Group #18019417 — subgroups

Subgroups — C₁₉×#13458 #18019417

Order 4864.

324 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 122 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	C₂×C₁₉×#499	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#365	C₂	1	2	yes	maximal
3	C₁₉×#1945	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1386	C₂	1	2	yes	maximal
5	C₁₉×#1376	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₉×#125	—	2	1	no	—
7	C₂×C₁₉×M₃₂	C₂²	1	2	yes	—
8	C₁₉×#499	—	2	4	no	—
9	C₂×C₁₉×#108	C₂²	1	1	yes	characteristic
10	C₁₉×#427	C₂²	1	1	yes	characteristic
11	C₂×C₁₉×#107	—	2	1	no	—
12	C₂×C₁₉×#106	C₂²	1	2	yes	—
13	C₁₉×#424	—	2	1	no	—
14	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
15	C₁₉×#365	—	2	4	no	—
16	C₁₉×#339	—	2	1	no	—
17	C₂×M₁₆×C₁₉	—	2	1	no	—
18	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
20	C₁₉×#125	—	2	4	no	—
21	C₁₉×#125	—	4	2	no	—
22	C₁₉×#117	—	2	1	no	—
23	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
24	C₂×C₄⋊C₄×C₁₉	—	4	2	no	—
25	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
26	C₁₉×M₃₂	—	2	2	no	—
27	C₁₉×M₃₂	Q₈	1	4	yes	—
28	C₂×C₁₆×C₁₉	—	2	2	no	—
29	C₁₉×#108	—	2	2	no	—
30	C₁₉×#107	—	2	4	no	—
31	C₁₉×#107	Q₈	1	2	yes	—
32	C₁₉×#106	—	2	2	no	—
33	C₁₉×#106	—	2	1	no	—
34	C₁₉×#106	D₄	1	4	yes	—
35	C₁₉×C₄⋊C₈	—	4	2	no	—
36	C₄×C₈×C₁₉	—	2	2	no	—
37	C₂²×C₄×C₁₉	—	2	1	no	—
38	C₂²×C₄×C₁₉	—	4	2	no	—
39	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
40	M₁₆×C₁₉	—	4	1	no	—
41	M₁₆×C₁₉	—	4	1	no	—
42	C₂×C₈×C₁₉	—	2	1	no	—
43	C₂×C₈×C₁₉	—	2	1	no	—
44	C₂×C₈×C₁₉	—	4	1	no	—
45	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
46	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
47	C₂×C₈×C₁₉	C₂×Q₈	1	2	yes	—
48	C₂×C₈×C₁₉	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
50	C₄⋊C₄×C₁₉	—	2	4	no	—
51	C₄⋊C₄×C₁₉	—	2	2	no	—
52	C₄⋊C₄×C₁₉	—	4	8	no	—
53	C₄⋊C₄×C₁₉	—	4	2	no	—
54	C₂²⋊C₄×C₁₉	—	4	1	no	—
55	C₄²×C₁₉	—	2	1	no	—
56	C₄²×C₁₉	—	4	1	no	—
57	C₄²×C₁₉	SD₁₆	1	2	yes	—
58	C₁₆×C₁₉	—	2	4	no	—
59	#13458	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂³×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₂×C₄×C₁₉	—	2	2	no	—
62	C₂×C₄×C₁₉	—	4	2	no	—
63	C₂×C₄×C₁₉	—	4	4	no	—
64	C₂×C₄×C₁₉	—	4	2	no	—
65	C₂×C₄×C₁₉	—	8	4	no	—
66	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
67	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
70	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
72	C₈×C₁₉	—	2	2	no	—
73	C₈×C₁₉	—	8	1	no	—
74	C₈×C₁₉	C₂²⋊Q₈	1	2	yes	—
75	C₈×C₁₉	C₄⋊Q₈	1	2	yes	—
76	C₂×#499	C₂×C₁₉	1	1	yes	characteristic
77	C₂×#365	C₂×C₁₉	1	2	yes	—
78	#1945	C₂×C₁₉	1	1	yes	characteristic
79	#1386	C₂×C₁₉	1	2	yes	—
80	#1376	C₂×C₁₉	1	1	yes	characteristic
81	C₂²×C₁₉	—	2	2	no	—
82	C₂²×C₁₉	#450	1	1	yes	characteristic
83	C₂²×C₁₉	#476	1	1	yes	characteristic
84	C₂²×C₁₉	#501	1	1	yes	characteristic
85	C₄×C₁₉	—	4	2	no	—
86	C₄×C₁₉	—	8	4	no	—
87	C₄×C₁₉	—	8	2	no	—
