Group #18013003 — subgroups

Subgroups — C₁₉×#7044 #18013003

Order 4864.

352 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 114 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×#1817	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#1797	C₂	1	2	yes	maximal
3	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
4	C₂×C₁₉×C₄⋊C₈	—	2	2	no	—
5	C₂×C₁₉×C₂²⋊C₈	—	2	2	no	—
6	C₂×C₁₉×C₂²⋊C₈	C₄	1	2	yes	—
7	C₁₉×#336	—	2	2	no	—
8	C₁₉×#336	C₄	1	4	yes	—
9	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
10	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
11	C₂³×C₄×C₁₉	Q₈	1	1	yes	characteristic
12	C₂²×C₈×C₁₉	—	2	4	no	—
13	C₂²×C₈×C₁₉	—	2	2	no	—
14	C₂×C₄²×C₁₉	—	4	2	no	—
15	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
16	C₂×C₄²×C₁₉	D₄	1	2	yes	—
17	C₁₉×C₄⋊C₈	—	4	4	no	—
18	C₁₉×C₂²⋊C₈	—	4	2	no	—
19	C₁₉×C₂²⋊C₈	—	4	2	no	—
20	C₁₉×C₂²⋊C₈	—	4	2	no	—
21	C₂⁴×C₁₉	C₄⋊C₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₉	—	2	2	no	—
23	C₂²×C₄×C₁₉	—	2	4	no	—
24	C₂²×C₄×C₁₉	—	4	1	no	—
25	C₂²×C₄×C₁₉	—	4	1	no	—
26	C₂²×C₄×C₁₉	—	4	2	no	—
27	C₂²×C₄×C₁₉	—	4	1	no	—
28	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
29	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
31	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
32	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
33	C₂²×C₄×C₁₉	SD₁₆	1	2	yes	—
34	C₂²×C₄×C₁₉	Q₁₆	1	1	yes	characteristic
35	C₂×C₈×C₁₉	—	4	4	no	—
36	C₂×C₈×C₁₉	—	4	4	no	—
37	C₂×C₈×C₁₉	—	4	4	no	—
38	C₂×C₈×C₁₉	—	4	2	no	—
39	C₂×C₈×C₁₉	—	4	4	no	—
40	C₄²×C₁₉	—	2	4	no	—
41	C₄²×C₁₉	—	4	2	no	—
42	#7044	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂³×C₁₉	—	2	1	no	—
44	C₂³×C₁₉	—	2	2	no	—
45	C₂³×C₁₉	—	2	1	no	—
46	C₂³×C₁₉	—	2	2	no	—
47	C₂³×C₁₉	#95	1	1	yes	characteristic
48	C₂³×C₁₉	#106	1	1	yes	characteristic
