Group #18013246 — subgroups

Subgroups — C₁₉×#7287 #18013246

Order 4864.

322 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 82 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×#1955	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#1459	C₂	1	2	yes	maximal
3	C₂×C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
4	C₂×C₁₉×C₄⋊C₈	C₄	1	1	yes	characteristic
5	C₂×C₁₉×C₂²⋊C₈	—	2	2	no	—
6	C₂×C₁₉×C₂²⋊C₈	—	2	2	no	—
7	C₂×C₁₉×#97	C₄	1	1	yes	characteristic
8	C₁₉×#324	—	2	2	no	—
9	C₁₉×#323	C₄	1	4	yes	—
10	C₂²×D₄×C₁₉	Q₈	1	1	yes	characteristic
11	C₂³×C₄×C₁₉	D₄	1	2	yes	—
12	C₂²×C₈×C₁₉	—	2	1	no	—
13	C₂²×C₈×C₁₉	—	2	1	no	—
14	C₂²×C₈×C₁₉	—	4	2	no	—
15	C₄×D₄×C₁₉	—	4	1	no	—
16	C₄×D₄×C₁₉	—	4	1	no	—
17	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
18	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
20	C₁₉×C₄⋊C₈	—	4	1	no	—
21	C₁₉×C₂²⋊C₈	—	2	4	no	—
22	C₁₉×C₂²⋊C₈	—	4	4	no	—
23	C₁₉×C₂²⋊C₈	—	4	2	no	—
24	C₁₉×#97	—	4	1	no	—
25	C₂⁴×C₁₉	C₄⋊C₄	1	2	yes	—
26	C₂×D₄×C₁₉	—	2	4	no	—
27	C₂×D₄×C₁₉	—	4	1	no	—
28	C₂²×C₄×C₁₉	—	2	2	no	—
29	C₂²×C₄×C₁₉	—	2	2	no	—
30	C₂²×C₄×C₁₉	—	4	2	no	—
31	C₂²×C₄×C₁₉	—	4	2	no	—
32	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
33	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
34	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₁₉	—	2	2	no	—
37	C₂×C₈×C₁₉	—	4	1	no	—
38	C₂×C₈×C₁₉	—	4	1	no	—
39	C₂×C₈×C₁₉	—	4	2	no	—
40	C₂×C₈×C₁₉	—	4	4	no	—
41	C₂×C₈×C₁₉	—	4	1	no	—
42	C₂×C₈×C₁₉	—	8	2	no	—
43	C₂×C₈×C₁₉	—	8	2	no	—
44	C₄⋊C₄×C₁₉	—	4	1	no	—
45	C₂²⋊C₄×C₁₉	—	4	2	no	—
46	C₄²×C₁₉	—	2	1	no	—
47	C₄²×C₁₉	—	2	1	no	—
48	#7287	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂³×C₁₉	—	2	2	no	—
50	C₂³×C₁₉	—	2	2	no	—
51	C₂³×C₁₉	—	4	2	no	—
52	C₂³×C₁₉	—	4	2	no	—
53	C₂³×C₁₉	#95	1	1	yes	characteristic
54	C₂³×C₁₉	#108	1	4	yes	—
55	D₄×C₁₉	—	4	4	no	—
56	C₂×C₄×C₁₉	—	2	1	no	—
57	C₂×C₄×C₁₉	—	2	1	no	—
58	C₂×C₄×C₁₉	—	2	2	no	—
59	C₂×C₄×C₁₉	—	2	1	no	—
60	C₂×C₄×C₁₉	—	2	1	no	—
61	C₂×C₄×C₁₉	—	2	2	no	—
62	C₂×C₄×C₁₉	—	4	2	no	—
63	C₂×C₄×C₁₉	—	4	2	no	—
64	C₂×C₄×C₁₉	—	4	2	no	—
65	C₂×C₄×C₁₉	—	4	1	no	—
66	C₂×C₄×C₁₉	—	8	2	no	—
67	C₂×C₄×C₁₉	#99	1	1	yes	characteristic
68	C₂×C₄×C₁₉	#99	1	1	yes	characteristic
69	C₈×C₁₉	—	4	2	no	—
70	C₈×C₁₉	—	8	4	no	—
71	C₈×C₁₉	—	8	1	no	—
72	#1955	C₂×C₁₉	1	1	yes	characteristic
73	#1459	C₂×C₁₉	1	2	yes	—
74	C₂²×C₁₉	—	2	1	no	—
75	C₂²×C₁₉	—	2	1	no	—
76	C₂²×C₁₉	—	2	4	no	—
77	C₂²×C₁₉	—	4	4	no	—
78	C₂²×C₁₉	—	4	2	no	—
79	C₂²×C₁₉	—	4	2	no	—
80	C₂²×C₁₉	—	8	2	no	—
81	C₂²×C₁₉	#341	1	2	yes	—
82	C₂²×C₁₉	#342	1	1	yes	characteristic
83	C₄×C₁₉	—	2	1	no	—
84	C₄×C₁₉	—	4	1	no	—
85	C₄×C₁₉	—	4	1	no	—
