Group #18070044 — subgroups

Subgroups — #18070044

Order 4864.

234 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 37 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	#14564756	C₂	1	1	yes	maximal, characteristic
2	#14561758	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1321	C₂	1	1	yes	maximal, characteristic
4	C₂×#10597397	—	2	1	no	—
5	C₂²×Dic₇₆	C₄	1	1	yes	characteristic
6	#11926528	—	2	1	no	—
7	C₂×#10597281	—	2	1	no	—
8	#11926309	C₄	1	1	yes	characteristic
9	C₁₉×#408	C₂²	1	1	yes	characteristic
10	C₁₉×#349	—	2	1	no	—
11	C₂×#41×C₁₉	D₄	1	1	yes	characteristic
12	C₂²×Dic₃₈	—	2	1	no	—
13	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
14	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
15	#10597397	—	4	2	no	—
16	#10597397	—	4	2	no	—
17	C₂×#64528	—	4	1	no	—
18	C₂×#64526	—	4	1	no	—
19	C₂×#64520	—	2	1	no	—
20	C₂×#64520	—	4	1	no	—
21	C₂×Dic₇₆	—	4	1	no	—
22	C₂×Dic₇₆	—	4	2	no	—
23	C₁₉×M₃₂	—	4	1	no	—
24	C₁₉×C₂²⋊C₈	—	4	1	no	—
25	#10597293	—	4	1	no	—
26	#10597290	—	2	2	no	—
27	#10597281	—	4	2	no	—
28	#41×C₁₉	—	4	1	no	—
29	#41×C₁₉	—	4	1	no	—
30	C₂×Q₈×C₁₉	D₈	1	1	yes	characteristic
31	C₂×D₄×C₁₉	—	2	1	no	—
32	C₂×D₄×C₁₉	SD₁₆	1	1	yes	characteristic
33	C₂²×C₄×C₁₉	—	2	1	no	—
34	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
35	C₂²×Dic₁₉	—	4	1	no	—
36	C₂×Dic₃₈	—	2	2	no	—
37	C₂×Dic₃₈	—	4	1	no	—
38	C₂×Dic₃₈	—	4	1	no	—
39	C₂×Dic₃₈	—	4	1	no	—
40	C₂×Dic₃₈	—	4	1	no	—
41	C₂×Dic₃₈	—	4	1	no	—
42	M₁₆×C₁₉	—	4	1	no	—
43	C₂×C₈×C₁₉	—	2	1	no	—
44	C₂×C₈×C₁₉	—	2	1	no	—
45	C₂×C₈×C₁₉	—	4	1	no	—
46	#64528	—	4	4	no	—
47	#64526	—	4	4	no	—
48	#64520	—	2	2	no	—
49	#64520	—	4	1	no	—
50	#64520	—	4	2	no	—
51	#64520	—	8	1	no	—
52	C₂×C₁₉⋊C₈	—	4	1	no	—
53	C₂×C₁₉⋊C₈	—	8	1	no	—
54	Dic₇₆	—	4	2	no	—
55	Dic₇₆	—	4	2	no	—
56	C₁₆×C₁₉	—	8	1	no	—
57	#7401	—	19	1	no	maximal, Sylow
58	C₂³×C₁₉	—	2	1	no	—
59	C₂³×C₁₉	#104	1	1	yes	characteristic
60	Q₈×C₁₉	—	4	1	no	—
61	D₄×C₁₉	—	4	1	no	—
62	D₄×C₁₉	—	8	1	no	—
63	C₂×C₄×C₁₉	—	2	1	no	—
64	C₂×C₄×C₁₉	—	2	1	no	—
65	C₂×C₄×C₁₉	—	2	1	no	—
66	C₂×C₄×C₁₉	—	4	1	no	—
67	C₂×C₄×C₁₉	—	4	1	no	—
68	C₂×C₄×C₁₉	#99	1	1	yes	characteristic
69	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
70	C₂×Dic₁₉	—	4	2	no	—
71	C₂×Dic₁₉	—	8	1	no	—
72	C₂×Dic₁₉	—	8	1	no	—
73	Dic₃₈	—	4	2	no	—
74	Dic₃₈	—	4	2	no	—
75	Dic₃₈	—	8	1	no	—
76	Dic₃₈	—	8	1	no	—
77	C₈×C₁₉	—	4	1	no	—
78	C₈×C₁₉	—	4	1	no	—
79	C₁₉⋊C₈	—	4	2	no	—
80	C₁₉⋊C₈	—	8	2	no	—
81	#2020	—	19	1	no	—
82	#1392	—	19	1	no	—
83	#1321	D₁₉	1	1	yes	characteristic
84	C₂²×C₁₉	—	2	1	no	—
85	C₂²×C₁₉	—	4	1	no	—
86	C₂²×C₁₉	—	4	1	no	—
87	C₂²×C₁₉	—	8	1	no	—
