Group #18069553 — subgroups

Subgroups — #18069553

Order 4864.

222 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 119 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	#14565434	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₁₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	C₈×M₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₉⋊C₁₆	C₄	1	2	yes	—
5	C₄⋊(C₁₉⋊C₁₆)	—	2	1	no	—
6	C₄⋊(C₁₉⋊C₁₆)	—	2	1	no	—
7	C₄⋊(C₁₉⋊C₁₆)	—	2	1	no	—
8	C₂²⋊(C₁₉⋊C₁₆)	C₄	1	2	yes	—
9	C₄×C₁₉⋊C₁₆	—	2	1	no	—
10	C₁₉×#431	C₄	1	1	yes	characteristic
11	C₄×M₁₆×C₁₉	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
13	C₁₉×#322	—	2	1	no	—
14	C₈²×C₁₉	—	2	1	no	—
15	C₂×M₁₆×C₁₉	C₈	1	2	yes	—
16	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
19	C₁₉×C₄⋊C₈	—	2	1	no	—
20	C₁₉×C₂²⋊C₈	C₈	1	2	yes	—
21	C₁₉×#97	—	2	1	no	—
22	C₄×C₈×C₁₉	—	2	1	no	—
23	C₄×C₈×C₁₉	—	2	1	no	—
24	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
25	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
26	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
27	C₄×C₈×C₁₉	Q₈	1	1	yes	characteristic
28	C₂×C₁₉⋊C₁₆	—	2	2	no	—
29	C₂×C₁₉⋊C₁₆	—	2	2	no	—
30	C₂×C₁₉⋊C₁₆	—	4	2	no	—
31	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
32	C₂²×C₄×C₁₉	C₂×C₈	1	1	yes	characteristic
33	C₂²×C₄×C₁₉	C₂×C₈	1	1	yes	characteristic
34	M₁₆×C₁₉	—	2	2	no	—
35	M₁₆×C₁₉	C₁₆	1	4	yes	complemented
36	C₂×C₈×C₁₉	—	2	2	no	—
37	C₂×C₈×C₁₉	—	2	1	no	—
38	C₂×C₈×C₁₉	—	2	1	no	—
39	C₂×C₈×C₁₉	—	2	2	no	—
40	C₂×C₈×C₁₉	—	4	1	no	—
41	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
45	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₄²×C₁₉	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₁₉	M₁₆	1	1	yes	characteristic
48	C₄²×C₁₉	M₁₆	1	1	yes	characteristic
49	C₁₉⋊C₁₆	—	4	2	no	—
50	C₁₉⋊C₁₆	—	4	2	no	—
51	#7209	—	19	1	no	maximal, Sylow
52	C₂³×C₁₉	C₄×C₈	1	1	yes	characteristic
53	C₂×C₄×C₁₉	—	2	1	no	—
54	C₂×C₄×C₁₉	—	2	1	no	—
55	C₂×C₄×C₁₉	—	4	1	no	—
56	C₂×C₄×C₁₉	#95	1	1	yes	characteristic
57	C₂×C₄×C₁₉	#97	1	1	yes	characteristic
58	C₂×C₄×C₁₉	C₂²⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₁₉	C₂²⋊C₈	1	1	yes	characteristic
60	C₂×C₄×C₁₉	C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₁₉	C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₁₉	C₂×C₁₆	1	2	yes	—
63	C₂×C₄×C₁₉	M₃₂	1	2	yes	—
64	C₈×C₁₉	—	2	1	no	—
65	C₈×C₁₉	—	2	1	no	—
66	C₈×C₁₉	—	2	1	no	—
67	C₈×C₁₉	—	2	4	no	—
68	C₈×C₁₉	—	4	2	no	—
69	C₈×C₁₉	#100	1	1	yes	characteristic
70	C₈×C₁₉	#101	1	1	yes	characteristic
71	#2162	—	19	1	no	—
72	C₂×C₄⋊C₁₆	—	19	1	no	—
73	C₈×M₁₆	D₁₉	1	1	yes	characteristic
74	C₂²×C₁₉	—	2	1	no	—
75	C₂²×C₁₉	—	2	1	no	—
76	C₂²×C₁₉	#336	1	1	yes	characteristic
77	C₂²×C₁₉	C₄×C₁₆	1	1	yes	characteristic
78	C₂²×C₁₉	#346	1	1	yes	characteristic
79	C₄×C₁₉	—	2	1	no	—
80	C₄×C₁₉	—	2	1	no	—
81	C₄×C₁₉	—	2	1	no	—
82	C₄×C₁₉	#343	1	1	yes	characteristic
83	C₄×C₁₉	#344	1	1	yes	characteristic
84	C₄×C₁₉	C₂²⋊C₁₆	1	2	yes	—
85	C₄×C₁₉	C₄⋊C₁₆	1	2	yes	—
86	C₂²×C₁₆	—	19	2	no	—
