Group #18069071 — subgroups

Subgroups — #18069071

Order 4864.

522 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 127 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	#14563703	C₂	1	2	yes	maximal
2	C₄×C₁₉×#95	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
4	#11926299	—	2	2	no	—
5	#11926299	C₄	1	4	yes	complemented
6	C₂²×C₄²×C₁₉	C₄	1	2	yes	—
7	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
8	C₂×C₁₉×#95	—	2	2	no	—
9	C₂³×C₄×C₁₉	—	2	2	no	—
10	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
11	C₂³×C₄×C₁₉	D₄	1	2	yes	—
12	C₂×C₄²×C₁₉	—	2	2	no	—
13	C₂×C₄²×C₁₉	—	4	2	no	—
14	C₂×C₄²×C₁₉	—	4	2	no	—
15	C₂×C₄²×C₁₉	—	4	2	no	—
16	C₂×C₄²×C₁₉	C₈	1	4	yes	complemented
17	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
19	C₂×C₄²×C₁₉	Q₈	1	1	yes	characteristic
20	C₂²×C₁₉⋊C₈	—	2	4	no	—
21	C₁₉×#95	—	2	8	no	—
22	C₂²⋊(C₁₉⋊C₈)	—	4	4	no	—
23	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₉	—	2	2	no	—
25	C₂²×C₄×C₁₉	—	2	1	no	—
26	C₂²×C₄×C₁₉	—	2	1	no	—
27	C₂²×C₄×C₁₉	—	2	8	no	—
28	C₂²×C₄×C₁₉	—	2	4	no	—
29	C₂²×C₄×C₁₉	—	2	4	no	—
30	C₂²×C₄×C₁₉	—	2	4	no	—
31	C₂²×C₄×C₁₉	—	4	2	no	—
32	C₂²×C₄×C₁₉	—	4	4	no	—
33	C₂²×C₄×C₁₉	—	4	2	no	—
34	C₂²×C₄×C₁₉	—	4	2	no	—
35	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
36	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
37	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
38	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₉	C₂×C₈	1	2	yes	—
40	C₂²×C₄×C₁₉	M₁₆	1	2	yes	—
41	C₄²×C₁₉	—	4	2	no	—
42	C₄²×C₁₉	—	4	1	no	—
43	C₄²×C₁₉	—	4	1	no	—
44	C₄²×C₁₉	—	4	4	no	—
45	C₄²×C₁₉	—	4	4	no	—
46	C₂×C₁₉⋊C₈	—	4	4	no	—
47	C₂×C₁₉⋊C₈	—	4	4	no	—
48	C₂×C₁₉⋊C₈	—	4	4	no	—
49	#7017	—	19	1	no	maximal, Sylow
50	C₂³×C₁₉	—	2	2	no	—
51	C₂³×C₁₉	—	2	2	no	—
52	C₂³×C₁₉	#95	1	1	yes	characteristic
53	C₂³×C₁₉	C₂²⋊C₈	1	2	yes	—
54	C₂³×C₁₉	#99	1	2	yes	—
55	C₂³×C₁₉	#100	1	1	yes	characteristic
56	C₂³×C₁₉	#101	1	1	yes	characteristic
57	C₂×C₄×C₁₉	—	2	2	no	—
58	C₂×C₄×C₁₉	—	2	2	no	—
59	C₂×C₄×C₁₉	—	2	2	no	—
60	C₂×C₄×C₁₉	—	2	2	no	—
61	C₂×C₄×C₁₉	—	2	2	no	—
62	C₂×C₄×C₁₉	—	2	2	no	—
63	C₂×C₄×C₁₉	—	2	1	no	—
64	C₂×C₄×C₁₉	—	2	1	no	—
65	C₂×C₄×C₁₉	—	2	1	no	—
66	C₂×C₄×C₁₉	—	2	1	no	—
67	C₂×C₄×C₁₉	—	2	8	no	—
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	—	4	8	no	—
72	C₂×C₄×C₁₉	—	4	8	no	—
73	C₂×C₄×C₁₉	—	4	8	no	—
74	C₂×C₄×C₁₉	—	4	4	no	—
75	C₂×C₄×C₁₉	—	4	4	no	—
76	C₂×C₄×C₁₉	C₄×C₈	1	1	yes	characteristic
77	C₂×C₄×C₁₉	#97	1	1	yes	characteristic
78	C₂×C₄×C₁₉	#104	1	4	yes	—
79	C₂×C₄×C₁₉	C₄⋊C₈	1	2	yes	—
80	C₁₉⋊C₈	—	8	4	no	—
81	#1817	—	19	2	no	—
82	C₄×#95	D₁₉	1	1	yes	characteristic
83	C₂²×C₁₉	—	2	4	no	—
84	C₂²×C₁₉	—	2	2	no	—
85	C₂²×C₁₉	—	2	2	no	—
86	C₂²×C₁₉	—	2	2	no	—
87	C₂²×C₁₉	—	2	2	no	—
88	C₂²×C₁₉	#323	1	2	yes	—
89	C₂²×C₁₉	#324	1	2	yes	—
90	C₂²×C₁₉	#336	1	1	yes	characteristic
91	C₂²×C₁₉	H(3,ℤ/4)	1	2	yes	—
92	C₂²×C₁₉	#339	1	2	yes	—
93	C₂²×C₁₉	#342	1	1	yes	characteristic
94	C₂²×C₁₉	#343	1	1	yes	characteristic
95	C₄×C₁₉	—	2	1	no	—
96	C₄×C₁₉	—	2	1	no	—
97	C₄×C₁₉	—	2	2	no	—
98	C₄×C₁₉	—	2	4	no	—
99	C₄×C₁₉	—	4	8	no	—
