Group #18015397 — subgroups

Subgroups — C₁₉×#9438 #18015397

Order 4864.

452 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 126 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×#3138	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2986	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#1561	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#1545	C₂	1	2	yes	maximal
5	C₁₉×#1299	C₂	1	2	yes	maximal, complemented
6	C₁₉×#533	C₂²	1	1	yes	characteristic
7	C₂×C₁₉×#127	C₂²	1	1	yes	characteristic
8	C₁₉×#490	C₄	1	4	yes	—
9	C₁₉×#486	—	2	2	no	—
10	C₁₉×#446	—	2	2	no	—
11	C₈×Q₈×C₁₉	—	2	2	no	—
12	C₁₉×#433	—	2	2	no	—
13	C₁₉×#431	C₂²	1	2	yes	—
14	C₁₉×#423	C₂²	1	2	yes	—
15	C₁₉×#418	—	2	2	no	—
16	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#332	—	2	2	no	—
18	C₁₉×#331	—	2	2	no	—
19	C₁₉×#331	—	2	2	no	—
20	C₂²×D₄×C₁₉	—	2	1	no	—
21	C₂²×D₄×C₁₉	—	2	2	no	—
22	C₂×M₁₆×C₁₉	—	2	1	no	—
23	C₂×M₁₆×C₁₉	—	2	2	no	—
24	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
25	C₁₉×#127	—	2	1	no	—
26	C₁₉×#127	—	2	2	no	—
27	C₁₉×#127	C₂×C₄	1	2	yes	—
28	C₁₉×#125	C₂×C₄	1	2	yes	—
29	C₁₉×C₂²⋊Q₈	—	2	2	no	—
30	C₄×Q₈×C₁₉	—	2	2	no	—
31	C₁₉×#117	D₄	1	2	yes	—
32	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
33	C₁₉×C₄⋊C₈	—	2	4	no	—
34	C₁₉×C₄⋊C₈	—	2	2	no	—
35	C₁₉×C₄⋊C₈	—	2	2	no	—
36	C₁₉×C₄⋊C₈	D₄	1	2	yes	—
37	C₁₉×C₄⋊C₈	D₄	1	2	yes	—
38	C₁₉×#102	—	2	8	no	—
39	C₁₉×C₂²⋊C₈	—	2	2	no	—
40	C₁₉×#97	—	2	2	no	—
41	C₁₉×#97	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₁₉	—	2	2	no	—
43	C₄×C₈×C₁₉	—	2	1	no	—
44	C₄×C₈×C₁₉	—	2	2	no	—
45	C₂⁴×C₁₉	—	4	1	no	—
46	C₂×Q₈×C₁₉	—	2	2	no	—
47	C₂×D₄×C₁₉	—	2	2	no	—
48	C₂×D₄×C₁₉	—	2	4	no	—
49	C₂×D₄×C₁₉	—	4	4	no	—
50	C₂×D₄×C₁₉	—	4	1	no	—
51	C₂×D₄×C₁₉	—	4	2	no	—
52	C₂×D₄×C₁₉	—	8	2	no	—
53	C₂×D₄×C₁₉	—	8	2	no	—
54	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	#41	1	2	yes	—
56	M₁₆×C₁₉	—	4	2	no	—
57	M₁₆×C₁₉	—	8	2	no	—
58	C₂×C₈×C₁₉	—	2	2	no	—
59	C₂×C₈×C₁₉	—	2	2	no	—
60	C₂×C₈×C₁₉	—	2	2	no	—
61	C₂×C₈×C₁₉	—	4	2	no	—
62	C₄⋊C₄×C₁₉	—	2	4	no	—
63	C₄⋊C₄×C₁₉	—	2	2	no	—