88	C₄×C₁₉	#395	1	1	yes	characteristic
89	C₄×C₁₉	#448	1	1	yes	characteristic
90	C₄×C₁₉	#475	1	1	yes	characteristic
91	C₄×C₁₉	#498	1	1	yes	characteristic
92	C₂×#125	—	2	1	no	—
93	C₂×M₃₂	C₂²×C₁₉	1	2	yes	—
94	#499	—	2	4	no	—
95	C₂×#108	C₂²×C₁₉	1	1	yes	characteristic
96	#427	C₂²×C₁₉	1	1	yes	characteristic
97	C₂×#107	—	2	1	no	—
98	C₂×#106	C₂²×C₁₉	1	2	yes	—
99	#424	—	2	1	no	—
100	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
101	#365	—	2	4	no	—
102	#339	—	2	1	no	—
103	C₂×C₁₉	—	2	1	no	—
104	C₂×C₁₉	—	2	1	no	—
105	C₂×C₁₉	#2058	1	1	yes	characteristic
106	C₂×C₁₉	#2231	1	2	yes	—
107	C₂×M₁₆	—	2	1	no	—
108	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
109	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
110	#125	—	2	4	no	—
111	#125	—	4	2	no	—
112	#117	—	2	1	no	—
113	C₂×C₄⋊C₄	—	2	2	no	—
114	C₂×C₄⋊C₄	—	4	2	no	—
115	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
116	M₃₂	—	2	2	no	—
117	M₃₂	Q₈×C₁₉	1	4	yes	—
118	C₂×C₁₆	—	2	2	no	—
119	#108	—	2	2	no	—
120	#107	—	2	4	no	—
121	#107	Q₈×C₁₉	1	2	yes	—
122	#106	—	2	2	no	—
123	#106	—	2	1	no	—
124	#106	D₄×C₁₉	1	4	yes	—
125	C₄⋊C₈	—	4	2	no	—
126	C₄×C₈	—	2	2	no	—
127	C₁₉	#13458	1	1	yes	characteristic, direct product factor, Sylow
128	C₂²×C₄	—	2	1	no	—
129	C₂²×C₄	—	4	2	no	—
130	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
131	M₁₆	—	4	1	no	—
132	M₁₆	—	4	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	4	1	no	—
136	C₂×C₈	SD₁₆×C₁₉	1	2	yes	—
137	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
138	C₂×C₈	C₂×Q₈×C₁₉	1	2	yes	—
139	C₂×C₈	C₂×Q₈×C₁₉	1	1	yes	characteristic
140	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
141	C₄⋊C₄	—	2	4	no	—
142	C₄⋊C₄	—	2	2	no	—
143	C₄⋊C₄	—	4	8	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₂²⋊C₄	—	4	1	no	—
146	C₄²	—	2	1	no	—
147	C₄²	—	4	1	no	—
148	C₄²	SD₁₆×C₁₉	1	2	yes	—
149	C₁₆	—	2	4	no	—
150	C₂³	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	4	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	8	4	no	—
156	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
157	C₂×C₄	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
158	C₂×C₄	C₁₉×C₄⋊Q₈	1	1	yes	characteristic
159	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
160	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
161	C₂×C₄	C₁₉×#137	1	1	yes	characteristic
162	C₈	—	2	2	no	—
163	C₈	—	8	1	no	—
164	C₈	C₁₉×C₂²⋊Q₈	1	2	yes	—
165	C₈	C₁₉×C₄⋊Q₈	1	2	yes	—
166	C₂²	—	2	2	no	—
167	C₂²	C₁₉×#450	1	1	yes	characteristic
168	C₂²	C₁₉×#476	1	1	yes	characteristic
169	C₂²	C₁₉×#501	1	1	yes	characteristic
170	C₄	—	4	2	no	—
171	C₄	—	8	4	no	—
172	C₄	—	8	2	no	—
173	C₄	C₁₉×#395	1	1	yes	characteristic
174	C₄	C₁₉×#448	1	1	yes	characteristic
175	C₄	C₁₉×#475	1	1	yes	characteristic
176	C₄	C₁₉×#498	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	C₁₉×#2058	1	1	yes	characteristic
180	C₂	C₁₉×#2231	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₉×#499

