Group #18069855 — subgroups

Subgroups — #18069855

Order 4864.

256 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 71 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	#14564576	C₂	1	1	yes	maximal, characteristic
2	#14563631	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1467	C₂	1	1	yes	maximal, characteristic
4	#11926568	—	2	1	no	—
5	C₂×C₄⋊(C₁₉⋊C₈)	C₄	1	1	yes	characteristic
6	C₂×#10597293	C₄	1	2	yes	—
7	C₂×#10597260	C₄	1	1	yes	characteristic
8	#11926313	—	2	1	no	—
9	#11926299	—	2	1	no	—
10	#11926299	—	2	1	no	—
11	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
12	C₁₉×#432	—	2	1	no	—
13	C₁₉×#324	—	2	1	no	—
14	C₁₉×#324	C₄	1	2	yes	—
15	C₂²×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
16	C₄×Q₈×C₁₉	—	4	1	no	—
17	C₄×Q₈×C₁₉	—	4	1	no	—
18	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₁₉	Q₈	1	1	yes	characteristic
21	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
24	C₂×#64520	—	2	2	no	—
25	C₂²×C₁₉⋊C₈	—	2	1	no	—
26	C₂²×C₁₉⋊C₈	—	2	1	no	—
27	C₁₉×C₄⋊C₈	—	4	1	no	—
28	C₁₉×C₂²⋊C₈	—	2	2	no	—
29	C₁₉×#97	—	4	1	no	—
30	#10597293	—	4	2	no	—
31	C₄⋊(C₁₉⋊C₈)	—	4	1	no	—
32	C₄⋊(C₁₉⋊C₈)	—	4	1	no	—
33	C₄⋊(C₁₉⋊C₈)	—	4	1	no	—
34	#10597260	—	4	1	no	—
35	C₂×Q₈×C₁₉	—	2	1	no	—
36	C₂×Q₈×C₁₉	—	2	1	no	—
37	C₂×Q₈×C₁₉	—	4	1	no	—
38	C₂×Q₈×C₁₉	—	4	1	no	—
39	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
40	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₉	—	4	2	no	—
47	C₄⋊C₄×C₁₉	—	4	1	no	—
48	C₄⋊C₄×C₁₉	—	4	1	no	—
49	C₄⋊C₄×C₁₉	—	4	1	no	—
50	C₄²×C₁₉	—	2	1	no	—
51	C₄²×C₁₉	—	2	1	no	—
52	C₄²×C₁₉	—	2	1	no	—
53	C₄²×C₁₉	—	2	1	no	—
54	C₄²×C₁₉	—	2	1	no	—
55	C₄²×C₁₉	—	2	1	no	—
56	#64520	—	4	2	no	—
57	#64520	—	4	2	no	—
58	C₂×C₁₉⋊C₈	—	2	2	no	—
59	C₂×C₁₉⋊C₈	—	4	1	no	—
60	C₂×C₁₉⋊C₈	—	4	1	no	—
61	C₂×C₁₉⋊C₈	—	4	2	no	—
62	C₂×C₁₉⋊C₈	—	4	2	no	—
63	C₂×C₁₉⋊C₈	—	4	1	no	—
64	#7323	—	19	1	no	maximal, Sylow
65	C₂³×C₁₉	#95	1	1	yes	characteristic
66	Q₈×C₁₉	—	4	2	no	—
67	Q₈×C₁₉	—	4	2	no	—
68	C₂×C₄×C₁₉	—	2	1	no	—
69	C₂×C₄×C₁₉	—	2	1	no	—
70	C₂×C₄×C₁₉	—	2	1	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	—	2	1	no	—
76	C₂×C₄×C₁₉	—	2	1	no	—
77	C₂×C₄×C₁₉	—	2	1	no	—
78	C₂×C₄×C₁₉	—	2	1	no	—
79	C₂×C₄×C₁₉	—	2	1	no	—
80	C₂×C₄×C₁₉	—	2	1	no	—
81	C₂×C₄×C₁₉	—	4	1	no	—
82	C₂×C₄×C₁₉	—	4	1	no	—
83	C₂×C₄×C₁₉	—	4	1	no	—
84	C₂×C₄×C₁₉	#99	1	1	yes	characteristic
85	C₂×C₄×C₁₉	#99	1	1	yes	characteristic
86	C₂×C₄×C₁₉	#100	1	1	yes	characteristic
87	C₂×C₄×C₁₉	#101	1	1	yes	characteristic
88	C₈×C₁₉	—	8	2	no	—
89	C₁₉⋊C₈	—	4	2	no	—
90	C₁₉⋊C₈	—	8	2	no	—
91	C₁₉⋊C₈	—	8	1	no	—
92	#1985	—	19	1	no	—
93	#1801	—	19	1	no	—
94	#1467	D₁₉	1	1	yes	characteristic
95	C₂²×C₁₉	—	2	1	no	—
96	C₂²×C₁₉	—	2	1	no	—
97	C₂²×C₁₉	#342	1	1	yes	characteristic
98	C₂²×C₁₉	#343	1	1	yes	characteristic
99	C₂²×C₁₉	#344	1	1	yes	characteristic
100	C₄×C₁₉	—	2	1	no	—
101	C₄×C₁₉	—	2	1	no	—
102	C₄×C₁₉	—	2	1	no	—
103	C₄×C₁₉	—	4	1	no	—
104	C₄×C₁₉	—	4	1	no	—
105	C₄×C₁₉	—	4	1	no	—
106	C₄×C₁₉	—	4	1	no	—
107	C₄×C₁₉	—	4	1	no	—
108	C₄×C₁₉	#355	1	1	yes	characteristic
109	C₄×C₁₉	#356	1	1	yes	characteristic
110	C₂×C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
