Group #18086703 — subgroups

Subgroups — #18086703

Order 4864.

326 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 77 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	#14563775	C₂	1	1	yes	maximal, characteristic
2	#14562365	C₂	1	2	yes	maximal
3	#14562250	C₂	1	2	yes	maximal
4	#14561546	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#3255	C₂	1	1	yes	maximal, characteristic
6	#11926601	C₂²	1	2	yes	—
7	#11926568	C₂²	1	1	yes	characteristic
8	C₂×#10597260	C₂²	1	1	yes	characteristic
9	#11926308	—	2	2	no	—
10	#11926293	—	2	2	no	—
11	#11926290	—	2	2	no	—
12	#11926286	—	2	4	no	—
13	C₁₉×#533	C₂²	1	2	yes	—
14	C₁₉×#475	—	2	2	no	—
15	C₁₉×#470	—	2	2	no	—
16	C₁₉×#462	—	2	2	no	—
17	C₁₉×#423	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₁₉	—	2	2	no	—
19	C₂×M₁₆×C₁₉	D₄	1	2	yes	—
20	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
21	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
22	C₁₉×#125	—	2	2	no	—
23	C₁₉×C₂²⋊Q₈	—	2	2	no	—
24	C₁₉×C₂²⋊Q₈	—	2	2	no	—
25	C₄×Q₈×C₁₉	—	2	2	no	—
26	C₄×Q₈×C₁₉	—	2	2	no	—
27	C₁₉×#117	—	2	2	no	—
28	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
29	C₂×#64520	—	2	1	no	—
30	C₂²×C₁₉⋊C₈	—	2	1	no	—
31	C₁₉×#106	—	2	2	no	—
32	C₁₉×C₄⋊C₈	—	2	2	no	—
33	C₁₉×#103	—	2	4	no	—
34	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
35	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
36	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
37	#10597260	—	2	1	no	—
38	#10597260	—	2	2	no	—
39	#10597260	D₄	1	2	yes	—
40	C₂×Q₈×C₁₉	—	2	2	no	—
41	C₂×Q₈×C₁₉	—	2	2	no	—
42	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₉	#41	1	2	yes	—
44	Q₁₆×C₁₉	—	8	2	no	—
45	M₁₆×C₁₉	—	4	2	no	—
46	C₂×C₈×C₁₉	—	2	2	no	—
47	C₄⋊C₄×C₁₉	—	2	2	no	—
48	C₄⋊C₄×C₁₉	—	2	2	no	—
49	C₄⋊C₄×C₁₉	—	2	2	no	—
50	C₄⋊C₄×C₁₉	—	2	2	no	—
51	C₄⋊C₄×C₁₉	—	4	2	no	—
52	C₄⋊C₄×C₁₉	—	4	2	no	—
53	C₂²⋊C₄×C₁₉	—	4	2	no	—
54	C₄²×C₁₉	—	4	2	no	—
55	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₁₉	C₂×D₄	1	2	yes	—
57	C₄²×C₁₉	#41	1	1	yes	characteristic
58	#64520	—	4	2	no	—
59	C₂×C₁₉⋊C₈	—	2	2	no	—
60	C₂×C₁₉⋊C₈	—	4	1	no	—
61	C₂×C₁₉⋊C₈	—	4	1	no	—
62	C₂×C₁₉⋊C₈	—	4	1	no	—
63	C₂×C₁₉⋊C₈	—	4	2	no	—
64	#10529	—	19	1	no	maximal, Sylow
65	C₂³×C₁₉	#124	1	1	yes	characteristic
66	Q₈×C₁₉	—	4	2	no	—
67	Q₈×C₁₉	—	4	2	no	—
68	Q₈×C₁₉	—	8	2	no	—
69	C₂×C₄×C₁₉	—	2	1	no	—
70	C₂×C₄×C₁₉	—	2	1	no	—
71	C₂×C₄×C₁₉	—	4	2	no	—
72	C₂×C₄×C₁₉	—	4	2	no	—
73	C₂×C₄×C₁₉	—	4	2	no	—
74	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
75	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
76	C₂×C₄×C₁₉	#121	1	2	yes	—
77	C₂×C₄×C₁₉	#123	1	2	yes	—
78	C₈×C₁₉	—	4	2	no	—
79	C₁₉⋊C₈	—	4	2	no	—
80	C₁₉⋊C₈	—	8	2	no	—
81	C₁₉⋊C₈	—	8	1	no	—
82	#3255	D₁₉	1	1	yes	characteristic
83	#1838	—	19	1	no	—
