Group #18070806 — subgroups

Subgroups — #18070806

Order 4864.

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

normal subgroups only

This group has 636 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	#14563384	C₂	1	2	yes	maximal, complemented
2	#14561456	C₂	1	4	yes	maximal, complemented
3	C₄×C₁₉×#127	C₂	1	1	yes	maximal, characteristic
4	C₂×#10597260	—	2	2	no	—
5	#11926299	—	2	2	no	—
6	#11926299	C₂²	1	4	yes	—
7	C₂×C₁₉×#127	C₄	1	1	yes	characteristic
8	C₂×C₄×D₄×C₁₉	—	2	2	no	—
9	C₂×C₄×D₄×C₁₉	C₄	1	2	yes	—
10	C₁₉×#390	—	2	2	no	—
11	C₁₉×#390	C₂²	1	2	yes	—
12	C₁₉×#384	C₄	1	1	yes	characteristic
13	C₄³×C₁₉	C₂²	1	1	yes	characteristic
14	C₂²×D₄×C₁₉	—	2	2	no	—
15	C₂²×D₄×C₁₉	C₂×C₄	1	2	yes	—
16	C₂³×C₄×C₁₉	—	2	2	no	—
17	C₁₉×#127	—	2	4	no	—
18	C₄×D₄×C₁₉	—	4	8	no	—
19	C₄×D₄×C₁₉	—	4	4	no	—
20	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
21	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
22	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
23	C₂×C₂²⋊C₄×C₁₉	—	2	4	no	—
24	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₉	D₄	1	2	yes	—
27	C₂×C₄²×C₁₉	D₄	1	2	yes	—
28	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
29	C₂²×C₁₉⋊C₈	—	2	4	no	—
30	#10597260	—	2	8	no	—
31	C₂⁴×C₁₉	—	2	2	no	—
32	C₂×D₄×C₁₉	—	2	8	no	—
33	C₂×D₄×C₁₉	—	2	2	no	—
34	C₂×D₄×C₁₉	—	4	2	no	—
35	C₂×D₄×C₁₉	—	4	2	no	—
36	C₂×D₄×C₁₉	—	4	4	no	—
37	C₂×D₄×C₁₉	—	4	4	no	—
38	C₂×D₄×C₁₉	M₁₆	1	4	yes	—
39	C₂²×C₄×C₁₉	—	2	2	no	—
40	C₂²×C₄×C₁₉	—	4	8	no	—
41	C₂²×C₄×C₁₉	—	4	4	no	—
42	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
47	C₄⋊C₄×C₁₉	—	4	2	no	—
48	C₄⋊C₄×C₁₉	—	4	4	no	—
49	C₂²⋊C₄×C₁₉	—	4	4	no	—
50	C₂²⋊C₄×C₁₉	—	4	8	no	—
51	C₄²×C₁₉	—	2	2	no	—
52	C₄²×C₁₉	—	2	4	no	—
53	C₄²×C₁₉	—	2	2	no	—
54	C₄²×C₁₉	—	2	4	no	—
55	C₄²×C₁₉	—	4	1	no	—
56	C₂×C₁₉⋊C₈	—	2	8	no	—
57	C₂×C₁₉⋊C₈	—	4	4	no	—
58	C₂×C₁₉⋊C₈	—	4	4	no	—
59	#7637	—	19	1	no	maximal, Sylow
60	C₂³×C₁₉	—	2	4	no	—
61	C₂³×C₁₉	—	4	2	no	—
62	C₂³×C₁₉	—	4	2	no	—
63	C₂³×C₁₉	—	4	4	no	—
64	C₂³×C₁₉	—	4	4	no	—
65	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	D₄×C₁₉	—	4	4	no	—
67	D₄×C₁₉	—	4	4	no	—
68	D₄×C₁₉	—	4	16	no	—
69	C₂×C₄×C₁₉	—	2	2	no	—
70	C₂×C₄×C₁₉	—	2	2	no	—
71	C₂×C₄×C₁₉	—	2	1	no	—
72	C₂×C₄×C₁₉	—	2	1	no	—
73	C₂×C₄×C₁₉	—	2	2	no	—
74	C₂×C₄×C₁₉	—	2	2	no	—
75	C₂×C₄×C₁₉	—	2	2	no	—
76	C₂×C₄×C₁₉	—	2	1	no	—
77	C₂×C₄×C₁₉	—	2	2	no	—
78	C₂×C₄×C₁₉	—	4	2	no	—
79	C₂×C₄×C₁₉	—	4	4	no	—
80	C₂×C₄×C₁₉	—	8	8	no	—
81	C₂×C₄×C₁₉	#100	1	2	yes	—
82	C₂×C₄×C₁₉	#104	1	4	yes	—
83	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
84	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	4	yes	—
85	C₁₉⋊C₈	—	4	8	no	—
86	C₄×#127	D₁₉	1	1	yes	characteristic
87	#1745	—	19	2	no	—
88	#1270	—	19	4	no	—
89	C₂²×C₁₉	—	4	4	no	—
90	C₂²×C₁₉	—	4	8	no	—
91	C₂²×C₁₉	—	8	2	no	—
92	C₂²×C₁₉	—	8	4	no	—
93	C₂²×C₁₉	—	8	2	no	—
94	C₂²×C₁₉	#407	1	1	yes	characteristic
95	C₂²×C₁₉	#410	1	1	yes	characteristic
96	C₂²×C₁₉	C₂×#100	1	1	yes	characteristic
97	C₂²×C₁₉	#418	1	2	yes	—
98	C₂²×C₁₉	C₂×#104	1	2	yes	—
99	C₄×C₁₉	—	2	2	no	—
100	C₄×C₁₉	—	2	2	no	—
101	C₄×C₁₉	—	2	4	no	—
