Group #18085274 — subgroups

Subgroups — #18085274

Order 4864.

528 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 111 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	#14574580	C₂	1	2	yes	maximal
2	#14572013	C₂	1	1	yes	maximal, characteristic
3	#14562317	C₂	1	2	yes	maximal
4	#14561468	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1994	C₂	1	1	yes	maximal, characteristic
6	C₂×#10597315	C₂²	1	2	yes	—
7	#11926923	—	2	2	no	—
8	#11926896	—	2	2	no	—
9	#11926754	—	2	4	no	—
10	#11926699	—	2	4	no	—
11	#11926597	C₂²	1	2	yes	—
12	#11926450	—	2	4	no	—
13	#11926410	—	2	2	no	—
14	#11926292	—	2	4	no	—
15	C₁₉×#533	C₂²	1	1	yes	characteristic
16	C₁₉×#423	C₂²	1	1	yes	characteristic
17	C₄³×C₁₉	C₂²	1	1	yes	characteristic
18	C₁₉×#339	—	2	2	no	—
19	C₂²×Dic₃₈	D₄	1	4	yes	—
20	C₂×M₁₆×C₁₉	—	2	1	no	—
21	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
22	C₁₉×#125	—	2	1	no	—
23	C₁₉×C₂²⋊Q₈	—	2	2	no	—
24	C₄×Q₈×C₁₉	—	2	2	no	—
25	C₁₉×#117	—	2	1	no	—
26	C₂×C₄²×C₁₉	—	2	2	no	—
27	C₂×C₄²×C₁₉	D₄	1	2	yes	—
28	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
29	C₂×#64528	—	2	8	no	—
30	C₂×C₄⋊Dic₁₉	—	2	2	no	—
31	C₂×C₄⋊Dic₁₉	—	2	2	no	—
32	#10597315	—	2	2	no	—
33	#10597315	—	2	2	no	—
34	#10597315	—	2	2	no	—
35	#10597315	—	2	2	no	—
36	C₁₉×C₄⋊C₈	—	2	2	no	—
37	C₁₉⋊#95	—	2	8	no	—
38	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
39	#10597267	—	2	8	no	—
40	C₄⋊(C₁₉⋊C₈)	D₄	1	4	yes	—
41	C₄×C₁₉⋊C₈	—	2	2	no	—
42	C₂×Q₈×C₁₉	—	2	2	no	—
43	C₂²×C₄×C₁₉	—	2	2	no	—
44	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
46	C₂²×Dic₁₉	—	2	4	no	—
47	C₂×Dic₃₈	—	2	4	no	—
48	C₂×Dic₃₈	—	2	4	no	—
49	C₂×Dic₃₈	—	4	8	no	—
50	M₁₆×C₁₉	—	4	2	no	—
51	C₂×C₈×C₁₉	—	4	1	no	—
52	C₄⋊C₄×C₁₉	—	2	2	no	—
53	C₄⋊C₄×C₁₉	—	2	2	no	—
54	C₄⋊C₄×C₁₉	—	4	2	no	—
55	C₂²⋊C₄×C₁₉	—	4	1	no	—
56	C₄²×C₁₉	—	2	2	no	—
57	C₄²×C₁₉	—	4	4	no	—
58	C₄²×C₁₉	—	4	1	no	—
59	C₄²×C₁₉	Q₁₆	1	4	yes	complemented
60	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₁₉	C₂×D₄	1	2	yes	—
62	C₄²×C₁₉	#41	1	1	yes	characteristic
63	#64528	—	8	8	no	—
64	C₄⋊Dic₁₉	—	4	2	no	—
65	C₄⋊Dic₁₉	—	4	2	no	—
66	C₄⋊Dic₁₉	—	4	2	no	—
67	C₄⋊Dic₁₉	—	4	2	no	—
68	C₂×C₁₉⋊C₈	—	2	4	no	—
69	#10297	—	19	1	no	maximal, Sylow
70	C₂³×C₁₉	#120	1	1	yes	characteristic
71	Q₈×C₁₉	—	4	4	no	—
72	C₂×C₄×C₁₉	—	2	4	no	—
73	C₂×C₄×C₁₉	—	2	4	no	—
74	C₂×C₄×C₁₉	—	4	4	no	—
75	C₂×C₄×C₁₉	—	4	2	no	—
76	C₂×C₄×C₁₉	#120	1	2	yes	—
77	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
78	C₂×C₄×C₁₉	#121	1	2	yes	—
79	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
80	C₂×C₄×C₁₉	C₂×Q₁₆	1	2	yes	—
81	C₂×C₄×C₁₉	#137	1	2	yes	—
82	C₂×Dic₁₉	—	4	8	no	—
83	C₂×Dic₁₉	—	4	4	no	—
84	Dic₃₈	—	4	8	no	—
85	Dic₃₈	—	8	4	no	—
86	C₈×C₁₉	—	8	1	no	—
87	C₁₉⋊C₈	—	4	4	no	—
88	#3209	—	19	2	no	—
89	#2862	—	19	1	no	—
90	#1994	D₁₉	1	1	yes	characteristic
91	#1548	—	19	2	no	—
92	#1276	—	19	1	no	—
93	C₂²×C₁₉	—	2	2	no	—
94	C₂²×C₁₉	#392	1	1	yes	characteristic
95	C₂²×C₁₉	#451	1	2	yes	—
