Group #18069240 — subgroups

Subgroups — #18069240

Order 4864.

196 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 45 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	#14565063	C₂	1	1	yes	maximal, characteristic
2	#14561512	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1314	C₂	1	1	yes	maximal, characteristic
4	C₂×#10597275	C₄	1	1	yes	characteristic
5	#11926508	—	2	1	no	—
6	#11926404	—	2	1	no	—
7	#11926307	—	2	1	no	—
8	#11926307	C₄	1	1	yes	characteristic
9	C₂×C₁₉×C₂²⋊C₈	C₂²	1	1	yes	characteristic
10	C₁₉×C₂²⋊C₁₆	—	2	1	no	—
11	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
12	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
14	C₂×C₂²⋊Dic₁₉	—	4	1	no	—
15	C₂×C₄⋊Dic₁₉	—	2	1	no	—
16	C₂×C₄⋊Dic₁₉	—	2	1	no	—
17	C₂×#64520	—	4	1	no	—
18	C₂×C₁₆×C₁₉	—	4	1	no	—
19	C₁₉×C₂²⋊C₈	—	4	1	no	—
20	C₁₉⋊#95	—	4	1	no	—
21	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
22	C₂²⋊(C₁₉⋊C₈)	—	4	1	no	—
23	#10597275	—	4	1	no	—
24	C₂⁴×C₁₉	SD₁₆	1	1	yes	characteristic
25	C₂²×C₄×C₁₉	—	2	1	no	—
26	C₂²×C₄×C₁₉	—	2	1	no	—
27	C₂²×C₄×C₁₉	—	4	1	no	—
28	C₂²×C₄×C₁₉	—	4	1	no	—
29	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
31	C₂²×Dic₁₉	—	4	1	no	—
32	C₂²×Dic₁₉	—	4	1	no	—
33	C₂×C₈×C₁₉	—	2	1	no	—
34	C₂×C₈×C₁₉	—	2	1	no	—
35	C₂×C₈×C₁₉	—	4	1	no	—
36	C₂×C₈×C₁₉	—	4	1	no	—
37	C₂²⋊Dic₁₉	—	8	2	no	—
38	C₄⋊Dic₁₉	—	4	1	no	—
39	C₄⋊Dic₁₉	—	4	1	no	—
40	C₄⋊Dic₁₉	—	4	1	no	—
41	C₄⋊Dic₁₉	—	4	1	no	—
42	#64520	—	8	1	no	—
43	#64520	—	8	1	no	—
44	C₂×C₁₉⋊C₈	—	4	2	no	—
45	C₁₆×C₁₉	—	8	1	no	—
46	#7085	—	19	1	no	maximal, Sylow
47	C₂³×C₁₉	—	2	1	no	—
48	C₂³×C₁₉	—	2	1	no	—
49	C₂³×C₁₉	—	2	1	no	—
50	C₂³×C₁₉	—	4	1	no	—
51	C₂³×C₁₉	—	4	1	no	—
52	C₂³×C₁₉	#102	1	1	yes	characteristic
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	—	4	1	no	—
57	C₂×C₄×C₁₉	—	4	1	no	—
58	C₂×C₄×C₁₉	—	4	1	no	—
59	C₂×C₄×C₁₉	—	8	1	no	—
60	C₂×C₄×C₁₉	#99	1	1	yes	characteristic
61	C₂×C₄×C₁₉	#104	1	1	yes	characteristic
62	C₂×Dic₁₉	—	8	1	no	—
63	C₂×Dic₁₉	—	8	1	no	—
64	C₂×Dic₁₉	—	8	1	no	—
65	C₂×Dic₁₉	—	8	1	no	—
66	C₂×Dic₁₉	—	8	1	no	—
67	C₂×Dic₁₉	—	8	1	no	—
68	C₈×C₁₉	—	4	1	no	—
69	C₈×C₁₉	—	4	1	no	—
70	C₁₉⋊C₈	—	8	2	no	—
71	#2076	—	19	1	no	—
72	#1314	D₁₉	1	1	yes	characteristic
73	#1293	—	19	1	no	—
74	C₂²×C₁₉	—	2	1	no	—
75	C₂²×C₁₉	—	2	1	no	—
76	C₂²×C₁₉	—	4	2	no	—
77	C₂²×C₁₉	—	4	1	no	—
78	C₂²×C₁₉	—	4	1	no	—
79	C₂²×C₁₉	—	4	1	no	—
80	C₂²×C₁₉	—	8	1	no	—
81	C₂²×C₁₉	#327	1	1	yes	characteristic
82	C₂²×C₁₉	#359	1	1	yes	characteristic
83	C₂²×C₁₉	#362	1	1	yes	characteristic
84	C₄×C₁₉	—	2	1	no	—
85	C₄×C₁₉	—	4	1	no	—
86	C₄×C₁₉	#352	1	1	yes	characteristic