49	C₂³×C₁₉	#107	1	1	yes	characteristic
50	C₂×C₄×C₁₉	—	2	2	no	—
51	C₂×C₄×C₁₉	—	2	2	no	—
52	C₂×C₄×C₁₉	—	2	2	no	—
53	C₂×C₄×C₁₉	—	2	2	no	—
54	C₂×C₄×C₁₉	—	2	2	no	—
55	C₂×C₄×C₁₉	—	2	1	no	—
56	C₂×C₄×C₁₉	—	2	2	no	—
57	C₂×C₄×C₁₉	—	4	1	no	—
58	C₂×C₄×C₁₉	—	4	2	no	—
59	C₂×C₄×C₁₉	—	4	1	no	—
60	C₂×C₄×C₁₉	—	4	4	no	—
61	C₂×C₄×C₁₉	—	4	4	no	—
62	C₂×C₄×C₁₉	#99	1	2	yes	—
63	C₂×C₄×C₁₉	#102	1	2	yes	—
64	C₂×C₄×C₁₉	#103	1	2	yes	—
65	C₂×C₄×C₁₉	#108	1	4	yes	—
66	C₈×C₁₉	—	8	4	no	—
67	C₈×C₁₉	—	8	2	no	—
68	#1817	C₂×C₁₉	1	1	yes	characteristic
69	#1797	C₂×C₁₉	1	2	yes	—
70	C₂²×C₁₉	—	2	1	no	—
71	C₂²×C₁₉	—	2	1	no	—
72	C₂²×C₁₉	—	2	2	no	—
73	C₂²×C₁₉	—	2	2	no	—
74	C₂²×C₁₉	—	4	2	no	—
75	C₂²×C₁₉	—	4	1	no	—
76	C₂²×C₁₉	—	4	1	no	—
77	C₂²×C₁₉	H(3,ℤ/4)	1	1	yes	characteristic
78	C₂²×C₁₉	#340	1	1	yes	characteristic
79	C₂²×C₁₉	#341	1	2	yes	—
80	C₂²×C₁₉	#342	1	1	yes	characteristic
81	C₂²×C₁₉	#344	1	2	yes	—
82	C₄×C₁₉	—	2	2	no	—
83	C₄×C₁₉	—	2	2	no	—
84	C₄×C₁₉	—	2	4	no	—
85	C₄×C₁₉	—	2	2	no	—
86	C₄×C₁₉	—	4	1	no	—
87	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
88	C₂×C₄⋊C₈	—	2	2	no	—
89	C₂×C₂²⋊C₈	—	2	2	no	—
90	C₂×C₂²⋊C₈	C₄×C₁₉	1	2	yes	—
91	#336	—	2	2	no	—
92	#336	C₄×C₁₉	1	4	yes	—
93	C₂×C₁₉	—	4	1	no	—
94	C₂×C₁₉	—	4	1	no	—
95	C₂×C₁₉	#1287	1	2	yes	—
96	C₂×C₁₉	#1289	1	1	yes	characteristic
97	C₂×C₁₉	#1298	1	1	yes	characteristic
98	C₂×C₁₉	#1301	1	2	yes	—
99	C₂×C₁₉	#1306	1	1	yes	characteristic
100	C₂³×C₄	C₂×C₄×C₁₉	1	1	yes	characteristic
101	C₂³×C₄	D₄×C₁₉	1	1	yes	characteristic
102	C₂³×C₄	Q₈×C₁₉	1	1	yes	characteristic
103	C₂²×C₈	—	2	2	no	—
104	C₂²×C₈	—	2	4	no	—
105	C₂×C₄²	—	4	2	no	—
106	C₂×C₄²	C₂×C₄×C₁₉	1	2	yes	—
107	C₂×C₄²	D₄×C₁₉	1	2	yes	—
108	C₄⋊C₈	—	4	4	no	—
109	C₂²⋊C₈	—	4	2	no	—
110	C₂²⋊C₈	—	4	2	no	—
111	C₂²⋊C₈	—	4	2	no	—