96	C₄×C₁₉	—	2	2	no	—
97	C₄×C₁₉	—	4	4	no	—
98	C₄×C₁₉	—	8	2	no	—
99	C₄×C₁₉	#455	1	4	yes	—
100	C₄×C₁₉	#462	1	2	yes	—
101	C₄×C₁₉	#494	1	1	yes	characteristic
102	C₄×C₁₉	#497	1	1	yes	characteristic
103	Dic₁₉	—	8	4	no	—
104	#533	C₂×D₁₉	1	1	yes	characteristic
105	C₂×C₄⋊Q₈	—	19	2	no	—
106	#494	—	38	2	no	—
107	#487	—	38	2	no	—
108	#462	—	38	4	no	—
109	#451	—	38	4	no	—
110	#431	—	19	2	no	—
111	#423	C₂×D₁₉	1	1	yes	characteristic
112	#391	—	38	4	no	—
113	#382	—	38	2	no	—
114	C₄³	C₂×D₁₉	1	1	yes	characteristic
115	#339	—	2	2	no	—
116	#333	—	38	4	no	—
117	C₂×C₁₉	—	2	2	no	—
118	C₂×C₁₉	#2004	1	2	yes	—
119	C₂×C₁₉	#2016	1	1	yes	characteristic
120	C₂²×Q₈	—	19	4	no	—
121	C₂×Q₁₆	—	38	8	no	—
122	C₂×M₁₆	—	2	1	no	—
123	C₄⋊Q₈	—	38	2	no	—
124	C₄⋊Q₈	—	38	2	no	—
125	C₄⋊Q₈	—	38	2	no	—
126	C₄⋊Q₈	—	38	2	no	—
127	C₄⋊Q₈	D₇₆	1	2	yes	—
128	#125	—	2	1	no	—
129	C₂²⋊Q₈	—	2	2	no	—
130	C₄×Q₈	—	2	2	no	—
131	#117	—	2	1	no	—
132	C₂×C₄⋊C₄	—	38	2	no	—
133	C₂×C₄⋊C₄	—	38	2	no	—
134	C₂×C₄²	—	2	2	no	—
135	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
136	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
137	C₄⋊C₈	—	2	2	no	—
138	C₄⋊C₈	—	19	4	no	—
139	#103	—	38	8	no	—
140	C₂²⋊C₈	—	38	2	no	—
141	C₄×C₈	—	38	2	no	—
142	#95	—	38	8	no	—
143	C₁₉	#10297	1	1	yes	characteristic, complemented, Sylow
144	C₂×Q₈	—	2	2	no	—
145	C₂×Q₈	—	38	4	no	—
146	C₂×Q₈	—	38	4	no	—
147	C₂×Q₈	—	76	8	no	—
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	—	38	4	no	—
150	C₂²×C₄	D₄×D₁₉	1	2	yes	—
151	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
152	Q₁₆	—	152	8	no	—
153	M₁₆	—	4	2	no	—
154	C₂×C₈	—	4	1	no	—
155	C₂×C₈	—	38	4	no	—
156	C₄⋊C₄	—	2	2	no	—
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	76	2	no	—
160	C₄⋊C₄	—	76	2	no	—
161	C₄⋊C₄	—	76	2	no	—
162	C₄⋊C₄	—	76	2	no	—
163	C₂²⋊C₄	—	4	1	no	—
164	C₄²	—	2	2	no	—
165	C₄²	—	4	4	no	—
166	C₄²	—	4	1	no	—
167	C₄²	#64528	1	4	yes	complemented
168	C₄²	C₂×D₇₆	1	1	yes	characteristic
169	C₄²	#64541	1	1	yes	characteristic
170	C₄²	D₄×D₁₉	1	2	yes	—
171	C₂³	#10597383	1	1	yes	characteristic
172	Q₈	—	4	4	no	—
173	Q₈	—	76	8	no	—
174	Q₈	—	152	4	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	4	4	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	76	8	no	—
180	C₂×C₄	—	76	4	no	—
181	C₂×C₄	#10597328	1	2	yes	—
182	C₂×C₄	#10597330	1	2	yes	—
183	C₂×C₄	#10597376	1	1	yes	characteristic
184	C₂×C₄	#10597386	1	1	yes	characteristic
185	C₂×C₄	C₂×#64528	1	2	yes	—
186	C₂×C₄	#10597397	1	2	yes	—
187	C₈	—	8	1	no	—
188	C₈	—	76	4	no	—
189	C₂²	—	2	2	no	—
190	C₂²	#11926453	1	1	yes	characteristic
191	C₂²	#11926699	1	2	yes	—
192	C₄	—	2	2	no	—
193	C₄	—	4	4	no	—
194	C₄	—	8	2	no	—
195	C₄	—	152	4	no	—
196	C₄	#11926723	1	4	yes	—
197	C₄	#11926754	1	2	yes	—
198	C₄	#11926923	1	1	yes	characteristic
199	C₄	#11926939	1	1	yes	characteristic
200	C₂	—	2	2	no	—
201	C₂	#14564663	1	2	yes	—
202	C₂	#14564734	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14574580

