Group #18069396 — subgroups

Subgroups — #18069396

Order 4864.

266 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 99 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	#14565438	C₂	1	1	yes	maximal, characteristic
2	C₄×#10597272	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#2163	C₂	1	1	yes	maximal, characteristic
4	C₂×#10597272	—	2	1	no	—
5	C₂×#10597272	—	2	1	no	—
6	C₂×C₄×C₁₉⋊C₈	C₄	1	1	yes	characteristic
7	C₄²×Dic₁₉	C₄	1	1	yes	characteristic
8	C₄⋊(C₁₉⋊C₁₆)	C₄	1	2	yes	complemented
9	C₂²⋊(C₁₉⋊C₁₆)	—	2	1	no	—
10	#11926320	—	2	1	no	—
11	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
12	C₁₉×C₄⋊C₁₆	C₄	1	2	yes	complemented
13	C₁₉×C₂²⋊C₁₆	—	2	1	no	—
14	C₁₉×#346	—	2	1	no	—
15	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
17	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×Dic₁₉	—	2	1	no	—
19	C₂×C₄×Dic₁₉	—	2	1	no	—
20	C₂×C₄×Dic₁₉	C₈	1	2	yes	—
21	C₂²×C₁₉⋊C₈	—	2	1	no	—
22	C₂×C₁₆×C₁₉	—	2	2	no	—
23	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
26	C₄×C₈×C₁₉	Q₈	1	1	yes	characteristic
27	#10597272	—	2	4	no	—
28	#10597272	—	2	4	no	—
29	C₂×C₁₉⋊C₁₆	—	2	2	no	—
30	C₄×C₁₉⋊C₈	—	2	1	no	—
31	C₄×C₁₉⋊C₈	C₈	1	2	yes	—
32	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₉	C₂×C₈	1	1	yes	characteristic
34	C₂²×C₄×C₁₉	M₁₆	1	1	yes	characteristic
35	C₂²×Dic₁₉	—	2	2	no	—
36	C₂×C₈×C₁₉	—	2	2	no	—
37	C₂×C₈×C₁₉	—	2	2	no	—
38	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
42	C₄²×C₁₉	C₄²	1	1	yes	characteristic
43	C₄²×C₁₉	C₄⋊C₄	1	1	yes	characteristic
44	C₄²×C₁₉	C₂×C₈	1	1	yes	characteristic
45	C₄²×C₁₉	M₁₆	1	1	yes	characteristic
46	C₄×Dic₁₉	—	2	2	no	—
47	C₄×Dic₁₉	—	2	2	no	—
48	C₄×Dic₁₉	—	4	2	no	—
49	C₄×Dic₁₉	—	4	2	no	—
50	C₂×C₁₉⋊C₈	—	2	2	no	—
51	C₂×C₁₉⋊C₈	—	2	2	no	—
52	C₂×C₁₉⋊C₈	—	2	2	no	—
53	C₁₆×C₁₉	—	4	2	no	—
54	C₁₉⋊C₁₆	—	4	2	no	—
55	#7149	—	19	1	no	maximal, Sylow
56	C₂³×C₁₉	C₂²⋊C₈	1	1	yes	characteristic
57	C₂×C₄×C₁₉	—	2	2	no	—
58	C₂×C₄×C₁₉	—	2	1	no	—
59	C₂×C₄×C₁₉	—	2	1	no	—
60	C₂×C₄×C₁₉	—	2	2	no	—
61	C₂×C₄×C₁₉	#95	1	1	yes	characteristic
62	C₂×C₄×C₁₉	C₄×C₈	1	1	yes	characteristic
63	C₂×C₄×C₁₉	#97	1	1	yes	characteristic
64	C₂×C₄×C₁₉	C₂²⋊C₈	1	1	yes	characteristic
65	C₂×C₄×C₁₉	C₄⋊C₈	1	1	yes	characteristic
66	C₂×C₄×C₁₉	C₄⋊C₈	1	1	yes	characteristic
67	C₂×Dic₁₉	—	2	4	no	—
68	C₂×Dic₁₉	—	4	2	no	—
69	C₂×Dic₁₉	—	4	2	no	—
70	C₈×C₁₉	—	2	1	no	—
71	C₈×C₁₉	—	2	1	no	—
72	C₈×C₁₉	#104	1	2	yes	—
73	C₈×C₁₉	#104	1	2	yes	—
74	C₁₉⋊C₈	—	2	4	no	—
75	#2163	—	19	1	no	—
76	#2163	D₁₉	1	1	yes	characteristic
77	C₄×#97	—	19	1	no	—
78	C₂²×C₁₉	—	2	1	no	—
79	C₂²×C₁₉	—	2	1	no	—
80	C₂²×C₁₉	#336	1	1	yes	characteristic
81	C₂²×C₁₉	#349	1	2	yes	—
82	C₄×C₁₉	—	2	1	no	—
83	C₄×C₁₉	—	2	1	no	—
84	C₄×C₁₉	—	2	1	no	—
85	C₄×C₁₉	—	2	1	no	—
86	C₄×C₁₉	#339	1	1	yes	characteristic
87	C₄×C₁₉	#339	1	1	yes	characteristic
88	C₄×C₁₉	#347	1	2	yes	—