100	C₄×C₁₉	—	4	2	no	—
101	C₂²×C₄²	Dic₁₉	1	2	yes	—
102	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
103	C₂×C₂²⋊C₈	—	38	2	no	—
104	C₂×#95	—	2	2	no	—
105	#336	—	19	4	no	—
106	#336	—	38	2	no	—
107	C₂×C₁₉	—	2	2	no	—
108	C₂×C₁₉	—	2	2	no	—
109	C₂×C₁₉	#1274	1	1	yes	characteristic
110	C₂×C₁₉	#1276	1	1	yes	characteristic
111	C₂×C₁₉	#1280	1	1	yes	characteristic
112	C₂×C₁₉	#1284	1	2	yes	—
113	C₂×C₁₉	#1285	1	2	yes	—
114	C₂³×C₄	—	2	2	no	—
115	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
116	C₂³×C₄	C₂²⋊D₁₉	1	2	yes	—
117	C₂²×C₈	—	38	4	no	—
118	C₂×C₄²	—	2	2	no	—
119	C₂×C₄²	—	4	2	no	—
120	C₂×C₄²	—	4	2	no	—
121	C₂×C₄²	—	4	2	no	—
122	C₂×C₄²	C₁₉⋊C₈	1	4	yes	complemented
123	C₂×C₄²	Dic₃₈	1	1	yes	characteristic
124	C₂×C₄²	C₄×D₁₉	1	2	yes	—
125	C₂×C₄²	D₇₆	1	1	yes	characteristic
126	C₂²⋊C₈	—	76	4	no	—
127	#95	—	2	8	no	—
128	C₁₉	#7017	1	1	yes	characteristic, complemented, Sylow
129	C₂⁴	C₂²⋊Dic₁₉	1	1	yes	characteristic
130	C₂²×C₄	—	2	1	no	—
131	C₂²×C₄	—	2	2	no	—
132	C₂²×C₄	—	2	1	no	—
133	C₂²×C₄	—	2	8	no	—
134	C₂²×C₄	—	2	4	no	—
135	C₂²×C₄	—	2	4	no	—
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	—	4	2	no	—
138	C₂²×C₄	—	4	4	no	—
139	C₂²×C₄	—	4	2	no	—
140	C₂²×C₄	—	4	2	no	—
141	C₂²×C₄	C₂×C₁₉⋊C₈	1	2	yes	—
142	C₂²×C₄	#64520	1	2	yes	—
143	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
144	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
145	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
146	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
147	C₂×C₈	—	76	4	no	—
148	C₂×C₈	—	76	4	no	—
149	C₂×C₈	—	76	4	no	—
150	C₄²	—	4	1	no	—
151	C₄²	—	4	2	no	—
152	C₄²	—	4	1	no	—
153	C₄²	—	4	4	no	—
154	C₄²	—	4	4	no	—
155	C₂³	—	2	2	no	—
156	C₂³	—	2	2	no	—
157	C₂³	C₂²⋊(C₁₉⋊C₈)	1	2	yes	—
158	C₂³	C₁₉⋊#95	1	1	yes	characteristic
159	C₂³	#10597290	1	1	yes	characteristic
160	C₂³	#10597291	1	2	yes	—
161	C₂³	#10597293	1	1	yes	characteristic
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	8	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	4	8	no	—
177	C₂×C₄	—	4	4	no	—
178	C₂×C₄	—	4	8	no	—
179	C₂×C₄	—	4	8	no	—
180	C₂×C₄	—	4	4	no	—
181	C₂×C₄	C₄×C₁₉⋊C₈	1	1	yes	characteristic
182	C₂×C₄	#10597260	1	1	yes	characteristic
183	C₂×C₄	C₄⋊(C₁₉⋊C₈)	1	2	yes	—
184	C₂×C₄	#10597262	1	4	yes	—
185	C₈	—	152	4	no	—
186	C₂²	—	2	2	no	—
187	C₂²	—	2	2	no	—
188	C₂²	—	2	4	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	2	no	—
191	C₂²	#11926268	1	2	yes	—
192	C₂²	#11926270	1	2	yes	—
193	C₂²	#11926299	1	1	yes	characteristic
194	C₂²	#11926300	1	2	yes	—
195	C₂²	#11926304	1	2	yes	—
196	C₂²	#11926311	1	1	yes	characteristic
197	C₂²	#11926313	1	1	yes	characteristic
198	C₄	—	2	1	no	—
199	C₄	—	2	1	no	—
200	C₄	—	2	2	no	—
201	C₄	—	2	4	no	—
202	C₄	—	4	8	no	—
203	C₄	—	4	2	no	—
204	C₂	—	2	2	no	—
205	C₂	—	2	2	no	—
206	C₂	#14561464	1	1	yes	characteristic
207	C₂	#14561468	1	1	yes	characteristic
208	C₂	#14561478	1	1	yes	characteristic
209	C₂	#14561488	1	2	yes	—
210	C₂	#14561491	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14563703

