Group #18075559 — subgroups

Subgroups — #18075559

Order 4864.

336 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 109 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	#14572570	C₂	1	1	yes	maximal, characteristic
2	#14562390	C₂	1	2	yes	maximal
3	#14562373	C₂	1	1	yes	maximal, characteristic
4	C₁₉⋊#1309	C₂	1	2	yes	maximal
5	C₁₉×#1561	C₂	1	1	yes	maximal, characteristic
6	#11926675	—	2	2	no	—
7	Q₈×C₁₉⋊C₈	—	2	2	no	—
8	#11926571	C₄	1	2	yes	—
9	#11926568	C₂²	1	2	yes	—
10	#11926568	C₂²	1	1	yes	characteristic
11	#11926500	C₄	1	2	yes	—
12	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
13	#11926294	—	2	2	no	—
14	#11926294	—	2	2	no	—
15	#11926290	—	2	2	no	—
16	C₁₉×#533	C₂²	1	1	yes	characteristic
17	C₁₉×#423	C₂²	1	2	yes	—
18	C₁₉×#332	—	2	2	no	—
19	C₂×M₁₆×C₁₉	—	2	2	no	—
20	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
21	C₁₉×#125	Q₈	1	2	yes	—
22	C₁₉×C₂²⋊Q₈	—	2	2	no	—
23	C₄×Q₈×C₁₉	—	2	2	no	—
24	C₁₉×#117	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
26	C₂×#64520	—	2	2	no	—
27	C₂×#64520	C₂×C₄	1	2	yes	—
28	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
30	C₁₉×C₄⋊C₈	—	2	2	no	—
31	C₁₉×C₄⋊C₈	—	2	2	no	—
32	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
33	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
34	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
35	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
36	#10597260	—	2	2	no	—
37	C₄×C₁₉⋊C₈	—	2	2	no	—
38	C₄×C₁₉⋊C₈	D₄	1	2	yes	—
39	C₄×C₁₉⋊C₈	D₄	1	2	yes	—
40	C₂×Q₈×C₁₉	—	2	2	no	—
41	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₉	#41	1	2	yes	—
43	M₁₆×C₁₉	—	8	2	no	—
44	C₂×C₈×C₁₉	—	4	2	no	—
45	C₄⋊C₄×C₁₉	—	2	2	no	—
46	C₄⋊C₄×C₁₉	—	2	4	no	—
47	C₄⋊C₄×C₁₉	C₄⋊C₄	1	4	yes	—
48	C₂²⋊C₄×C₁₉	—	2	2	no	—
49	C₄²×C₁₉	—	2	2	no	—
50	C₄²×C₁₉	C₂×D₄	1	2	yes	—
51	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
52	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
53	#64520	—	4	2	no	—
54	#64520	—	8	2	no	—
55	C₂×C₁₉⋊C₈	—	2	2	no	—
56	C₂×C₁₉⋊C₈	—	2	2	no	—
57	C₂×C₁₉⋊C₈	—	4	1	no	—
58	C₂×C₁₉⋊C₈	—	4	2	no	—
59	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	2	yes	—
61	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
62	#8637	—	19	1	no	maximal, Sylow
63	C₂³×C₁₉	#117	1	1	yes	characteristic
64	Q₈×C₁₉	—	4	2	no	—
65	Q₈×C₁₉	—	4	2	no	—
66	C₂×C₄×C₁₉	—	2	1	no	—
67	C₂×C₄×C₁₉	—	2	1	no	—
68	C₂×C₄×C₁₉	—	2	2	no	—
69	C₂×C₄×C₁₉	—	2	2	no	—
70	C₂×C₄×C₁₉	—	2	4	no	—
71	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₉	#121	1	2	yes	—
74	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
75	C₈×C₁₉	—	8	2	no	—
76	C₁₉⋊C₈	—	2	1	no	—
77	C₁₉⋊C₈	—	2	2	no	—
78	C₁₉⋊C₈	—	4	2	no	—
79	C₁₉⋊C₈	—	8	2	no	—
80	C₁₉⋊C₈	#101	1	2	yes	—
81	#2960	—	19	1	no	—
82	#1566	—	19	2	no	—
83	#1561	—	19	1	no	—
84	#1561	D₁₉	1	1	yes	characteristic
85	#1309	—	19	2	no	—
86	C₂²×C₁₉	—	2	1	no	—
87	C₂²×C₁₉	—	2	2	no	—
88	C₂²×C₁₉	#380	1	1	yes	characteristic
89	C₄×C₁₉	—	2	1	no	—
90	C₄×C₁₉	—	2	2	no	—
91	C₄×C₁₉	—	4	4	no	—
92	C₄×C₁₉	C₂×#101	1	1	yes	characteristic
93	C₄×C₁₉	#413	1	1	yes	characteristic
94	C₄×C₁₉	#471	1	2	yes	—
95	C₄×C₁₉	#471	1	2	yes	—
96	#533	C₂×D₁₉	1	1	yes	characteristic
97	#446	—	38	2	no	—
98	C₈×Q₈	—	38	2	no	—
99	#424	—	19	2	no	—
100	#423	—	19	1	no	—
101	#423	—	19	2	no	—
102	#423	C₂×D₁₉	1	2	yes	—
103	#405	—	19	2	no	—
104	C₂×C₄×C₈	—	19	1	no	—
105	#334	—	38	2	no	—
106	#334	—	38	2	no	—
107	#332	—	2	2	no	—
108	#332	—	38	2	no	—
109	C₂×C₁₉	—	4	1	no	—
110	C₂×C₁₉	#1801	1	1	yes	characteristic
111	C₂×C₁₉	#1815	1	2	yes	—
112	C₂×M₁₆	—	2	2	no	—
113	C₂×M₁₆	—	19	2	no	—
114	C₂×M₁₆	—	38	2	no	—
115	C₂²×C₈	—	19	1	no	—
116	C₂²×C₈	—	19	1	no	—
117	C₄⋊Q₈	C₂²⋊D₁₉	1	2	yes	—
118	#125	Dic₃₈	1	2	yes	—
119	C₂²⋊Q₈	—	2	2	no	—
120	C₄×Q₈	—	2	2	no	—
121	#117	C₄×D₁₉	1	2	yes	—
122	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	38	4	no	—
126	C₄⋊C₈	—	38	2	no	—
127	C₄⋊C₈	—	38	2	no	—
128	C₄⋊C₈	—	38	2	no	—
129	#97	—	38	2	no	—
130	C₄×C₈	—	19	2	no	—
131	C₄×C₈	—	19	2	no	—
132	C₄×C₈	—	38	2	no	—
133	C₁₉	#8637	1	1	yes	characteristic, complemented, Sylow
134	C₂×Q₈	—	2	2	no	—
135	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
136	C₂²×C₄	#64543	1	2	yes	—
137	M₁₆	—	8	2	no	—
138	M₁₆	—	76	2	no	—
139	M₁₆	—	152	2	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	19	1	no	—
142	C₂×C₈	—	19	1	no	—
143	C₂×C₈	—	19	2	no	—
144	C₂×C₈	—	38	2	no	—
145	C₂×C₈	—	38	2	no	—
146	C₂×C₈	—	76	1	no	—
147	C₂×C₈	—	76	2	no	—
148	C₄⋊C₄	—	2	2	no	—
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	Dic₁₉⋊C₄	1	4	yes	—
151	C₂²⋊C₄	—	2	2	no	—
152	C₄²	—	2	2	no	—
153	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
154	C₄²	D₄×D₁₉	1	2	yes	—
155	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
156	C₂³	#10597323	1	1	yes	characteristic
157	Q₈	—	4	2	no	—
158	Q₈	—	4	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	#10597324	1	2	yes	—
165	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
166	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
167	C₂×C₄	#10597385	1	2	yes	—
168	C₈	—	8	2	no	—
169	C₈	—	19	2	no	—
170	C₈	—	38	1	no	—
171	C₈	—	38	2	no	—
172	C₈	—	76	2	no	—
173	C₈	—	152	2	no	—
174	C₂²	—	2	1	no	—
175	C₂²	—	2	2	no	—
176	C₂²	#11926402	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	2	2	no	—
179	C₄	—	4	4	no	—
180	C₄	#101×D₁₉	1	1	yes	characteristic
181	C₄	#11926527	1	1	yes	characteristic
182	C₄	#11926805	1	2	yes	—
183	C₄	#11926809	1	2	yes	—
184	C₂	—	4	1	no	—
185	C₂	#14563629	1	1	yes	characteristic
186	C₂	#14563692	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14572570