89	Dic₁₉	—	4	4	no	—
90	C₂×#97	—	38	1	no	—
91	C₂×#97	—	38	1	no	—
92	C₂×C₄×C₈	—	19	1	no	—
93	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
94	C₄³	—	19	1	no	—
95	C₄⋊C₁₆	—	19	2	no	—
96	C₄⋊C₁₆	Dic₁₉	1	2	yes	complemented
97	C₂²⋊C₁₆	—	2	1	no	—
98	C₂²⋊C₁₆	—	38	1	no	—
99	#346	—	2	1	no	—
100	#346	—	38	1	no	—
101	C₂×C₁₉	—	2	2	no	—
102	C₂×C₁₉	#1276	1	1	yes	characteristic
103	C₂×C₁₉	#1375	1	1	yes	characteristic
104	C₂×C₁₉	#1377	1	1	yes	characteristic
105	C₂²×C₈	—	38	1	no	—
106	C₂²×C₈	D₇₆	1	1	yes	characteristic
107	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
108	C₂×C₄²	—	19	2	no	—
109	C₂×C₄²	—	38	1	no	—
110	C₂×C₄²	—	38	1	no	—
111	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
112	C₂×C₁₆	—	2	2	no	—
113	C₂×C₁₆	—	38	2	no	—
114	#97	—	38	4	no	—
115	#97	—	38	4	no	—
116	C₄×C₈	—	19	2	no	—
117	C₄×C₈	—	38	1	no	—
118	C₄×C₈	Dic₃₈	1	1	yes	characteristic
119	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
120	C₄×C₈	C₂×Dic₁₉	1	1	yes	characteristic
121	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
122	C₁₉	#7149	1	1	yes	characteristic, complemented, Sylow
123	C₂²×C₄	—	38	2	no	—
124	C₂²×C₄	C₈×D₁₉	1	1	yes	characteristic
125	C₂²×C₄	#64515	1	1	yes	characteristic
126	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	38	2	no	—
130	C₂×C₈	—	38	2	no	—
131	C₂×C₈	—	38	2	no	—
132	C₂×C₈	Dic₁₉⋊C₄	1	1	yes	characteristic
133	C₂×C₈	C₄⋊Dic₁₉	1	1	yes	characteristic
134	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
135	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
136	C₄²	—	38	2	no	—
137	C₄²	—	38	2	no	—
138	C₄²	—	76	2	no	—
139	C₄²	—	76	2	no	—
140	C₄²	C₈×D₁₉	1	1	yes	characteristic
141	C₄²	#64515	1	1	yes	characteristic
142	C₄²	C₄×Dic₁₉	1	1	yes	characteristic
143	C₄²	Dic₁₉⋊C₄	1	1	yes	characteristic
144	C₁₆	—	4	2	no	—
145	C₁₆	—	76	2	no	—
146	C₂³	(C₂×D₁₉)⋊C₈	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	38	4	no	—
152	C₂×C₄	—	76	2	no	—
153	C₂×C₄	—	76	2	no	—
154	C₂×C₄	C₈×Dic₁₉	1	1	yes	characteristic
155	C₂×C₄	Dic₁₉⋊C₈	1	1	yes	characteristic
156	C₂×C₄	Dic₁₉⋊C₈	1	1	yes	characteristic
157	C₂×C₄	#10597272	1	1	yes	characteristic
158	C₂×C₄	(C₂×D₁₉)⋊C₈	1	1	yes	characteristic
159	C₂×C₄	C₁₉⋊#95	1	1	yes	characteristic
160	C₈	—	2	1	no	—
161	C₈	—	2	1	no	—
162	C₈	—	38	4	no	—
163	C₈	#10597282	1	2	yes	—
164	C₈	#10597294	1	2	yes	—
165	C₂²	—	2	1	no	—
166	C₂²	—	2	1	no	—
167	C₂²	#11926298	1	1	yes	characteristic
168	C₂²	#11926325	1	2	yes	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	2	1	no	—
173	C₄	—	76	4	no	—
174	C₄	#11926303	1	1	yes	characteristic
175	C₄	#11926305	1	1	yes	characteristic
176	C₄	#11926321	1	2	yes	—
177	C₂	—	2	2	no	—
178	C₂	#14561467	1	1	yes	characteristic
179	C₂	#14561714	1	1	yes	characteristic
180	C₂	#14561718	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14565438