C₂

yes

maximal

#14572013

C₂

yes

maximal, characteristic

#14562317

C₂

yes

maximal

#14561468

C₂

yes

maximal, characteristic

C₁₉×#1994

C₂

yes

maximal, characteristic

C₂×#10597315

C₂²

yes

—

—

—

—

—

C₂²

yes

—

#11926450

—

#11926410

—

#11926292

—

C₁₉×#533

C₂²

yes

characteristic

C₁₉×#423

C₂²

yes

characteristic

C₄³×C₁₉

C₂²

yes

characteristic

C₁₉×#339

—

C₂²×Dic₃₈

D₄

yes

—

C₂×M₁₆×C₁₉

—

C₁₉×C₄⋊Q₈

D₄

yes

—

C₁₉×#125

—

C₁₉×C₂²⋊Q₈

—

C₄×Q₈×C₁₉

—

C₁₉×#117

—

C₂×C₄²×C₁₉

—

C₂×C₄²×C₁₉

D₄

yes

—

C₂×C₄²×C₁₉

C₂³

yes

characteristic

C₂×#64528

—

C₂×C₄⋊Dic₁₉

—

C₂×C₄⋊Dic₁₉

—

—

—

—

—

C₁₉×C₄⋊C₈

—

C₁₉⋊#95

—

C₂²⋊(C₁₉⋊C₈)

—

#10597267

—

C₄⋊(C₁₉⋊C₈)

