Group #18085570 — subgroups

Subgroups — #18085570

Order 4864.

416 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 121 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	#14568877	C₂	1	1	yes	maximal, characteristic
2	#14564582	C₂	1	2	yes	maximal
3	#14564436	C₂	1	1	yes	maximal, characteristic
4	#14561474	C₂	1	2	yes	maximal
5	C₁₉×#1296	C₂	1	1	yes	maximal, characteristic
6	C₂×#10597315	C₂²	1	1	yes	characteristic
7	C₂×#10597267	—	2	2	no	—
8	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
9	#11926483	—	2	2	no	—
10	#11926443	—	2	2	no	—
11	#11926409	—	2	2	no	—
12	C₄×C₄⋊Dic₁₉	C₂²	1	2	yes	—
13	#11926307	—	2	2	no	—
14	#11926307	—	2	2	no	—
15	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
16	C₁₉×#384	C₂²	1	1	yes	characteristic
17	C₂²×Dic₃₈	—	2	1	no	—
18	C₂²×C₈×C₁₉	—	2	2	no	—
19	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
20	C₂×C₄⋊C₄×C₁₉	D₄	1	2	yes	—
21	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
22	C₂×C₄⋊Dic₁₉	—	2	2	no	—
23	C₂×C₄⋊Dic₁₉	—	2	2	no	—
24	C₂×C₄⋊Dic₁₉	D₄	1	2	yes	—
25	C₂×C₄⋊Dic₁₉	D₄	1	2	yes	—
26	C₂×C₄×Dic₁₉	—	2	2	no	—
27	C₂²×C₁₉⋊C₈	—	2	1	no	—
28	#10597315	—	2	2	no	—
29	#10597315	—	2	1	no	—
30	#10597315	Q₈	1	2	yes	—
31	C₁₉×C₄⋊C₈	—	2	2	no	—
32	C₁₉×C₄⋊C₈	—	2	2	no	—
33	C₁₉⋊#95	—	2	2	no	—
34	C₁₉⋊#95	—	2	4	no	—
35	C₁₉⋊#95	—	2	2	no	—
36	#10597267	—	4	4	no	—
37	C₄×C₁₉⋊C₈	—	2	2	no	—
38	C₂²×C₄×C₁₉	—	2	2	no	—
39	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
41	C₂²×Dic₁₉	—	2	2	no	—
42	C₂²×Dic₁₉	—	2	2	no	—
43	C₂²×Dic₁₉	—	4	1	no	—
44	C₂×Dic₃₈	—	2	2	no	—
45	C₂×Dic₃₈	—	4	2	no	—
46	C₂×Dic₃₈	—	4	2	no	—
47	C₂×Dic₃₈	—	4	1	no	—
48	C₂×C₈×C₁₉	—	4	2	no	—
49	C₂×C₈×C₁₉	—	4	2	no	—
50	C₂×C₈×C₁₉	—	4	2	no	—
51	C₄⋊C₄×C₁₉	—	4	2	no	—
52	C₄⋊C₄×C₁₉	—	4	2	no	—
53	C₄⋊C₄×C₁₉	—	4	2	no	—
54	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
55	C₄²×C₁₉	#41	1	2	yes	—
56	C₄²×C₁₉	#41	1	1	yes	characteristic
57	C₄⋊Dic₁₉	—	2	4	no	—
58	C₄⋊Dic₁₉	—	2	2	no	—
59	C₄⋊Dic₁₉	—	2	2	no	—
60	C₄⋊Dic₁₉	—	2	2	no	—
61	C₄⋊Dic₁₉	—	2	2	no	—
62	C₄⋊Dic₁₉	—	4	2	no	—
63	C₄⋊Dic₁₉	—	8	2	no	—
64	C₄×Dic₁₉	—	4	2	no	—
65	C₄×Dic₁₉	—	4	2	no	—
66	C₂×C₁₉⋊C₈	—	2	2	no	—
67	C₂×C₁₉⋊C₈	—	4	1	no	—
68	C₂×C₁₉⋊C₈	—	4	1	no	—