111	#432	—	2	1	no	—
112	#423	—	38	1	no	—
113	C₂×C₄⋊C₈	—	19	1	no	—
114	C₂×#101	—	19	2	no	—
115	C₂×#97	—	19	1	no	—
116	#343	—	38	1	no	—
117	#336	—	38	1	no	—
118	#336	—	38	1	no	—
119	#324	—	2	1	no	—
120	#324	Dic₁₉	1	2	yes	—
121	C₂×C₁₉	—	2	1	no	—
122	C₂×C₁₉	—	2	1	no	—
123	C₂×C₁₉	#1305	1	1	yes	characteristic
124	C₂×C₁₉	#1394	1	1	yes	characteristic
125	C₂×C₁₉	#1395	1	1	yes	characteristic
126	C₂²×Q₈	C₄×D₁₉	1	1	yes	characteristic
127	C₂×M₁₆	—	38	2	no	—
128	C₂²×C₈	—	38	1	no	—
129	C₂²×C₈	—	38	1	no	—
130	C₄×Q₈	—	4	1	no	—
131	C₄×Q₈	—	4	1	no	—
132	C₂×C₄⋊C₄	Dic₃₈	1	1	yes	characteristic
133	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
134	C₂×C₄⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
135	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
136	C₂×C₄²	D₇₆	1	1	yes	characteristic
137	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
138	C₄⋊C₈	—	4	1	no	—
139	C₄⋊C₈	—	76	1	no	—
140	C₄⋊C₈	—	76	1	no	—
141	C₄⋊C₈	—	76	1	no	—
142	#101	—	76	2	no	—
143	C₂²⋊C₈	—	2	2	no	—
144	#97	—	4	1	no	—
145	#97	—	76	1	no	—
146	C₁₉	#7323	1	1	yes	characteristic, complemented, Sylow
147	C₂×Q₈	—	2	1	no	—
148	C₂×Q₈	—	2	1	no	—
149	C₂×Q₈	—	4	1	no	—
150	C₂×Q₈	—	4	1	no	—
151	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
152	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
153	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
154	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
155	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
156	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
157	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
158	M₁₆	—	76	2	no	—
159	M₁₆	—	76	2	no	—
160	C₂×C₈	—	4	2	no	—
161	C₂×C₈	—	38	2	no	—
162	C₂×C₈	—	76	1	no	—
163	C₂×C₈	—	76	1	no	—
164	C₂×C₈	—	76	2	no	—
165	C₂×C₈	—	76	2	no	—
166	C₂×C₈	—	76	1	no	—
167	C₄⋊C₄	—	4	1	no	—
168	C₄⋊C₄	—	4	1	no	—
169	C₄⋊C₄	—	4	1	no	—
170	C₄²	—	2	1	no	—
171	C₄²	—	2	1	no	—
172	C₄²	—	2	1	no	—
173	C₄²	—	2	1	no	—
174	C₄²	—	2	1	no	—
175	C₄²	—	2	1	no	—
176	C₂³	C₁₉⋊#95	1	1	yes	characteristic
177	Q₈	—	4	2	no	—
178	Q₈	—	4	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	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	4	1	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	#10597263	1	1	yes	characteristic
196	C₂×C₄	#10597263	1	1	yes	characteristic
197	C₂×C₄	#10597280	1	1	yes	characteristic
198	C₂×C₄	#10597281	1	1	yes	characteristic
199	C₈	—	8	2	no	—
200	C₈	—	76	2	no	—
201	C₈	—	152	2	no	—
202	C₈	—	152	1	no	—
203	C₂²	—	2	1	no	—
204	C₂²	—	2	1	no	—
205	C₂²	#11926310	1	1	yes	characteristic
206	C₂²	#11926312	1	1	yes	characteristic
207	C₂²	#11926315	1	1	yes	characteristic
208	C₄	—	2	1	no	—
209	C₄	—	2	1	no	—
210	C₄	—	2	1	no	—
211	C₄	—	4	1	no	—
212	C₄	—	4	1	no	—
213	C₄	—	4	1	no	—
214	C₄	—	4	1	no	—
215	C₄	—	4	1	no	—
216	C₄	#11926340	1	1	yes	characteristic
217	C₄	#11926342	1	1	yes	characteristic
218	C₂	—	2	1	no	—
219	C₂	—	2	1	no	—
220	C₂	#14561542	1	1	yes	characteristic
221	C₂	#14561761	1	1	yes	characteristic
222	C₂	#14561763	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14564576