C₂

yes

maximal

C₄×C₁₉×#95

C₂

yes

maximal, characteristic

C₂×C₂²⋊(C₁₉⋊C₈)

—

#11926299

—

#11926299

C₄

yes

complemented

C₂²×C₄²×C₁₉

C₄

yes

—

C₂²×C₄²×C₁₉

C₂²

yes

characteristic

C₂×C₁₉×#95

—

C₂³×C₄×C₁₉

—

C₂³×C₄×C₁₉

C₂×C₄

yes

characteristic

C₂³×C₄×C₁₉

D₄

yes

—

C₂×C₄²×C₁₉

—

C₂×C₄²×C₁₉

—

C₂×C₄²×C₁₉

—

C₂×C₄²×C₁₉

—

C₂×C₄²×C₁₉

C₈

yes

complemented

C₂×C₄²×C₁₉

C₂×C₄

yes

—

C₂×C₄²×C₁₉

D₄

yes

characteristic

C₂×C₄²×C₁₉

Q₈

yes

characteristic

C₂²×C₁₉⋊C₈

—

C₁₉×#95

—

C₂²⋊(C₁₉⋊C₈)

—

C₂⁴×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₄²

yes

characteristic

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₉

C₄⋊C₄

yes

—

C₂²×C₄×C₁₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

C₂×C₈

yes

—

C₂²×C₄×C₁₉

M₁₆

yes

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

#7017

—

maximal, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

#95

yes

characteristic

C₂³×C₁₉

C₂²⋊C₈

yes

—

C₂³×C₁₉

#99

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

—

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

C₄×C₈

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

—

C₂×C₄×C₁₉

C₄⋊C₈

yes

—

C₁₉⋊C₈

—

#1817

—

C₄×#95

D₁₉

yes

characteristic

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#323

yes

—

C₂²×C₁₉

#324

yes

—

C₂²×C₁₉

#336

yes

characteristic

C₂²×C₁₉

H(3,ℤ/4)