84	#1559	—	19	2	no	—
85	#1531	—	19	2	no	—
86	#1307	—	19	1	no	—
87	C₂²×C₁₉	—	2	2	no	—
88	C₂²×C₁₉	—	2	1	no	—
89	C₂²×C₁₉	#394	1	1	yes	characteristic
90	C₄×C₁₉	—	2	2	no	—
91	C₄×C₁₉	—	2	2	no	—
92	C₄×C₁₉	—	2	1	no	—
93	C₄×C₁₉	—	8	2	no	—
94	C₄×C₁₉	—	8	2	no	—
95	C₄×C₁₉	#454	1	1	yes	characteristic
96	C₄×C₁₉	#455	1	1	yes	characteristic
97	#533	C₂×D₁₉	1	2	yes	—
98	#475	—	2	2	no	—
99	#470	—	2	2	no	—
100	#462	—	2	2	no	—
101	#432	—	19	2	no	—
102	#423	—	19	1	no	—
103	#423	C₂×D₁₉	1	1	yes	characteristic
104	C₂×#97	—	19	1	no	—
105	#341	—	38	2	no	—
106	#333	—	38	2	no	—
107	#332	—	38	2	no	—
108	#330	—	38	4	no	—
109	C₂×C₁₉	—	4	1	no	—
110	C₂×C₁₉	#2040	1	1	yes	characteristic
111	C₂×C₁₉	#2049	1	2	yes	—
112	C₂×Q₁₆	—	2	2	no	—
113	C₂×M₁₆	—	38	1	no	—
114	C₂×M₁₆	C₂²⋊D₁₉	1	2	yes	—
115	C₂²×C₈	—	38	1	no	—
116	C₄⋊Q₈	C₂²⋊D₁₉	1	2	yes	—
117	C₄⋊Q₈	C₂²⋊D₁₉	1	2	yes	—
118	#125	—	2	2	no	—
119	C₂²⋊Q₈	—	2	2	no	—
120	C₂²⋊Q₈	—	2	2	no	—
121	C₄×Q₈	—	2	2	no	—
122	C₄×Q₈	—	2	2	no	—
123	#117	—	2	2	no	—
124	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
125	#106	—	2	2	no	—
126	C₄⋊C₈	—	2	2	no	—
127	C₄⋊C₈	—	38	2	no	—
128	C₄⋊C₈	—	38	2	no	—
129	#103	—	2	4	no	—
130	C₂²⋊C₈	—	38	2	no	—
131	#97	—	19	2	no	—
132	#97	—	38	2	no	—
133	#97	—	38	1	no	—
134	C₁₉	#10529	1	1	yes	characteristic, complemented, Sylow
135	C₂×Q₈	—	2	2	no	—
136	C₂×Q₈	—	2	2	no	—
137	C₂²×C₄	#64543	1	2	yes	—
138	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
139	Q₁₆	—	8	2	no	—
140	M₁₆	—	4	2	no	—
141	M₁₆	—	76	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	38	2	no	—
144	C₂×C₈	—	76	1	no	—
145	C₂×C₈	—	76	1	no	—
146	C₂×C₈	—	76	2	no	—
147	C₂×C₈	—	76	1	no	—
148	C₄⋊C₄	—	2	2	no	—
149	C₄⋊C₄	—	2	2	no	—
150	C₄⋊C₄	—	2	2	no	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	4	2	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₂²⋊C₄	—	4	2	no	—
155	C₄²	—	4	2	no	—
156	C₄²	D₄×D₁₉	1	1	yes	characteristic
157	C₄²	#64543	1	1	yes	characteristic
158	C₄²	C₂×C₂²⋊D₁₉	1	2	yes	—
159	C₂³	#10597382	1	1	yes	characteristic
160	Q₈	—	4	2	no	—
161	Q₈	—	4	2	no	—
162	Q₈	—	8	2	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	#10597381	1	2	yes	—
169	C₂×C₄	#10597384	1	1	yes	characteristic
170	C₂×C₄	#10597385	1	2	yes	—
171	C₂×C₄	#10597399	1	1	yes	characteristic
172	C₈	—	4	2	no	—
173	C₈	—	76	2	no	—
174	C₈	—	152	2	no	—
175	C₈	—	152	1	no	—
176	C₂²	—	2	2	no	—
177	C₂²	—	2	1	no	—
178	C₂²	#11926458	1	1	yes	characteristic
179	C₄	—	2	2	no	—
180	C₄	—	2	2	no	—
181	C₄	—	2	1	no	—
182	C₄	—	8	2	no	—
183	C₄	—	8	2	no	—
184	C₄	#11926717	1	1	yes	characteristic
185	C₄	#11926722	1	1	yes	characteristic
186	C₂	—	4	1	no	—
187	C₂	#14564861	1	1	yes	characteristic
188	C₂	#14564912	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14563775