64	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
65	C₂²⋊C₄×C₁₉	—	2	2	no	—
66	C₄²×C₁₉	—	2	2	no	—
67	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₁₉	C₂×D₄	1	2	yes	—
70	#9438	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂³×C₁₉	—	4	2	no	—
72	C₂³×C₁₉	—	8	2	no	—
73	C₂³×C₁₉	—	8	4	no	—
74	C₂³×C₁₉	#123	1	1	yes	characteristic
75	Q₈×C₁₉	—	4	2	no	—
76	Q₈×C₁₉	—	4	2	no	—
77	D₄×C₁₉	—	4	4	no	—
78	D₄×C₁₉	—	8	4	no	—
79	D₄×C₁₉	—	8	2	no	—
80	D₄×C₁₉	—	8	2	no	—
81	D₄×C₁₉	—	8	2	no	—
82	C₂×C₄×C₁₉	—	2	2	no	—
83	C₂×C₄×C₁₉	—	2	2	no	—
84	C₂×C₄×C₁₉	—	2	2	no	—
85	C₂×C₄×C₁₉	—	2	4	no	—
86	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
88	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
89	C₂×C₄×C₁₉	#121	1	2	yes	—
90	C₈×C₁₉	—	4	2	no	—
91	C₈×C₁₉	—	4	2	no	—
92	C₈×C₁₉	—	4	2	no	—
93	C₈×C₁₉	—	8	2	no	—
94	#3138	C₂×C₁₉	1	1	yes	characteristic
95	#2986	C₂×C₁₉	1	1	yes	characteristic, complemented
96	#1561	C₂×C₁₉	1	1	yes	characteristic, complemented
97	#1545	C₂×C₁₉	1	2	yes	—
98	#1299	C₂×C₁₉	1	2	yes	complemented
99	C₂²×C₁₉	—	2	1	no	—
100	C₂²×C₁₉	—	2	2	no	—
101	C₂²×C₁₉	—	8	2	no	—
102	C₂²×C₁₉	—	8	4	no	—
103	C₂²×C₁₉	—	16	4	no	—
104	C₂²×C₁₉	#386	1	1	yes	characteristic
105	C₄×C₁₉	—	2	1	no	—
106	C₄×C₁₉	—	2	2	no	—
107	C₄×C₁₉	—	4	4	no	—
108	C₄×C₁₉	#454	1	2	yes	—
109	C₄×C₁₉	#471	1	2	yes	—
110	C₄×C₁₉	#472	1	2	yes	—
111	#533	C₂²×C₁₉	1	1	yes	characteristic
112	C₂×#127	C₂²×C₁₉	1	1	yes	characteristic
113	#490	C₄×C₁₉	1	4	yes	—
114	#486	—	2	2	no	—
115	#446	—	2	2	no	—
116	C₈×Q₈	—	2	2	no	—
117	#433	—	2	2	no	—
118	#431	C₂²×C₁₉	1	2	yes	—
119	#423	C₂²×C₁₉	1	2	yes	—
120	#418	—	2	2	no	—
121	C₄×M₁₆	C₂²×C₁₉	1	1	yes	characteristic
122	#332	—	2	2	no	—
123	#331	—	2	2	no	—
124	#331	—	2	2	no	—
125	C₂×C₁₉	—	4	1	no	—
126	C₂×C₁₉	—	16	2	no	—
127	C₂×C₁₉	#1888	1	1	yes	characteristic
128	C₂×C₁₉	#1907	1	2	yes	—
129	C₂²×D₄	—	2	1	no	—
130	C₂²×D₄	—	2	2	no	—
131	C₂×M₁₆	—	2	1	no	—
132	C₂×M₁₆	—	2	2	no	—
133	C₄⋊Q₈	D₄×C₁₉	1	2	yes	—
134	#127	—	2	2	no	—
135	#127	—	2	1	no	—
136	#127	C₂×C₄×C₁₉	1	2	yes	—