88	C₂²×C₁₉	#327	1	1	yes	characteristic
89	C₄×C₁₉	—	2	1	no	—
90	C₄×C₁₉	—	4	1	no	—
91	C₄×C₁₉	AΣL(1,𝔽₁₆)	1	1	yes	characteristic
92	C₄×C₁₉	#353	1	1	yes	characteristic
93	Dic₁₉	—	8	2	no	—
94	C₂×#137	—	38	1	no	—
95	C₂²×Q₁₆	—	19	1	no	—
96	#413	—	38	1	no	—
97	C₂×#101	—	38	1	no	—
98	#408	C₂×D₁₉	1	1	yes	characteristic
99	#349	—	2	1	no	—
100	#341	—	19	1	no	—
101	C₂×C₁₉	—	2	1	no	—
102	C₂×C₁₉	—	4	1	no	—
103	C₂×C₁₉	—	8	1	no	—
104	C₂×C₁₉	#1343	1	1	yes	characteristic
105	C₂×#41	D₇₆	1	1	yes	characteristic
106	C₂²×Q₈	—	38	1	no	—
107	#137	—	76	2	no	—
108	#137	—	76	2	no	—
109	C₂×Q₁₆	—	76	1	no	—
110	C₂×Q₁₆	—	76	2	no	—
111	C₂×Q₁₆	—	76	1	no	—
112	C₂×SD₁₆	—	76	1	no	—
113	C₂×M₁₆	—	38	1	no	—
114	C₂×M₁₆	—	76	1	no	—
115	C₂×M₁₆	C₂²⋊D₁₉	1	1	yes	characteristic
116	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
117	M₃₂	—	4	1	no	—
118	#101	—	76	2	no	—
119	#101	—	76	1	no	—
120	#100	—	38	2	no	—
121	C₂²⋊C₈	—	4	1	no	—
122	C₁₉	#7401	1	1	yes	characteristic, complemented, Sylow
123	#41	—	4	1	no	—
124	#41	—	4	1	no	—
125	C₂×Q₈	—	38	2	no	—
126	C₂×Q₈	—	76	1	no	—
127	C₂×Q₈	—	76	1	no	—
128	C₂×Q₈	—	76	1	no	—
129	C₂×Q₈	—	76	1	no	—
130	C₂×Q₈	—	76	1	no	—
131	C₂×Q₈	D₁₅₂	1	1	yes	characteristic
132	C₂×D₄	—	2	1	no	—
133	C₂×D₄	#64516	1	1	yes	characteristic
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	76	1	no	—
136	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
137	Q₁₆	—	76	2	no	—
138	Q₁₆	—	76	2	no	—
139	Q₁₆	—	76	4	no	—
140	SD₁₆	—	76	4	no	—
141	M₁₆	—	4	1	no	—
142	M₁₆	—	38	2	no	—
143	M₁₆	—	76	1	no	—
144	M₁₆	—	76	2	no	—
145	M₁₆	—	152	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	2	1	no	—
148	C₂×C₈	—	4	1	no	—
149	C₂×C₈	—	76	1	no	—
150	C₂×C₈	—	152	1	no	—
151	C₁₆	—	8	1	no	—
152	C₂³	—	2	1	no	—
153	C₂³	#10597282	1	1	yes	characteristic
154	Q₈	—	4	1	no	—
155	Q₈	—	76	2	no	—
156	Q₈	—	76	2	no	—
157	Q₈	—	152	1	no	—
158	Q₈	—	152	1	no	—
159	D₄	—	4	1	no	—
160	D₄	—	8	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	4	1	no	—
165	C₂×C₄	—	4	1	no	—
166	C₂×C₄	—	76	2	no	—
167	C₂×C₄	—	152	1	no	—
168	C₂×C₄	—	152	1	no	—
169	C₂×C₄	#10597263	1	1	yes	characteristic
170	C₂×C₄	#10597278	1	1	yes	characteristic
171	C₈	—	4	1	no	—
172	C₈	—	4	1	no	—
173	C₈	—	76	2	no	—
174	C₈	—	152	2	no	—
175	C₂²	—	2	1	no	—
176	C₂²	—	4	1	no	—
177	C₂²	—	4	1	no	—
178	C₂²	—	8	1	no	—
179	C₂²	#11926279	1	1	yes	characteristic
180	C₄	—	2	1	no	—
181	C₄	—	4	1	no	—
182	C₄	—	152	2	no	—
183	C₄	#11926332	1	1	yes	characteristic
184	C₄	#11926336	1	1	yes	characteristic
185	C₂	—	2	1	no	—
186	C₂	—	4	1	no	—
187	C₂	—	8	1	no	—
188	C₂	#14561633	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14564756

