Group #18069162 — subgroups

Subgroups — #18069162

Order 4864.

248 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 59 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	#14561575	C₂	1	2	yes	maximal, complemented
2	C₁₉×#1954	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊(C₁₉⋊C₁₆)	—	2	2	no	—
4	C₂×C₄×D₄×C₁₉	C₄	1	1	yes	characteristic
5	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
6	C₂×C₁₉×C₂²⋊C₈	—	2	2	no	—
7	C₂×C₄×C₈×C₁₉	C₄	1	1	yes	characteristic
8	C₂²×D₄×C₁₉	C₈	1	1	yes	characteristic
9	C₂³×C₄×C₁₉	—	2	1	no	—
10	C₂²×C₈×C₁₉	—	2	1	no	—
11	C₂²×C₈×C₁₉	D₄	1	2	yes	—
12	C₄×D₄×C₁₉	—	4	1	no	—
13	C₄×D₄×C₁₉	—	4	1	no	—
14	C₂×C₄⋊C₄×C₁₉	C₈	1	1	yes	characteristic
15	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
16	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
17	C₁₉×C₄⋊C₈	—	4	1	no	—
18	C₁₉×C₂²⋊C₈	—	4	4	no	—
19	C₄×C₈×C₁₉	—	4	1	no	—
20	C₂×C₁₉⋊C₁₆	—	4	2	no	—
21	C₂⁴×C₁₉	—	2	1	no	—
22	C₂×D₄×C₁₉	—	2	1	no	—
23	C₂×D₄×C₁₉	—	4	1	no	—
24	C₂×D₄×C₁₉	—	4	1	no	—
25	C₂×D₄×C₁₉	C₁₆	1	2	yes	complemented
26	C₂²×C₄×C₁₉	—	2	1	no	—
27	C₂²×C₄×C₁₉	—	4	2	no	—
28	C₂²×C₄×C₁₉	—	4	2	no	—
29	C₂²×C₄×C₁₉	—	4	1	no	—
30	C₂²×C₄×C₁₉	—	4	1	no	—
31	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₉	C₂×C₈	1	1	yes	characteristic
33	C₂²×C₄×C₁₉	M₁₆	1	1	yes	characteristic
34	C₂×C₈×C₁₉	—	2	2	no	—
35	C₂×C₈×C₁₉	—	2	2	no	—
36	C₂×C₈×C₁₉	—	4	1	no	—
37	C₂×C₈×C₁₉	—	4	1	no	—
38	C₂×C₈×C₁₉	—	4	2	no	—
39	C₄⋊C₄×C₁₉	—	4	1	no	—
40	C₂²⋊C₄×C₁₉	—	4	2	no	—
41	C₄²×C₁₉	—	2	1	no	—
42	C₄²×C₁₉	—	2	1	no	—
43	C₁₉⋊C₁₆	—	8	2	no	—
44	#7056	—	19	1	no	maximal, Sylow
45	C₂³×C₁₉	—	2	2	no	—
46	C₂³×C₁₉	—	4	2	no	—
47	C₂³×C₁₉	—	4	2	no	—
48	C₂³×C₁₉	—	4	1	no	—
49	C₂³×C₁₉	—	4	1	no	—
50	C₂³×C₁₉	C₂²⋊C₈	1	1	yes	characteristic
51	D₄×C₁₉	—	4	2	no	—
52	D₄×C₁₉	—	4	2	no	—
53	C₂×C₄×C₁₉	—	2	1	no	—
54	C₂×C₄×C₁₉	—	2	1	no	—
55	C₂×C₄×C₁₉	—	2	1	no	—
56	C₂×C₄×C₁₉	—	2	1	no	—
57	C₂×C₄×C₁₉	—	2	1	no	—
58	C₂×C₄×C₁₉	—	4	1	no	—
59	C₂×C₄×C₁₉	—	4	2	no	—
60	C₂×C₄×C₁₉	—	4	1	no	—
61	C₂×C₄×C₁₉	—	8	2	no	—
62	C₂×C₄×C₁₉	—	8	1	no	—
63	C₂×C₄×C₁₉	—	8	1	no	—
64	C₂×C₄×C₁₉	#99	1	1	yes	characteristic
65	C₂×C₄×C₁₉	#100	1	1	yes	characteristic
66	C₂×C₄×C₁₉	C₂×C₁₆	1	1	yes	characteristic
67	C₂×C₄×C₁₉	M₃₂	1	1	yes	characteristic
68	C₈×C₁₉	—	4	2	no	—
69	C₈×C₁₉	—	4	2	no	—
70	#1954	D₁₉	1	1	yes	characteristic
71	#1322	—	19	2	no	—
72	C₂²×C₁₉	—	2	1	no	—
73	C₂²×C₁₉	—	2	1	no	—
74	C₂²×C₁₉	—	4	4	no	—
75	C₂²×C₁₉	—	4	2	no	—
76	C₂²×C₁₉	—	8	2	no	—
77	C₂²×C₁₉	—	8	1	no	—
78	C₂²×C₁₉	—	8	1	no	—
79	C₂²×C₁₉	#323	1	1	yes	characteristic
80	C₂²×C₁₉	C₂²⋊C₁₆	1	1	yes	characteristic
81	C₂²×C₁₉	#349	1	1	yes	characteristic
82	C₄×C₁₉	—	2	1	no	—
83	C₄×C₁₉	—	2	1	no	—