137	#125	C₂×C₄×C₁₉	1	2	yes	—
138	C₂²⋊Q₈	—	2	2	no	—
139	C₄×Q₈	—	2	2	no	—
140	#117	D₄×C₁₉	1	2	yes	—
141	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
142	C₄⋊C₈	—	2	4	no	—
143	C₄⋊C₈	—	2	2	no	—
144	C₄⋊C₈	—	2	2	no	—
145	C₄⋊C₈	D₄×C₁₉	1	2	yes	—
146	C₄⋊C₈	D₄×C₁₉	1	2	yes	—
147	#102	—	2	8	no	—
148	C₂²⋊C₈	—	2	2	no	—
149	#97	—	2	2	no	—
150	#97	C₂×C₄×C₁₉	1	2	yes	—
151	C₄×C₈	—	2	2	no	—
152	C₄×C₈	—	2	1	no	—
153	C₄×C₈	—	2	2	no	—
154	C₁₉	#9438	1	1	yes	characteristic, direct product factor, Sylow
155	C₂⁴	—	4	1	no	—
156	C₂×Q₈	—	2	2	no	—
157	C₂×D₄	—	2	2	no	—
158	C₂×D₄	—	2	4	no	—
159	C₂×D₄	—	4	4	no	—
160	C₂×D₄	—	4	1	no	—
161	C₂×D₄	—	4	2	no	—
162	C₂×D₄	—	8	2	no	—
163	C₂×D₄	—	8	2	no	—
164	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
165	C₂²×C₄	#41×C₁₉	1	2	yes	—
166	M₁₆	—	4	2	no	—
167	M₁₆	—	8	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	4	2	no	—
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	C₂²⋊C₄×C₁₉	1	4	yes	—
175	C₂²⋊C₄	—	2	2	no	—
176	C₄²	—	2	2	no	—
177	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
178	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
179	C₄²	C₂×D₄×C₁₉	1	2	yes	—
180	C₂³	—	4	2	no	—
181	C₂³	—	8	2	no	—
182	C₂³	—	8	4	no	—
183	C₂³	C₁₉×#123	1	1	yes	characteristic
184	Q₈	—	4	2	no	—
185	Q₈	—	4	2	no	—
186	D₄	—	4	4	no	—
187	D₄	—	8	4	no	—
188	D₄	—	8	2	no	—
189	D₄	—	8	2	no	—
190	D₄	—	8	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
196	C₂×C₄	C₄×D₄×C₁₉	1	2	yes	—
197	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
198	C₂×C₄	C₁₉×#121	1	2	yes	—
199	C₈	—	4	2	no	—
200	C₈	—	4	2	no	—
201	C₈	—	4	2	no	—
202	C₈	—	8	2	no	—
203	C₂²	—	2	1	no	—
204	C₂²	—	2	2	no	—
205	C₂²	—	8	2	no	—
206	C₂²	—	8	4	no	—
207	C₂²	—	16	4	no	—
208	C₂²	C₁₉×#386	1	1	yes	characteristic
209	C₄	—	2	1	no	—
210	C₄	—	2	2	no	—
211	C₄	—	4	4	no	—
212	C₄	C₁₉×#454	1	2	yes	—
213	C₄	C₁₉×#471	1	2	yes	—
214	C₄	C₁₉×#472	1	2	yes	—
215	C₂	—	4	1	no	—
216	C₂	—	16	2	no	—
217	C₂	C₁₉×#1888	1	1	yes	characteristic
218	C₂	C₁₉×#1907	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#3138

