Group #18069968 — subgroups

Subgroups — #18069968

Order 4864.

202 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 47 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	#14561569	C₂	1	1	yes	maximal, characteristic
2	#14561461	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1857	C₂	1	1	yes	maximal, characteristic
4	#11926299	—	2	1	no	—
5	#11926270	—	2	1	no	—
6	C₁₉×#519	C₄	1	1	yes	characteristic
7	C₁₉×#428	—	2	1	no	—
8	C₂×C₁₉×#101	C₄	1	1	yes	characteristic
9	C₁₉×#391	C₂²	1	1	yes	characteristic
10	C₁₉×#339	—	2	1	no	—
11	C₂²×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
12	C₂×M₁₆×C₁₉	—	2	1	no	—
13	C₄×Q₈×C₁₉	—	4	1	no	—
14	C₄×Q₈×C₁₉	—	4	1	no	—
15	C₁₉×#117	—	2	1	no	—
16	C₁₉×#117	—	4	1	no	—
17	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
18	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
19	C₂²×C₁₉⋊C₈	—	4	1	no	—
20	C₁₉×#107	—	4	1	no	—
21	C₁₉×#106	—	4	1	no	—
22	C₁₉×#101	—	2	1	no	—
23	C₁₉×#101	C₈	1	2	yes	complemented
24	C₁₉×#95	—	4	1	no	—
25	C₂²⋊(C₁₉⋊C₈)	—	4	1	no	—
26	C₂×Q₈×C₁₉	—	2	1	no	—
27	C₂×Q₈×C₁₉	—	4	1	no	—
28	C₂×Q₈×C₁₉	—	4	1	no	—
29	C₂×Q₈×C₁₉	C₂×C₈	1	1	yes	characteristic
30	C₂×Q₈×C₁₉	M₁₆	1	1	yes	characteristic
31	C₂²×C₄×C₁₉	—	2	1	no	—
32	C₂²×C₄×C₁₉	—	2	1	no	—
33	C₂²×C₄×C₁₉	—	2	1	no	—
34	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
35	M₁₆×C₁₉	—	2	2	no	—
36	M₁₆×C₁₉	—	4	1	no	—
37	C₂×C₈×C₁₉	—	4	1	no	—
38	C₄⋊C₄×C₁₉	—	2	1	no	—
39	C₄⋊C₄×C₁₉	—	2	1	no	—
40	C₄⋊C₄×C₁₉	—	4	1	no	—
41	C₄⋊C₄×C₁₉	—	4	1	no	—
42	C₄⋊C₄×C₁₉	—	4	1	no	—
43	C₂²⋊C₄×C₁₉	—	4	1	no	—
44	C₄²×C₁₉	—	4	1	no	—
45	C₄²×C₁₉	—	4	1	no	—
46	C₄²×C₁₉	—	4	2	no	—
47	C₂×C₁₉⋊C₈	—	4	2	no	—
48	C₂×C₁₉⋊C₈	—	8	1	no	—
49	C₂×C₁₉⋊C₈	—	8	1	no	—
50	C₂×C₁₉⋊C₈	—	8	1	no	—
51	#7376	—	19	1	no	maximal, Sylow
52	C₂³×C₁₉	#99	1	1	yes	characteristic
53	Q₈×C₁₉	—	4	1	no	—
54	Q₈×C₁₉	—	4	1	no	—
55	Q₈×C₁₉	—	4	2	no	—
56	C₂×C₄×C₁₉	—	2	1	no	—
57	C₂×C₄×C₁₉	—	2	1	no	—
58	C₂×C₄×C₁₉	—	2	1	no	—
59	C₂×C₄×C₁₉	—	2	1	no	—
60	C₂×C₄×C₁₉	—	2	1	no	—
61	C₂×C₄×C₁₉	—	2	1	no	—
62	C₂×C₄×C₁₉	—	4	1	no	—
63	C₂×C₄×C₁₉	—	4	1	no	—
64	C₂×C₄×C₁₉	—	4	1	no	—
65	C₂×C₄×C₁₉	—	4	1	no	—
66	C₂×C₄×C₁₉	—	4	1	no	—
67	C₂×C₄×C₁₉	—	4	1	no	—
68	C₂×C₄×C₁₉	—	4	1	no	—
69	C₂×C₄×C₁₉	—	8	1	no	—
70	C₂×C₄×C₁₉	C₂²⋊C₈	1	1	yes	characteristic
71	C₂×C₄×C₁₉	#101	1	1	yes	characteristic
72	C₈×C₁₉	—	4	2	no	—
73	C₁₉⋊C₈	—	8	2	no	—
74	C₁₉⋊C₈	—	16	1	no	—
75	#1857	D₁₉	1	1	yes	characteristic
76	#1319	—	19	1	no	—
77	#1272	—	19	1	no	—
78	C₂²×C₁₉	—	2	1	no	—
79	C₂²×C₁₉	—	2	1	no	—
80	C₂²×C₁₉	#324	1	1	yes	characteristic
81	C₂²×C₁₉	AΣL(1,𝔽₁₆)	1	1	yes	characteristic
82	C₂²×C₁₉	#352	1	1	yes	characteristic