C₂

yes

maximal, characteristic

#14561758

C₂

yes

maximal, characteristic

C₁₉×#1321

C₂

yes

maximal, characteristic

C₂×#10597397

—

C₂²×Dic₇₆

C₄

yes

—

—

C₄

yes

characteristic

C₁₉×#408

C₂²

yes

characteristic

C₁₉×#349

—

C₂×#41×C₁₉

D₄

yes

characteristic

C₂²×Dic₃₈

—

C₂×M₁₆×C₁₉

D₄

yes

characteristic

C₂²×C₈×C₁₉

C₂×C₄

yes

—

—

—

—

—

—

C₂×Dic₇₆

—

C₂×Dic₇₆

—

C₁₉×M₃₂

—

C₁₉×C₂²⋊C₈

—

#10597293

—

#10597290

—

#10597281

—

#41×C₁₉

—

#41×C₁₉

—

C₂×Q₈×C₁₉

D₈

yes

characteristic

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

SD₁₆

yes

characteristic

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×Dic₁₉

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

—

—

—

—

—

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

Dic₇₆

—

Dic₇₆

—

C₁₆×C₁₉

—

#7401

—

maximal, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

#104

yes

characteristic

Q₈×C₁₉

—

D₄×C₁₉

—

D₄×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×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

Dic₃₈

—

Dic₃₈

—

Dic₃₈

—

Dic₃₈

—

C₈×C₁₉

—

C₈×C₁₉

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

#2020

—

#1392

—

#1321

D₁₉

yes

characteristic

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#327

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

AΣL(1,𝔽₁₆)

yes

characteristic

C₄×C₁₉

#353

yes

characteristic

Dic₁₉

—

C₂×#137

—

C₂²×Q₁₆

—

#413

—

C₂×#101

—

#408

C₂×D₁₉

yes

characteristic

#349

—

100

#341

—

101

C₂×C₁₉

—

102

C₂×C₁₉

—

103

C₂×C₁₉

—

104

C₂×C₁₉

#1343

yes

characteristic

105

C₂×#41

D₇₆

yes

characteristic

106

C₂²×Q₈

—

107

#137

—

108

#137

—

109

C₂×Q₁₆

—

110

C₂×Q₁₆

—

111

C₂×Q₁₆

—

112

C₂×SD₁₆

—

113

C₂×M₁₆

—

114

C₂×M₁₆

—

115

C₂×M₁₆

C₂²⋊D₁₉

yes

characteristic

116

C₂²×C₈

C₄×D₁₉

yes

characteristic

117

M₃₂

—

118

#101

—

119

#101

—

120

#100

—

121

C₂²⋊C₈

—

122

C₁₉

#7401

yes

characteristic, complemented, Sylow

123

#41

—

124

#41

—

125

C₂×Q₈

—

126

C₂×Q₈

—

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

D₁₅₂

yes

characteristic

132

C₂×D₄

—

133

C₂×D₄

#64516

yes

characteristic

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

(C₂×D₁₉)⋊C₄

yes

characteristic

137

Q₁₆

—

138

Q₁₆

—

139

Q₁₆

—

140

SD₁₆

—

141

M₁₆

—

142

M₁₆

—

143

M₁₆

—

144

M₁₆

—

145

M₁₆

—

152

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

152

—

151

C₁₆

—

152

C₂³

—

153

C₂³

#10597282

yes

characteristic

154

Q₈

—

155

Q₈

—

156

Q₈

—

157

Q₈

—

152

—

158

Q₈

—

152

—

159

D₄

—

160

D₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

152

—

168

C₂×C₄

—

152

—

169

C₂×C₄

#10597263

yes

characteristic

170

C₂×C₄

#10597278

yes

characteristic

171

C₈

—

172

C₈

—

173

C₈

—

174

C₈

—

152

—

175

C₂²

—

176

C₂²

—

177

C₂²

—

178

C₂²

—

179

C₂²

#11926279

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

152

—

183

C₄

#11926332

yes

characteristic

184

C₄

#11926336

yes

characteristic

185

C₂

—

186

C₂

—

187

C₂

—

188

C₂

#14561633

yes

characteristic

	subgroup	G/H
center	C₂	#14561633
commutator	C₂²×C₈×C₁₉	C₂×C₄
Frattini	#408	C₂×D₁₉
Fitting	C₁₉×#1321	C₂
socle	C₂×C₁₉	#1343
prime radical	C₂×#41×C₁₉	D₄
cosocle	C₁₉×#408	C₂²
hypercenter	#1321	D₁₉
hypocenter	C₁₉	#7401
LCS	G ▷ C₂²×C₈×C₁₉ ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #408 ◁ #1321

subgroup

G/H

center

C₂

#14561633

commutator

C₂²×C₈×C₁₉

C₂×C₄

Frattini

#408

C₂×D₁₉

Fitting

C₁₉×#1321

C₂

socle

C₂×C₁₉

#1343

prime radical

C₂×#41×C₁₉

D₄

cosocle

C₁₉×#408

C₂²

D₁₉

C₁₉

G ▷ C₂²×C₈×C₁₉ ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #408 ◁ #1321

N	H ≅ G/N	type
122. C₁₉	#7401	semidirect

H ≅ G/N

type

122. C₁₉

#7401

semidirect

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

subgroup	index
#14564756	2
#14561758	2
C₁₉×#1321	2
#7401	19

subgroup

index

#14564756

#14561758

C₁₉×#1321

#7401

Subgroups — #18070044

Conjugacy classes of subgroups

Complements

Maximal subgroups