C₂

yes

maximal, characteristic

#14563631

C₂

yes

maximal, characteristic

C₁₉×#1467

C₂

yes

maximal, characteristic

#11926568

—

C₂×C₄⋊(C₁₉⋊C₈)

C₄

yes

characteristic

C₂×#10597293

C₄

yes

—

C₂×#10597260

C₄

yes

—

—

—

C₂×C₄×Q₈×C₁₉

C₂²

yes

characteristic

C₁₉×#432

—

C₁₉×#324

—

C₁₉×#324

C₄

yes

—

C₂²×Q₈×C₁₉

C₂×C₄

yes

characteristic

C₄×Q₈×C₁₉

—

C₄×Q₈×C₁₉

—

C₂×C₄⋊C₄×C₁₉

C₂×C₄

yes

characteristic

C₂×C₄⋊C₄×C₁₉

D₄

yes

characteristic

C₂×C₄⋊C₄×C₁₉

Q₈

yes

characteristic

C₂×C₄²×C₁₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₉

D₄

yes

characteristic

C₂×C₄²×C₁₉

D₄

yes

characteristic

C₂×#64520

—

C₂²×C₁₉⋊C₈

—

C₂²×C₁₉⋊C₈

—

C₁₉×C₄⋊C₈

—

C₁₉×C₂²⋊C₈

—

C₁₉×#97

—

#10597293

—

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

—

#10597260

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂²×C₄×C₁₉

C₄²

yes

characteristic

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

#64520

—

#64520

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

—

C₂³×C₁₉

yes

Q₈×C₁₉

—

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

—

C₂×C₄×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

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₈×C₁₉

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

#1985

—

#1801

—

#1467

D₁₉

yes

characteristic

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

100

C₄×C₁₉

—

101

C₄×C₁₉

—

102

C₄×C₁₉

—

103

C₄×C₁₉

—

104

C₄×C₁₉

—

105

C₄×C₁₉

—

106

C₄×C₁₉

—

107

C₄×C₁₉

—

108

C₄×C₁₉

#355

yes

characteristic

109

C₄×C₁₉

#356

yes

characteristic

110

C₂×C₄×Q₈

C₂×D₁₉

yes

characteristic

111

#432

—

112

#423

—

113

C₂×C₄⋊C₈

—

114

C₂×#101

—

115

C₂×#97

—

116

#343

—

117

#336

—

118

#336

—

119

#324

—

120

#324

Dic₁₉

yes

—

121

C₂×C₁₉

—

122

C₂×C₁₉

—

123

C₂×C₁₉

#1305

yes

characteristic

124

C₂×C₁₉

#1394

yes

characteristic

125

C₂×C₁₉

#1395

yes

characteristic

126

C₂²×Q₈

C₄×D₁₉

yes

characteristic

127

C₂×M₁₆

—

128

C₂²×C₈

—

129

C₂²×C₈

—

130

C₄×Q₈

—

131

C₄×Q₈

—

132

C₂×C₄⋊C₄

Dic₃₈

yes

characteristic

133

C₂×C₄⋊C₄

C₂×Dic₁₉

yes

characteristic

134

C₂×C₄⋊C₄

C₂²⋊D₁₉

yes

characteristic

135

C₂×C₄²

C₄×D₁₉

yes

characteristic

136

C₂×C₄²

D₇₆

yes

characteristic

137

C₂×C₄²

C₂²⋊D₁₉

yes

characteristic

138

C₄⋊C₈

—

139

C₄⋊C₈

—

140

C₄⋊C₈

—

141

C₄⋊C₈

—

142

#101

—

143