69	#10352	—	19	1	no	maximal, Sylow
70	C₂³×C₁₉	#120	1	1	yes	characteristic
71	C₂×C₄×C₁₉	—	2	1	no	—
72	C₂×C₄×C₁₉	—	2	1	no	—
73	C₂×C₄×C₁₉	—	4	4	no	—
74	C₂×C₄×C₁₉	—	4	2	no	—
75	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
76	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
78	C₂×C₄×C₁₉	#124	1	2	yes	—
79	C₂×C₄×C₁₉	#135	1	2	yes	—
80	C₂×C₄×C₁₉	#135	1	2	yes	—
81	C₂×C₄×C₁₉	#137	1	2	yes	—
82	C₂×C₄×C₁₉	#137	1	2	yes	—
83	C₂×Dic₁₉	—	4	2	no	—
84	C₂×Dic₁₉	—	4	2	no	—
85	C₂×Dic₁₉	—	4	2	no	—
86	C₂×Dic₁₉	—	4	2	no	—
87	C₂×Dic₁₉	—	4	2	no	—
88	C₂×Dic₁₉	—	4	2	no	—
89	C₂×Dic₁₉	—	4	2	no	—
90	C₂×Dic₁₉	—	8	1	no	—
91	C₂×Dic₁₉	—	8	1	no	—
92	Dic₃₈	—	4	4	no	—
93	Dic₃₈	—	8	2	no	—
94	C₈×C₁₉	—	8	2	no	—
95	C₁₉⋊C₈	—	4	2	no	—
96	#2571	—	19	1	no	—
97	#1987	—	19	2	no	—
98	#1956	—	19	1	no	—
99	#1296	D₁₉	1	1	yes	characteristic
100	#1279	—	19	2	no	—
101	C₂²×C₁₉	#393	1	1	yes	characteristic
102	C₂²×C₁₉	#452	1	2	yes	—
103	C₂²×C₁₉	#452	1	2	yes	—
104	C₂²×C₁₉	#453	1	1	yes	characteristic
105	C₂²×C₁₉	#454	1	1	yes	characteristic
106	C₄×C₁₉	—	2	1	no	—
107	C₄×C₁₉	—	2	1	no	—
108	C₄×C₁₉	—	8	2	no	—
109	C₄×C₁₉	#479	1	2	yes	—
110	C₄×C₁₉	#479	1	2	yes	—
111	C₄×C₁₉	#488	1	2	yes	—
112	C₄×C₁₉	#491	1	2	yes	—
113	Dic₁₉	—	8	2	no	—
114	Dic₁₉	—	8	2	no	—
115	Dic₁₉	—	8	2	no	—
116	C₂×C₄⋊Q₈	—	19	1	no	—
117	C₂×C₄⋊C₈	C₂×D₁₉	1	2	yes	—
118	C₂×#103	—	38	2	no	—
119	C₂×C₄×C₈	—	19	1	no	—
120	#400	—	38	2	no	—
121	#389	—	38	2	no	—
122	#384	C₂×D₁₉	1	1	yes	characteristic
123	#382	—	38	2	no	—
124	C₄×C₄⋊C₄	—	19	2	no	—
125	#340	—	38	2	no	—
126	#340	—	38	2	no	—
127	C₂×C₁₉	#1630	1	1	yes	characteristic
128	C₂×C₁₉	#1634	1	1	yes	characteristic
129	C₂×C₁₉	#1638	1	2	yes	—
130	C₂×C₁₉	#2026	1	2	yes	—
131	C₂×C₁₉	#2027	1	1	yes	characteristic
132	C₂²×Q₈	—	38	1	no	—
133	C₂²×C₈	—	2	2	no	—
134	C₂²×C₈	—	38	1	no	—
135	C₄⋊Q₈	—	19	2	no	—
136	C₄⋊Q₈	—	38	2	no	—
137	C₄⋊Q₈	—	38	1	no	—
138	C₂×C₄⋊C₄	—	2	2	no	—
139	C₂×C₄⋊C₄	—	19	2	no	—
140	C₂×C₄⋊C₄	—	19	2	no	—
141	C₂×C₄⋊C₄	—	38	2	no	—
142	C₂×C₄⋊C₄	—	38	2	no	—
143	C₂×C₄⋊C₄	D₇₆	1	2	yes	—
144	C₂×C₄²	—	38	2	no	—
145	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	—	2	2	no	—
148	#103	—	76	4	no	—
149	C₄×C₈	—	38	2	no	—
