Group #18069325 — subgroups

Subgroups — #18069325

Order 4864.

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

normal subgroups only

This group has 212 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	#14564383	C₂	1	1	yes	maximal, characteristic
2	C₁₉⋊#1273	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1376	C₂	1	1	yes	maximal, characteristic
4	C₂×#10597276	C₄	1	1	yes	characteristic
5	#11926582	C₄	1	1	yes	characteristic
6	#11926570	—	2	1	no	—
7	C₂×C₄×C₁₉⋊C₈	—	2	1	no	—
8	#11926304	—	2	1	no	—
9	C₂×C₁₉×M₃₂	—	2	1	no	—
10	C₂×C₄×C₈×C₁₉	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	#10597372	—	2	1	no	—
15	C₂×#64520	—	2	1	no	—
16	C₂²×C₁₉⋊C₈	—	2	2	no	—
17	C₁₉×M₃₂	—	2	2	no	—
18	C₁₉×M₃₂	—	4	1	no	—
19	C₂×C₁₆×C₁₉	—	4	1	no	—
20	C₄×C₈×C₁₉	—	4	1	no	—
21	#10597276	—	4	1	no	—
22	#10597275	C₈	1	1	yes	characteristic, complemented
23	#10597275	C₈	1	1	yes	characteristic, complemented
24	#10597274	—	2	1	no	—
25	Dic₁₉⋊C₈	—	4	1	no	—
26	Dic₁₉⋊C₈	—	4	1	no	—
27	C₄×C₁₉⋊C₈	—	4	1	no	—
28	C₄×C₁₉⋊C₈	—	4	1	no	—
29	C₂²×C₄×C₁₉	—	2	1	no	—
30	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₁₉	—	2	1	no	—
32	C₂×C₈×C₁₉	—	2	1	no	—
33	C₂×C₈×C₁₉	—	2	1	no	—
34	C₂×C₈×C₁₉	—	2	1	no	—
35	C₂×C₈×C₁₉	C₂×C₈	1	1	yes	characteristic
36	C₂×C₈×C₁₉	M₁₆	1	1	yes	characteristic
37	C₂×C₈×C₁₉	D₈	1	1	yes	characteristic
38	C₂×C₈×C₁₉	SD₁₆	1	1	yes	characteristic
39	C₄²×C₁₉	—	2	1	no	—
40	C₄²×C₁₉	SD₁₆	1	1	yes	characteristic
41	C₄²×C₁₉	Q₁₆	1	1	yes	characteristic
42	C₂²⋊Dic₁₉	—	4	1	no	—
43	C₄⋊Dic₁₉	—	2	1	no	—
44	C₄⋊Dic₁₉	—	2	1	no	—
45	C₄×Dic₁₉	—	4	1	no	—
46	#64520	—	4	1	no	—
47	#64520	—	4	1	no	—
48	C₂×C₁₉⋊C₈	—	2	4	no	—
49	C₂×C₁₉⋊C₈	—	4	2	no	—
50	C₂×C₁₉⋊C₈	—	4	2	no	—
51	C₂×C₁₉⋊C₈	—	4	1	no	—
52	C₁₆×C₁₉	—	4	2	no	—
53	#7119	—	19	1	no	maximal, Sylow
54	C₂³×C₁₉	#104	1	1	yes	characteristic
55	C₂×C₄×C₁₉	—	2	2	no	—
56	C₂×C₄×C₁₉	—	2	2	no	—
57	C₂×C₄×C₁₉	—	2	1	no	—
58	C₂×C₄×C₁₉	—	2	1	no	—
59	C₂×C₄×C₁₉	C₂²⋊C₈	1	1	yes	characteristic
60	C₂×C₄×C₁₉	#99	1	1	yes	characteristic
61	C₂×C₄×C₁₉	#101	1	1	yes	characteristic
62	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
63	C₂×C₄×C₁₉	#103	1	1	yes	characteristic
64	C₂×C₄×C₁₉	#104	1	1	yes	characteristic
65	C₂×Dic₁₉	—	4	1	no	—
66	C₂×Dic₁₉	—	4	1	no	—
67	C₈×C₁₉	—	2	1	no	—
68	C₈×C₁₉	—	2	1	no	—
69	C₈×C₁₉	—	2	1	no	—
70	C₈×C₁₉	—	2	1	no	—
71	C₁₉⋊C₈	—	4	4	no	—
72	C₁₉⋊C₈	—	8	1	no	—
73	#1945	—	19	1	no	—
74	#1376	D₁₉	1	1	yes	characteristic
75	#1273	—	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	#328	1	1	yes	characteristic
80	C₂²×C₁₉	#330	1	1	yes	characteristic
81	C₄×C₁₉	—	2	2	no	—
82	C₄×C₁₉	—	2	2	no	—
83	C₄×C₁₉	#324	1	1	yes	characteristic