C₂

yes

maximal, characteristic

C₁₉×#2986

C₂

yes

maximal, characteristic, complemented

C₁₉×#1561

C₂

yes

maximal, characteristic, complemented

C₁₉×#1545

C₂

yes

maximal

C₁₉×#1299

C₂

yes

maximal, complemented

C₁₉×#533

C₂²

yes

characteristic

C₂×C₁₉×#127

C₂²

yes

characteristic

C₁₉×#490

C₄

yes

—

C₁₉×#486

—

C₁₉×#446

—

C₈×Q₈×C₁₉

—

C₁₉×#433

—

C₁₉×#431

C₂²

yes

—

C₁₉×#423

C₂²

yes

—

C₁₉×#418

—

C₄×M₁₆×C₁₉

C₂²

yes

characteristic

C₁₉×#332

—

C₁₉×#331

—

C₁₉×#331

—

C₂²×D₄×C₁₉

—

C₂²×D₄×C₁₉

—

C₂×M₁₆×C₁₉

—

C₂×M₁₆×C₁₉

—

C₁₉×C₄⋊Q₈

D₄

yes

—

C₁₉×#127

—

C₁₉×#127

—

C₁₉×#127

C₂×C₄

yes

—

C₁₉×#125

C₂×C₄

yes

—

C₁₉×C₂²⋊Q₈

—

C₄×Q₈×C₁₉

—

C₁₉×#117

D₄

yes

—

C₂×C₄²×C₁₉

C₂³

yes

characteristic

C₁₉×C₄⋊C₈

—

C₁₉×C₄⋊C₈

—

C₁₉×C₄⋊C₈

—

C₁₉×C₄⋊C₈

D₄

yes

—

C₁₉×C₄⋊C₈

D₄

yes

—

C₁₉×#102

—

C₁₉×C₂²⋊C₈

—

C₁₉×#97

—

C₁₉×#97

C₂×C₄

yes

—

C₄×C₈×C₁₉

—

C₄×C₈×C₁₉

—

C₄×C₈×C₁₉

—

C₂⁴×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂²×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₉