102	C₄×C₁₉	—	2	4	no	—
103	C₄×C₁₉	—	4	1	no	—
104	C₄×C₁₉	—	8	2	no	—
105	C₂×#127	Dic₁₉	1	1	yes	characteristic
106	C₂×C₄×D₄	—	2	2	no	—
107	C₂×C₄×D₄	Dic₁₉	1	2	yes	—
108	C₂×#97	—	38	2	no	—
109	#390	—	2	2	no	—
110	#390	C₂×D₁₉	1	2	yes	—
111	#384	Dic₁₉	1	1	yes	characteristic
112	C₄³	C₂×D₁₉	1	1	yes	characteristic
113	#336	—	19	4	no	—
114	#336	—	38	2	no	—
115	C₂×C₁₉	—	8	4	no	—
116	C₂×C₁₉	#1466	1	1	yes	characteristic
117	C₂×C₁₉	#1490	1	2	yes	—
118	C₂×C₁₉	#1505	1	2	yes	—
119	C₂×C₁₉	#1525	1	1	yes	characteristic
120	C₂×C₁₉	#1555	1	1	yes	characteristic
121	C₂²×D₄	—	2	2	no	—
122	C₂²×D₄	C₂×Dic₁₉	1	2	yes	—
123	C₂³×C₄	—	2	2	no	—
124	C₂²×C₈	—	38	4	no	—
125	#127	—	2	4	no	—
126	C₄×D₄	—	4	8	no	—
127	C₄×D₄	—	4	4	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
130	C₂×C₂²⋊C₄	—	2	2	no	—
131	C₂×C₂²⋊C₄	—	2	4	no	—
132	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
133	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
134	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
135	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
136	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
137	#97	—	38	8	no	—
138	C₁₉	#7637	1	1	yes	characteristic, complemented, Sylow
139	C₂⁴	—	2	2	no	—
140	C₂×D₄	—	2	8	no	—
141	C₂×D₄	—	2	2	no	—
142	C₂×D₄	—	4	2	no	—
143	C₂×D₄	—	4	2	no	—
144	C₂×D₄	—	4	4	no	—
145	C₂×D₄	—	4	4	no	—
146	C₂×D₄	#64520	1	4	yes	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	—	4	8	no	—
149	C₂²×C₄	—	4	4	no	—
150	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
151	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
152	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
153	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
154	C₂²×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
155	C₂×C₈	—	38	8	no	—
156	C₂×C₈	—	76	4	no	—
157	C₂×C₈	—	76	4	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	4	no	—
160	C₂²⋊C₄	—	4	4	no	—
161	C₂²⋊C₄	—	4	8	no	—
162	C₄²	—	2	4	no	—
163	C₄²	—	2	2	no	—
164	C₄²	—	2	4	no	—
165	C₄²	—	2	2	no	—
166	C₄²	—	4	1	no	—
167	C₂³	—	2	4	no	—
168	C₂³	—	4	4	no	—
169	C₂³	—	4	4	no	—
170	C₂³	—	4	2	no	—
171	C₂³	—	4	2	no	—
172	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
173	D₄	—	4	4	no	—
174	D₄	—	4	16	no	—
175	D₄	—	4	4	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	8	8	no	—
188	C₂×C₄	#10597290	1	2	yes	—
189	C₂×C₄	#10597294	1	4	yes	—
190	C₂×C₄	C₂×#64520	1	2	yes	—
191	C₂×C₄	#10597378	1	4	yes	—
192	C₈	—	76	8	no	—
193	C₂²	—	4	8	no	—
194	C₂²	—	4	4	no	—
195	C₂²	—	8	2	no	—
196	C₂²	—	8	4	no	—
197	C₂²	—	8	2	no	—
198	C₂²	#11926508	1	1	yes	characteristic
199	C₂²	#11926518	1	1	yes	characteristic
200	C₂²	C₂×#10597290	1	1	yes	characteristic
201	C₂²	#11926549	1	2	yes	—
202	C₂²	C₂×#10597294	1	2	yes	—
203	C₄	—	2	4	no	—
204	C₄	—	2	2	no	—
205	C₄	—	2	4	no	—
206	C₄	—	2	2	no	—
207	C₄	—	4	1	no	—
208	C₄	—	8	2	no	—
209	C₂	—	8	4	no	—
210	C₂	#14561976	1	1	yes	characteristic
211	C₂	#14562078	1	2	yes	—
212	C₂	#14562143	1	2	yes	—
213	C₂	#14562227	1	1	yes	characteristic
214	C₂	#14562349	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14563384

