Group #18017475 — subgroups

Subgroups — C₁₉×#11516 #18017475

Order 4864.

418 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 110 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	C₁₉×#2317	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#1464	C₂	1	2	yes	maximal
3	C₁₉×#1325	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₉×#117	C₄	1	1	yes	characteristic
5	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
6	C₂²×C₄⋊C₄×C₁₉	C₄	1	1	yes	characteristic
7	C₁₉×#432	—	2	2	no	—
8	C₁₉×#389	—	2	2	no	—
9	C₁₉×#385	—	2	2	no	—
10	C₁₉×#384	C₂²	1	2	yes	—
11	C₁₉×#382	C₄	1	2	yes	—
12	C₁₉×#324	C₂²	1	4	yes	—
13	C₁₉×#323	—	2	2	no	—
14	C₂³×C₄×C₁₉	—	2	1	no	—
15	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
16	C₁₉×#117	—	2	4	no	—
17	C₁₉×#117	—	4	2	no	—
18	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
19	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
20	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
21	C₂×C₄⋊C₄×C₁₉	—	4	2	no	—
22	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
23	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
26	C₂×C₂²⋊C₄×C₁₉	D₄	1	2	yes	—
27	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₁₉	D₄	1	2	yes	—
29	C₁₉×C₄⋊C₈	—	4	2	no	—
30	C₁₉×C₂²⋊C₈	—	2	4	no	—
31	C₁₉×#97	—	4	2	no	—
32	C₁₉×#95	—	2	4	no	—
33	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₉	—	2	2	no	—
35	C₂²×C₄×C₁₉	—	2	2	no	—
36	C₂²×C₄×C₁₉	—	2	2	no	—
37	C₂²×C₄×C₁₉	—	2	2	no	—
38	C₂²×C₄×C₁₉	—	2	2	no	—
39	C₂²×C₄×C₁₉	—	4	2	no	—
40	C₂²×C₄×C₁₉	—	8	2	no	—
41	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
45	C₂×C₈×C₁₉	—	4	4	no	—
46	C₄⋊C₄×C₁₉	—	2	4	no	—
47	C₄⋊C₄×C₁₉	—	4	1	no	—
48	C₄⋊C₄×C₁₉	—	4	1	no	—
49	C₄⋊C₄×C₁₉	—	4	2	no	—
50	C₄⋊C₄×C₁₉	—	4	2	no	—
51	C₄⋊C₄×C₁₉	—	8	4	no	—
52	C₂²⋊C₄×C₁₉	—	2	4	no	—
53	C₂²⋊C₄×C₁₉	—	4	2	no	—
54	C₄²×C₁₉	—	2	2	no	—
55	C₄²×C₁₉	—	2	2	no	—
56	C₄²×C₁₉	—	2	2	no	—
57	C₄²×C₁₉	M₁₆	1	4	yes	—
58	#11516	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂³×C₁₉	—	2	1	no	—
60	C₂³×C₁₉	—	2	1	no	—
61	C₂³×C₁₉	—	4	1	no	—
62	C₂³×C₁₉	—	4	1	no	—
63	C₂³×C₁₉	#101	1	2	yes	—
64	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₉	—	2	8	no	—
66	C₂×C₄×C₁₉	—	2	1	no	—
67	C₂×C₄×C₁₉	—	2	1	no	—
68	C₂×C₄×C₁₉	—	2	1	no	—
69	C₂×C₄×C₁₉	—	2	1	no	—
70	C₂×C₄×C₁₉	—	2	4	no	—
71	C₂×C₄×C₁₉	—	2	2	no	—
72	C₂×C₄×C₁₉	—	2	2	no	—
73	C₂×C₄×C₁₉	—	4	2	no	—
74	C₂×C₄×C₁₉	—	4	2	no	—
75	C₂×C₄×C₁₉	—	4	2	no	—
76	C₂×C₄×C₁₉	—	4	2	no	—
77	C₂×C₄×C₁₉	—	4	4	no	—
78	C₂×C₄×C₁₉	—	4	2	no	—
79	C₂×C₄×C₁₉	—	8	4	no	—
80	C₂×C₄×C₁₉	#99	1	2	yes	—
81	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
82	C₈×C₁₉	—	8	4	no	—
83	#2317	C₂×C₁₉	1	1	yes	characteristic
84	#1464	C₂×C₁₉	1	2	yes	—
85	#1325	C₂×C₁₉	1	4	yes	complemented
86	C₂²×C₁₉	—	2	1	no	—
87	C₂²×C₁₉	—	2	1	no	—
88	C₂²×C₁₉	—	2	2	no	—
89	C₂²×C₁₉	—	4	2	no	—
90	C₂²×C₁₉	—	4	2	no	—
91	C₂²×C₁₉	—	4	1	no	—
92	C₂²×C₁₉	—	4	1	no	—
93	C₂²×C₁₉	#407	1	1	yes	characteristic
94	C₂²×C₁₉	C₂×#99	1	1	yes	characteristic
95	C₂²×C₁₉	C₂×#101	1	1	yes	characteristic