87	#431	Dic₁₉	1	1	yes	characteristic
88	C₄×M₁₆	Dic₁₉	1	1	yes	characteristic
89	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
90	C₄⋊C₁₆	—	38	1	no	—
91	C₄⋊C₁₆	—	38	1	no	—
92	C₄⋊C₁₆	—	38	1	no	—
93	C₂²⋊C₁₆	—	19	2	no	—
94	C₄×C₁₆	—	38	1	no	—
95	#322	—	2	1	no	—
96	C₈²	—	2	1	no	—
97	C₂×C₁₉	—	2	1	no	—
98	C₂×C₁₉	—	2	1	no	—
99	C₂×C₁₉	#1283	1	1	yes	characteristic
100	C₂×C₁₉	#1374	1	1	yes	characteristic
101	C₂×C₁₉	#1378	1	1	yes	characteristic
102	C₂×M₁₆	C₁₉⋊C₈	1	2	yes	—
103	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
104	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
105	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
106	C₂×C₁₆	—	38	2	no	—
107	C₂×C₁₆	—	38	2	no	—
108	C₂×C₁₆	—	76	2	no	—
109	C₄⋊C₈	—	2	1	no	—
110	C₂²⋊C₈	C₁₉⋊C₈	1	2	yes	—
111	#97	—	2	1	no	—
112	C₄×C₈	—	2	1	no	—
113	C₄×C₈	—	2	1	no	—
114	C₄×C₈	Dic₃₈	1	1	yes	characteristic
115	C₄×C₈	D₇₆	1	1	yes	characteristic
116	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
117	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
118	C₁₉	#7209	1	1	yes	characteristic, complemented, Sylow
119	C₂²×C₄	C₂×C₁₉⋊C₈	1	1	yes	characteristic
120	C₂²×C₄	C₂×C₁₉⋊C₈	1	1	yes	characteristic
121	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
122	M₁₆	—	2	2	no	—
123	M₁₆	C₁₉⋊C₁₆	1	4	yes	complemented
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	4	1	no	—
129	C₂×C₈	Dic₁₉⋊C₄	1	1	yes	characteristic
130	C₂×C₈	Dic₁₉⋊C₄	1	1	yes	characteristic
131	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
132	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
133	C₄²	#64520	1	1	yes	characteristic
134	C₄²	#64520	1	1	yes	characteristic
135	C₄²	C₄⋊Dic₁₉	1	1	yes	characteristic
136	C₄²	C₂²⋊Dic₁₉	1	1	yes	characteristic
137	C₁₆	—	76	2	no	—
138	C₁₆	—	76	2	no	—
139	C₂³	C₄×C₁₉⋊C₈	1	1	yes	characteristic
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	#10597260	1	1	yes	characteristic
144	C₂×C₄	C₄⋊(C₁₉⋊C₈)	1	1	yes	characteristic
145	C₂×C₄	C₄⋊(C₁₉⋊C₈)	1	1	yes	characteristic
146	C₂×C₄	C₂×C₁₉⋊C₁₆	1	2	yes	—
147	C₂×C₄	#10597269	1	2	yes	—
148	C₂×C₄	C₂²⋊(C₁₉⋊C₈)	1	1	yes	characteristic
149	C₂×C₄	C₂²⋊(C₁₉⋊C₈)	1	1	yes	characteristic
150	C₂×C₄	C₁₉⋊#95	1	1	yes	characteristic
151	C₈	—	2	1	no	—
152	C₈	—	2	1	no	—
153	C₈	—	2	1	no	—
154	C₈	—	2	4	no	—
155	C₈	—	4	2	no	—
156	C₈	#10597280	1	1	yes	characteristic
157	C₈	#10597281	1	1	yes	characteristic
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	#11926299	1	1	yes	characteristic
161	C₂²	C₄×C₁₉⋊C₁₆	1	1	yes	characteristic
162	C₂²	#11926320	1	1	yes	characteristic
163	C₄	—	2	1	no	—
164	C₄	—	2	1	no	—
165	C₄	—	2	1	no	—
166	C₄	#11926312	1	1	yes	characteristic
167	C₄	#11926316	1	1	yes	characteristic
168	C₄	C₂²⋊(C₁₉⋊C₁₆)	1	2	yes	—
169	C₄	C₄⋊(C₁₉⋊C₁₆)	1	2	yes	—
170	C₂	—	2	1	no	—
171	C₂	—	2	1	no	—
172	C₂	#14561486	1	1	yes	characteristic
173	C₂	#14561713	1	1	yes	characteristic
174	C₂	#14561721	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14565434

