Group #18069344 — subgroups

Subgroups — #18069344

Order 4864.

202 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 67 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	#14564378	C₂	1	1	yes	maximal, characteristic
2	#14561466	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1377	C₂	1	1	yes	maximal, characteristic
4	#11926597	—	2	1	no	—
5	C₂×#10597275	C₄	1	1	yes	characteristic
6	C₂×#10597274	C₄	1	1	yes	characteristic
7	#11926442	—	2	1	no	—
8	#11926307	—	2	1	no	—
9	C₂×C₁₉×M₃₂	—	2	1	no	—
10	C₂×C₁₉×#97	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
13	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
14	C₂×C₄⋊Dic₁₉	—	2	1	no	—
15	C₂×C₄⋊Dic₁₉	—	2	1	no	—
16	C₂×Dic₁₉⋊C₄	—	4	1	no	—
17	C₁₉×M₃₂	—	2	2	no	—
18	C₁₉×M₃₂	—	4	1	no	—
19	C₂×C₁₆×C₁₉	—	4	1	no	—
20	C₁₉×#97	—	4	1	no	—
21	C₁₉⋊#95	—	4	1	no	—
22	C₂²⋊(C₁₉⋊C₈)	—	4	1	no	—
23	#10597275	—	4	1	no	—
24	#10597274	—	2	1	no	—
25	#10597274	C₈	1	2	yes	complemented
26	C₄⋊(C₁₉⋊C₈)	—	4	1	no	—
27	C₄×C₁₉⋊C₈	—	2	2	no	—
28	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
30	C₂²×C₄×C₁₉	SD₁₆	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	—	2	1	no	—
36	C₂×C₈×C₁₉	—	2	1	no	—
37	C₂×C₈×C₁₉	C₂×C₈	1	1	yes	characteristic
38	C₂×C₈×C₁₉	M₁₆	1	1	yes	characteristic
39	C₂×C₈×C₁₉	SD₁₆	1	1	yes	characteristic
40	C₂×C₈×C₁₉	Q₁₆	1	1	yes	characteristic
41	C₄²×C₁₉	—	2	1	no	—
42	C₄²×C₁₉	—	2	1	no	—
43	C₄⋊Dic₁₉	—	2	2	no	—
44	C₄⋊Dic₁₉	—	4	1	no	—
45	C₄⋊Dic₁₉	—	4	1	no	—
46	C₄⋊Dic₁₉	—	4	1	no	—
47	Dic₁₉⋊C₄	—	8	2	no	—
48	C₂×C₁₉⋊C₈	—	4	2	no	—
49	C₁₆×C₁₉	—	4	2	no	—
50	#7126	—	19	1	no	maximal, Sylow
51	C₂³×C₁₉	#102	1	1	yes	characteristic
52	C₂×C₄×C₁₉	—	2	1	no	—
53	C₂×C₄×C₁₉	—	2	1	no	—
54	C₂×C₄×C₁₉	—	4	1	no	—
55	C₂×C₄×C₁₉	—	4	1	no	—
56	C₂×C₄×C₁₉	C₂²⋊C₈	1	1	yes	characteristic
57	C₂×C₄×C₁₉	#99	1	1	yes	characteristic
58	C₂×C₄×C₁₉	#101	1	1	yes	characteristic
59	C₂×C₄×C₁₉	#103	1	1	yes	characteristic
60	C₂×C₄×C₁₉	#104	1	1	yes	characteristic
61	C₂×C₄×C₁₉	#104	1	1	yes	characteristic
62	C₂×Dic₁₉	—	4	2	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	—	2	1	no	—
69	C₈×C₁₉	—	2	1	no	—
70	C₈×C₁₉	—	2	1	no	—
71	C₈×C₁₉	—	2	1	no	—
72	C₁₉⋊C₈	—	8	2	no	—
73	#1944	—	19	1	no	—
74	#1377	D₁₉	1	1	yes	characteristic
75	#1275	—	19	1	no	—
76	C₂²×C₁₉	—	2	1	no	—
77	C₂²×C₁₉	—	2	1	no	—
78	C₂²×C₁₉	D₄⋊C₈	1	1	yes	characteristic
79	C₂²×C₁₉	#327	1	1	yes	characteristic
80	C₂²×C₁₉	#332	1	1	yes	characteristic
81	C₄×C₁₉	—	4	1	no	—
82	C₄×C₁₉	—	4	1	no	—
83	C₄×C₁₉	#324	1	1	yes	characteristic
84	C₄×C₁₉	Q₈⋊C₈	1	1	yes	characteristic
85	C₄×C₁₉	#328	1	1	yes	characteristic
86	C₄×C₁₉	#330	1	1	yes	characteristic
87	Dic₁₉	—	8	2	no	—
88	Dic₁₉	—	16	1	no	—