#41

yes

—

M₁₆×C₁₉

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

C₂²⋊C₄

yes

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

C₂²×C₄

yes

characteristic

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₄²×C₁₉

C₂×D₄

yes

—

#9438

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

#123

yes

characteristic

Q₈×C₁₉

—

Q₈×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

D₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₉

C₄×D₄

yes

—

C₂×C₄×C₁₉

#120

yes

characteristic

C₂×C₄×C₁₉

#121

yes

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

#3138

C₂×C₁₉

yes

characteristic

#2986

C₂×C₁₉

yes

characteristic, complemented

#1561

C₂×C₁₉

yes

characteristic, complemented

#1545

C₂×C₁₉

yes

—

#1299

C₂×C₁₉

yes

complemented

C₂²×C₁₉

—

100

C₂²×C₁₉

—

101

C₂²×C₁₉

—

102

C₂²×C₁₉

—

103

C₂²×C₁₉

—

104

C₂²×C₁₉

#386

yes

characteristic

105

C₄×C₁₉

—

106

C₄×C₁₉

—

107

C₄×C₁₉

—

108

C₄×C₁₉

#454

yes

—

109

C₄×C₁₉

#471

yes

—

110

C₄×C₁₉

#472

yes

—

111

#533

C₂²×C₁₉

yes

characteristic

112

C₂×#127

C₂²×C₁₉

yes

characteristic

113

#490

C₄×C₁₉

yes

—

114

#486

—

115

#446

—

116

C₈×Q₈

—

117

#433

—

118

#431

C₂²×C₁₉

yes

—

119

#423

C₂²×C₁₉

yes

—

120

#418

—

121

C₄×M₁₆

C₂²×C₁₉

yes

characteristic

122

#332

—

123

#331

—

124

#331

—

125

C₂×C₁₉

—

126

C₂×C₁₉

—

127

C₂×C₁₉

#1888

yes

characteristic

128

C₂×C₁₉

#1907

yes

—

129

C₂²×D₄

—

130

C₂²×D₄

—

131

C₂×M₁₆

—

132

C₂×M₁₆

—

133

C₄⋊Q₈

D₄×C₁₉

yes

—

134

#127

—

135

#127

—

136

#127

C₂×C₄×C₁₉

yes

—

137

#125

C₂×C₄×C₁₉

yes

—

138

C₂²⋊Q₈

—

139

C₄×Q₈

—

140

#117

D₄×C₁₉

yes

—

141

C₂×C₄²

C₂³×C₁₉

yes

characteristic

142

C₄⋊C₈

—

143

C₄⋊C₈

—

144

C₄⋊C₈

—

145

C₄⋊C₈

D₄×C₁₉

yes

—

146

C₄⋊C₈

D₄×C₁₉

yes

—

147

#102

—

148

C₂²⋊C₈

—

149

#97

—

150

#97

C₂×C₄×C₁₉

yes

—

151

C₄×C₈

—

152

C₄×C₈

—

153

C₄×C₈

—

154

C₁₉

#9438

yes

characteristic, direct product factor, Sylow

155

C₂⁴

—

156

C₂×Q₈

—

157

C₂×D₄

—

158

C₂×D₄

—

159

C₂×D₄

—

160

C₂×D₄

—

161

C₂×D₄

—

162

C₂×D₄

—

163

C₂×D₄

—

164

C₂²×C₄

C₂×D₄×C₁₉

yes

characteristic

165

C₂²×C₄

#41×C₁₉

yes

—

166

M₁₆

—

167

M₁₆

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

—

171

C₂×C₈

—

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₄⋊C₄

C₂²⋊C₄×C₁₉

yes

—

175

C₂²⋊C₄

—

176

C₄²

—

177

C₄²

C₂²×C₄×C₁₉

yes

characteristic

178

C₄²

C₂×D₄×C₁₉

yes

characteristic

179

C₄²

C₂×D₄×C₁₉

yes

—

180

C₂³

—

181

C₂³

—

182

C₂³

—

183

C₂³

C₁₉×#123

yes

characteristic

184

Q₈

—

185

Q₈

—

186

D₄

—

187

D₄

—

188

D₄

—

189

D₄

—

190

D₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

C₂×C₂²⋊C₄×C₁₉

yes

characteristic

196

C₂×C₄

C₄×D₄×C₁₉

yes

—

197

C₂×C₄

C₁₉×#120

yes

characteristic

198

C₂×C₄

C₁₉×#121

yes

—

199

C₈

—

200

C₈

—

201

C₈

—

202

C₈

—

203

C₂²

—

204

C₂²

—

205

C₂²

—

206

C₂²

—

207

C₂²

—

208

C₂²

C₁₉×#386

yes

characteristic

209

C₄

—

210

C₄

—

211

C₄

—

212

C₄

C₁₉×#454

yes

—

213

C₄

C₁₉×#471

yes

—

214

C₄

C₁₉×#472

yes

—

215

C₂

—

216

C₂

—

217

C₂

C₁₉×#1888

yes

characteristic

218

C₂

C₁₉×#1907

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₉	#386
commutator	C₄²	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
prime radical	C₂×C₁₉×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₉

#386

commutator

C₄²

C₂²×C₄×C₁₉

Frattini, cosocle

C₂×C₄²

C₂³×C₁₉

prime radical

C₂×C₁₉×#127

C₂²

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

N	H ≅ G/N	type
2. C₁₉×#2986	C₂	semidirect
3. C₁₉×#1561	C₂	semidirect
5. C₁₉×#1299	C₂	semidirect
70. #9438	C₁₉	direct (N × H)
95. #2986	C₂×C₁₉	semidirect
96. #1561	C₂×C₁₉	semidirect
98. #1299	C₂×C₁₉	semidirect
154. C₁₉	#9438	direct (N × H)

H ≅ G/N

type

2. C₁₉×#2986

C₂

semidirect

3. C₁₉×#1561

C₂

semidirect

5. C₁₉×#1299

C₂

semidirect

70. #9438

C₁₉

direct (N × H)

95. #2986

C₂×C₁₉

semidirect

96. #1561

C₂×C₁₉

semidirect

98. #1299

C₂×C₁₉

semidirect

154. C₁₉

#9438

direct (N × H)

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

subgroup	index
C₁₉×#3138	2
C₁₉×#2986	2
C₁₉×#1561	2
C₁₉×#1545	2
C₁₉×#1299	2
#9438	19

subgroup

index

C₁₉×#3138

C₁₉×#2986

C₁₉×#1561

C₁₉×#1545

C₁₉×#1299

#9438

Subgroups — C₁₉×#9438 #18015397

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#9438 #18015397

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#9438 #18015397