84	C₄×C₁₉	—	2	1	no	—
85	C₄×C₁₉	—	4	1	no	—
86	C₄×C₁₉	—	8	1	no	—
87	C₄×C₁₉	#353	1	1	yes	characteristic
88	C₄×C₁₉	#355	1	1	yes	characteristic
89	C₂×C₄×D₄	Dic₁₉	1	1	yes	characteristic
90	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
91	C₂×C₂²⋊C₈	—	2	2	no	—
92	C₂×C₄×C₈	Dic₁₉	1	1	yes	characteristic
93	C₂²⋊C₁₆	—	38	2	no	—
94	C₂×C₁₉	—	2	1	no	—
95	C₂×C₁₉	—	2	1	no	—
96	C₂×C₁₉	—	8	2	no	—
97	C₂×C₁₉	#1314	1	1	yes	characteristic
98	C₂×C₁₉	#1318	1	1	yes	characteristic
99	C₂×C₁₉	#1321	1	1	yes	characteristic
100	C₂²×D₄	C₁₉⋊C₈	1	1	yes	characteristic
101	C₂³×C₄	—	2	1	no	—
102	C₂²×C₈	—	2	1	no	—
103	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
104	C₄×D₄	—	4	1	no	—
105	C₄×D₄	—	4	1	no	—
106	C₂×C₄⋊C₄	C₁₉⋊C₈	1	1	yes	characteristic
107	C₂×C₂²⋊C₄	—	2	1	no	—
108	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
109	C₂×C₁₆	—	76	2	no	—
110	C₄⋊C₈	—	4	1	no	—
111	C₂²⋊C₈	—	4	4	no	—
112	C₄×C₈	—	4	1	no	—
113	C₁₉	#7056	1	1	yes	characteristic, complemented, Sylow
114	C₂⁴	—	2	1	no	—
115	C₂×D₄	—	2	1	no	—
116	C₂×D₄	—	4	1	no	—
117	C₂×D₄	—	4	1	no	—
118	C₂×D₄	C₁₉⋊C₁₆	1	2	yes	complemented
119	C₂²×C₄	—	2	1	no	—
120	C₂²×C₄	—	4	2	no	—
121	C₂²×C₄	—	4	2	no	—
122	C₂²×C₄	—	4	1	no	—
123	C₂²×C₄	—	4	1	no	—
124	C₂²×C₄	C₂×C₁₉⋊C₈	1	1	yes	characteristic
125	C₂²×C₄	#64520	1	1	yes	characteristic
126	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	4	1	no	—
130	C₂×C₈	—	4	1	no	—
131	C₂×C₈	—	4	2	no	—
132	C₄⋊C₄	—	4	1	no	—
133	C₂²⋊C₄	—	4	2	no	—
134	C₄²	—	2	1	no	—
135	C₄²	—	2	1	no	—
136	C₁₆	—	152	2	no	—
137	C₂³	—	2	2	no	—
138	C₂³	—	4	2	no	—
139	C₂³	—	4	2	no	—
140	C₂³	—	4	1	no	—
141	C₂³	—	4	1	no	—
142	C₂³	C₂²⋊(C₁₉⋊C₈)	1	1	yes	characteristic
143	D₄	—	4	2	no	—
144	D₄	—	4	2	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	—	8	2	no	—
154	C₂×C₄	—	8	1	no	—
155	C₂×C₄	—	8	1	no	—
156	C₂×C₄	C₂×C₁₉⋊C₁₆	1	1	yes	characteristic
157	C₂×C₄	#10597269	1	1	yes	characteristic
158	C₂×C₄	#10597290	1	1	yes	characteristic
159	C₂×C₄	#10597291	1	1	yes	characteristic
160	C₈	—	4	2	no	—
161	C₈	—	4	2	no	—
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	4	4	no	—
165	C₂²	—	4	2	no	—
166	C₂²	—	8	2	no	—
167	C₂²	—	8	1	no	—
168	C₂²	—	8	1	no	—
169	C₂²	#11926268	1	1	yes	characteristic
170	C₂²	C₂²⋊(C₁₉⋊C₁₆)	1	1	yes	characteristic
171	C₂²	#11926326	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	4	1	no	—
176	C₄	—	8	1	no	—
177	C₄	#11926337	1	1	yes	characteristic
178	C₄	#11926341	1	1	yes	characteristic
179	C₂	—	2	1	no	—
180	C₂	—	2	1	no	—
181	C₂	—	8	2	no	—
182	C₂	#14561559	1	1	yes	characteristic
183	C₂	#14561567	1	1	yes	characteristic
184	C₂	#14561573	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14561575