C₂

yes

maximal, characteristic

#14562390

C₂

yes

maximal

#14562373

C₂

yes

maximal, characteristic

C₁₉⋊#1309

C₂

yes

maximal

C₁₉×#1561

C₂

yes

maximal, characteristic

#11926675

—

Q₈×C₁₉⋊C₈

—

#11926571

C₄

yes

—

#11926568

C₂²

yes

—

#11926568

C₂²

yes

characteristic

#11926500

C₄

yes

—

C₂×C₄×C₁₉⋊C₈

C₂²

yes

—

—

—

C₁₉×#533

C₂²

yes

characteristic

C₁₉×#423

C₂²

yes

—

C₁₉×#332

—

C₂×M₁₆×C₁₉

—

C₁₉×C₄⋊Q₈

D₄

yes

—

C₁₉×#125

Q₈

yes

—

C₁₉×C₂²⋊Q₈

—

C₄×Q₈×C₁₉

—

C₁₉×#117

C₂×C₄

yes

—

C₂×C₄²×C₁₉

C₂³

yes

characteristic

C₂×#64520

—

C₂×#64520

C₂×C₄

yes

—

C₂²×C₁₉⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₁₉⋊C₈

C₂×C₄

yes

characteristic

C₁₉×C₄⋊C₈

—

C₁₉×C₄⋊C₈

—

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

—

C₄⋊(C₁₉⋊C₈)