C₂

yes

maximal, characteristic

C₄×#10597272

C₂

yes

maximal, characteristic

C₁₉×#2163

C₂

yes

maximal, characteristic

C₂×#10597272

—

C₂×#10597272

—

C₂×C₄×C₁₉⋊C₈

C₄

yes

characteristic

C₄²×Dic₁₉

C₄

yes

characteristic

C₄⋊(C₁₉⋊C₁₆)

C₄

yes

complemented

C₂²⋊(C₁₉⋊C₁₆)

—

#11926320

—

C₂×C₄×C₈×C₁₉

C₂²

yes

characteristic

C₁₉×C₄⋊C₁₆

C₄

yes

complemented

C₁₉×C₂²⋊C₁₆

—

C₁₉×#346

—

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₂×C₄²×C₁₉

C₂×C₄

yes

characteristic

C₂×C₄×Dic₁₉

—

C₂×C₄×Dic₁₉

—

C₂×C₄×Dic₁₉

C₈

yes

—

C₂²×C₁₉⋊C₈

—

C₂×C₁₆×C₁₉

—

C₄×C₈×C₁₉

C₂×C₄

yes

characteristic

C₄×C₈×C₁₉

C₂×C₄

yes

characteristic

C₄×C₈×C₁₉

D₄

yes

characteristic

C₄×C₈×C₁₉

Q₈

yes

characteristic

#10597272

—

#10597272

—

C₂×C₁₉⋊C₁₆

—

C₄×C₁₉⋊C₈

—

C₄×C₁₉⋊C₈

C₈

yes

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

C₂×C₈

yes

characteristic

C₂²×C₄×C₁₉

M₁₆

yes

characteristic

C₂²×Dic₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₉

C₄⋊C₄

yes

characteristic

C₄²×C₁₉

C₄²

yes

characteristic

C₄²×C₁₉

C₄⋊C₄

yes

characteristic

C₄²×C₁₉

C₂×C₈

yes

characteristic

C₄²×C₁₉

M₁₆

yes

characteristic

C₄×Dic₁₉

—

C₄×Dic₁₉

—

C₄×Dic₁₉

—

C₄×Dic₁₉

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₁₆×C₁₉

—

C₁₉⋊C₁₆

—

#7149

—

maximal, Sylow

C₂³×C₁₉

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

#95

yes

characteristic

C₂×C₄×C₁₉

C₄×C₈

yes

characteristic

C₂×C₄×C₁₉

#97

yes

characteristic

C₂×C₄×C₁₉

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₁₉

C₄⋊C₈

yes

characteristic

C₂×C₄×C₁₉

C₄⋊C₈

yes

characteristic

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

#104

yes

—

C₈×C₁₉

#104

yes

—

C₁₉⋊C₈

—

#2163

—

#2163

D₁₉

yes

characteristic

C₄×#97

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#336

yes

characteristic

C₂²×C₁₉

#349

yes

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

yes

C₄×C₁₉

yes

C₄×C₁₉

yes

—

Dic₁₉

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₁₉

yes

characteristic

C₄³

—

C₄⋊C₁₆

—

C₄⋊C₁₆

Dic₁₉

yes

complemented

C₂²⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

100

#346

—

101

C₂×C₁₉

—

102

C₂×C₁₉

#1276

yes

characteristic

103

C₂×C₁₉

#1375

yes

characteristic

104

C₂×C₁₉

#1377

yes

characteristic

105

C₂²×C₈

—

106

C₂²×C₈

D₇₆

yes

characteristic

107

C₂²×C₈

C₂²⋊D₁₉

yes

characteristic

108

C₂×C₄²

—

109

C₂×C₄²

—

110

C₂×C₄²

—

111

C₂×C₄²

C₄×D₁₉

yes

characteristic

112

C₂×C₁₆

—

113

C₂×C₁₆

—

114

#97

—

115

#97

—

116

C₄×C₈

—

117

C₄×C₈

—

118

C₄×C₈

Dic₃₈

yes

characteristic

119

C₄×C₈

C₄×D₁₉

yes

characteristic

120

C₄×C₈

C₂×Dic₁₉