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₁₉⋊C₁₆)

C₂

yes

maximal, characteristic

C₈×M₁₆×C₁₉

C₂

yes

maximal, characteristic

C₂²×C₁₉⋊C₁₆

C₄

yes

—

C₄⋊(C₁₉⋊C₁₆)

—

C₄⋊(C₁₉⋊C₁₆)

—

C₄⋊(C₁₉⋊C₁₆)

—

C₂²⋊(C₁₉⋊C₁₆)

C₄

yes

—

C₄×C₁₉⋊C₁₆

—

C₁₉×#431

C₄

yes

characteristic

C₄×M₁₆×C₁₉

C₄

yes

characteristic

C₂×C₄×C₈×C₁₉

C₂²

yes

characteristic

C₁₉×#322

—

C₈²×C₁₉

—

C₂×M₁₆×C₁₉

C₈

yes

—

C₂²×C₈×C₁₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₉

C₂×C₄

yes

characteristic

C₁₉×C₄⋊C₈

—

C₁₉×C₂²⋊C₈

C₈

yes

—

C₁₉×#97

—

C₄×C₈×C₁₉

—

C₄×C₈×C₁₉

—

C₄×C₈×C₁₉

D₄

yes

characteristic

C₄×C₈×C₁₉

D₄

yes

characteristic

C₄×C₈×C₁₉

D₄

yes

characteristic

C₄×C₈×C₁₉

Q₈

yes

characteristic

C₂×C₁₉⋊C₁₆

—

C₂×C₁₉⋊C₁₆

—

C₂×C₁₉⋊C₁₆

—

C₂²×C₄×C₁₉

C₄²

yes

characteristic

C₂²×C₄×C₁₉

C₂×C₈

yes

characteristic

C₂²×C₄×C₁₉

C₂×C₈

yes

characteristic

M₁₆×C₁₉

—

M₁₆×C₁₉

C₁₆

yes

complemented

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

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

C₂²⋊C₄

yes

characteristic

C₄²×C₁₉

C₄⋊C₄

yes

characteristic

C₄²×C₁₉

M₁₆

yes

characteristic

C₄²×C₁₉

M₁₆

yes

characteristic

C₁₉⋊C₁₆

—

C₁₉⋊C₁₆

—

#7209

—

maximal, Sylow

C₂³×C₁₉

C₄×C₈

yes

characteristic

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₁₉

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₁₉

C₄⋊C₈

yes

characteristic

C₂×C₄×C₁₉

C₄⋊C₈

yes

characteristic

C₂×C₄×C₁₉

C₂×C₁₆

yes

—

C₂×C₄×C₁₉

M₃₂

yes

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

yes

C₈×C₁₉

yes

—

C₂×C₄⋊C₁₆

—

C₈×M₁₆

D₁₉

yes

characteristic

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#336

yes

characteristic

C₂²×C₁₉

C₄×C₁₆

yes

characteristic

C₂²×C₁₉

#346

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

yes

C₄×C₁₉

yes

C₄×C₁₉

C₂²⋊C₁₆

yes

—

C₄×C₁₉

C₄⋊C₁₆

yes

—

C₂²×C₁₆

—

#431

Dic₁₉

yes

characteristic

C₄×M₁₆

Dic₁₉

yes

characteristic

C₂×C₄×C₈

C₂×D₁₉

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

#322

—

C₈²

—

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

#1283

yes

characteristic

100

C₂×C₁₉

#1374

yes

characteristic

101

C₂×C₁₉

#1378

yes

characteristic

102

C₂×M₁₆

C₁₉⋊C₈

yes

—

103

C₂²×C₈

C₄×D₁₉

yes

characteristic

104

C₂²×C₈

C₄×D₁₉

yes

characteristic

105

C₂×C₄²

C₂×Dic₁₉

yes

characteristic

106

C₂×C₁₆

—

107

C₂×C₁₆

—

108

C₂×C₁₆

—

109

C₄⋊C₈

—

110

C₂²⋊C₈

C₁₉⋊C₈

yes

—

111

#97

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

Dic₃₈

yes

characteristic

115

C₄×C₈

D₇₆

yes

characteristic

116

C₄×C₈

C₂²⋊D₁₉

yes

characteristic

117

C₄×C₈

C₂²⋊D₁₉

yes

characteristic

118

C₁₉

#7209

yes

characteristic, complemented, Sylow