150	#95	—	38	2	no	—
151	#95	—	38	4	no	—
152	#95	—	38	2	no	—
153	C₁₉	#10352	1	1	yes	characteristic, complemented, Sylow
154	C₂×Q₈	—	38	2	no	—
155	C₂×Q₈	—	76	2	no	—
156	C₂×Q₈	—	76	2	no	—
157	C₂×Q₈	—	76	1	no	—
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	—	38	2	no	—
160	C₂²×C₄	—	38	2	no	—
161	C₂²×C₄	—	76	1	no	—
162	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
163	C₂²×C₄	D₄×D₁₉	1	2	yes	—
164	C₂×C₈	—	4	2	no	—
165	C₂×C₈	—	4	2	no	—
166	C₂×C₈	—	4	2	no	—
167	C₂×C₈	—	38	2	no	—
168	C₂×C₈	—	76	1	no	—
169	C₂×C₈	—	76	1	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄⋊C₄	—	38	4	no	—
174	C₄⋊C₄	—	38	2	no	—
175	C₄⋊C₄	—	38	2	no	—
176	C₄⋊C₄	—	38	2	no	—
177	C₄⋊C₄	—	38	2	no	—
178	C₄⋊C₄	—	76	2	no	—
179	C₄⋊C₄	—	152	2	no	—
180	C₄²	—	76	2	no	—
181	C₄²	—	76	2	no	—
182	C₄²	#64541	1	2	yes	—
183	C₄²	#64543	1	1	yes	characteristic
184	C₄²	Q₈×D₁₉	1	1	yes	characteristic
185	C₂³	#10597328	1	1	yes	characteristic
186	Q₈	—	76	4	no	—
187	Q₈	—	152	2	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	76	2	no	—
193	C₂×C₄	—	76	2	no	—
194	C₂×C₄	—	76	2	no	—
195	C₂×C₄	—	76	2	no	—
196	C₂×C₄	—	76	2	no	—
197	C₂×C₄	—	76	2	no	—
198	C₂×C₄	—	76	2	no	—
199	C₂×C₄	—	152	1	no	—
200	C₂×C₄	—	152	1	no	—
201	C₂×C₄	#10597321	1	1	yes	characteristic
202	C₂×C₄	#10597331	1	2	yes	—
203	C₂×C₄	#10597342	1	2	yes	—
204	C₂×C₄	#10597343	1	1	yes	characteristic
205	C₂×C₄	#10597358	1	2	yes	—
206	C₂×C₄	#10597361	1	2	yes	—
207	C₂×C₄	#10597362	1	2	yes	—
208	C₂×C₄	#10597364	1	2	yes	—
209	C₈	—	8	2	no	—
210	C₈	—	76	2	no	—
211	C₂²	#11926457	1	1	yes	characteristic
212	C₂²	#11926707	1	2	yes	—
213	C₂²	#11926709	1	2	yes	—
214	C₂²	#11926714	1	1	yes	characteristic
215	C₂²	#11926719	1	1	yes	characteristic
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	8	2	no	—
219	C₄	—	152	2	no	—
220	C₄	—	152	2	no	—
221	C₄	—	152	2	no	—
222	C₄	#11926850	1	2	yes	—
223	C₄	#11926854	1	2	yes	—
224	C₄	#11926897	1	2	yes	—
225	C₄	#11926911	1	2	yes	—
226	C₂	#14562757	1	1	yes	characteristic
227	C₂	#14562781	1	1	yes	characteristic
228	C₂	#14562808	1	2	yes	—
229	C₂	#14564785	1	2	yes	—
230	C₂	#14564792	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14568877