84	C₄×C₁₉	Q₈⋊C₈	1	1	yes	characteristic
85	C₄×C₁₉	#327	1	1	yes	characteristic
86	C₄×C₁₉	#332	1	1	yes	characteristic
87	Dic₁₉	—	8	1	no	—
88	Dic₁₉	—	8	1	no	—
89	C₂×M₃₂	—	2	1	no	—
90	C₂×#108	—	19	1	no	—
91	#427	—	19	1	no	—
92	#424	—	38	1	no	—
93	C₂×C₄×C₈	—	38	1	no	—
94	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
95	#339	—	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	#1335	1	1	yes	characteristic
100	C₂×C₁₉	#1336	1	1	yes	characteristic
101	C₂×M₁₆	—	38	1	no	—
102	C₂²×C₈	—	38	2	no	—
103	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
104	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
105	#117	—	38	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	#108	—	76	1	no	—
111	#107	—	19	1	no	—
112	#107	—	19	1	no	—
113	#106	—	38	1	no	—
114	C₄⋊C₈	—	76	1	no	—
115	C₄⋊C₈	—	76	1	no	—
116	C₄×C₈	—	4	1	no	—
117	C₄×C₈	—	76	1	no	—
118	C₄×C₈	—	76	1	no	—
119	C₁₉	#7119	1	1	yes	characteristic, complemented, Sylow
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
122	M₁₆	—	76	1	no	—
123	M₁₆	—	76	1	no	—
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	—	38	4	no	—
129	C₂×C₈	—	76	2	no	—
130	C₂×C₈	—	76	2	no	—
131	C₂×C₈	—	76	1	no	—
132	C₂×C₈	C₈×D₁₉	1	1	yes	characteristic
133	C₂×C₈	#64515	1	1	yes	characteristic
134	C₂×C₈	#64525	1	1	yes	characteristic
135	C₂×C₈	#64527	1	1	yes	characteristic
136	C₄⋊C₄	—	38	1	no	—
137	C₄⋊C₄	—	38	1	no	—
138	C₂²⋊C₄	—	76	1	no	—
139	C₄²	—	2	1	no	—
140	C₄²	—	76	1	no	—
141	C₄²	#64516	1	1	yes	characteristic
142	C₄²	Dic₇₆	1	1	yes	characteristic
143	C₁₆	—	4	2	no	—
144	C₂³	#10597282	1	1	yes	characteristic
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	76	1	no	—
150	C₂×C₄	—	76	1	no	—
151	C₂×C₄	#10597262	1	1	yes	characteristic
152	C₂×C₄	#10597263	1	1	yes	characteristic
153	C₂×C₄	#10597266	1	1	yes	characteristic
154	C₂×C₄	#10597273	1	1	yes	characteristic
155	C₂×C₄	(C₂×D₁₉)⋊C₈	1	1	yes	characteristic
156	C₂×C₄	#10597281	1	1	yes	characteristic
157	C₈	—	2	1	no	—
158	C₈	—	2	1	no	—
159	C₈	—	2	1	no	—
160	C₈	—	2	1	no	—
161	C₈	—	76	4	no	—
162	C₈	—	152	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	#11926271	1	1	yes	characteristic
166	C₂²	#11926282	1	1	yes	characteristic
167	C₂²	#11926285	1	1	yes	characteristic
168	C₄	—	2	2	no	—
169	C₄	—	2	2	no	—
170	C₄	—	152	1	no	—
171	C₄	—	152	1	no	—
172	C₄	#11926269	1	1	yes	characteristic
173	C₄	#11926276	1	1	yes	characteristic
174	C₄	#11926277	1	1	yes	characteristic
175	C₄	#11926291	1	1	yes	characteristic
176	C₂	—	2	1	no	—
177	C₂	—	2	1	no	—
178	C₂	#14561459	1	1	yes	characteristic
179	C₂	#14561607	1	1	yes	characteristic
180	C₂	#14561610	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14564383

