Group #18085225 — subgroups

Subgroups — #18085225

Order 4864.

614 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 113 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	#14574756	C₂	1	2	yes	maximal
2	#14564463	C₂	1	1	yes	maximal, characteristic
3	#14562307	C₂	1	2	yes	maximal, complemented
4	#14561468	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#2868	C₂	1	1	yes	maximal, characteristic, complemented
6	#11927094	C₂²	1	1	yes	characteristic
7	#11926947	—	2	2	no	—
8	#11926892	—	2	2	no	—
9	#11926857	—	2	4	no	—
10	#11926821	—	2	4	no	—
11	#11926597	C₂²	1	2	yes	—
12	#11926568	C₂²	1	1	yes	characteristic
13	C₂×#10597262	—	2	2	no	—
14	#11926290	—	2	4	no	—
15	C₂×C₁₉×C₄⋊Q₈	—	2	2	no	—
16	C₂×C₁₉×C₄⋊Q₈	C₂²	1	2	yes	—
17	C₁₉×#384	—	2	4	no	—
18	C₄³×C₁₉	C₂²	1	1	yes	characteristic
19	C₂²×Q₈×C₁₉	—	2	2	no	—
20	C₂×#64541	—	2	1	no	—
21	C₁₉×C₄⋊Q₈	—	2	2	no	—
22	C₁₉×C₄⋊Q₈	—	2	2	no	—
23	C₁₉×C₄⋊Q₈	—	2	2	no	—
24	C₁₉×C₄⋊Q₈	—	2	2	no	—
25	C₁₉×C₄⋊Q₈	—	4	4	no	—
26	C₁₉×C₄⋊Q₈	—	4	4	no	—
27	C₂×C₄⋊C₄×C₁₉	—	2	8	no	—
28	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
29	C₂×C₄⋊C₄×C₁₉	D₄	1	4	yes	—
30	C₂×C₄²×C₁₉	—	2	2	no	—
31	C₂×C₄²×C₁₉	D₄	1	2	yes	—
32	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
33	#10597376	—	2	2	no	—
34	C₂×#64520	—	2	1	no	—
35	#10597321	—	2	1	no	—
36	#10597320	D₄	1	1	yes	characteristic
37	C₄×D₇₆	—	2	2	no	—
38	#10597315	D₄	1	1	yes	characteristic
39	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
40	#10597266	—	2	8	no	—
41	#10597264	—	2	8	no	—
42	#10597262	—	4	4	no	—
43	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
44	C₄⋊(C₁₉⋊C₈)	D₄	1	4	yes	—
45	C₄×C₁₉⋊C₈	—	2	2	no	—
46	C₂×Q₈×C₁₉	—	4	2	no	—
47	C₂×Q₈×C₁₉	—	4	2	no	—
48	C₂×Q₈×C₁₉	—	4	4	no	—
49	C₂×Q₈×C₁₉	—	4	4	no	—
50	C₂²×C₄×C₁₉	—	2	2	no	—
51	C₂²×C₄×C₁₉	—	2	4	no	—
52	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₂²⋊D₁₉	—	4	1	no	—
55	#64541	—	4	2	no	—
56	#64541	—	4	2	no	—
57	C₂×D₇₆	—	2	2	no	—
58	C₂×D₇₆	—	2	1	no	—
59	C₂×C₄×D₁₉	—	4	1	no	—
60	C₂×Dic₃₈	—	2	1	no	—
61	C₄⋊C₄×C₁₉	—	2	4	no	—
62	C₄⋊C₄×C₁₉	—	2	4	no	—
63	C₄⋊C₄×C₁₉	—	4	16	no	—
64	C₄⋊C₄×C₁₉	—	4	2	no	—
65	C₄⋊C₄×C₁₉	—	4	2	no	—
66	C₄²×C₁₉	—	2	8	no	—
67	C₄²×C₁₉	—	2	2	no	—
68	C₄²×C₁₉	D₈	1	4	yes	—
69	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₁₉	C₂×D₄	1	2	yes	—