C₂

yes

maximal, characteristic

#14564582

C₂

yes

maximal

#14564436

C₂

yes

maximal, characteristic

#14561474

C₂

yes

maximal

C₁₉×#1296

C₂

yes

maximal, characteristic

C₂×#10597315

C₂²

yes

characteristic

C₂×#10597267

—

C₂×C₄×C₁₉⋊C₈

C₂²

yes

—

—

—

C₄×C₄⋊Dic₁₉

C₂²

yes

—

#11926307

—

#11926307

—

C₂×C₁₉×C₄⋊C₈

C₂²

yes

—

C₁₉×#384

C₂²

yes

characteristic

C₂²×Dic₃₈

—

C₂²×C₈×C₁₉

—

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

D₄

yes

—

C₂×C₄²×C₁₉

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₉

—

C₂×C₄⋊Dic₁₉

—

C₂×C₄⋊Dic₁₉

D₄

yes

—

C₂×C₄⋊Dic₁₉

D₄

yes

—

C₂×C₄×Dic₁₉

—

C₂²×C₁₉⋊C₈

—

#10597315

—

#10597315

—

#10597315

Q₈

yes

—

C₁₉×C₄⋊C₈

—

C₁₉×C₄⋊C₈

—

C₁₉⋊#95

—

C₁₉⋊#95

—

C₁₉⋊#95

—

#10597267

—

C₄×C₁₉⋊C₈

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₉

C₂×D₄

yes

—

C₂²×Dic₁₉

—

C₂²×Dic₁₉

—

C₂²×Dic₁₉

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄²×C₁₉

C₂×Q₈

yes

characteristic

C₄²×C₁₉

#41

yes

—

C₄²×C₁₉

#41

yes

characteristic

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄×Dic₁₉

—

C₄×Dic₁₉

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

—

C₂³×C₁₉

yes

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₉

C₂²⋊Q₈

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

—

C₂×C₄×C₁₉

#135

yes

—

C₂×C₄×C₁₉

#135

yes

—

C₂×C₄×C₁₉

#137

yes

—

C₂×C₄×C₁₉

#137

yes

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

Dic₃₈

—

Dic₃₈

—

C₈×C₁₉

—

C₁₉⋊C₈

—

—

—

—

D₁₉

yes

characteristic

100

#1279

—

101

C₂²×C₁₉

#393

yes

characteristic

102

C₂²×C₁₉

#452

yes

—

103

C₂²×C₁₉

#452

yes

—

104

C₂²×C₁₉

#453

yes

characteristic

105

C₂²×C₁₉

#454

yes

characteristic

106

C₄×C₁₉

—

107

C₄×C₁₉

—

108

C₄×C₁₉

—

109

C₄×C₁₉

#479

yes

—

110

C₄×C₁₉

#479

yes

—

111

C₄×C₁₉

#488

yes

—

112

C₄×C₁₉

#491

yes

—

113

Dic₁₉

—

114

Dic₁₉

—

115

Dic₁₉

—

116

C₂×C₄⋊Q₈

—

117

C₂×C₄⋊C₈

C₂×D₁₉

yes

—

118

C₂×#103

—

119

C₂×C₄×C₈

—

120

#400

—

121

#389

—

122

#384

C₂×D₁₉

yes

characteristic

123

#382

—

124

C₄×C₄⋊C₄

—

125

#340

—

126

#340

—

127

C₂×C₁₉

#1630

yes

characteristic

128

C₂×C₁₉

#1634

yes

characteristic

129

C₂×C₁₉

#1638

yes

—

130

C₂×C₁₉

#2026

yes

—

131

C₂×C₁₉

#2027

yes

characteristic

132

C₂²×Q₈

—

133

C₂²×C₈

—

134

C₂²×C₈

—

135

C₄⋊Q₈

—

136

C₄⋊Q₈

—

137

C₄⋊Q₈

—

138

C₂×C₄⋊C₄

—

139

C₂×C₄⋊C₄

—

140

C₂×C₄⋊C₄

—

141

C₂×C₄⋊C₄

—

142

C₂×C₄⋊C₄

—

143

C₂×C₄⋊C₄

D₇₆

yes

—

144

C₂×C₄²

—

145

C₂×C₄²