112	C₁₉	#7044	1	1	yes	characteristic, direct product factor, Sylow
113	C₂⁴	C₄⋊C₄×C₁₉	1	1	yes	characteristic
114	C₂²×C₄	—	2	2	no	—
115	C₂²×C₄	—	2	4	no	—
116	C₂²×C₄	—	4	1	no	—
117	C₂²×C₄	—	4	1	no	—
118	C₂²×C₄	—	4	2	no	—
119	C₂²×C₄	—	4	1	no	—
120	C₂²×C₄	C₄²×C₁₉	1	1	yes	characteristic
121	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
122	C₂²×C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
123	C₂²×C₄	C₄⋊C₄×C₁₉	1	2	yes	—
124	C₂²×C₄	D₈×C₁₉	1	1	yes	characteristic
125	C₂²×C₄	SD₁₆×C₁₉	1	2	yes	—
126	C₂²×C₄	Q₁₆×C₁₉	1	1	yes	characteristic
127	C₂×C₈	—	4	4	no	—
128	C₂×C₈	—	4	4	no	—
129	C₂×C₈	—	4	4	no	—
130	C₂×C₈	—	4	2	no	—
131	C₂×C₈	—	4	4	no	—
132	C₄²	—	2	4	no	—
133	C₄²	—	4	2	no	—
134	C₂³	—	2	2	no	—
135	C₂³	—	2	1	no	—
136	C₂³	—	2	2	no	—
137	C₂³	—	2	1	no	—
138	C₂³	C₁₉×#95	1	1	yes	characteristic
139	C₂³	C₁₉×#106	1	1	yes	characteristic
140	C₂³	C₁₉×#107	1	1	yes	characteristic
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	4	4	no	—
152	C₂×C₄	—	4	4	no	—
153	C₂×C₄	C₁₉×#99	1	2	yes	—
154	C₂×C₄	C₁₉×#102	1	2	yes	—
155	C₂×C₄	C₁₉×#103	1	2	yes	—
156	C₂×C₄	C₁₉×#108	1	4	yes	—
157	C₈	—	8	4	no	—
158	C₈	—	8	2	no	—
159	C₂²	—	2	1	no	—
160	C₂²	—	2	1	no	—
161	C₂²	—	2	2	no	—
162	C₂²	—	2	2	no	—
163	C₂²	—	4	2	no	—
164	C₂²	—	4	1	no	—
165	C₂²	—	4	1	no	—
166	C₂²	C₁₉×H(3,ℤ/4)	1	1	yes	characteristic
167	C₂²	C₁₉×#340	1	1	yes	characteristic
168	C₂²	C₁₉×#341	1	2	yes	—
169	C₂²	C₁₉×#342	1	1	yes	characteristic
170	C₂²	C₁₉×#344	1	2	yes	—
171	C₄	—	2	2	no	—
172	C₄	—	2	2	no	—
173	C₄	—	2	4	no	—
174	C₄	—	2	2	no	—
175	C₄	—	4	1	no	—
176	C₂	—	4	1	no	—
177	C₂	—	4	1	no	—
178	C₂	C₁₉×#1287	1	2	yes	—
179	C₂	C₁₉×#1289	1	1	yes	characteristic
180	C₂	C₁₉×#1298	1	1	yes	characteristic
181	C₂	C₁₉×#1301	1	2	yes	—
182	C₂	C₁₉×#1306	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#1817