71	C₄²×C₁₉	#41	1	1	yes	characteristic
72	(C₂×D₁₉)⋊C₄	—	4	2	no	—
73	C₄⋊Dic₁₉	—	2	2	no	—
74	#64520	—	4	2	no	—
75	C₂×C₁₉⋊C₈	—	2	4	no	—
76	C₂×C₁₉⋊C₈	—	4	1	no	—
77	#10289	—	19	1	no	maximal, Sylow
78	C₂³×C₁₉	#120	1	1	yes	characteristic
79	C₂²×D₁₉	—	4	1	no	—
80	Q₈×C₁₉	—	8	4	no	—
81	Q₈×C₁₉	—	8	2	no	—
82	Q₈×C₁₉	—	8	2	no	—
83	C₂×C₄×C₁₉	—	2	8	no	—
84	C₂×C₄×C₁₉	—	2	4	no	—
85	C₂×C₄×C₁₉	—	2	4	no	—
86	C₂×C₄×C₁₉	—	4	4	no	—
87	C₂×C₄×C₁₉	—	4	8	no	—
88	C₂×C₄×C₁₉	#120	1	2	yes	—
89	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
90	C₂×C₄×C₁₉	#121	1	2	yes	—
91	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
92	C₂×C₄×C₁₉	C₂×D₈	1	2	yes	—
93	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
94	C₂²⋊D₁₉	—	8	2	no	—
95	C₂×Dic₁₉	—	4	1	no	—
96	D₇₆	—	4	4	no	—
97	D₇₆	—	4	2	no	—
98	C₄×D₁₉	—	8	1	no	—
99	C₄×D₁₉	—	8	1	no	—
100	Dic₃₈	—	4	2	no	—
101	C₁₉⋊C₈	—	4	4	no	—
102	C₁₉⋊C₈	—	8	1	no	—
103	#3226	—	19	2	no	—
104	#2868	D₁₉	1	1	yes	characteristic, complemented
105	#1962	—	19	1	no	—
106	#1546	—	19	2	no	—
107	#1276	—	19	1	no	—
108	C₂²×C₁₉	—	2	2	no	—
109	C₂²×C₁₉	#392	1	1	yes	characteristic
110	C₂²×C₁₉	#447	1	2	yes	—
111	C₂×D₁₉	—	8	2	no	—
112	C₂×D₁₉	—	8	1	no	—
113	C₄×C₁₉	—	2	8	no	—
114	C₄×C₁₉	—	2	2	no	—
115	C₄×C₁₉	—	8	4	no	—
116	C₄×C₁₉	#456	1	4	yes	—
117	C₄×C₁₉	#459	1	2	yes	—
118	C₄×C₁₉	#493	1	1	yes	characteristic
119	C₄×C₁₉	#496	1	1	yes	characteristic
120	Dic₁₉	—	8	1	no	—
121	#532	—	19	1	no	—
122	C₂×C₄⋊Q₈	—	2	2	no	—
123	C₂×C₄⋊Q₈	C₂×D₁₉	1	2	yes	—
124	#500	—	38	2	no	—
125	#486	—	38	2	no	—
126	#480	—	38	4	no	—
127	#474	—	38	4	no	—
128	#431	—	19	2	no	—
129	#423	—	19	1	no	—
130	C₂×#104	—	38	2	no	—
131	#384	—	2	4	no	—
132	C₄³	C₂×D₁₉	1	1	yes	characteristic
133	#332	—	38	4	no	—
134	C₂×C₁₉	—	2	2	no	—
135	C₂×C₁₉	#2005	1	2	yes	—
136	C₂×C₁₉	#2011	1	1	yes	characteristic
137	D₁₉	—	16	1	no	—
138	C₂×#41	—	38	1	no	—
139	C₂²×Q₈	—	2	2	no	—
140	C₂×M₁₆	—	38	1	no	—
141	C₄⋊Q₈	—	2	2	no	—
142	C₄⋊Q₈	—	2	2	no	—
143	C₄⋊Q₈	—	2	2	no	—
144	C₄⋊Q₈	—	2	2	no	—
145	C₄⋊Q₈	—	4	4	no	—
146	C₄⋊Q₈	—	4	4	no	—
147	C₄⋊Q₈	—	19	1	no	—
148	#127	—	19	1	no	—
149	#124	—	38	1	no	—
150	#121	—	38	2	no	—
151	C₄×D₄	—	38	2	no	—
152	C₂×C₄⋊C₄	—	2	8	no	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	C₂²⋊D₁₉	1	4	yes	—
155	C₂×C₄²	—	2	2	no	—