C₂

yes

maximal, complemented

C₁₉×#1954

C₂

yes

maximal, characteristic

C₂²⋊(C₁₉⋊C₁₆)

—

C₂×C₄×D₄×C₁₉

C₄

yes

characteristic

C₂×C₁₉×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₁₉×C₂²⋊C₈

—

C₂×C₄×C₈×C₁₉

C₄

yes

characteristic

C₂²×D₄×C₁₉

C₈

yes

characteristic

C₂³×C₄×C₁₉

—

C₂²×C₈×C₁₉

—

C₂²×C₈×C₁₉

D₄

yes

—

C₄×D₄×C₁₉

—

C₄×D₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

C₈

yes

characteristic

C₂×C₂²⋊C₄×C₁₉

—

C₂×C₄²×C₁₉

C₂×C₄

yes

characteristic

C₁₉×C₄⋊C₈

—

C₁₉×C₂²⋊C₈

—

C₄×C₈×C₁₉

—

C₂×C₁₉⋊C₁₆

—

C₂⁴×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

C₁₆

yes

complemented

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

C₂×C₈

yes

characteristic

C₂²×C₄×C₁₉

M₁₆

yes

characteristic

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

—

C₁₉⋊C₁₆

—

#7056

—

maximal, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

C₂²⋊C₈

yes

characteristic

D₄×C₁₉

—

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

C₂×C₄×C₁₉

C₂×C₁₆

yes

characteristic

C₂×C₄×C₁₉

M₃₂

yes

characteristic

C₈×C₁₉

—

C₈×C₁₉

—

#1954

D₁₉

yes

characteristic

#1322

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#323

yes

characteristic

C₂²×C₁₉

C₂²⋊C₁₆

yes

characteristic

C₂²×C₁₉

#349

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

yes

C₄×C₁₉

yes

C₂×C₄×D₄

Dic₁₉

yes

characteristic

C₂×C₄⋊C₈

C₂×D₁₉

yes

characteristic

C₂×C₂²⋊C₈

—

C₂×C₄×C₈

Dic₁₉

yes

characteristic

C₂²⋊C₁₆

—

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

100

C₂²×D₄

C₁₉⋊C₈

yes

characteristic

101

C₂³×C₄

—

102

C₂²×C₈

—

103

C₂²×C₈

C₂²⋊D₁₉

yes

—

104

C₄×D₄

—

105

C₄×D₄

—

106

C₂×C₄⋊C₄

C₁₉⋊C₈

yes

characteristic

107

C₂×C₂²⋊C₄

—

108

C₂×C₄²

C₂×Dic₁₉

yes

characteristic

109

C₂×C₁₆

—

110

C₄⋊C₈

—

111

C₂²⋊C₈

—

112

C₄×C₈

—

113

C₁₉

#7056

yes

characteristic, complemented, Sylow

114

C₂⁴

—

115

C₂×D₄

—

116

C₂×D₄

—

117

C₂×D₄

—

118

C₂×D₄

C₁₉⋊C₁₆

yes

complemented

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

C₂×C₁₉⋊C₈

yes

characteristic

125

C₂²×C₄

#64520

yes

characteristic

126

C₂²×C₄

C₂²⋊Dic₁₉