96	C₄×C₁₉	—	2	2	no	—
97	C₄×C₁₉	—	2	2	no	—
98	C₄×C₁₉	—	4	4	no	—
99	C₄×C₁₉	—	4	2	no	—
100	C₄×C₁₉	—	4	2	no	—
101	C₄×C₁₉	—	8	2	no	—
102	C₂×#117	C₄×C₁₉	1	1	yes	characteristic
103	C₂×#117	C₂²×C₁₉	1	1	yes	characteristic
104	C₂²×C₄⋊C₄	C₄×C₁₉	1	1	yes	characteristic
105	#432	—	2	2	no	—
106	#389	—	2	2	no	—
107	#385	—	2	2	no	—
108	#384	C₂²×C₁₉	1	2	yes	—
109	#382	C₄×C₁₉	1	2	yes	—
110	#324	C₂²×C₁₉	1	4	yes	—
111	#323	—	2	2	no	—
112	C₂×C₁₉	—	2	1	no	—
113	C₂×C₁₉	—	2	1	no	—
114	C₂×C₁₉	—	4	2	no	—
115	C₂×C₁₉	#1463	1	1	yes	characteristic
116	C₂×C₁₉	#2128	1	1	yes	characteristic
117	C₂×C₁₉	#2135	1	1	yes	characteristic
118	C₂³×C₄	—	2	1	no	—
119	C₂³×C₄	C₂×C₄×C₁₉	1	1	yes	characteristic
120	#117	—	2	4	no	—
121	#117	—	4	2	no	—
122	C₂×C₄⋊C₄	—	2	1	no	—
123	C₂×C₄⋊C₄	—	2	1	no	—
124	C₂×C₄⋊C₄	—	2	4	no	—
125	C₂×C₄⋊C₄	—	4	2	no	—
126	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
127	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	1	yes	characteristic
128	C₂×C₄⋊C₄	C₂³×C₁₉	1	1	yes	characteristic
129	C₂×C₂²⋊C₄	—	2	1	no	—
130	C₂×C₂²⋊C₄	D₄×C₁₉	1	2	yes	—
131	C₂×C₄²	C₂×C₄×C₁₉	1	2	yes	—
132	C₂×C₄²	D₄×C₁₉	1	2	yes	—
133	C₄⋊C₈	—	4	2	no	—
134	C₂²⋊C₈	—	2	4	no	—
135	#97	—	4	2	no	—
136	#95	—	2	4	no	—
137	C₁₉	#11516	1	1	yes	characteristic, direct product factor, Sylow
138	C₂⁴	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
139	C₂²×C₄	—	2	2	no	—
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	—	2	2	no	—
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	—	4	2	no	—
145	C₂²×C₄	—	8	2	no	—
146	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
147	C₂²×C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
148	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
149	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
150	C₂×C₈	—	4	4	no	—
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	—	4	1	no	—
153	C₄⋊C₄	—	4	1	no	—
154	C₄⋊C₄	—	4	2	no	—
155	C₄⋊C₄	—	4	2	no	—
156	C₄⋊C₄	—	8	4	no	—
157	C₂²⋊C₄	—	2	4	no	—
158	C₂²⋊C₄	—	4	2	no	—
159	C₄²	—	2	2	no	—
160	C₄²	—	2	2	no	—
161	C₄²	—	2	2	no	—
162	C₄²	M₁₆×C₁₉	1	4	yes	—
163	C₂³	—	2	1	no	—
164	C₂³	—	2	1	no	—
165	C₂³	—	4	1	no	—
166	C₂³	—	4	1	no	—
167	C₂³	C₁₉×#101	1	2	yes	—
168	C₂³	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
169	C₂×C₄	—	2	8	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	8	4	no	—
184	C₂×C₄	C₁₉×#99	1	2	yes	—
185	C₂×C₄	C₂×M₁₆×C₁₉	1	2	yes	—
186	C₈	—	8	4	no	—
187	C₂²	—	2	1	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	4	2	no	—
191	C₂²	—	4	2	no	—
192	C₂²	—	4	1	no	—
193	C₂²	—	4	1	no	—
194	C₂²	C₁₉×#407	1	1	yes	characteristic
195	C₂²	C₂×C₁₉×#99	1	1	yes	characteristic
196	C₂²	C₂×C₁₉×#101	1	1	yes	characteristic
197	C₄	—	2	2	no	—
198	C₄	—	2	2	no	—
199	C₄	—	4	4	no	—
200	C₄	—	4	2	no	—
201	C₄	—	4	2	no	—
202	C₄	—	8	2	no	—
203	C₂	—	2	1	no	—
204	C₂	—	2	1	no	—
205	C₂	—	4	2	no	—
206	C₂	C₁₉×#1463	1	1	yes	characteristic
207	C₂	C₁₉×#2128	1	1	yes	characteristic
208	C₂	C₁₉×#2135	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#2317