83	C₄×C₁₉	—	2	1	no	—
84	C₄×C₁₉	—	2	1	no	—
85	C₄×C₁₉	—	4	1	no	—
86	C₄×C₁₉	—	4	1	no	—
87	C₄×C₁₉	—	4	1	no	—
88	C₄×C₁₉	—	4	1	no	—
89	C₄×C₁₉	—	8	1	no	—
90	C₄×C₁₉	—	8	1	no	—
91	C₄×C₁₉	—	8	1	no	—
92	#519	Dic₁₉	1	1	yes	characteristic
93	#428	—	2	1	no	—
94	C₂×#101	Dic₁₉	1	1	yes	characteristic
95	#391	C₂×D₁₉	1	1	yes	characteristic
96	#339	—	2	1	no	—
97	#336	—	38	1	no	—
98	#324	—	38	1	no	—
99	C₂×C₁₉	—	2	1	no	—
100	C₂×C₁₉	—	2	1	no	—
101	C₂×C₁₉	#1326	1	1	yes	characteristic
102	C₂×C₁₉	#1404	1	1	yes	characteristic
103	C₂×C₁₉	#1405	1	1	yes	characteristic
104	C₂²×Q₈	C₂×Dic₁₉	1	1	yes	characteristic
105	C₂×M₁₆	—	2	1	no	—
106	C₂²×C₈	—	76	1	no	—
107	C₄×Q₈	—	4	1	no	—
108	C₄×Q₈	—	4	1	no	—
109	#117	—	2	1	no	—
110	#117	—	4	1	no	—
111	C₂×C₄⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
112	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
113	#107	—	4	1	no	—
114	#106	—	4	1	no	—
115	#101	—	2	1	no	—
116	#101	C₁₉⋊C₈	1	2	yes	complemented
117	C₂²⋊C₈	—	76	1	no	—
118	#95	—	4	1	no	—
119	C₁₉	#7376	1	1	yes	characteristic, complemented, Sylow
120	C₂×Q₈	—	2	1	no	—
121	C₂×Q₈	—	4	1	no	—
122	C₂×Q₈	—	4	1	no	—
123	C₂×Q₈	C₂×C₁₉⋊C₈	1	1	yes	characteristic
124	C₂×Q₈	#64520	1	1	yes	characteristic
125	C₂²×C₄	—	2	1	no	—
126	C₂²×C₄	—	2	1	no	—
127	C₂²×C₄	—	2	1	no	—
128	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
129	M₁₆	—	2	2	no	—
130	M₁₆	—	4	1	no	—
131	C₂×C₈	—	4	1	no	—
132	C₂×C₈	—	76	2	no	—
133	C₂×C₈	—	152	1	no	—
134	C₂×C₈	—	152	1	no	—
135	C₂×C₈	—	152	1	no	—
136	C₄⋊C₄	—	2	1	no	—
137	C₄⋊C₄	—	2	1	no	—
138	C₄⋊C₄	—	4	1	no	—
139	C₄⋊C₄	—	4	1	no	—
140	C₄⋊C₄	—	4	1	no	—
141	C₂²⋊C₄	—	4	1	no	—
142	C₄²	—	4	1	no	—
143	C₄²	—	4	1	no	—
144	C₄²	—	4	2	no	—
145	C₂³	#10597291	1	1	yes	characteristic
146	Q₈	—	4	1	no	—
147	Q₈	—	4	1	no	—
148	Q₈	—	4	2	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	4	1	no	—
158	C₂×C₄	—	4	1	no	—
159	C₂×C₄	—	4	1	no	—
160	C₂×C₄	—	4	1	no	—
161	C₂×C₄	—	4	1	no	—
162	C₂×C₄	—	8	1	no	—
163	C₂×C₄	C₂²⋊(C₁₉⋊C₈)	1	1	yes	characteristic
164	C₂×C₄	#10597293	1	1	yes	characteristic
165	C₈	—	4	2	no	—
166	C₈	—	304	1	no	—
167	C₈	—	152	2	no	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	#11926270	1	1	yes	characteristic
171	C₂²	#11926331	1	1	yes	characteristic
172	C₂²	#11926334	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	4	1	no	—
176	C₄	—	4	1	no	—
177	C₄	—	4	1	no	—
178	C₄	—	4	1	no	—
179	C₄	—	8	1	no	—
180	C₄	—	8	1	no	—
181	C₄	—	8	1	no	—
182	C₂	—	2	1	no	—
183	C₂	—	2	1	no	—
184	C₂	#14561583	1	1	yes	characteristic
185	C₂	#14561786	1	1	yes	characteristic
186	C₂	#14561789	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14561569

