Group #18081621 — subgroups

Subgroups — #18081621

Order 4864.

538 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 175 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	#14563752	C₂	1	1	yes	maximal, characteristic
2	#14563488	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926293	C₂	1	4	yes	maximal
4	C₁₉×#2868	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
6	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×#10597260	—	2	1	no	—
8	#11926299	—	2	1	no	—
9	#11926299	—	2	2	no	—
10	#11926293	—	2	8	no	—
11	C₂×C₁₉×C₄⋊Q₈	C₄	1	4	yes	—
12	C₂×C₁₉×C₄⋊Q₈	C₂²	1	2	yes	—
13	C₁₉×#384	—	2	1	no	—
14	C₁₉×#384	—	2	2	no	—
15	C₁₉×#384	—	2	1	no	—
16	C₄³×C₁₉	C₂²	1	1	yes	characteristic
17	C₂²×Q₈×C₁₉	C₂×C₄	1	4	yes	—
18	C₁₉×C₄⋊Q₈	—	2	4	no	—
19	C₁₉×C₄⋊Q₈	—	2	4	no	—
20	C₁₉×C₄⋊Q₈	—	4	8	no	—
21	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
22	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
23	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
24	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
25	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₉	D₄	1	2	yes	—
28	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
30	C₂²×C₁₉⋊C₈	—	2	2	no	—
31	C₂²×C₁₉⋊C₈	—	2	2	no	—
32	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
33	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
34	C₄⋊(C₁₉⋊C₈)	D₄	1	4	yes	—
35	#10597260	—	4	2	no	—
36	C₂×Q₈×C₁₉	—	2	8	no	—
37	C₂×Q₈×C₁₉	—	4	8	no	—
38	C₂²×C₄×C₁₉	—	2	4	no	—
39	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
44	C₄⋊C₄×C₁₉	—	4	8	no	—
45	C₄⋊C₄×C₁₉	—	4	8	no	—
46	C₄⋊C₄×C₁₉	—	4	4	no	—
47	C₄⋊C₄×C₁₉	—	4	2	no	—
48	C₄⋊C₄×C₁₉	—	4	2	no	—
49	C₄²×C₁₉	—	2	4	no	—
50	C₄²×C₁₉	—	2	2	no	—
51	C₄²×C₁₉	—	2	4	no	—
52	C₄²×C₁₉	SD₁₆	1	2	yes	—
53	C₄²×C₁₉	Q₁₆	1	2	yes	—
54	C₄²×C₁₉	C₂×D₄	1	2	yes	—
55	C₄²×C₁₉	#41	1	2	yes	—
56	C₂×C₁₉⋊C₈	—	2	4	no	—
57	C₂×C₁₉⋊C₈	—	4	2	no	—
58	C₂×C₁₉⋊C₈	—	4	2	no	—
59	C₂×C₁₉⋊C₈	—	4	2	no	—
60	C₂×C₁₉⋊C₈	—	4	4	no	—
61	#9702	—	19	1	no	maximal, Sylow
62	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	Q₈×C₁₉	—	8	8	no	—
64	C₂×C₄×C₁₉	—	2	2	no	—
65	C₂×C₄×C₁₉	—	2	2	no	—
66	C₂×C₄×C₁₉	—	2	2	no	—
67	C₂×C₄×C₁₉	—	2	2	no	—
68	C₂×C₄×C₁₉	—	2	2	no	—
69	C₂×C₄×C₁₉	—	4	4	no	—
70	C₂×C₄×C₁₉	—	4	8	no	—
71	C₂×C₄×C₁₉	#100	1	4	yes	—
72	C₂×C₄×C₁₉	#103	1	4	yes	—
73	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
74	C₂×C₄×C₁₉	#121	1	2	yes	—
75	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
76	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
77	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
79	C₂×C₄×C₁₉	#137	1	2	yes	—
80	C₂×C₄×C₁₉	#137	1	4	yes	—
81	C₁₉⋊C₈	—	4	4	no	—
82	C₁₉⋊C₈	—	8	2	no	—
83	#2868	D₁₉	1	1	yes	characteristic
84	#1831	—	19	1	no	—
85	#1767	—	19	1	no	—
86	C₂×#333	—	19	4	no	—
87	C₂²×C₁₉	#390	1	1	yes	characteristic
88	C₂²×C₁₉	C₂×#100	1	2	yes	—
89	C₂²×C₁₉	C₂×#103	1	1	yes	characteristic
90	C₂²×C₁₉	#419	1	2	yes	—
91	C₂²×C₁₉	#419	1	1	yes	characteristic
92	C₄×C₁₉	—	2	4	no	—
93	C₄×C₁₉	—	2	4	no	—
94	C₄×C₁₉	—	2	2	no	—
95	C₄×C₁₉	—	8	4	no	—
96	C₄×C₁₉	#461	1	2	yes	—
97	C₄×C₁₉	#462	1	2	yes	—
98	C₄×C₁₉	#491	1	2	yes	—