156	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
157	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
158	#107	—	38	8	no	—
159	C₄⋊C₈	—	19	4	no	—
160	C₄⋊C₈	—	38	2	no	—
161	#104	—	76	4	no	—
162	#102	—	38	8	no	—
163	C₂²⋊C₈	—	38	2	no	—
164	C₄×C₈	—	38	2	no	—
165	C₁₉	#10289	1	1	yes	characteristic, complemented, Sylow
166	#41	—	76	2	no	—
167	#41	—	76	2	no	—
168	C₂×Q₈	—	4	4	no	—
169	C₂×Q₈	—	4	4	no	—
170	C₂×Q₈	—	4	2	no	—
171	C₂×Q₈	—	4	2	no	—
172	C₂×Q₈	—	38	1	no	—
173	C₂×D₄	—	38	2	no	—
174	C₂×D₄	—	38	1	no	—
175	C₂×D₄	—	76	1	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	—	76	1	no	—
179	C₂²×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
180	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
181	M₁₆	—	76	2	no	—
182	C₂×C₈	—	38	4	no	—
183	C₂×C₈	—	76	1	no	—
184	C₄⋊C₄	—	2	4	no	—
185	C₄⋊C₄	—	2	4	no	—
186	C₄⋊C₄	—	4	16	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	—	38	2	no	—
190	C₂²⋊C₄	—	76	2	no	—
191	C₄²	—	2	8	no	—
192	C₄²	—	2	2	no	—
193	C₄²	#64525	1	4	yes	—
194	C₄²	D₄×D₁₉	1	1	yes	characteristic
195	C₄²	D₄×D₁₉	1	2	yes	—
196	C₄²	#64543	1	1	yes	characteristic
197	C₂³	—	76	1	no	—
198	C₂³	#10597399	1	1	yes	characteristic
199	Q₈	—	8	2	no	—
200	Q₈	—	8	2	no	—
201	Q₈	—	8	4	no	—
202	Q₈	—	76	2	no	—
203	D₄	—	76	4	no	—
204	D₄	—	76	2	no	—
205	D₄	—	152	2	no	—
206	C₂×C₄	—	2	8	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	4	8	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	—	76	1	no	—
212	C₂×C₄	—	152	1	no	—
213	C₂×C₄	—	152	1	no	—
214	C₂×C₄	C₂×#64525	1	2	yes	—
215	C₂×C₄	#10597378	1	2	yes	—
216	C₂×C₄	#10597383	1	2	yes	—
217	C₂×C₄	#10597384	1	1	yes	characteristic
218	C₂×C₄	#10597385	1	2	yes	—
219	C₂×C₄	#10597386	1	1	yes	characteristic
220	C₈	—	76	4	no	—
221	C₈	—	152	1	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	152	2	no	—
224	C₂²	—	152	1	no	—
225	C₂²	#11926452	1	1	yes	characteristic
226	C₂²	#11926678	1	2	yes	—
227	C₄	—	2	8	no	—
228	C₄	—	2	2	no	—
229	C₄	—	8	4	no	—
230	C₄	—	152	1	no	—
231	C₄	#11926726	1	4	yes	—
232	C₄	#11926736	1	2	yes	—
233	C₄	#11926920	1	1	yes	characteristic
234	C₄	#11926931	1	1	yes	characteristic
235	C₂	—	2	2	no	—
236	C₂	—	304	1	no	—
237	C₂	#14564671	1	2	yes	—
238	C₂	#14564706	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14574756