C₂

yes

maximal, characteristic

#14561461

C₂

yes

maximal, characteristic

C₁₉×#1857

C₂

yes

maximal, characteristic

#11926299

—

#11926270

—

C₁₉×#519

C₄

yes

characteristic

C₁₉×#428

—

C₂×C₁₉×#101

C₄

yes

characteristic

C₁₉×#391

C₂²

yes

characteristic

C₁₉×#339

—

C₂²×Q₈×C₁₉

C₂×C₄

yes

characteristic

C₂×M₁₆×C₁₉

—

C₄×Q₈×C₁₉

—

C₄×Q₈×C₁₉

—

C₁₉×#117

—

C₁₉×#117

—

C₂×C₄⋊C₄×C₁₉

D₄

yes

characteristic

C₂×C₄²×C₁₉

D₄

yes

characteristic

C₂²×C₁₉⋊C₈

—

C₁₉×#107

—

C₁₉×#106

—

C₁₉×#101

—

C₁₉×#101

C₈

yes

complemented

C₁₉×#95

—

C₂²⋊(C₁₉⋊C₈)

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

C₂×C₈

yes

characteristic

C₂×Q₈×C₁₉

M₁₆

yes

characteristic

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

M₁₆×C₁₉

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

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

—

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

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₁₉

#101

yes

characteristic

C₈×C₁₉

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

#1857

D₁₉

yes

characteristic

#1319

—

#1272

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#324

yes

characteristic

C₂²×C₁₉

AΣL(1,𝔽₁₆)