C₂

yes

maximal, characteristic

#14562365

C₂

yes

maximal

#14562250

C₂

yes

maximal

#14561546

C₂

yes

maximal, characteristic

C₁₉×#3255

C₂

yes

maximal, characteristic

#11926601

C₂²

yes

—

#11926568

C₂²

yes

characteristic

C₂×#10597260

C₂²

yes

—

—

—

—

C₁₉×#533

C₂²

yes

—

C₁₉×#475

—

C₁₉×#470

—

C₁₉×#462

—

C₁₉×#423

C₂²

yes

characteristic

C₂×Q₁₆×C₁₉

—

C₂×M₁₆×C₁₉

D₄

yes

—

C₁₉×C₄⋊Q₈

D₄

yes

—

C₁₉×C₄⋊Q₈

D₄

yes

—

C₁₉×#125

—

C₁₉×C₂²⋊Q₈

—

C₁₉×C₂²⋊Q₈

—

C₄×Q₈×C₁₉

—

C₄×Q₈×C₁₉

—

C₁₉×#117

—

C₂×C₄²×C₁₉

C₂³

yes

characteristic

C₂×#64520

—

C₂²×C₁₉⋊C₈

—

C₁₉×#106

—

C₁₉×C₄⋊C₈

—

C₁₉×#103

—

C₂²⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

—

#10597260

—

#10597260

—

#10597260

D₄

yes

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂²×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₉

#41

yes

—

Q₁₆×C₁₉

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₄²×C₁₉

C₂×D₄

yes

—

C₄²×C₁₉

#41

yes

characteristic

#64520

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

—

C₂³×C₁₉

yes

Q₈×C₁₉

—

Q₈×C₁₉

—

Q₈×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

yes

—

C₂×C₄×C₁₉

#123

yes

—

C₈×C₁₉

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

#3255

D₁₉

yes

—

—

—

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#394

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

yes

C₄×C₁₉

yes

C₂×D₁₉

yes

—

#475

—

#470

—

100

#462

—

101

#432

—

102

#423

—

103

#423

C₂×D₁₉

yes

characteristic

104

C₂×#97

—

105

#341

—

106

#333

—

107

#332

—

108

#330

—

109

C₂×C₁₉

—

110

C₂×C₁₉

#2040

yes

characteristic

111

C₂×C₁₉

#2049

yes

—

112

C₂×Q₁₆

—

113

C₂×M₁₆

—

114

C₂×M₁₆

C₂²⋊D₁₉

yes

—

115

C₂²×C₈

—

116

C₄⋊Q₈

C₂²⋊D₁₉

yes

—

117

C₄⋊Q₈

C₂²⋊D₁₉

yes

—

118

#125

—

119

C₂²⋊Q₈

—

120

C₂²⋊Q₈

—

121

C₄×Q₈

—

122

C₄×Q₈

—

123

#117

—

124

C₂×C₄²

C₂²×D₁₉

yes

characteristic

125

#106

—

126

C₄⋊C₈

—

127

C₄⋊C₈

—

128

C₄⋊C₈

—

129

#103

—

130

C₂²⋊C₈

—

131

#97

—

132

#97

—

133

#97

—

134

C₁₉

#10529

yes

characteristic, complemented, Sylow

135

C₂×Q₈

—

136

C₂×Q₈

—

137

C₂²×C₄

#64543

yes

—

138

C₂²×C₄

C₂×C₂²⋊D₁₉

yes

characteristic

139

Q₁₆

—

140

M₁₆

—

141

M₁₆

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₂²⋊C₄

—

155

C₄²

—

156

C₄²

D₄×D₁₉

yes

characteristic

157

C₄²

#64543

yes

characteristic

158

C₄²

C₂×C₂²⋊D₁₉

yes

—

159

C₂³

#10597382

yes

characteristic

160

Q₈

—

161

Q₈

—

162

Q₈

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

#10597381

yes

—

169

C₂×C₄

#10597384

yes

characteristic

170

C₂×C₄

#10597385

yes

—

171

C₂×C₄

#10597399

yes

characteristic

172

C₈

—

173

C₈

—

174

C₈

—

152

—

175

C₈

—

152

—

176

C₂²

—

177

C₂²

—

178

C₂²

#11926458

yes

characteristic

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

#11926717

yes

characteristic

185

C₄

#11926722

yes

characteristic

186

C₂

—

187

C₂

#14564861

yes

characteristic

188

C₂

#14564912

yes

—

	subgroup	G/H
center	C₂²	#11926458
commutator, cosocle	C₂×C₄²×C₁₉	C₂³
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#3255	C₂
socle	C₂²×C₁₉	#394
prime radical	C₂³×C₁₉	#124
hypercenter	#3255	D₁₉
hypocenter	C₁₉	#10529
LCS	G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3255

subgroup

G/H

center

C₂²

#11926458

commutator, cosocle

C₂×C₄²×C₁₉

C₂³

Frattini

C₂×C₄²

C₂²×D₁₉

Fitting

C₁₉×#3255

C₂

C₂²×C₁₉

C₂³×C₁₉

D₁₉

C₁₉

G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3255

N	H ≅ G/N	type
134. C₁₉	#10529	semidirect

H ≅ G/N

type

134. C₁₉

#10529

semidirect

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

subgroup	index
#14563775	2
#14562365	2
#14562250	2
#14561546	2
C₁₉×#3255	2
#10529	19

subgroup

index

C₁₉×#3255

Subgroups — #18086703

Conjugacy classes of subgroups

Complements

Maximal subgroups