yes

—

C₂²×C₁₉

#339

yes

—

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

100

C₄×C₁₉

—

101

C₂²×C₄²

Dic₁₉

yes

—

102

C₂²×C₄²

C₂×D₁₉

yes

characteristic

103

C₂×C₂²⋊C₈

—

104

C₂×#95

—

105

#336

—

106

#336

—

107

C₂×C₁₉

—

108

C₂×C₁₉

—

109

C₂×C₁₉

#1274

yes

characteristic

110

C₂×C₁₉

#1276

yes

characteristic

111

C₂×C₁₉

#1280

yes

characteristic

112

C₂×C₁₉

#1284

yes

—

113

C₂×C₁₉

#1285

yes

—

114

C₂³×C₄

—

115

C₂³×C₄

C₂×Dic₁₉

yes

characteristic

116

C₂³×C₄

C₂²⋊D₁₉

yes

—

117

C₂²×C₈

—

118

C₂×C₄²

—

119

C₂×C₄²

—

120

C₂×C₄²

—

121

C₂×C₄²

—

122

C₂×C₄²

C₁₉⋊C₈

yes

complemented

123

C₂×C₄²

Dic₃₈

yes

characteristic

124

C₂×C₄²

C₄×D₁₉

yes

—

125

C₂×C₄²

D₇₆

yes

characteristic

126

C₂²⋊C₈

—

127

#95

—

128

C₁₉

#7017

yes

characteristic, complemented, Sylow

129

C₂⁴

C₂²⋊Dic₁₉

yes

characteristic

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

C₂×C₁₉⋊C₈

yes

—

142

C₂²×C₄

#64520

yes

—

143

C₂²×C₄

C₄×Dic₁₉

yes

characteristic

144

C₂²×C₄

Dic₁₉⋊C₄

yes

—

145

C₂²×C₄

C₄⋊Dic₁₉

yes

characteristic

146

C₂²×C₄

(C₂×D₁₉)⋊C₄

yes

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₄²

—

151

C₄²

—

152

C₄²

—

153

C₄²

—

154

C₄²

—

155

C₂³

—

156

C₂³

—

157

C₂³

C₂²⋊(C₁₉⋊C₈)

yes

—

158

C₂³

C₁₉⋊#95

yes

characteristic

159

C₂³

#10597290

yes

characteristic

160

C₂³

#10597291

yes

—

161

C₂³

#10597293

yes

characteristic

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

C₄×C₁₉⋊C₈

yes

characteristic

182

C₂×C₄

#10597260

yes

characteristic

183

C₂×C₄

C₄⋊(C₁₉⋊C₈)

yes

—

184

C₂×C₄

#10597262

yes

—

185

C₈

—

152

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

189

C₂²

—

190

C₂²

—

191

C₂²

#11926268

yes

—

192

C₂²

#11926270

yes

—

193

C₂²

#11926299

yes

characteristic

194

C₂²

#11926300

yes

—

195

C₂²

#11926304

yes

—

196

C₂²

#11926311

yes

characteristic

197

C₂²

#11926313

yes

characteristic

198

C₄

—

199

C₄

—

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

—

204

C₂

—

205

C₂

—

206

C₂

#14561464

yes

characteristic

207

C₂

#14561468

yes

characteristic

208

C₂

#14561478

yes

characteristic

209

C₂

#14561488

yes

—

210

C₂

#14561491

yes

—

	subgroup	G/H
center	C₂³	C₁₉⋊#95
commutator	C₂×C₄×C₁₉	C₄×C₈
Frattini	C₂²×C₄²	C₂×D₁₉
Fitting	C₄×C₁₉×#95	C₂
socle	C₂³×C₁₉	#95
prime radical	C₂⁴×C₁₉	C₂²⋊C₄
cosocle	C₂²×C₄²×C₁₉	C₂²
hypercenter	C₄×#95	D₁₉
hypocenter	C₁₉	#7017
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₄×#95

subgroup

G/H

center

C₂³

C₁₉⋊#95

commutator

C₂×C₄×C₁₉

C₄×C₈

Frattini

C₂²×C₄²

C₂×D₁₉

Fitting

C₄×C₁₉×#95

C₂

socle

C₂³×C₁₉

#95

prime radical

C₂⁴×C₁₉

C₂²⋊C₄

cosocle

C₂²×C₄²×C₁₉

C₂²

hypercenter

C₄×#95

D₁₉

hypocenter

C₁₉

#7017

LCS

G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₄×#95

N	H ≅ G/N	type
5. #11926299	C₄	semidirect
16. C₂×C₄²×C₁₉	C₈	semidirect
122. C₂×C₄²	C₁₉⋊C₈	semidirect
128. C₁₉	#7017	semidirect

H ≅ G/N

type

5. #11926299

C₄

semidirect

16. C₂×C₄²×C₁₉

C₈

semidirect

122. C₂×C₄²

C₁₉⋊C₈

semidirect

128. C₁₉

#7017

semidirect

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

subgroup	index
#14563703	2
C₄×C₁₉×#95	2
#7017	19

subgroup

index

#14563703

C₄×C₁₉×#95

#7017

Subgroups — #18069071

Conjugacy classes of subgroups

Complements

Maximal subgroups