C₂

yes

maximal, complemented

#14561456

C₂

yes

maximal, complemented

C₄×C₁₉×#127

C₂

yes

maximal, characteristic

C₂×#10597260

—

#11926299

—

#11926299

C₂²

yes

—

C₂×C₁₉×#127

C₄

yes

characteristic

C₂×C₄×D₄×C₁₉

—

C₂×C₄×D₄×C₁₉

C₄

yes

—

C₁₉×#390

—

C₁₉×#390

C₂²

yes

—

C₁₉×#384

C₄

yes

characteristic

C₄³×C₁₉

C₂²

yes

characteristic

C₂²×D₄×C₁₉

—

C₂²×D₄×C₁₉

C₂×C₄

yes

—

C₂³×C₄×C₁₉

—

C₁₉×#127

—

C₄×D₄×C₁₉

—

C₄×D₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

C₂×C₄

yes

—

C₂×C₂²⋊C₄×C₁₉

—

C₂×C₂²⋊C₄×C₁₉

—

C₂×C₄²×C₁₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₉

D₄

yes

—

C₂×C₄²×C₁₉

D₄

yes

—

C₂×C₄²×C₁₉

C₂³

yes

characteristic

C₂²×C₁₉⋊C₈

—

#10597260

—

C₂⁴×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

M₁₆

yes

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₉

C₂×D₄

yes

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

#7637

—

maximal, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

C₂×C₂²⋊C₄

yes

characteristic

D₄×C₁₉

—

D₄×C₁₉

—

D₄×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

#100

yes

—

C₂×C₄×C₁₉

#104

yes

—

C₂×C₄×C₁₉

C₂×M₁₆

yes

—

C₂×C₄×C₁₉

AGL(1,ℤ/8)