yes

characteristic

C₂²×C₁₉

#352

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

#519

Dic₁₉

yes

characteristic

#428

—

C₂×#101

Dic₁₉

yes

characteristic

#391

C₂×D₁₉

yes

—

—

—

C₂×C₁₉

—

100

C₂×C₁₉

—

101

C₂×C₁₉

#1326

yes

characteristic

102

C₂×C₁₉

#1404

yes

characteristic

103

C₂×C₁₉

#1405

yes

characteristic

104

C₂²×Q₈

C₂×Dic₁₉

yes

characteristic

105

C₂×M₁₆

—

106

C₂²×C₈

—

107

C₄×Q₈

—

108

C₄×Q₈

—

109

#117

—

110

#117

—

111

C₂×C₄⋊C₄

C₂²⋊D₁₉

yes

characteristic

112

C₂×C₄²

C₂²⋊D₁₉

yes

characteristic

113

#107

—

114

#106

—

115

#101

—

116

#101

C₁₉⋊C₈

yes

complemented

117

C₂²⋊C₈

—

118

#95

—

119

C₁₉

#7376

yes

characteristic, complemented, Sylow

120

C₂×Q₈

—

121

C₂×Q₈

—

122

C₂×Q₈

—

123

C₂×Q₈

C₂×C₁₉⋊C₈

yes

characteristic

124

C₂×Q₈

#64520

yes

characteristic

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂²×C₄

C₂²⋊Dic₁₉

yes

characteristic

129

M₁₆

—

130

M₁₆

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

152

—

134

C₂×C₈

—

152

—

135

C₂×C₈

—

152

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₂²⋊C₄

—

142

C₄²

—

143

C₄²

—

144

C₄²

—

145

C₂³

#10597291

yes

characteristic

146

Q₈

—

147

Q₈

—

148

Q₈

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

C₂²⋊(C₁₉⋊C₈)

yes

characteristic

164

C₂×C₄

#10597293

yes

characteristic

165

C₈

—

166

C₈

—

304

—

167

C₈

—

152

—

168

C₂²

—

169

C₂²

—

170

C₂²

#11926270

yes

characteristic

171

C₂²

#11926331

yes

characteristic

172

C₂²

#11926334

yes

characteristic

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

—

182

C₂

—

183

C₂

—

184

C₂

#14561583

yes

characteristic

185

C₂

#14561786

yes

characteristic

186

C₂

#14561789

yes

characteristic

	subgroup	G/H
center	C₂²	#11926331
commutator	C₂×Q₈×C₁₉	C₂×C₈
Frattini	#391	C₂×D₁₉
Fitting	C₁₉×#1857	C₂
socle	C₂²×C₁₉	AΣL(1,𝔽₁₆)
prime radical	C₂³×C₁₉	#99
cosocle	C₁₉×#391	C₂²
hypercenter	#1857	D₁₉
hypocenter	C₁₉	#7376
LCS	G ▷ C₂×Q₈×C₁₉ ▷ C₂×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #391 ◁ #1857

subgroup

G/H

center

C₂²

#11926331

commutator

C₂×Q₈×C₁₉

C₂×C₈

Frattini

#391

C₂×D₁₉

Fitting

C₁₉×#1857

C₂

socle

C₂²×C₁₉

AΣL(1,𝔽₁₆)

prime radical

C₂³×C₁₉

#99

cosocle

C₁₉×#391

C₂²

D₁₉

C₁₉

G ▷ C₂×Q₈×C₁₉ ▷ C₂×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #391 ◁ #1857

N	H ≅ G/N	type
23. C₁₉×#101	C₈	semidirect
116. #101	C₁₉⋊C₈	semidirect
119. C₁₉	#7376	semidirect

H ≅ G/N

type

23. C₁₉×#101

C₈

semidirect

116. #101

C₁₉⋊C₈

semidirect

119. C₁₉

#7376

semidirect

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

subgroup	index
#14561569	2
#14561461	2
C₁₉×#1857	2
#7376	19

subgroup

index

#14561569

#14561461

C₁₉×#1857

#7376

Subgroups — #18069968

Conjugacy classes of subgroups

Complements

Maximal subgroups