yes

characteristic

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₄⋊C₄

—

133

C₂²⋊C₄

—

134

C₄²

—

135

C₄²

—

136

C₁₆

—

152

—

137

C₂³

—

138

C₂³

—

139

C₂³

—

140

C₂³

—

141

C₂³

—

142

C₂³

C₂²⋊(C₁₉⋊C₈)

yes

characteristic

143

D₄

—

144

D₄

—

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

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

C₂×C₁₉⋊C₁₆

yes

characteristic

157

C₂×C₄

#10597269

yes

characteristic

158

C₂×C₄

#10597290

yes

characteristic

159

C₂×C₄

#10597291

yes

characteristic

160

C₈

—

161

C₈

—

162

C₂²

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

—

169

C₂²

#11926268

yes

characteristic

170

C₂²

C₂²⋊(C₁₉⋊C₁₆)

yes

characteristic

171

C₂²

#11926326

yes

characteristic

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

#11926337

yes

characteristic

178

C₄

#11926341

yes

characteristic

179

C₂

—

180

C₂

—

181

C₂

—

182

C₂

#14561559

yes

characteristic

183

C₂

#14561567

yes

characteristic

184

C₂

#14561573

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#10597291
commutator	C₂×C₄×C₁₉	C₂×C₁₆
Frattini	C₂×C₄⋊C₈	C₂×D₁₉
Fitting	C₁₉×#1954	C₂
socle	C₂²×C₁₉	#323
prime radical	C₂²×D₄×C₁₉	C₈
cosocle	C₂×C₁₉×C₄⋊C₈	C₂²
hypercenter	#1954	D₁₉
hypocenter	C₁₉	#7056
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈ ◁ #1954

subgroup

G/H

center

C₂×C₄

#10597291

commutator

C₂×C₄×C₁₉

C₂×C₁₆

Frattini

C₂×C₄⋊C₈

C₂×D₁₉

Fitting

C₁₉×#1954

C₂

socle

C₂²×C₁₉

#323

prime radical

C₂²×D₄×C₁₉

C₈

cosocle

C₂×C₁₉×C₄⋊C₈

C₂²

D₁₉

C₁₉

G ▷ C₂×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈ ◁ #1954

N	H ≅ G/N	type
1. #14561575	C₂	semidirect
25. C₂×D₄×C₁₉	C₁₆	semidirect
113. C₁₉	#7056	semidirect
118. C₂×D₄	C₁₉⋊C₁₆	semidirect

H ≅ G/N

type

1. #14561575

C₂

semidirect

25. C₂×D₄×C₁₉

C₁₆

semidirect

113. C₁₉

#7056

semidirect

118. C₂×D₄

C₁₉⋊C₁₆

semidirect

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

subgroup	index
#14561575	2
C₁₉×#1954	2
#7056	19

subgroup

index

#14561575

C₁₉×#1954

#7056

Subgroups — #18069162

Conjugacy classes of subgroups

Complements

Maximal subgroups