87	C₄×C₁₉	#354	1	1	yes	characteristic
88	Dic₁₉	—	16	1	no	—
89	Dic₁₉	—	16	1	no	—
90	C₂×#107	—	19	1	no	—
91	#407	—	38	1	no	—
92	C₂×C₂²⋊C₈	C₂×D₁₉	1	1	yes	characteristic
93	#380	—	38	1	no	—
94	C₂²⋊C₁₆	—	2	1	no	—
95	#340	—	19	1	no	—
96	#340	—	38	1	no	—
97	C₂×C₁₉	—	2	1	no	—
98	C₂×C₁₉	—	2	1	no	—
99	C₂×C₁₉	—	8	1	no	—
100	C₂×C₁₉	#1341	1	1	yes	characteristic
101	C₂×C₁₉	#1344	1	1	yes	characteristic
102	C₂×C₁₉	#1350	1	1	yes	characteristic
103	C₂³×C₄	D₇₆	1	1	yes	characteristic
104	C₂×M₁₆	—	76	1	no	—
105	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
106	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
107	C₂×C₄⋊C₄	—	38	1	no	—
108	C₂×C₄⋊C₄	—	38	1	no	—
109	C₂×C₂²⋊C₄	—	76	1	no	—
110	C₂×C₁₆	—	4	1	no	—
111	#107	—	76	1	no	—
112	C₂²⋊C₈	—	4	1	no	—
113	C₂²⋊C₈	—	38	2	no	—
114	C₂²⋊C₈	—	76	1	no	—
115	#95	—	76	1	no	—
116	C₁₉	#7085	1	1	yes	characteristic, complemented, Sylow
117	C₂⁴	#64516	1	1	yes	characteristic
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	—	2	1	no	—
120	C₂²×C₄	—	4	1	no	—
121	C₂²×C₄	—	4	1	no	—
122	C₂²×C₄	—	76	1	no	—
123	C₂²×C₄	—	76	1	no	—
124	C₂²×C₄	D₁₅₂	1	1	yes	characteristic
125	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
126	M₁₆	—	152	1	no	—
127	M₁₆	—	152	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	4	1	no	—
131	C₂×C₈	—	4	1	no	—
132	C₂×C₈	—	76	2	no	—
133	C₄⋊C₄	—	76	1	no	—
134	C₄⋊C₄	—	76	1	no	—
135	C₄⋊C₄	—	76	1	no	—
136	C₄⋊C₄	—	76	1	no	—
137	C₂²⋊C₄	—	152	2	no	—
138	C₁₆	—	8	1	no	—
139	C₂³	—	2	1	no	—
140	C₂³	—	2	1	no	—
141	C₂³	—	2	1	no	—
142	C₂³	—	4	1	no	—
143	C₂³	—	4	1	no	—
144	C₂³	#10597278	1	1	yes	characteristic
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	8	1	no	—
152	C₂×C₄	—	152	1	no	—
153	C₂×C₄	—	152	1	no	—
154	C₂×C₄	—	152	1	no	—
155	C₂×C₄	—	152	1	no	—
156	C₂×C₄	—	152	1	no	—
157	C₂×C₄	—	152	1	no	—
158	C₂×C₄	#10597263	1	1	yes	characteristic
159	C₂×C₄	#10597282	1	1	yes	characteristic
160	C₈	—	4	1	no	—
161	C₈	—	4	1	no	—
162	C₈	—	152	2	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	4	2	no	—
166	C₂²	—	4	1	no	—
167	C₂²	—	4	1	no	—
168	C₂²	—	4	1	no	—
169	C₂²	—	8	1	no	—
170	C₂²	#11926279	1	1	yes	characteristic
171	C₂²	#11926352	1	1	yes	characteristic
172	C₂²	#11926361	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	4	1	no	—
175	C₄	—	304	1	no	—
176	C₄	—	304	1	no	—
177	C₄	#11926335	1	1	yes	characteristic
178	C₄	#11926338	1	1	yes	characteristic
179	C₂	—	2	1	no	—
180	C₂	—	2	1	no	—
181	C₂	—	8	1	no	—
182	C₂	#14561627	1	1	yes	characteristic
183	C₂	#14561636	1	1	yes	characteristic
184	C₂	#14561654	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14565063