C₂²×D₁₉

yes

characteristic

146

C₄⋊C₈

—

147

C₄⋊C₈

—

148

#103

—

149

C₄×C₈

—

150

#95

—

151

#95

—

152

#95

—

153

C₁₉

#10352

yes

characteristic, complemented, Sylow

154

C₂×Q₈

—

155

C₂×Q₈

—

156

C₂×Q₈

—

157

C₂×Q₈

—

158

C₂²×C₄

—

159

C₂²×C₄

—

160

C₂²×C₄

—

161

C₂²×C₄

—

162

C₂²×C₄

C₂×D₇₆

yes

characteristic

163

C₂²×C₄

D₄×D₁₉

yes

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₄⋊C₄

—

171

C₄⋊C₄

—

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₄⋊C₄

—

175

C₄⋊C₄

—

176

C₄⋊C₄

—

177

C₄⋊C₄

—

178

C₄⋊C₄

—

179

C₄⋊C₄

—

152

—

180

C₄²

—

181

C₄²

—

182

C₄²

#64541

yes

—

183

C₄²

#64543

yes

characteristic

184

C₄²

Q₈×D₁₉

yes

characteristic

185

C₂³

#10597328

yes

characteristic

186

Q₈

—

187

Q₈

—

152

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

152

—

200

C₂×C₄

—

152

—

201

C₂×C₄

#10597321

yes

characteristic

202

C₂×C₄

#10597331

yes

—

203

C₂×C₄

#10597342

yes

—

204

C₂×C₄

#10597343

yes

characteristic

205

C₂×C₄

#10597358

yes

—

206

C₂×C₄

#10597361

yes

—

207

C₂×C₄

#10597362

yes

—

208

C₂×C₄

#10597364

yes

—

209

C₈

—

210

C₈

—

211

C₂²

#11926457

yes

characteristic

212

C₂²

#11926707

yes

—

213

C₂²

#11926709

yes

—

214

C₂²

#11926714

yes

characteristic

215

C₂²

#11926719

yes

characteristic

216

C₄

—

217

C₄

—

218

C₄

—

219

C₄

—

152

—

220

C₄

—

152

—

221

C₄

—

152

—

222

C₄

#11926850

yes

—

223

C₄

#11926854

yes

—

224

C₄

#11926897

yes

—

225

C₄

#11926911

yes

—

226

C₂

#14562757

yes

characteristic

227

C₂

#14562781

yes

characteristic

228

C₂

#14562808

yes

—

229

C₂

#14564785

yes

—

230

C₂

#14564792

yes

characteristic

	subgroup	G/H
center	C₂³	#10597328
commutator, cosocle	C₂×C₄²×C₁₉	C₂³
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1296	C₂
socle, prime radical	C₂³×C₁₉	#120
hypercenter	#1296	D₁₉
hypocenter	C₁₉	#10352
LCS	G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1296

subgroup

G/H

center

C₂³

#10597328

commutator, cosocle

C₂×C₄²×C₁₉

C₂³

Frattini

C₂×C₄²

C₂²×D₁₉

Fitting

C₁₉×#1296

C₂

C₂³×C₁₉

D₁₉

C₁₉

G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1296

N	H ≅ G/N	type
153. C₁₉	#10352	semidirect

H ≅ G/N

type

153. C₁₉

#10352

semidirect

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

subgroup	index
#14568877	2
#14564582	2
#14564436	2
#14561474	2
C₁₉×#1296	2
#10352	19

subgroup

index

C₁₉×#1296

Subgroups — #18085570

Conjugacy classes of subgroups

Complements

Maximal subgroups