yes

characteristic

121

C₄×C₈

C₂²⋊D₁₉

yes

characteristic

122

C₁₉

#7149

yes

characteristic, complemented, Sylow

123

C₂²×C₄

—

124

C₂²×C₄

C₈×D₁₉

yes

characteristic

125

C₂²×C₄

#64515

yes

characteristic

126

C₂²×C₄

(C₂×D₁₉)⋊C₄

yes

characteristic

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

Dic₁₉⋊C₄

yes

characteristic

133

C₂×C₈

C₄⋊Dic₁₉

yes

characteristic

134

C₂×C₈

(C₂×D₁₉)⋊C₄

yes

characteristic

135

C₂×C₈

C₂²⋊Dic₁₉

yes

characteristic

136

C₄²

—

137

C₄²

—

138

C₄²

—

139

C₄²

—

140

C₄²

C₈×D₁₉

yes

characteristic

141

C₄²

#64515

yes

characteristic

142

C₄²

C₄×Dic₁₉

yes

characteristic

143

C₄²

Dic₁₉⋊C₄

yes

characteristic

144

C₁₆

—

145

C₁₆

—

146

C₂³

(C₂×D₁₉)⋊C₈

yes

characteristic

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

C₈×Dic₁₉

yes

characteristic

155

C₂×C₄

Dic₁₉⋊C₈

yes

characteristic

156

C₂×C₄

Dic₁₉⋊C₈

yes

characteristic

157

C₂×C₄

#10597272

yes

characteristic

158

C₂×C₄

(C₂×D₁₉)⋊C₈

yes

characteristic

159

C₂×C₄

C₁₉⋊#95

yes

characteristic

160

C₈

—

161

C₈

—

162

C₈

—

163

C₈

#10597282

yes

—

164

C₈

#10597294

yes

—

165

C₂²

—

166

C₂²

—

167

C₂²

#11926298

yes

characteristic

168

C₂²

#11926325

yes

—

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

#11926303

yes

characteristic

175

C₄

#11926305

yes

characteristic

176

C₄

#11926321

yes

—

177

C₂

—

178

C₂

#14561467

yes

characteristic

179

C₂

#14561714

yes

characteristic

180

C₂

#14561718

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₁₉⋊#95
commutator	C₂×C₄×C₁₉	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₁₉
Fitting	C₁₉×#2163	C₂
socle	C₂²×C₁₉	#336
prime radical	C₂³×C₁₉	C₂²⋊C₈
cosocle	C₂×C₄×C₈×C₁₉	C₂²
hypercenter	#2163	D₁₉
hypocenter	C₁₉	#7149
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2163

subgroup

G/H

center

C₂×C₄

C₁₉⋊#95

commutator

C₂×C₄×C₁₉

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₁₉

Fitting

C₁₉×#2163

C₂

socle

C₂²×C₁₉

#336

prime radical

C₂³×C₁₉

C₂²⋊C₈

cosocle

C₂×C₄×C₈×C₁₉

C₂²

D₁₉

C₁₉

G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2163

N	H ≅ G/N	type
8. C₄⋊(C₁₉⋊C₁₆)	C₄	semidirect
12. C₁₉×C₄⋊C₁₆	C₄	semidirect
96. C₄⋊C₁₆	Dic₁₉	semidirect
122. C₁₉	#7149	semidirect

H ≅ G/N

type

8. C₄⋊(C₁₉⋊C₁₆)

C₄

semidirect

12. C₁₉×C₄⋊C₁₆

C₄

semidirect

96. C₄⋊C₁₆

Dic₁₉

semidirect

122. C₁₉

#7149

semidirect

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

subgroup	index
#14565438	2
C₄×#10597272	2
C₁₉×#2163	2
#7149	19

subgroup

index

#14565438

C₄×#10597272

C₁₉×#2163

#7149

Subgroups — #18069396

Conjugacy classes of subgroups

Complements

Maximal subgroups