C₂

yes

maximal, characteristic

C₁₉×#1464

C₂

yes

maximal

C₁₉×#1325

C₂

yes

maximal, complemented

C₂×C₁₉×#117

C₄

yes

characteristic

C₂×C₁₉×#117

C₂²

yes

characteristic

C₂²×C₄⋊C₄×C₁₉

C₄

yes

characteristic

C₁₉×#432

—

C₁₉×#389

—

C₁₉×#385

—

C₁₉×#384

C₂²

yes

—

C₁₉×#382

C₄

yes

—

C₁₉×#324

C₂²

yes

—

C₁₉×#323

—

C₂³×C₄×C₁₉

—

C₂³×C₄×C₁₉

C₂×C₄

yes

characteristic

C₁₉×#117

—

C₁₉×#117

—

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₁₉

C₂×C₄

yes

characteristic

C₂×C₄⋊C₄×C₁₉

C₂³

yes

characteristic

C₂×C₂²⋊C₄×C₁₉

—

C₂×C₂²⋊C₄×C₁₉

D₄

yes

—

C₂×C₄²×C₁₉

C₂×C₄

yes

—

C₂×C₄²×C₁₉

D₄

yes

—

C₁₉×C₄⋊C₈

—

C₁₉×C₂²⋊C₈

—

C₁₉×#97

—

C₁₉×#95

—

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

yes

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₉

C₂×D₄

yes

—

C₂×C₈×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

M₁₆

yes

—

#11516

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

#101

yes

—

C₂³×C₁₉

C₂×C₂²⋊C₄

yes

characteristic

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

#99

yes

—

C₂×C₄×C₁₉

C₂×M₁₆

yes

—

C₈×C₁₉

—

#2317

C₂×C₁₉

yes

characteristic

#1464

C₂×C₁₉

yes

—

#1325

C₂×C₁₉

yes

complemented

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#407

yes

characteristic

C₂²×C₁₉

C₂×#99

yes

characteristic

C₂²×C₁₉

C₂×#101

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

100

C₄×C₁₉

—

101

C₄×C₁₉

—

102

C₂×#117

C₄×C₁₉

yes

characteristic

103

C₂×#117

C₂²×C₁₉

yes

characteristic

104

C₂²×C₄⋊C₄

C₄×C₁₉

yes

characteristic

105

#432

—

106

#389

—

107

#385

—

108

#384

C₂²×C₁₉

yes

—

109

#382

C₄×C₁₉

yes

—

110

#324

C₂²×C₁₉

yes

—

111

#323

—

112

C₂×C₁₉

—

113

C₂×C₁₉

—

114

C₂×C₁₉

—

115

C₂×C₁₉

#1463

yes

characteristic

116

C₂×C₁₉

#2128

yes

characteristic

117

C₂×C₁₉

#2135

yes

characteristic

118

C₂³×C₄

—

119

C₂³×C₄

C₂×C₄×C₁₉

yes

characteristic

120

#117

—

121

#117

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄⋊C₄

—

126

C₂×C₄⋊C₄

C₂×C₄×C₁₉

yes

—

127

C₂×C₄⋊C₄

C₂×C₄×C₁₉

yes

characteristic

128

C₂×C₄⋊C₄

C₂³×C₁₉

yes

characteristic

129

C₂×C₂²⋊C₄

—

130

C₂×C₂²⋊C₄

D₄×C₁₉

yes

—

131

C₂×C₄²

C₂×C₄×C₁₉

yes

—

132

C₂×C₄²

D₄×C₁₉

yes

—

133

C₄⋊C₈

—

134

C₂²⋊C₈

—

135

#97