89	C₂×M₃₂	—	2	1	no	—
90	#431	—	38	1	no	—
91	C₂×#107	—	19	1	no	—
92	C₂×#106	—	19	1	no	—
93	C₂×#97	C₂×D₁₉	1	1	yes	characteristic
94	#389	—	38	1	no	—
95	#340	—	38	1	no	—
96	C₂×C₁₉	—	2	1	no	—
97	C₂×C₁₉	—	2	1	no	—
98	C₂×C₁₉	#1271	1	1	yes	characteristic
99	C₂×C₁₉	#1340	1	1	yes	characteristic
100	C₂×C₁₉	#1341	1	1	yes	characteristic
101	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
102	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
103	C₂×C₄⋊C₄	—	38	1	no	—
104	C₂×C₄⋊C₄	—	38	1	no	—
105	C₂×C₄⋊C₄	—	76	1	no	—
106	C₂×C₄²	D₇₆	1	1	yes	characteristic
107	M₃₂	—	2	2	no	—
108	M₃₂	—	4	1	no	—
109	C₂×C₁₆	—	4	1	no	—
110	#107	—	76	1	no	—
111	#106	—	19	2	no	—
112	#106	—	38	1	no	—
113	C₄⋊C₈	—	76	1	no	—
114	C₂²⋊C₈	—	76	1	no	—
115	#97	—	4	1	no	—
116	C₄×C₈	—	38	2	no	—
117	#95	—	76	1	no	—
118	C₁₉	#7126	1	1	yes	characteristic, complemented, Sylow
119	C₂²×C₄	—	76	1	no	—
120	C₂²×C₄	—	76	1	no	—
121	C₂²×C₄	#64516	1	1	yes	characteristic
122	C₂²×C₄	D₁₅₂	1	1	yes	characteristic
123	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	76	2	no	—
129	C₂×C₈	C₈×D₁₉	1	1	yes	characteristic
130	C₂×C₈	#64515	1	1	yes	characteristic
131	C₂×C₈	#64526	1	1	yes	characteristic
132	C₂×C₈	#64528	1	1	yes	characteristic
133	C₄⋊C₄	—	38	2	no	—
134	C₄⋊C₄	—	76	1	no	—
135	C₄⋊C₄	—	76	1	no	—
136	C₄⋊C₄	—	76	1	no	—
137	C₄⋊C₄	—	152	2	no	—
138	C₄²	—	2	1	no	—
139	C₄²	—	2	1	no	—
140	C₁₆	—	4	2	no	—
141	C₂³	#10597278	1	1	yes	characteristic
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	4	1	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	76	2	no	—
147	C₂×C₄	—	152	1	no	—
148	C₂×C₄	—	152	1	no	—
149	C₂×C₄	—	152	1	no	—
150	C₂×C₄	—	152	1	no	—
151	C₂×C₄	—	152	1	no	—
152	C₂×C₄	#10597262	1	1	yes	characteristic
153	C₂×C₄	#10597263	1	1	yes	characteristic
154	C₂×C₄	#10597267	1	1	yes	characteristic
155	C₂×C₄	(C₂×D₁₉)⋊C₈	1	1	yes	characteristic
156	C₂×C₄	#10597281	1	1	yes	characteristic
157	C₂×C₄	#10597282	1	1	yes	characteristic
158	C₈	—	2	1	no	—
159	C₈	—	2	1	no	—
160	C₈	—	2	1	no	—
161	C₈	—	2	1	no	—
162	C₈	—	152	2	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	#11926273	1	1	yes	characteristic
166	C₂²	#11926279	1	1	yes	characteristic
167	C₂²	#11926289	1	1	yes	characteristic
168	C₄	—	4	1	no	—
169	C₄	—	4	1	no	—
170	C₄	—	304	1	no	—
171	C₄	—	152	2	no	—
172	C₄	#11926269	1	1	yes	characteristic
173	C₄	#11926274	1	1	yes	characteristic
174	C₄	#11926280	1	1	yes	characteristic
175	C₄	#11926285	1	1	yes	characteristic
176	C₂	—	2	1	no	—
177	C₂	—	2	1	no	—
178	C₂	#14561457	1	1	yes	characteristic
179	C₂	#14561624	1	1	yes	characteristic
180	C₂	#14561627	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14564378