99	C₄×C₁₉	#497	1	2	yes	—
100	C₂×C₄⋊Q₈	Dic₁₉	1	4	yes	—
101	C₂×C₄⋊Q₈	C₂×D₁₉	1	2	yes	—
102	C₂×C₄⋊C₈	—	19	2	no	—
103	C₂×C₄⋊C₈	—	19	2	no	—
104	C₂×#97	—	38	1	no	—
105	#384	—	2	1	no	—
106	#384	—	2	2	no	—
107	#384	—	2	1	no	—
108	C₄³	C₂×D₁₉	1	1	yes	characteristic
109	#336	—	38	2	no	—
110	#336	—	38	1	no	—
111	#333	—	38	8	no	—
112	C₂×C₁₉	#1548	1	2	yes	—
113	C₂×C₁₉	#1558	1	2	yes	—
114	C₂×C₁₉	#1960	1	1	yes	characteristic
115	C₂×C₁₉	#1965	1	1	yes	characteristic
116	C₂×C₁₉	#1981	1	1	yes	characteristic
117	C₂²×Q₈	C₂×Dic₁₉	1	4	yes	—
118	C₂²×C₈	—	38	2	no	—
119	C₂²×C₈	—	38	2	no	—
120	C₄⋊Q₈	—	2	4	no	—
121	C₄⋊Q₈	—	2	4	no	—
122	C₄⋊Q₈	—	4	8	no	—
123	C₂×C₄⋊C₄	—	2	4	no	—
124	C₂×C₄⋊C₄	—	2	4	no	—
125	C₂×C₄⋊C₄	—	2	2	no	—
126	C₂×C₄⋊C₄	—	2	2	no	—
127	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
128	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
129	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
130	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
131	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
132	C₄⋊C₈	—	19	4	no	—
133	C₄⋊C₈	—	38	4	no	—
134	C₄⋊C₈	—	38	2	no	—
135	#97	—	76	2	no	—
136	C₁₉	#9702	1	1	yes	characteristic, complemented, Sylow
137	C₂×Q₈	—	2	8	no	—
138	C₂×Q₈	—	4	8	no	—
139	C₂²×C₄	—	2	4	no	—
140	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
141	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
142	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
143	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
144	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
145	C₂×C₈	—	38	4	no	—
146	C₂×C₈	—	76	2	no	—
147	C₂×C₈	—	76	2	no	—
148	C₂×C₈	—	76	2	no	—
149	C₂×C₈	—	76	4	no	—
150	C₄⋊C₄	—	4	8	no	—
151	C₄⋊C₄	—	4	2	no	—
152	C₄⋊C₄	—	4	4	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₄⋊C₄	—	4	8	no	—
155	C₄²	—	2	4	no	—
156	C₄²	—	2	2	no	—
157	C₄²	—	2	4	no	—
158	C₄²	#64527	1	2	yes	—
159	C₄²	#64528	1	2	yes	—
160	C₄²	D₄×D₁₉	1	2	yes	—
161	C₄²	#64543	1	2	yes	—
162	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
163	Q₈	—	8	8	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	—	4	8	no	—
171	C₂×C₄	#10597290	1	4	yes	—
172	C₂×C₄	#10597292	1	4	yes	—
173	C₂×C₄	D₄×Dic₁₉	1	2	yes	—
174	C₂×C₄	#10597382	1	1	yes	characteristic
175	C₂×C₄	#10597384	1	2	yes	—
176	C₂×C₄	#10597386	1	1	yes	characteristic
177	C₂×C₄	C₂×#64527	1	1	yes	characteristic
178	C₂×C₄	#10597388	1	4	yes	—
179	C₂×C₄	#10597388	1	2	yes	—
180	C₂×C₄	C₂×#64528	1	1	yes	characteristic
181	C₈	—	76	4	no	—
182	C₈	—	152	2	no	—
183	C₂²	#11926447	1	1	yes	characteristic
184	C₂²	C₂×#10597290	1	2	yes	—
185	C₂²	C₂×#10597292	1	1	yes	characteristic
186	C₂²	#11926553	1	2	yes	—
187	C₂²	#11926553	1	1	yes	characteristic
188	C₄	—	2	4	no	—
189	C₄	—	2	4	no	—
190	C₄	—	2	2	no	—
191	C₄	—	8	4	no	—
192	C₄	#11926746	1	2	yes	—
193	C₄	#11926751	1	2	yes	—
194	C₄	#11926912	1	2	yes	—
195	C₄	#11926936	1	2	yes	—
196	C₂	#14562316	1	2	yes	—
197	C₂	#14562361	1	2	yes	—
198	C₂	#14564456	1	1	yes	characteristic
199	C₂	#14564476	1	1	yes	characteristic
200	C₂	#14564559	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14563752