—

136

#95

—

137

C₁₉

#11516

yes

characteristic, direct product factor, Sylow

138

C₂⁴

C₂²⋊C₄×C₁₉

yes

characteristic

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

—

146

C₂²×C₄

C₂²⋊C₄×C₁₉

yes

characteristic

147

C₂²×C₄

C₂²⋊C₄×C₁₉

yes

—

148

C₂²×C₄

C₂²×C₄×C₁₉

yes

characteristic

149

C₂²×C₄

C₂×D₄×C₁₉

yes

—

150

C₂×C₈

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₂²⋊C₄

—

158

C₂²⋊C₄

—

159

C₄²

—

160

C₄²

—

161

C₄²

—

162

C₄²

M₁₆×C₁₉

yes

—

163

C₂³

—

164

C₂³

—

165

C₂³

—

166

C₂³

—

167

C₂³

C₁₉×#101

yes

—

168

C₂³

C₂×C₂²⋊C₄×C₁₉

yes

characteristic

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

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

C₁₉×#99

yes

—

185

C₂×C₄

C₂×M₁₆×C₁₉

yes

—

186

C₈

—

187

C₂²

—

188

C₂²

—

189

C₂²

—

190

C₂²

—

191

C₂²

—

192

C₂²

—

193

C₂²

—

194

C₂²

C₁₉×#407

yes

characteristic

195

C₂²

C₂×C₁₉×#99

yes

characteristic

196

C₂²

C₂×C₁₉×#101

yes

characteristic

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

—

201

C₄

—

202

C₄

—

203

C₂

—

204

C₂

—

205

C₂

—

206

C₂

C₁₉×#1463

yes

characteristic

207

C₂

C₁₉×#2128

yes

characteristic

208

C₂

C₁₉×#2135

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×#99
commutator	C₂²×C₄	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×C₄⋊C₄	C₂³×C₁₉
prime radical	C₂⁴×C₁₉	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂×C₁₉×#117 ◁ G

subgroup

G/H

center, socle

C₂²×C₁₉

C₂×#99

commutator

C₂²×C₄

C₂²×C₄×C₁₉

Frattini, cosocle

C₂×C₄⋊C₄

C₂³×C₁₉

prime radical

C₂⁴×C₁₉

C₂²⋊C₄

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂×C₁₉×#117 ◁ G

N	H ≅ G/N	type
3. C₁₉×#1325	C₂	semidirect
58. #11516	C₁₉	direct (N × H)
85. #1325	C₂×C₁₉	semidirect
137. C₁₉	#11516	direct (N × H)

H ≅ G/N

type

3. C₁₉×#1325

C₂

semidirect

58. #11516

C₁₉

direct (N × H)

85. #1325

C₂×C₁₉

semidirect

137. C₁₉

#11516

direct (N × H)

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

subgroup	index
C₁₉×#2317	2
C₁₉×#1464	2
C₁₉×#1325	2
#11516	19

subgroup

index

C₁₉×#2317

C₁₉×#1464

C₁₉×#1325

#11516

Subgroups — C₁₉×#11516 #18017475

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#11516 #18017475

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#11516 #18017475