C₂

yes

maximal, characteristic

C₁₉×#1797

C₂

yes

maximal

C₂²×C₄²×C₁₉

C₂²

yes

characteristic

C₂×C₁₉×C₄⋊C₈

—

C₂×C₁₉×C₂²⋊C₈

—

C₂×C₁₉×C₂²⋊C₈

C₄

yes

—

C₁₉×#336

—

C₁₉×#336

C₄

yes

—

C₂³×C₄×C₁₉

C₂×C₄

yes

characteristic

C₂³×C₄×C₁₉

D₄

yes

characteristic

C₂³×C₄×C₁₉

Q₈

yes

characteristic

C₂²×C₈×C₁₉

—

C₂²×C₈×C₁₉

—

C₂×C₄²×C₁₉

—

C₂×C₄²×C₁₉

C₂×C₄

yes

—

C₂×C₄²×C₁₉

D₄

yes

—

C₁₉×C₄⋊C₈

—

C₁₉×C₂²⋊C₈

—

C₁₉×C₂²⋊C₈

—

C₁₉×C₂²⋊C₈

—

C₂⁴×C₁₉

C₄⋊C₄

yes

characteristic

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

characteristic

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₉

C₄⋊C₄

yes

—

C₂²×C₄×C₁₉

D₈

yes

characteristic

C₂²×C₄×C₁₉

SD₁₆

yes

—

C₂²×C₄×C₁₉

Q₁₆

yes

characteristic

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

#7044

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

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

#99

yes

—

C₂×C₄×C₁₉

#102

yes

—

C₂×C₄×C₁₉

#103

yes

—

C₂×C₄×C₁₉

#108

yes

—

C₈×C₁₉

—

C₈×C₁₉

—

#1817

C₂×C₁₉

yes

characteristic

#1797

C₂×C₁₉

yes

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

—

C₂²×C₁₉

#342

yes

characteristic

C₂²×C₁₉

#344

yes

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₂²×C₄²

C₂²×C₁₉

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₂²⋊C₈

—

C₂×C₂²⋊C₈

C₄×C₁₉

yes

—

#336

—

#336

C₄×C₁₉

yes

—

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

#1287

yes

—

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

—

C₂×C₁₉

#1306

yes

characteristic

100

C₂³×C₄

C₂×C₄×C₁₉

yes

characteristic

101

C₂³×C₄

D₄×C₁₉

yes

characteristic

102

C₂³×C₄

Q₈×C₁₉

yes

characteristic

103

C₂²×C₈

—

104

C₂²×C₈

—

105

C₂×C₄²

—

106

C₂×C₄²

C₂×C₄×C₁₉

yes

—

107

C₂×C₄²

D₄×C₁₉

yes

—

108

C₄⋊C₈

—

109

C₂²⋊C₈

—

110

C₂²⋊C₈

—

111

C₂²⋊C₈

—

112

C₁₉

#7044

yes

characteristic, direct product factor, Sylow

113

C₂⁴

C₄⋊C₄×C₁₉

yes

characteristic

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

C₄²×C₁₉

yes

characteristic

121

C₂²×C₄

C₂²⋊C₄×C₁₉

yes

characteristic

122

C₂²×C₄

C₂²⋊C₄×C₁₉

yes

—

123

C₂²×C₄

C₄⋊C₄×C₁₉

yes

—

124

C₂²×C₄

D₈×C₁₉

yes

characteristic

125

C₂²×C₄

SD₁₆×C₁₉

yes

—

126

C₂²×C₄

Q₁₆×C₁₉

yes

characteristic

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₄²

—

133

C₄²

—

134

C₂³

—

135

C₂³

—

136

C₂³

—

137

C₂³

—

138

C₂³

C₁₉×#95

yes

characteristic

139

C₂³

C₁₉×#106

yes

characteristic

140

C₂³

C₁₉×#107

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

C₁₉×#99

yes

—

154

C₂×C₄

C₁₉×#102

yes

—

155

C₂×C₄

C₁₉×#103

yes

—

156

C₂×C₄

C₁₉×#108

yes

—

157

C₈

—

158

C₈

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

162

C₂²

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

C₁₉×H(3,ℤ/4)

yes

characteristic

167

C₂²

C₁₉×#340

yes

characteristic

168

C₂²

C₁₉×#341

yes

—

169

C₂²

C₁₉×#342

yes

characteristic

170

C₂²

C₁₉×#344

yes

—

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₂

—

177

C₂

—

178

C₂

C₁₉×#1287

yes

—

179

C₂

C₁₉×#1289

yes

characteristic

180

C₂

C₁₉×#1298

yes

characteristic

181

C₂

C₁₉×#1301

yes

—

182

C₂

C₁₉×#1306

yes

characteristic

	subgroup	G/H
center, socle	C₂³×C₁₉	#95
commutator	C₂²×C₄	C₄²×C₁₉
Frattini, cosocle	C₂²×C₄²	C₂²×C₁₉
prime radical	C₂⁴×C₁₉	C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₂²×C₄²×C₁₉ ◁ G

subgroup

G/H

center, socle

C₂³×C₁₉

#95

commutator

C₂²×C₄

C₄²×C₁₉

Frattini, cosocle

C₂²×C₄²

C₂²×C₁₉

prime radical

C₂⁴×C₁₉

C₄⋊C₄

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₁₉ ◁ C₂²×C₄²×C₁₉ ◁ G

N	H ≅ G/N	type
42. #7044	C₁₉	direct (N × H)
112. C₁₉	#7044	direct (N × H)

H ≅ G/N

type

42. #7044

C₁₉

direct (N × H)

112. C₁₉

#7044

direct (N × H)

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

subgroup	index
C₁₉×#1817	2
C₁₉×#1797	2
#7044	19

subgroup

index

C₁₉×#1817

C₁₉×#1797

#7044

Subgroups — C₁₉×#7044 #18013003

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#7044 #18013003

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#7044 #18013003