86	C₄×C₁₉	—	4	2	no	—
87	C₄×C₁₉	#353	1	1	yes	characteristic
88	C₄×C₁₉	#354	1	1	yes	characteristic
89	C₂×C₄×D₄	C₂²×C₁₉	1	1	yes	characteristic
90	C₂×C₄⋊C₈	C₄×C₁₉	1	1	yes	characteristic
91	C₂×C₂²⋊C₈	—	2	2	no	—
92	C₂×C₂²⋊C₈	—	2	2	no	—
93	C₂×#97	C₄×C₁₉	1	1	yes	characteristic
94	#324	—	2	2	no	—
95	#323	C₄×C₁₉	1	4	yes	—
96	C₂×C₁₉	—	2	2	no	—
97	C₂×C₁₉	—	4	4	no	—
98	C₂×C₁₉	#1298	1	1	yes	characteristic
99	C₂×C₁₉	#1392	1	1	yes	characteristic
100	C₂×C₁₉	#1393	1	1	yes	characteristic
101	C₂²×D₄	Q₈×C₁₉	1	1	yes	characteristic
102	C₂³×C₄	D₄×C₁₉	1	2	yes	—
103	C₂²×C₈	—	2	1	no	—
104	C₂²×C₈	—	2	1	no	—
105	C₂²×C₈	—	4	2	no	—
106	C₄×D₄	—	4	1	no	—
107	C₄×D₄	—	4	1	no	—
108	C₂×C₄⋊C₄	D₄×C₁₉	1	1	yes	characteristic
109	C₂×C₂²⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
110	C₂×C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
111	C₄⋊C₈	—	4	1	no	—
112	C₂²⋊C₈	—	2	4	no	—
113	C₂²⋊C₈	—	4	4	no	—
114	C₂²⋊C₈	—	4	2	no	—
115	#97	—	4	1	no	—
116	C₁₉	#7287	1	1	yes	characteristic, direct product factor, Sylow
117	C₂⁴	C₄⋊C₄×C₁₉	1	2	yes	—
118	C₂×D₄	—	2	4	no	—
119	C₂×D₄	—	4	1	no	—
120	C₂²×C₄	—	2	2	no	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	4	2	no	—
123	C₂²×C₄	—	4	2	no	—
124	C₂²×C₄	C₄²×C₁₉	1	1	yes	characteristic
125	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
126	C₂²×C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
127	C₂²×C₄	C₄⋊C₄×C₁₉	1	1	yes	characteristic
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	4	1	no	—
130	C₂×C₈	—	4	1	no	—
131	C₂×C₈	—	4	4	no	—
132	C₂×C₈	—	4	2	no	—
133	C₂×C₈	—	4	1	no	—
134	C₂×C₈	—	8	2	no	—
135	C₂×C₈	—	8	2	no	—
136	C₄⋊C₄	—	4	1	no	—
137	C₂²⋊C₄	—	4	2	no	—
138	C₄²	—	2	1	no	—
139	C₄²	—	2	1	no	—
140	C₂³	—	2	2	no	—
141	C₂³	—	2	2	no	—
142	C₂³	—	4	2	no	—
143	C₂³	—	4	2	no	—
144	C₂³	C₁₉×#95	1	1	yes	characteristic
145	C₂³	C₁₉×#108	1	4	yes	—
146	D₄	—	4	4	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	8	2	no	—
158	C₂×C₄	C₁₉×#99	1	1	yes	characteristic
159	C₂×C₄	C₁₉×#99	1	1	yes	characteristic
160	C₈	—	4	2	no	—
161	C₈	—	8	4	no	—
162	C₈	—	8	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	4	no	—
166	C₂²	—	4	4	no	—
167	C₂²	—	4	2	no	—
168	C₂²	—	4	2	no	—
169	C₂²	—	8	2	no	—
170	C₂²	C₁₉×#341	1	2	yes	—
171	C₂²	C₁₉×#342	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	4	1	no	—
174	C₄	—	4	2	no	—
175	C₄	—	4	1	no	—
176	C₄	C₁₉×#353	1	1	yes	characteristic
177	C₄	C₁₉×#354	1	1	yes	characteristic
178	C₂	—	2	2	no	—
179	C₂	—	4	4	no	—
180	C₂	C₁₉×#1298	1	1	yes	characteristic
181	C₂	C₁₉×#1392	1	1	yes	characteristic
182	C₂	C₁₉×#1393	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉×#1955