C₂

yes

maximal, characteristic

#14563488

C₂

yes

maximal, characteristic

C₂×#11926293

C₂

yes

maximal

C₁₉×#2868

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₁₉⋊C₈)

C₂²

yes

—

C₂×C₄⋊(C₁₉⋊C₈)

C₂²

yes

—

—

—

—

—

C₂×C₁₉×C₄⋊Q₈

C₄

yes

—

C₂×C₁₉×C₄⋊Q₈

C₂²

yes

—

C₁₉×#384

—

C₁₉×#384

—

C₁₉×#384

—

C₄³×C₁₉

C₂²

yes

characteristic

C₂²×Q₈×C₁₉

C₂×C₄

yes

—

C₁₉×C₄⋊Q₈

—

C₁₉×C₄⋊Q₈

—

C₁₉×C₄⋊Q₈

—

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄²×C₁₉

C₂×C₄

yes

—

C₂×C₄²×C₁₉

D₄

yes

characteristic

C₂×C₄²×C₁₉

D₄

yes

—

C₂×C₄²×C₁₉

D₄

yes

characteristic

C₂×C₄²×C₁₉

C₂³

yes

characteristic

C₂²×C₁₉⋊C₈

—

C₂²×C₁₉⋊C₈

—

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

D₄

yes

—

#10597260

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

SD₁₆

yes

—

C₄²×C₁₉

Q₁₆

yes

—

C₄²×C₁₉

C₂×D₄

yes

—

C₄²×C₁₉

#41

yes

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

#9702

—

maximal, Sylow

C₂³×C₁₉

C₂×C₂²⋊C₄

yes

characteristic

Q₈×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

#100

yes

—

C₂×C₄×C₁₉

#103

yes

—

C₂×C₄×C₁₉

C₄×D₄

yes

—

C₂×C₄×C₁₉

#121

yes

—

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₁₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₁₉