C₂

yes

maximal

#14564463

C₂

yes

maximal, characteristic

#14562307

C₂

yes

maximal, complemented

#14561468

C₂

yes

maximal, characteristic, complemented

C₁₉×#2868

C₂

yes

maximal, characteristic, complemented

#11927094

C₂²

yes

—

—

—

—

C₂²

yes

—

#11926568

C₂²

yes

characteristic

C₂×#10597262

—

#11926290

—

C₂×C₁₉×C₄⋊Q₈

—

C₂×C₁₉×C₄⋊Q₈

C₂²

yes

—

C₁₉×#384

—

C₄³×C₁₉

C₂²

yes

characteristic

C₂²×Q₈×C₁₉

—

C₂×#64541

—

C₁₉×C₄⋊Q₈

—

C₁₉×C₄⋊Q₈

—

C₁₉×C₄⋊Q₈

—

C₁₉×C₄⋊Q₈

—

C₁₉×C₄⋊Q₈

—

C₁₉×C₄⋊Q₈

—

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

D₄

yes

—

C₂×C₄²×C₁₉

—

C₂×C₄²×C₁₉

D₄

yes

—

C₂×C₄²×C₁₉

C₂³

yes

—

—

—

D₄

yes

characteristic

C₄×D₇₆

—

#10597315

D₄

yes

characteristic

C₂²⋊(C₁₉⋊C₈)

—

#10597266

—

#10597264

—

#10597262

—

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

D₄

yes

—

C₄×C₁₉⋊C₈

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂×D₄

yes

—

C₂²×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂×C₂²⋊D₁₉

—

#64541

—

#64541

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×C₄×D₁₉

—

C₂×Dic₃₈

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

D₈

yes

—

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₄²×C₁₉

C₂×D₄

yes

—

C₄²×C₁₉

#41

yes

characteristic

(C₂×D₁₉)⋊C₄

—

C₄⋊Dic₁₉

—

#64520

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

—

C₂³×C₁₉

yes

C₂²×D₁₉

—

Q₈×C₁₉

—

Q₈×C₁₉

—

Q₈×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

#120

yes

—

C₂×C₄×C₁₉

#121

yes

characteristic

C₂×C₄×C₁₉

#121

yes

—

C₂×C₄×C₁₉

#127

yes

characteristic

C₂×C₄×C₁₉

C₂×D₈

yes

—

C₂×C₄×C₁₉

AGL(1,ℤ/8)