D₄

yes

—

C₄×C₁₉⋊C₈

—

C₂×Q₈×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂×D₄

yes

—

C₂²×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂²×Dic₁₉

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

Q₁₆

yes

complemented

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₄²×C₁₉

C₂×D₄

yes

—

C₄²×C₁₉

#41

yes

characteristic

#64528

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₂×C₁₉⋊C₈

—

—

C₂³×C₁₉

yes

Q₈×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

#120

yes

—

C₂×C₄×C₁₉

#121

yes

characteristic

C₂×C₄×C₁₉

#121

yes

—

C₂×C₄×C₁₉

#127

yes

characteristic

C₂×C₄×C₁₉

C₂×Q₁₆

yes

—

C₂×C₄×C₁₉

#137

yes

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

Dic₃₈

—

Dic₃₈

—

C₈×C₁₉

—

C₁₉⋊C₈

—

#3209

—

#2862

—

#1994

D₁₉

yes

characteristic

#1548

—

#1276

—

C₂²×C₁₉

—

C₂²×C₁₉

#392

yes

characteristic

C₂²×C₁₉

#451

yes

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

#455

yes

—

100

C₄×C₁₉

#462

yes

—

101

C₄×C₁₉

#494

yes

characteristic

102

C₄×C₁₉

#497

yes

characteristic

103

Dic₁₉

—

104

#533

C₂×D₁₉

yes

characteristic

105

C₂×C₄⋊Q₈

—

106

#494

—

107

#487

—

108

#462

—

109

#451

—

110

#431

—

111

#423

C₂×D₁₉

yes

characteristic

112

#391

—

113

#382

—

114

C₄³

C₂×D₁₉

yes

characteristic

115

#339

—

116

#333

—

117

C₂×C₁₉

—

118

C₂×C₁₉

#2004

yes

—

119

C₂×C₁₉

#2016

yes

characteristic

120

C₂²×Q₈

—

121

C₂×Q₁₆

—

122

C₂×M₁₆

—

123

C₄⋊Q₈

—

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

126

C₄⋊Q₈

—

127

C₄⋊Q₈

D₇₆

yes

—

128

#125

—

129

C₂²⋊Q₈

—

130

C₄×Q₈

—

131

#117

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

—

134

C₂×C₄²

—

135

C₂×C₄²

C₂²⋊D₁₉

yes

—

136

C₂×C₄²

C₂²×D₁₉

yes

characteristic

137

C₄⋊C₈

—

138

C₄⋊C₈

—

139

#103

—

140

C₂²⋊C₈

—

141

C₄×C₈

—

142

#95

—

143

C₁₉

#10297

yes

characteristic, complemented, Sylow

144

C₂×Q₈

—

145

C₂×Q₈

—

146

C₂×Q₈

—

147

C₂×Q₈

—

148

C₂²×C₄

—

149

C₂²×C₄

—

150

C₂²×C₄

D₄×D₁₉

yes

—

151

C₂²×C₄

C₂×C₂²⋊D₁₉

yes

characteristic

152

Q₁₆

—

152

—

153

M₁₆

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₂²⋊C₄

—

164

C₄²

—

165

C₄²

—

166

C₄²

—

167

C₄²

#64528

yes

complemented

168

C₄²

C₂×D₇₆

yes

characteristic

169

C₄²

#64541

yes

characteristic

170

C₄²

D₄×D₁₉

yes

—

171

C₂³

#10597383

yes

characteristic

172

Q₈

—

173

Q₈

—

174

Q₈

—

152

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

#10597328

yes

—

182

C₂×C₄

#10597330

yes

—

183

C₂×C₄

#10597376

yes

characteristic

184

C₂×C₄

#10597386

yes

characteristic

185

C₂×C₄

C₂×#64528

yes

—

186

C₂×C₄

#10597397

yes

—

187

C₈

—

188

C₈

—

189

C₂²

—

190

C₂²

#11926453

yes

characteristic

191

C₂²

#11926699

yes

—

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

152

—

196

C₄

#11926723

yes

—

197

C₄

#11926754

yes

—

198

C₄

#11926923

yes

characteristic

199

C₄

#11926939

yes

characteristic

200

C₂

—

201

C₂

#14564663

yes

—

202

C₂

#14564734

yes

characteristic

	subgroup	G/H
center	C₂²	#11926453
commutator, cosocle	C₂×C₄²×C₁₉	C₂³
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1994	C₂
socle	C₂²×C₁₉	#392
prime radical	C₂³×C₁₉	#120
hypercenter	#1994	D₁₉
hypocenter	C₁₉	#10297
LCS	G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1994

subgroup

G/H

center

C₂²

#11926453

commutator, cosocle

C₂×C₄²×C₁₉

C₂³

Frattini

C₂×C₄²

C₂²×D₁₉

Fitting

C₁₉×#1994

C₂

C₂²×C₁₉

C₂³×C₁₉

D₁₉

C₁₉

G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1994

N	H ≅ G/N	type
59. C₄²×C₁₉	Q₁₆	semidirect
143. C₁₉	#10297	semidirect
167. C₄²	#64528	semidirect

H ≅ G/N

type

59. C₄²×C₁₉

Q₁₆

semidirect

143. C₁₉

#10297

semidirect

167. C₄²

#64528

semidirect

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

subgroup	index
#14574580	2
#14572013	2
#14562317	2
#14561468	2
C₁₉×#1994	2
#10297	19

subgroup

index

C₁₉×#1994

Subgroups — #18085274

Conjugacy classes of subgroups

Complements

Maximal subgroups