#137

yes

—

C₂×C₄×C₁₉

#137

yes

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

#2868

D₁₉

yes

—

—

—

C₂²×C₁₉

#390

yes

characteristic

C₂²×C₁₉

C₂×#100

yes

—

C₂²×C₁₉

C₂×#103

yes

characteristic

C₂²×C₁₉

#419

yes

—

C₂²×C₁₉

#419

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

#461

yes

—

C₄×C₁₉

#462

yes

—

C₄×C₁₉

#491

yes

—

C₄×C₁₉

#497

yes

—

100

C₂×C₄⋊Q₈

Dic₁₉

yes

—

101

C₂×C₄⋊Q₈

C₂×D₁₉

yes

—

102

C₂×C₄⋊C₈

—

103

C₂×C₄⋊C₈

—

104

C₂×#97

—

105

#384

—

106

#384

—

107

#384

—

108

C₄³

C₂×D₁₉

yes

characteristic

109

#336

—

110

#336

—

111

#333

—

112

C₂×C₁₉

#1548

yes

—

113

C₂×C₁₉

#1558

yes

—

114

C₂×C₁₉

#1960

yes

characteristic

115

C₂×C₁₉

#1965

yes

characteristic

116

C₂×C₁₉

#1981

yes

characteristic

117

C₂²×Q₈

C₂×Dic₁₉

yes

—

118

C₂²×C₈

—

119

C₂²×C₈

—

120

C₄⋊Q₈

—

121

C₄⋊Q₈

—

122

C₄⋊Q₈

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄⋊C₄

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄²

C₂×Dic₁₉

yes

—

128

C₂×C₄²

C₂²⋊D₁₉

yes

characteristic

129

C₂×C₄²

C₂²⋊D₁₉

yes

—

130

C₂×C₄²

C₂²⋊D₁₉

yes

characteristic

131

C₂×C₄²

C₂²×D₁₉

yes

characteristic

132

C₄⋊C₈

—

133

C₄⋊C₈

—

134

C₄⋊C₈

—

135

#97

—

136

C₁₉

#9702

yes

characteristic, complemented, Sylow

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂²×C₄

—

140

C₂²×C₄

C₂²⋊Dic₁₉

yes

—

141

C₂²×C₄

C₂²⋊Dic₁₉

yes

—

142

C₂²×C₄

C₂²×Dic₁₉

yes

characteristic

143

C₂²×C₄

C₂×C₂²⋊D₁₉

yes

characteristic

144

C₂²×C₄

C₂×C₂²⋊D₁₉

yes

characteristic

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄²

—

156

C₄²

—

157

C₄²

—

158

C₄²

#64527

yes

—

159

C₄²

#64528

yes

—

160

C₄²

D₄×D₁₉

yes

—

161

C₄²

#64543

yes

—

162

C₂³

C₂×C₂²⋊Dic₁₉

yes

characteristic

163

Q₈

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

#10597290

yes

—

172

C₂×C₄

#10597292

yes

—

173

C₂×C₄

D₄×Dic₁₉

yes

—

174

C₂×C₄

#10597382

yes

characteristic

175

C₂×C₄

#10597384

yes

—

176

C₂×C₄

#10597386

yes

characteristic

177

C₂×C₄

C₂×#64527

yes

characteristic

178

C₂×C₄

#10597388

yes

—

179

C₂×C₄

#10597388

yes

—

180

C₂×C₄

C₂×#64528

yes

characteristic

181

C₈

—

182

C₈

—

152

—

183

C₂²

#11926447

yes

characteristic

184

C₂²

C₂×#10597290

yes

—

185

C₂²

C₂×#10597292

yes

characteristic

186

C₂²

#11926553

yes

—

187

C₂²

#11926553

yes

characteristic

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

#11926746

yes

—

193

C₄

#11926751

yes

—

194

C₄

#11926912

yes

—

195

C₄

#11926936

yes

—

196

C₂

#14562316

yes

—

197

C₂

#14562361

yes

—

198

C₂

#14564456

yes

characteristic

199

C₂

#14564476

yes

characteristic

200

C₂

#14564559

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₉
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#2868	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#2868	D₁₉
hypocenter	C₁₉	#9702
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₁₉

commutator

C₂²×C₄×C₁₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₉

Fitting

C₁₉×#2868

C₂

socle, prime radical

C₂³×C₁₉

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₁₉

C₂³

D₁₉

C₁₉

G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

N	H ≅ G/N	type
136. C₁₉	#9702	semidirect

H ≅ G/N

type

136. C₁₉

#9702

semidirect

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

subgroup	index
#14563752	2
#14563488	2
C₂×#11926293	2
C₁₉×#2868	2
#9702	19

subgroup

index

C₁₉×#2868

Subgroups — #18081621

Conjugacy classes of subgroups

Complements

Maximal subgroups