C₂

yes

maximal, characteristic

#14561466

C₂

yes

maximal, characteristic

C₁₉×#1377

C₂

yes

maximal, characteristic

#11926597

—

C₂×#10597275

C₄

yes

characteristic

C₂×#10597274

C₄

yes

characteristic

#11926442

—

#11926307

—

C₂×C₁₉×M₃₂

—

C₂×C₁₉×#97

C₂²

yes

characteristic

C₂²×C₈×C₁₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₂×C₄²×C₁₉

D₄

yes

characteristic

C₂×C₄⋊Dic₁₉

—

C₂×C₄⋊Dic₁₉

—

C₂×Dic₁₉⋊C₄

—

C₁₉×M₃₂

—

C₁₉×M₃₂

—

C₂×C₁₆×C₁₉

—

C₁₉×#97

—

C₁₉⋊#95

—

C₂²⋊(C₁₉⋊C₈)

—

#10597275

—

#10597274

—

#10597274

C₈

yes

complemented

C₄⋊(C₁₉⋊C₈)

—

C₄×C₁₉⋊C₈

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

D₈

yes

characteristic

C₂²×C₄×C₁₉

SD₁₆

yes

characteristic

C₂²×Dic₁₉

—

C₂²×Dic₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

C₂×C₈

yes

characteristic

C₂×C₈×C₁₉

M₁₆

yes

characteristic

C₂×C₈×C₁₉

SD₁₆

yes

characteristic

C₂×C₈×C₁₉

Q₁₆

yes

characteristic

C₄²×C₁₉

—

C₄²×C₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

Dic₁₉⋊C₄

—

C₂×C₁₉⋊C₈

—

C₁₆×C₁₉

—

—

C₂³×C₁₉

yes

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂²⋊C₈

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×C₄×C₁₉

yes

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₁₉⋊C₈

—

#1944

—

#1377

D₁₉

yes

characteristic

#1275

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

D₄⋊C₈

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

#324

yes

characteristic

C₄×C₁₉

Q₈⋊C₈

yes

C₄×C₁₉

yes

C₄×C₁₉

yes

Dic₁₉

—

Dic₁₉

—

C₂×M₃₂

—

#431

—

C₂×#107

—

C₂×#106

—

C₂×#97

C₂×D₁₉

yes

characteristic

#389

—

#340

—

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

yes

C₂×C₁₉

yes

100

C₂×C₁₉

#1341

yes

characteristic

101

C₂²×C₈

C₄×D₁₉

yes

characteristic

102

C₂²×C₈

C₂²⋊D₁₉

yes

characteristic

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄²

D₇₆

yes

characteristic

107

M₃₂

—

108

M₃₂

—

109

C₂×C₁₆

—

110

#107

—

111

#106

—

112

#106

—

113

C₄⋊C₈

—

114

C₂²⋊C₈

—

115

#97

—

116

C₄×C₈

—

117

#95

—

118

C₁₉

#7126

yes

characteristic, complemented, Sylow

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

#64516

yes

characteristic

122

C₂²×C₄

D₁₅₂

yes

characteristic

123

C₂²×C₄

(C₂×D₁₉)⋊C₄

yes

characteristic

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

C₈×D₁₉

yes

characteristic

130

C₂×C₈

#64515

yes

characteristic

131

C₂×C₈

#64526

yes

characteristic

132

C₂×C₈

#64528

yes

characteristic

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

152

—

138

C₄²

—

139

C₄²

—

140

C₁₆

—

141

C₂³

#10597278

yes

characteristic

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

152

—

148

C₂×C₄

—

152

—

149

C₂×C₄

—

152

—

150

C₂×C₄

—

152

—

151

C₂×C₄

—

152

—

152

C₂×C₄

#10597262

yes

characteristic

153

C₂×C₄

#10597263

yes

characteristic

154

C₂×C₄

#10597267

yes

characteristic

155

C₂×C₄

(C₂×D₁₉)⋊C₈

yes

characteristic

156

C₂×C₄

#10597281

yes

characteristic

157

C₂×C₄

#10597282

yes

characteristic

158

C₈

—

159

C₈

—

160

C₈

—

161

C₈

—

162

C₈

—

152

—

163

C₂²

—

164

C₂²

—

165

C₂²

#11926273

yes

characteristic

166

C₂²

#11926279

yes

characteristic

167

C₂²

#11926289

yes

characteristic

168

C₄

—

169

C₄

—

170

C₄

—

304

—

171

C₄

—

152

—

172

C₄

#11926269

yes

characteristic

173

C₄

#11926274

yes

characteristic

174

C₄

#11926280

yes

characteristic

175

C₄

#11926285

yes

characteristic

176

C₂

—

177

C₂

—

178

C₂

#14561457

yes

characteristic

179

C₂

#14561624

yes

characteristic

180

C₂

#14561627

yes

characteristic

	subgroup	G/H
center	C₂²	#11926279
commutator	C₂×C₈×C₁₉	C₂×C₈
Frattini	C₂×#97	C₂×D₁₉
Fitting	C₁₉×#1377	C₂
socle	C₂²×C₁₉	#327
prime radical	C₂³×C₁₉	#102
cosocle	C₂×C₁₉×#97	C₂²
hypercenter	#1377	D₁₉
hypocenter	C₁₉	#7126
LCS	G ▷ C₂×C₈×C₁₉ ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1377

subgroup

G/H

center

C₂²

#11926279

commutator

C₂×C₈×C₁₉

C₂×C₈

Frattini

C₂×#97

C₂×D₁₉

Fitting

C₁₉×#1377

C₂

C₂²×C₁₉

C₂³×C₁₉

C₂×C₁₉×#97

C₂²

D₁₉

C₁₉

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1377

N	H ≅ G/N	type
25. #10597274	C₈	semidirect
118. C₁₉	#7126	semidirect

H ≅ G/N

type

25. #10597274

C₈

semidirect

118. C₁₉

#7126

semidirect

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

subgroup	index
#14564378	2
#14561466	2
C₁₉×#1377	2
#7126	19

subgroup

index

#14564378

#14561466

C₁₉×#1377

#7126

Subgroups — #18069344

Conjugacy classes of subgroups

Complements

Maximal subgroups