C₂

yes

maximal, characteristic

C₂×C₁₉×#365

C₂

yes

maximal

C₁₉×#1945

C₂

yes

maximal, characteristic

C₁₉×#1386

C₂

yes

maximal

C₁₉×#1376

C₂

yes

maximal, characteristic

C₂×C₁₉×#125

—

C₂×C₁₉×M₃₂

C₂²

yes

—

C₁₉×#499

—

C₂×C₁₉×#108

C₂²

yes

characteristic

C₁₉×#427

C₂²

yes

characteristic

C₂×C₁₉×#107

—

C₂×C₁₉×#106

C₂²

yes

—

C₁₉×#424

—

C₂×C₄×C₈×C₁₉

C₂²

yes

characteristic

C₁₉×#365

—

C₁₉×#339

—

C₂×M₁₆×C₁₉

—

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₂²×C₈×C₁₉

C₂³

yes

characteristic

C₁₉×#125

—

C₁₉×#125

—

C₁₉×#117

—

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄²×C₁₉

D₄

yes

characteristic

C₁₉×M₃₂

—

C₁₉×M₃₂

Q₈

yes

—

C₂×C₁₆×C₁₉

—

C₁₉×#108

—

C₁₉×#107

—

C₁₉×#107

Q₈

yes

—

C₁₉×#106

—

C₁₉×#106

—

C₁₉×#106

D₄

yes

—

C₁₉×C₄⋊C₈

—

C₄×C₈×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂×D₄

yes

characteristic

M₁₆×C₁₉

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

SD₁₆

yes

—

C₂×C₈×C₁₉

C₂×D₄

yes

—

C₂×C₈×C₁₉

C₂×Q₈

yes

—

C₂×C₈×C₁₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₉

#41

yes

characteristic

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

SD₁₆

yes

—

C₁₆×C₁₉

—

#13458

C₁₉

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₉

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

#120

yes

characteristic

C₂×C₄×C₁₉

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₁₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₁₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₁₉