yes

—

C₂²⋊D₁₉

—

C₂×Dic₁₉

—

D₇₆

—

D₇₆

—

C₄×D₁₉

—

C₄×D₁₉

—

100

Dic₃₈

—

101

C₁₉⋊C₈

—

102

C₁₉⋊C₈

—

103

#3226

—

104

#2868

D₁₉

yes

characteristic, complemented

105

#1962

—

106

#1546

—

107

#1276

—

108

C₂²×C₁₉

—

109

C₂²×C₁₉

#392

yes

characteristic

110

C₂²×C₁₉

#447

yes

—

111

C₂×D₁₉

—

112

C₂×D₁₉

—

113

C₄×C₁₉

—

114

C₄×C₁₉

—

115

C₄×C₁₉

—

116

C₄×C₁₉

#456

yes

—

117

C₄×C₁₉

#459

yes

—

118

C₄×C₁₉

#493

yes

characteristic

119

C₄×C₁₉

#496

yes

characteristic

120

Dic₁₉

—

121

#532

—

122

C₂×C₄⋊Q₈

—

123

C₂×C₄⋊Q₈

C₂×D₁₉

yes

—

124

#500

—

125

#486

—

126

#480

—

127

#474

—

128

#431

—

129

#423

—

130

C₂×#104

—

131

#384

—

132

C₄³

C₂×D₁₉

yes

characteristic

133

#332

—

134

C₂×C₁₉

—

135

C₂×C₁₉

#2005

yes

—

136

C₂×C₁₉

#2011

yes

characteristic

137

D₁₉

—

138

C₂×#41

—

139

C₂²×Q₈

—

140

C₂×M₁₆

—

141

C₄⋊Q₈

—

142

C₄⋊Q₈

—

143

C₄⋊Q₈

—

144

C₄⋊Q₈

—

145

C₄⋊Q₈

—

146

C₄⋊Q₈

—

147

C₄⋊Q₈

—

148

#127

—

149

#124

—

150

#121

—

151

C₄×D₄

—

152

C₂×C₄⋊C₄

—

153

C₂×C₄⋊C₄

—

154

C₂×C₄⋊C₄

C₂²⋊D₁₉

yes

—

155

C₂×C₄²

—

156

C₂×C₄²

C₂²⋊D₁₉

yes

—

157

C₂×C₄²

C₂²×D₁₉

yes

characteristic

158

#107

—

159

C₄⋊C₈

—

160

C₄⋊C₈

—

161

#104

—

162

#102

—

163

C₂²⋊C₈

—

164

C₄×C₈

—

165

C₁₉

#10289

yes

characteristic, complemented, Sylow

166

#41

—

167

#41

—

168

C₂×Q₈

—

169

C₂×Q₈

—

170

C₂×Q₈

—

171

C₂×Q₈

—

172

C₂×Q₈

—

173

C₂×D₄

—

174

C₂×D₄

—

175

C₂×D₄

—

176

C₂²×C₄

—

177

C₂²×C₄

—

178

C₂²×C₄

—

179

C₂²×C₄

C₂×C₂²⋊D₁₉

yes

—

180

C₂²×C₄

C₂×C₂²⋊D₁₉

yes

characteristic

181

M₁₆

—

182

C₂×C₈

—

183

C₂×C₈

—

184

C₄⋊C₄

—

185

C₄⋊C₄

—

186

C₄⋊C₄

—

187

C₄⋊C₄

—

188

C₄⋊C₄

—

189

C₄⋊C₄

—

190

C₂²⋊C₄

—

191

C₄²

—

192

C₄²

—

193

C₄²

#64525

yes

—

194

C₄²

D₄×D₁₉

yes

characteristic

195

C₄²

D₄×D₁₉

yes

—

196

C₄²

#64543

yes

characteristic

197

C₂³

—

198

C₂³

#10597399

yes

characteristic

199

Q₈

—

200

Q₈

—

201

Q₈

—

202

Q₈

—

203

D₄

—

204

D₄

—

205

D₄

—

152

—

206

C₂×C₄

—

207

C₂×C₄

—

208

C₂×C₄

—

209

C₂×C₄

—

210

C₂×C₄

—

211

C₂×C₄

—

212

C₂×C₄

—

152

—

213

C₂×C₄

—

152

—

214

C₂×C₄

C₂×#64525

yes

—

215

C₂×C₄

#10597378

yes

—

216

C₂×C₄

#10597383

yes

—

217

C₂×C₄

#10597384

yes

characteristic

218

C₂×C₄

#10597385

yes

—

219

C₂×C₄

#10597386

yes

characteristic

220

C₈

—

221

C₈

—

152

—

222

C₂²

—

223

C₂²

—

152

—

224

C₂²

—

152

—

225

C₂²

#11926452

yes

characteristic

226

C₂²

#11926678

yes

—

227

C₄

—

228

C₄

—

229

C₄

—

230

C₄

—

152

—

231

C₄

#11926726

yes

—

232

C₄

#11926736

yes

—

233

C₄

#11926920

yes

characteristic

234

C₄

#11926931

yes

characteristic

235

C₂

—

236

C₂

—

304

—

237

C₂

#14564671

yes

—

238

C₂

#14564706

yes

characteristic

	subgroup	G/H
center	C₂²	#11926452
commutator, cosocle	C₂×C₄²×C₁₉	C₂³
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#2868	C₂
socle	C₂²×C₁₉	#392
prime radical	#11927094	C₂²
hypercenter	#2868	D₁₉
hypocenter	C₁₉	#10289
LCS	G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2868

subgroup

G/H

center

C₂²

#11926452

commutator, cosocle

C₂×C₄²×C₁₉

C₂³

Frattini

C₂×C₄²

C₂²×D₁₉

Fitting

C₁₉×#2868

C₂

C₂²×C₁₉

C₂²

D₁₉

C₁₉

G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2868

N	H ≅ G/N	type
3. #14562307	C₂	semidirect
4. #14561468	C₂	semidirect
5. C₁₉×#2868	C₂	semidirect
104. #2868	D₁₉	semidirect
165. C₁₉	#10289	semidirect

H ≅ G/N

type

3. #14562307

C₂

semidirect

4. #14561468

C₂

semidirect

5. C₁₉×#2868

C₂

semidirect

104. #2868

D₁₉

semidirect

165. C₁₉

#10289

semidirect

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

subgroup	index
#14574756	2
#14564463	2
#14562307	2
#14561468	2
C₁₉×#2868	2
#10289	19

subgroup

index

C₁₉×#2868

Subgroups — #18085225

Conjugacy classes of subgroups

Complements

Maximal subgroups