C₂

yes

maximal, characteristic

C₁₉×#1459

C₂

yes

maximal

C₂×C₄×D₄×C₁₉

C₂²

yes

characteristic

C₂×C₁₉×C₄⋊C₈

C₄

yes

characteristic

C₂×C₁₉×C₂²⋊C₈

—

C₂×C₁₉×C₂²⋊C₈

—

C₂×C₁₉×#97

C₄

yes

characteristic

C₁₉×#324

—

C₁₉×#323

C₄

yes

—

C₂²×D₄×C₁₉

Q₈

yes

characteristic

C₂³×C₄×C₁₉

D₄

yes

—

C₂²×C₈×C₁₉

—

C₂²×C₈×C₁₉

—

C₂²×C₈×C₁₉

—

C₄×D₄×C₁₉

—

C₄×D₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

D₄

yes

characteristic

C₂×C₂²⋊C₄×C₁₉

C₂×C₄

yes

—

C₂×C₄²×C₁₉

C₂×C₄

yes

characteristic

C₁₉×C₄⋊C₈

—

C₁₉×C₂²⋊C₈

—

C₁₉×C₂²⋊C₈

—

C₁₉×C₂²⋊C₈

—

C₁₉×#97

—

C₂⁴×C₁₉

C₄⋊C₄

yes

—

C₂×D₄×C₁₉

—

C₂×D₄×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

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

#7287

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

#95

yes

characteristic

C₂³×C₁₉

#108

yes

—

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

yes

C₂×C₄×C₁₉

yes

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

#1955

C₂×C₁₉

yes

characteristic

#1459

C₂×C₁₉

yes

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#341

yes

—

C₂²×C₁₉

#342

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

yes

C₄×C₁₉

yes

C₂×C₄×D₄

C₂²×C₁₉

yes

characteristic

C₂×C₄⋊C₈

C₄×C₁₉

yes

characteristic

C₂×C₂²⋊C₈

—

C₂×C₂²⋊C₈

—

C₂×#97

C₄×C₁₉

yes

characteristic

#324

—

#323

C₄×C₁₉

yes

—

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

yes

C₂×C₁₉

yes

100

C₂×C₁₉

#1393

yes

characteristic

101

C₂²×D₄

Q₈×C₁₉

yes

characteristic

102

C₂³×C₄

D₄×C₁₉

yes

—

103

C₂²×C₈

—

104

C₂²×C₈

—

105

C₂²×C₈

—

106

C₄×D₄

—

107

C₄×D₄

—

108

C₂×C₄⋊C₄

D₄×C₁₉

yes

characteristic

109

C₂×C₂²⋊C₄

C₂×C₄×C₁₉

yes

—

110

C₂×C₄²

C₂×C₄×C₁₉

yes

characteristic

111

C₄⋊C₈

—

112

C₂²⋊C₈

—

113

C₂²⋊C₈

—

114

C₂²⋊C₈

—

115

#97

—

116

C₁₉

#7287

yes

characteristic, direct product factor, Sylow

117

C₂⁴

C₄⋊C₄×C₁₉

yes

—

118

C₂×D₄

—

119

C₂×D₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

C₄²×C₁₉

yes

characteristic

125

C₂²×C₄

C₂²⋊C₄×C₁₉

yes

characteristic

126

C₂²×C₄

C₂²⋊C₄×C₁₉

yes

—

127

C₂²×C₄

C₄⋊C₄×C₁₉

yes

characteristic

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₄⋊C₄

—

137

C₂²⋊C₄

—

138

C₄²

—

139

C₄²

—

140

C₂³

—

141

C₂³

—

142

C₂³

—

143

C₂³

—

144

C₂³

C₁₉×#95

yes

characteristic

145

C₂³

C₁₉×#108

yes

—

146

D₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

C₁₉×#99

yes

characteristic

159

C₂×C₄

C₁₉×#99

yes

characteristic

160

C₈

—

161

C₈

—

162

C₈

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

—

169

C₂²

—

170

C₂²

C₁₉×#341

yes

—

171

C₂²

C₁₉×#342

yes

characteristic

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

C₁₉×#353

yes

characteristic

177

C₄

C₁₉×#354

yes

characteristic

178

C₂

—

179

C₂

—

180

C₂

C₁₉×#1298

yes

characteristic

181

C₂

C₁₉×#1392

yes

characteristic

182

C₂

C₁₉×#1393

yes

characteristic

	subgroup	G/H
center	C₂×C₄×C₁₉	#99
commutator	C₂²×C₄	C₄²×C₁₉
Frattini, cosocle	C₂×C₄×D₄	C₂²×C₁₉
socle	C₂²×C₁₉	#342
prime radical	C₂²×D₄×C₁₉	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂×C₄×D₄×C₁₉ ◁ G

subgroup

G/H

center

C₂×C₄×C₁₉

#99

commutator

C₂²×C₄

C₄²×C₁₉

Frattini, cosocle

C₂×C₄×D₄

C₂²×C₁₉

socle

C₂²×C₁₉

#342

prime radical

C₂²×D₄×C₁₉

Q₈

LCS

G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂×C₄×D₄×C₁₉ ◁ G

N	H ≅ G/N	type
48. #7287	C₁₉	direct (N × H)
116. C₁₉	#7287	direct (N × H)

H ≅ G/N

type

48. #7287

C₁₉

direct (N × H)

116. C₁₉

#7287

direct (N × H)

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

subgroup	index
C₁₉×#1955	2
C₁₉×#1459	2
#7287	19

subgroup

index

C₁₉×#1955

C₁₉×#1459

#7287

Subgroups — C₁₉×#7287 #18013246

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#7287 #18013246

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#7287 #18013246