119

C₂²×C₄

C₂×C₁₉⋊C₈

yes

characteristic

120

C₂²×C₄

C₂×C₁₉⋊C₈

yes

characteristic

121

C₂²×C₄

C₄×Dic₁₉

yes

characteristic

122

M₁₆

—

123

M₁₆

C₁₉⋊C₁₆

yes

complemented

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

Dic₁₉⋊C₄

yes

characteristic

130

C₂×C₈

Dic₁₉⋊C₄

yes

characteristic

131

C₂×C₈

(C₂×D₁₉)⋊C₄

yes

characteristic

132

C₂×C₈

(C₂×D₁₉)⋊C₄

yes

characteristic

133

C₄²

#64520

yes

characteristic

134

C₄²

#64520

yes

characteristic

135

C₄²

C₄⋊Dic₁₉

yes

characteristic

136

C₄²

C₂²⋊Dic₁₉

yes

characteristic

137

C₁₆

—

138

C₁₆

—

139

C₂³

C₄×C₁₉⋊C₈

yes

characteristic

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

#10597260

yes

characteristic

144

C₂×C₄

C₄⋊(C₁₉⋊C₈)

yes

characteristic

145

C₂×C₄

C₄⋊(C₁₉⋊C₈)

yes

characteristic

146

C₂×C₄

C₂×C₁₉⋊C₁₆

yes

—

147

C₂×C₄

#10597269

yes

—

148

C₂×C₄

C₂²⋊(C₁₉⋊C₈)

yes

characteristic

149

C₂×C₄

C₂²⋊(C₁₉⋊C₈)

yes

characteristic

150

C₂×C₄

C₁₉⋊#95

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

—

155

C₈

—

156

C₈

#10597280

yes

characteristic

157

C₈

#10597281

yes

characteristic

158

C₂²

—

159

C₂²

—

160

C₂²

#11926299

yes

characteristic

161

C₂²

C₄×C₁₉⋊C₁₆

yes

characteristic

162

C₂²

#11926320

yes

characteristic

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

#11926312

yes

characteristic

167

C₄

#11926316

yes

characteristic

168

C₄

C₂²⋊(C₁₉⋊C₁₆)

yes

—

169

C₄

C₄⋊(C₁₉⋊C₁₆)

yes

—

170

C₂

—

171

C₂

—

172

C₂

#14561486

yes

characteristic

173

C₂

#14561713

yes

characteristic

174

C₂

#14561721

yes

characteristic

	subgroup	G/H
center	C₂×C₈	(C₂×D₁₉)⋊C₄
commutator	C₂²×C₁₉	C₄×C₁₆
Frattini	C₂×C₄×C₈	C₂×D₁₉
Fitting	C₈×M₁₆×C₁₉	C₂
socle	C₂²×C₁₉	#336
prime radical	C₂³×C₁₉	C₄×C₈
cosocle	C₂×C₄×C₈×C₁₉	C₂²
hypercenter	C₈×M₁₆	D₁₉
hypocenter	C₁₉	#7209
LCS	G ▷ C₂²×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₈×M₁₆

subgroup

G/H

center

C₂×C₈

(C₂×D₁₉)⋊C₄

commutator

C₂²×C₁₉

C₄×C₁₆

Frattini

C₂×C₄×C₈

C₂×D₁₉

Fitting

C₈×M₁₆×C₁₉

C₂

socle

C₂²×C₁₉

#336

prime radical

C₂³×C₁₉

C₄×C₈

cosocle

C₂×C₄×C₈×C₁₉

C₂²

hypercenter

C₈×M₁₆

D₁₉

hypocenter

C₁₉

#7209

LCS

G ▷ C₂²×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₈×M₁₆

N	H ≅ G/N	type
35. M₁₆×C₁₉	C₁₆	semidirect
118. C₁₉	#7209	semidirect
123. M₁₆	C₁₉⋊C₁₆	semidirect

H ≅ G/N

type

35. M₁₆×C₁₉

C₁₆

semidirect

118. C₁₉

#7209

semidirect

123. M₁₆

C₁₉⋊C₁₆

semidirect

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

subgroup	index
#14565434	2
C₂×C₄⋊(C₁₉⋊C₁₆)	2
C₈×M₁₆×C₁₉	2
#7209	19

subgroup

index

#14565434

C₂×C₄⋊(C₁₉⋊C₁₆)

C₈×M₁₆×C₁₉

#7209

Subgroups — #18069553

Conjugacy classes of subgroups

Complements

Maximal subgroups