C₂

yes

maximal, characteristic

#14561512

C₂

yes

maximal, characteristic

C₁₉×#1314

C₂

yes

maximal, characteristic

C₂×#10597275

C₄

yes

—

—

—

C₄

yes

characteristic

C₂×C₁₉×C₂²⋊C₈

C₂²

yes

characteristic

C₁₉×C₂²⋊C₁₆

—

C₂³×C₄×C₁₉

D₄

yes

characteristic

C₂²×C₈×C₁₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₂×C₂²⋊Dic₁₉

—

C₂×C₄⋊Dic₁₉

—

C₂×C₄⋊Dic₁₉

—

C₂×#64520

—

C₂×C₁₆×C₁₉

—

C₁₉×C₂²⋊C₈

—

C₁₉⋊#95

—

C₂²⋊(C₁₉⋊C₈)

—

C₂²⋊(C₁₉⋊C₈)

—

#10597275

—

C₂⁴×C₁₉

SD₁₆

yes

characteristic

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

D₈

yes

characteristic

C₂²×Dic₁₉

—

C₂²×Dic₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂²⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

#64520

—

#64520

—

C₂×C₁₉⋊C₈

—

C₁₆×C₁₉

—

#7085

—

maximal, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

#102

yes

characteristic

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×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₁₉⋊C₈

—

#2076

—

#1314

D₁₉

yes

characteristic

#1293

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

yes

C₄×C₁₉

yes

Dic₁₉

—

Dic₁₉

—

C₂×#107

—

#407

—

C₂×C₂²⋊C₈

C₂×D₁₉

yes

characteristic

#380

—

C₂²⋊C₁₆

—

#340

—

#340

—

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

—

100

C₂×C₁₉

#1341

yes

characteristic

101

C₂×C₁₉

#1344

yes

characteristic

102

C₂×C₁₉

#1350

yes

characteristic

103

C₂³×C₄

D₇₆

yes

characteristic

104

C₂×M₁₆

—

105

C₂²×C₈

C₄×D₁₉

yes

characteristic

106

C₂²×C₈

C₂²⋊D₁₉

yes

characteristic

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₂²⋊C₄

—

110

C₂×C₁₆

—

111

#107

—

112

C₂²⋊C₈

—

113

C₂²⋊C₈

—

114

C₂²⋊C₈

—

115

#95

—

116

C₁₉

#7085

yes

characteristic, complemented, Sylow

117

C₂⁴

#64516

yes

characteristic

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

D₁₅₂

yes

characteristic

125

C₂²×C₄

(C₂×D₁₉)⋊C₄

yes

characteristic

126

M₁₆

—

152

—

127

M₁₆

—

152

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₂²⋊C₄

—

152

—

138

C₁₆

—

139

C₂³

—

140

C₂³

—

141

C₂³

—

142

C₂³

—

143

C₂³

—

144

C₂³

#10597278

yes

characteristic

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

152

—

153

C₂×C₄

—

152

—

154

C₂×C₄

—

152

—

155

C₂×C₄

—

152

—

156

C₂×C₄

—

152

—

157

C₂×C₄

—

152

—

158

C₂×C₄

#10597263

yes

characteristic

159

C₂×C₄

#10597282

yes

characteristic

160

C₈

—

161

C₈

—

162

C₈

—

152

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

—

169

C₂²

—

170

C₂²

#11926279

yes

characteristic

171

C₂²

#11926352

yes

characteristic

172

C₂²

#11926361

yes

characteristic

173

C₄

—

174

C₄

—

175

C₄

—

304

—

176

C₄

—

304

—

177

C₄

#11926335

yes

characteristic

178

C₄

#11926338

yes

characteristic

179

C₂

—

180

C₂

—

181

C₂

—

182

C₂

#14561627

yes

characteristic

183

C₂

#14561636

yes

characteristic

184

C₂

#14561654

yes

characteristic

	subgroup	G/H
center	C₂²	#11926279
commutator	C₂²×C₈×C₁₉	C₂×C₄
Frattini	C₂×C₂²⋊C₈	C₂×D₁₉
Fitting	C₁₉×#1314	C₂
socle	C₂²×C₁₉	#327
prime radical	C₂⁴×C₁₉	SD₁₆
cosocle	C₂×C₁₉×C₂²⋊C₈	C₂²
hypercenter	#1314	D₁₉
hypocenter	C₁₉	#7085
LCS	G ▷ C₂²×C₈×C₁₉ ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈ ◁ #1314

subgroup

G/H

center

C₂²

#11926279

commutator

C₂²×C₈×C₁₉

C₂×C₄

Frattini

C₂×C₂²⋊C₈

C₂×D₁₉

Fitting

C₁₉×#1314

C₂

socle

C₂²×C₁₉

#327

prime radical

C₂⁴×C₁₉

SD₁₆

cosocle

C₂×C₁₉×C₂²⋊C₈

C₂²

D₁₉

C₁₉

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈ ◁ #1314

N	H ≅ G/N	type
116. C₁₉	#7085	semidirect

H ≅ G/N

type

116. C₁₉

#7085

semidirect

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

subgroup	index
#14565063	2
#14561512	2
C₁₉×#1314	2
#7085	19

subgroup

index

#14565063

#14561512

C₁₉×#1314

#7085

Subgroups — #18069240

Conjugacy classes of subgroups

Complements

Maximal subgroups