—

#10597260

—

C₄×C₁₉⋊C₈

—

C₄×C₁₉⋊C₈

D₄

yes

—

C₄×C₁₉⋊C₈

D₄

yes

—

C₂×Q₈×C₁₉

—

C₂²×C₄×C₁₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₉

#41

yes

—

M₁₆×C₁₉

—

C₂×C₈×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

C₄⋊C₄

yes

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

C₂×D₄

yes

—

C₄²×C₁₉

C₂×D₄

yes

characteristic

C₄²×C₁₉

C₂×Q₈

yes

characteristic

#64520

—

#64520

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

C₂²⋊C₄

yes

characteristic

C₂×C₁₉⋊C₈

C₂²⋊C₄

yes

—

C₂×C₁₉⋊C₈

C₂²⋊C₄

yes

—

C₂³×C₁₉

yes

Q₈×C₁₉

—

Q₈×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₉

#121

yes

—

C₂×C₄×C₁₉

C₂²⋊Q₈

yes

—

C₈×C₁₉

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

C₁₉⋊C₈

#101

yes

—

—

—

—

D₁₉

yes

characteristic

#1309

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#380

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

yes

C₄×C₁₉

yes

C₄×C₁₉

yes

—

C₄×C₁₉

#471

yes

—

#533

C₂×D₁₉

yes

characteristic

#446

—

C₈×Q₈

—

#424

—

100

#423

—

101

#423

—

102

#423

C₂×D₁₉

yes

—

103

#405

—

104

C₂×C₄×C₈

—

105

#334

—

106

#334

—

107

#332

—

108

#332

—

109

C₂×C₁₉

—

110

C₂×C₁₉

#1801

yes

characteristic

111

C₂×C₁₉

#1815

yes

—

112

C₂×M₁₆

—

113

C₂×M₁₆

—

114

C₂×M₁₆

—

115

C₂²×C₈

—

116

C₂²×C₈

—

117

C₄⋊Q₈

C₂²⋊D₁₉

yes

—

118

#125

Dic₃₈

yes

—

119

C₂²⋊Q₈

—

120

C₄×Q₈

—

121

#117

C₄×D₁₉

yes

—

122

C₂×C₄²

C₂²×D₁₉

yes

characteristic

123

C₄⋊C₈

—

124

C₄⋊C₈

—

125

C₄⋊C₈

—

126

C₄⋊C₈

—

127

C₄⋊C₈

—

128

C₄⋊C₈

—

129

#97

—

130

C₄×C₈

—

131

C₄×C₈

—

132

C₄×C₈

—

133

C₁₉

#8637

yes

characteristic, complemented, Sylow

134

C₂×Q₈

—

135

C₂²×C₄

C₂×C₄×D₁₉

yes

characteristic

136

C₂²×C₄

#64543

yes

—

137

M₁₆

—

138

M₁₆

—

139

M₁₆

—

152

—

140

C₂×C₈

—

141

C₂×C₈

—

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

Dic₁₉⋊C₄

yes

—

151

C₂²⋊C₄

—

152

C₄²

—

153

C₄²

C₂×Dic₃₈

yes

characteristic

154

C₄²

D₄×D₁₉

yes

—

155

C₄²

C₂×C₂²⋊D₁₉

yes

characteristic

156

C₂³

#10597323

yes

characteristic

157

Q₈

—

158

Q₈

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

#10597324

yes

—

165

C₂×C₄

C₂²⋊C₄×D₁₉

yes

characteristic

166

C₂×C₄

C₂×Dic₁₉⋊C₄

yes

characteristic

167

C₂×C₄

#10597385

yes

—

168

C₈

—

169

C₈

—

170

C₈

—

171

C₈

—

172

C₈

—

173

C₈

—

152

—

174

C₂²

—

175

C₂²

—

176

C₂²

#11926402

yes

characteristic

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

#101×D₁₉

yes

characteristic

181

C₄

#11926527

yes

characteristic

182

C₄

#11926805

yes

—

183

C₄

#11926809

yes

—

184

C₂

—

185

C₂

#14563629

yes

characteristic

186

C₂

#14563692

yes

—

	subgroup	G/H
center	C₂²	#11926402
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1561	C₂
socle	C₂²×C₁₉	#380
prime radical	C₂³×C₁₉	#117
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1561	D₁₉
hypocenter	C₁₉	#8637
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1561

subgroup

G/H

center

C₂²

#11926402

commutator

C₂²×C₄×C₁₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₉

Fitting

C₁₉×#1561

C₂

C₂²×C₁₉

C₂³×C₁₉

C₂×C₄²×C₁₉

C₂³

D₁₉

C₁₉

G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1561

N	H ≅ G/N	type
133. C₁₉	#8637	semidirect

H ≅ G/N

type

133. C₁₉

#8637

semidirect

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

subgroup	index
#14572570	2
#14562390	2
#14562373	2
C₁₉⋊#1309	2
C₁₉×#1561	2
#8637	19

subgroup

index

#14572570

#14562390

#14562373

C₁₉⋊#1309

C₁₉×#1561

#8637

Subgroups — #18075559

Conjugacy classes of subgroups

Complements

Maximal subgroups