#137

yes

characteristic

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

C₂²⋊Q₈

yes

—

C₈×C₁₉

C₄⋊Q₈

yes

—

C₂×#499

C₂×C₁₉

yes

characteristic

C₂×#365

C₂×C₁₉

yes

—

#1945

C₂×C₁₉

yes

characteristic

#1386

C₂×C₁₉

yes

—

#1376

C₂×C₁₉

yes

characteristic

C₂²×C₁₉

—

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

yes

C₄×C₁₉

yes

C₄×C₁₉

yes

C₄×C₁₉

yes

—

C₂×M₃₂

C₂²×C₁₉

yes

—

#499

—

C₂×#108

C₂²×C₁₉

yes

characteristic

#427

C₂²×C₁₉

yes

characteristic

C₂×#107

—

C₂×#106

C₂²×C₁₉

yes

—

#424

—

100

C₂×C₄×C₈

C₂²×C₁₉

yes

characteristic

101

#365

—

102

#339

—

103

C₂×C₁₉

—

104

C₂×C₁₉

—

105

C₂×C₁₉

#2058

yes

characteristic

106

C₂×C₁₉

#2231

yes

—

107

C₂×M₁₆

—

108

C₂²×C₈

D₄×C₁₉

yes

characteristic

109

C₂²×C₈

C₂³×C₁₉

yes

characteristic

110

#125

—

111

#125

—

112

#117

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄²

D₄×C₁₉

yes

characteristic

116

M₃₂

—

117

M₃₂

Q₈×C₁₉

yes

—

118

C₂×C₁₆

—

119

#108

—

120

#107

—

121

#107

Q₈×C₁₉

yes

—

122

#106

—

123

#106

—

124

#106

D₄×C₁₉

yes

—

125

C₄⋊C₈

—

126

C₄×C₈

—

127

C₁₉

#13458

yes

characteristic, direct product factor, Sylow

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

C₂×D₄×C₁₉

yes

characteristic

131

M₁₆

—

132

M₁₆

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

SD₁₆×C₁₉

yes

—

137

C₂×C₈

C₂×D₄×C₁₉

yes

—

138

C₂×C₈

C₂×Q₈×C₁₉

yes

—

139

C₂×C₈

C₂×Q₈×C₁₉

yes

characteristic

140

C₂×C₈

#41×C₁₉

yes

characteristic

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₂²⋊C₄

—

146

C₄²

—

147

C₄²

—

148

C₄²

SD₁₆×C₁₉

yes

—

149

C₁₆

—

150

C₂³

C₁₉×AGL(1,ℤ/8)

yes

characteristic

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

C₁₉×#120

yes

characteristic

157

C₂×C₄

C₁₉×C₂²⋊Q₈

yes

characteristic

158

C₂×C₄

C₁₉×C₄⋊Q₈

yes

characteristic

159

C₂×C₄

C₂×SD₁₆×C₁₉

yes

characteristic

160

C₂×C₄

C₂×SD₁₆×C₁₉

yes

characteristic

161

C₂×C₄

C₁₉×#137

yes

characteristic

162

C₈

—

163

C₈

—

164

C₈

C₁₉×C₂²⋊Q₈

yes

—

165

C₈

C₁₉×C₄⋊Q₈

yes

—

166

C₂²

—

167

C₂²

C₁₉×#450

yes

characteristic

168

C₂²

C₁₉×#476

yes

characteristic

169

C₂²

C₁₉×#501

yes

characteristic

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

C₁₉×#395

yes

characteristic

174

C₄

C₁₉×#448

yes

characteristic

175

C₄

C₁₉×#475

yes

characteristic

176

C₄

C₁₉×#498

yes

characteristic

177

C₂

—

178

C₂

—

179

C₂

C₁₉×#2058

yes

characteristic

180

C₂

C₁₉×#2231

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₉	#476
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
prime radical	C₂³×C₁₉	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂×C₄×C₈×C₁₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₉

#476

commutator, Frattini, cosocle

C₂²×C₈

C₂³×C₁₉

prime radical

C₂³×C₁₉

AGL(1,ℤ/8)

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂×C₄×C₈×C₁₉ ◁ G

N	H ≅ G/N	type
59. #13458	C₁₉	direct (N × H)
127. C₁₉	#13458	direct (N × H)

H ≅ G/N

type

59. #13458

C₁₉

direct (N × H)

127. C₁₉

#13458

direct (N × H)

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

subgroup	index
C₂×C₁₉×#499	2
C₂×C₁₉×#365	2
C₁₉×#1945	2
C₁₉×#1386	2
C₁₉×#1376	2
#13458	19

subgroup

index

C₂×C₁₉×#499

C₂×C₁₉×#365

C₁₉×#1945

C₁₉×#1386

C₁₉×#1376

#13458

Subgroups — C₁₉×#13458 #18019417

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C19×#13458 #18019417

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#13458 #18019417