C₂²⋊C₈

—

144

#97

—

145

#97

—

146

C₁₉

#7323

yes

characteristic, complemented, Sylow

147

C₂×Q₈

—

148

C₂×Q₈

—

149

C₂×Q₈

—

150

C₂×Q₈

—

151

C₂²×C₄

C₄×Dic₁₉

yes

characteristic

152

C₂²×C₄

Dic₁₉⋊C₄

yes

characteristic

153

C₂²×C₄

Dic₁₉⋊C₄

yes

characteristic

154

C₂²×C₄

C₄⋊Dic₁₉

yes

characteristic

155

C₂²×C₄

(C₂×D₁₉)⋊C₄

yes

characteristic

156

C₂²×C₄

(C₂×D₁₉)⋊C₄

yes

characteristic

157

C₂²×C₄

C₂²⋊Dic₁₉

yes

characteristic

158

M₁₆

—

159

M₁₆

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₄⋊C₄

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₄²

—

171

C₄²

—

172

C₄²

—

173

C₄²

—

174

C₄²

—

175

C₄²

—

176

C₂³

C₁₉⋊#95

yes

characteristic

177

Q₈

—

178

Q₈

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

#10597263

yes

characteristic

196

C₂×C₄

#10597263

yes

characteristic

197

C₂×C₄

#10597280

yes

characteristic

198

C₂×C₄

#10597281

yes

characteristic

199

C₈

—

200

C₈

—

201

C₈

—

152

—

202

C₈

—

152

—

203

C₂²

—

204

C₂²

—

205

C₂²

#11926310

yes

characteristic

206

C₂²

#11926312

yes

characteristic

207

C₂²

#11926315

yes

characteristic

208

C₄

—

209

C₄

—

210

C₄

—

211

C₄

—

212

C₄

—

213

C₄

—

214

C₄

—

215

C₄

—

216

C₄

#11926340

yes

characteristic

217

C₄

#11926342

yes

characteristic

218

C₂

—

219

C₂

—

220

C₂

#14561542

yes

characteristic

221

C₂

#14561761

yes

characteristic

222

C₂

#14561763

yes

characteristic

	subgroup	G/H
center	C₂²	#11926310
commutator	C₂²×C₄×C₁₉	C₄²
Frattini	C₂×C₄×Q₈	C₂×D₁₉
Fitting	C₁₉×#1467	C₂
socle	C₂²×C₁₉	#342
prime radical	C₂³×C₁₉	#95
cosocle	C₂×C₄×Q₈×C₁₉	C₂²
hypercenter	#1467	D₁₉
hypocenter	C₁₉	#7323
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂³×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ #1467

subgroup

G/H

center

C₂²

#11926310

commutator

C₂²×C₄×C₁₉

C₄²

Frattini

C₂×C₄×Q₈

C₂×D₁₉

Fitting

C₁₉×#1467

C₂

C₂²×C₁₉

C₂³×C₁₉

C₂×C₄×Q₈×C₁₉

C₂²

D₁₉

C₁₉

G ▷ C₂²×C₄×C₁₉ ▷ C₂³×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ #1467

N	H ≅ G/N	type
146. C₁₉	#7323	semidirect

H ≅ G/N

type

146. C₁₉

#7323

semidirect

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

subgroup	index
#14564576	2
#14563631	2
C₁₉×#1467	2
#7323	19

subgroup

index

#14564576

#14563631

C₁₉×#1467

#7323

Subgroups — #18069855

Conjugacy classes of subgroups

Complements

Maximal subgroups