yes

—

C₁₉⋊C₈

—

C₄×#127

D₁₉

yes

characteristic

#1745

—

#1270

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

—

C₂²×C₁₉

C₂×#104

yes

—

C₄×C₁₉

—

100

C₄×C₁₉

—

101

C₄×C₁₉

—

102

C₄×C₁₉

—

103

C₄×C₁₉

—

104

C₄×C₁₉

—

105

C₂×#127

Dic₁₉

yes

characteristic

106

C₂×C₄×D₄

—

107

C₂×C₄×D₄

Dic₁₉

yes

—

108

C₂×#97

—

109

#390

—

110

#390

C₂×D₁₉

yes

—

111

#384

Dic₁₉

yes

characteristic

112

C₄³

C₂×D₁₉

yes

characteristic

113

#336

—

114

#336

—

115

C₂×C₁₉

—

116

C₂×C₁₉

#1466

yes

characteristic

117

C₂×C₁₉

#1490

yes

—

118

C₂×C₁₉

#1505

yes

—

119

C₂×C₁₉

#1525

yes

characteristic

120

C₂×C₁₉

#1555

yes

characteristic

121

C₂²×D₄

—

122

C₂²×D₄

C₂×Dic₁₉

yes

—

123

C₂³×C₄

—

124

C₂²×C₈

—

125

#127

—

126

C₄×D₄

—

127

C₄×D₄

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

C₂×Dic₁₉

yes

—

130

C₂×C₂²⋊C₄

—

131

C₂×C₂²⋊C₄

—

132

C₂×C₄²

C₂×Dic₁₉

yes

characteristic

133

C₂×C₄²

C₂×Dic₁₉

yes

characteristic

134

C₂×C₄²

C₂²⋊D₁₉

yes

—

135

C₂×C₄²

C₂²⋊D₁₉

yes

—

136

C₂×C₄²

C₂²×D₁₉

yes

characteristic

137

#97

—

138

C₁₉

#7637

yes

characteristic, complemented, Sylow

139

C₂⁴

—

140

C₂×D₄

—

141

C₂×D₄

—

142

C₂×D₄

—

143

C₂×D₄

—

144

C₂×D₄

—

145

C₂×D₄

—

146

C₂×D₄

#64520

yes

—

147

C₂²×C₄

—

148

C₂²×C₄

—

149

C₂²×C₄

—

150

C₂²×C₄

C₂²⋊Dic₁₉

yes

characteristic

151

C₂²×C₄

C₂²⋊Dic₁₉

yes

characteristic

152

C₂²×C₄

C₂²⋊Dic₁₉

yes

—

153

C₂²×C₄

C₂²×Dic₁₉

yes

characteristic

154

C₂²×C₄

C₂×C₂²⋊D₁₉

yes

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₂²⋊C₄

—

161

C₂²⋊C₄

—

162

C₄²

—

163

C₄²

—

164

C₄²

—

165

C₄²

—

166

C₄²

—

167

C₂³

—

168

C₂³

—

169

C₂³

—

170

C₂³

—

171

C₂³

—

172

C₂³

C₂×C₂²⋊Dic₁₉

yes

characteristic

173

D₄

—

174

D₄

—

175

D₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

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

#10597290

yes

—

189

C₂×C₄

#10597294

yes

—

190

C₂×C₄

C₂×#64520

yes

—

191

C₂×C₄

#10597378

yes

—

192

C₈

—

193

C₂²

—

194

C₂²

—

195

C₂²

—

196

C₂²

—

197

C₂²

—

198

C₂²

#11926508

yes

characteristic

199

C₂²

#11926518

yes

characteristic

200

C₂²

C₂×#10597290

yes

characteristic

201

C₂²

#11926549

yes

—

202

C₂²

C₂×#10597294

yes

—

203

C₄

—

204

C₄

—

205

C₄

—

206

C₄

—

207

C₄

—

208

C₄

—

209

C₂

—

210

C₂

#14561976

yes

characteristic

211

C₂

#14562078

yes

—

212

C₂

#14562143

yes

—

213

C₂

#14562227

yes

characteristic

214

C₂

#14562349

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₉
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₄×C₁₉×#127	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂×C₁₉×#127	C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₄×#127	D₁₉
hypocenter	C₁₉	#7637
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#127

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₁₉

commutator

C₂²×C₄×C₁₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₉

Fitting

C₄×C₁₉×#127

C₂

socle

C₂³×C₁₉

C₂×C₂²⋊C₄

prime radical

C₂×C₁₉×#127

C₄

cosocle

C₂×C₄²×C₁₉

C₂³

D₁₉

C₁₉

G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#127

N	H ≅ G/N	type
1. #14563384	C₂	semidirect
2. #14561456	C₂	semidirect
138. C₁₉	#7637	semidirect

H ≅ G/N

type

1. #14563384

C₂

semidirect

2. #14561456

C₂

semidirect

138. C₁₉

#7637

semidirect

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

subgroup	index
#14563384	2
#14561456	2
C₄×C₁₉×#127	2
#7637	19

subgroup

index

#14563384

#14561456

C₄×C₁₉×#127

#7637

Subgroups — #18070806

Conjugacy classes of subgroups

Complements

Maximal subgroups