dic₁₉⋊C₄

yes

characteristic

167

C₂×C₈

C₄⋊Dic₁₉

yes

characteristic

168

C₂×C₈

(C₂×D₁₉)⋊C₄

yes

characteristic

169

C₁₆

—

170

C₂³

—

171

C₂³

—

172

C₂³

—

173

C₂³

#10597264

yes

characteristic

174

C₂³

#10597267

yes

characteristic

175

C₂³

#10597292

yes

characteristic

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

#10597265

yes

characteristic

184

C₂×C₄

#10597266

yes

characteristic

185

C₂×C₄

C₁₉⋊#95

yes

characteristic

186

C₂×C₄

#10597289

yes

characteristic

187

C₂×C₄

#10597291

yes

—

188

C₂×C₄

#10597294

yes

—

189

C₈

—

190

C₈

—

191

C₈

—

192

C₈

—

193

C₈

—

194

C₈

—

195

C₂²

—

196

C₂²

—

197

C₂²

—

198

C₂²

—

199

C₂²

—

200

C₂²

—

201

C₂²

—

202

C₂²

—

203

C₂²

#11926281

yes

—

204

C₂²

#11926306

yes

characteristic

205

C₂²

#11926373

yes

—

206

C₄

—

207

C₄

—

208

C₄

#11926278

yes

—

209

C₄

#11926303

yes

characteristic

210

C₄

#11926311

yes

characteristic

211

C₂

—

212

C₂

—

213

C₂

—

214

C₂

#14561513

yes

characteristic

215

C₂

#14561728

yes

characteristic

216

C₂

#14561746

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#10597289
commutator	C₂×C₈×C₁₉	C₄²
Frattini	C₂³×C₈	C₂×D₁₉
Fitting	C₁₉×#1810	C₂
socle	C₂²×C₁₉	#340
prime radical	C₂⁴×C₁₉	Q₁₆
cosocle	C₂³×C₈×C₁₉	C₂²
hypercenter	#1810	D₁₉
hypocenter	C₁₉	#7261
LCS	G ▷ C₂×C₈×C₁₉ ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #1810

subgroup

G/H

center

C₂×C₄

#10597289

commutator

C₂×C₈×C₁₉

C₄²

Frattini

C₂³×C₈

C₂×D₁₉

Fitting

C₁₉×#1810

C₂

socle

C₂²×C₁₉

#340

prime radical

C₂⁴×C₁₉

Q₁₆

cosocle

C₂³×C₈×C₁₉

C₂²

D₁₉

C₁₉

G ▷ C₂×C₈×C₁₉ ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #1810

N	H ≅ G/N	type
141. C₁₉	#7261	semidirect

H ≅ G/N

type

141. C₁₉

#7261

semidirect

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

subgroup	index
C₂×C₂²⋊(C₁₉⋊C₁₆)	2
C₂×#11926309	2
C₁₉×#1810	2
#7261	19

subgroup

index

C₂×C₂²⋊(C₁₉⋊C₁₆)

C₂×#11926309

C₁₉×#1810

#7261

Subgroups — #18069691

Conjugacy classes of subgroups

Complements

Maximal subgroups