C₂

yes

maximal, characteristic

C₁₉⋊#1273

C₂

yes

maximal, characteristic

C₁₉×#1376

C₂

yes

maximal, characteristic

C₂×#10597276

C₄

yes

characteristic

#11926582

C₄

yes

characteristic

#11926570

—

C₂×C₄×C₁₉⋊C₈

—

#11926304

—

C₂×C₁₉×M₃₂

—

C₂×C₄×C₈×C₁₉

C₂²

yes

characteristic

C₂²×C₈×C₁₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₂×C₄²×C₁₉

D₄

yes

characteristic

#10597372

—

C₂×#64520

—

C₂²×C₁₉⋊C₈

—

C₁₉×M₃₂

—

C₁₉×M₃₂

—

C₂×C₁₆×C₁₉

—

C₄×C₈×C₁₉

—

#10597276

—

#10597275

C₈

yes

characteristic, complemented

#10597275

C₈

yes

characteristic, complemented

#10597274

—

Dic₁₉⋊C₈

—

Dic₁₉⋊C₈

—

C₄×C₁₉⋊C₈

—

C₄×C₁₉⋊C₈

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

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

D₈

yes

characteristic

C₂×C₈×C₁₉

SD₁₆

yes

characteristic

C₄²×C₁₉

—

C₄²×C₁₉

SD₁₆

yes

characteristic

C₄²×C₁₉

Q₁₆

yes

characteristic

C₂²⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄×Dic₁₉

—

#64520

—

#64520

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

C₂×C₁₉⋊C₈

—

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

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

C₁₉⋊C₈

—

C₁₉⋊C₈

—

#1945

—

#1376

D₁₉

yes

characteristic

#1273

—

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

—

C₂×#108

—

#427

—

#424

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₁₉

yes

characteristic

#339

—

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

yes

C₂×C₁₉

yes

100

C₂×C₁₉

#1336

yes

characteristic

101

C₂×M₁₆

—

102

C₂²×C₈

—

103

C₂²×C₈

C₄×D₁₉

yes

characteristic

104

C₂²×C₈

C₂²⋊D₁₉

yes

characteristic

105

#117

—

106

C₂×C₄²

D₇₆

yes

characteristic

107

M₃₂

—

108

M₃₂

—

109

C₂×C₁₆

—

110

#108

—

111

#107

—

112

#107

—

113

#106

—

114

C₄⋊C₈

—

115

C₄⋊C₈

—

116

C₄×C₈

—

117

C₄×C₈

—

118

C₄×C₈

—

119

C₁₉

#7119

yes

characteristic, complemented, Sylow

120

C₂²×C₄

—

121

C₂²×C₄

(C₂×D₁₉)⋊C₄

yes

characteristic

122

M₁₆

—

123

M₁₆

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

C₈×D₁₉

yes

characteristic

133

C₂×C₈

#64515

yes

characteristic

134

C₂×C₈

#64525

yes

characteristic

135

C₂×C₈

#64527

yes

characteristic

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₂²⋊C₄

—

139

C₄²

—

140

C₄²

—

141

C₄²

#64516

yes

characteristic

142

C₄²

Dic₇₆

yes

characteristic

143

C₁₆

—

144

C₂³

#10597282

yes

characteristic

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

#10597262

yes

characteristic

152

C₂×C₄

#10597263

yes

characteristic

153

C₂×C₄

#10597266

yes

characteristic

154

C₂×C₄

#10597273

yes

characteristic

155

C₂×C₄

(C₂×D₁₉)⋊C₈

yes

characteristic

156

C₂×C₄

#10597281

yes

characteristic

157

C₈

—

158

C₈

—

159

C₈

—

160

C₈

—

161

C₈

—

162

C₈

—

152

—

163

C₂²

—

164

C₂²

—

165

C₂²

#11926271

yes

characteristic

166

C₂²

#11926282

yes

characteristic

167

C₂²

#11926285

yes

characteristic

168

C₄

—

169

C₄

—

170

C₄

—

152

—

171

C₄

—

152

—

172

C₄

#11926269

yes

characteristic

173

C₄

#11926276

yes

characteristic

174

C₄

#11926277

yes

characteristic

175

C₄

#11926291

yes

characteristic

176

C₂

—

177

C₂

—

178

C₂

#14561459

yes

characteristic

179

C₂

#14561607

yes

characteristic

180

C₂

#14561610

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#10597266
commutator	C₂×C₈×C₁₉	C₂×C₈
Frattini	C₂×C₄×C₈	C₂×D₁₉
Fitting	C₁₉×#1376	C₂
socle	C₂²×C₁₉	D₄⋊C₈
prime radical	C₂³×C₁₉	#104
cosocle	C₂×C₄×C₈×C₁₉	C₂²
hypercenter	#1376	D₁₉
hypocenter	C₁₉	#7119
LCS	G ▷ C₂×C₈×C₁₉ ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ #1376

subgroup

G/H

center

C₂×C₄

#10597266

commutator

C₂×C₈×C₁₉

C₂×C₈

Frattini

C₂×C₄×C₈

C₂×D₁₉

Fitting

C₁₉×#1376

C₂

socle

C₂²×C₁₉

D₄⋊C₈

prime radical

C₂³×C₁₉

#104

cosocle

C₂×C₄×C₈×C₁₉

C₂²

D₁₉

C₁₉

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

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ #1376

N	H ≅ G/N	type
22. #10597275	C₈	semidirect
23. #10597275	C₈	semidirect
119. C₁₉	#7119	semidirect

H ≅ G/N

type

22. #10597275

C₈

semidirect

23. #10597275

C₈

semidirect

119. C₁₉

#7119

semidirect

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

subgroup	index
#14564383	2
C₁₉⋊#1273	2
C₁₉×#1376	2
#7119	19

subgroup

index

#14564383

C₁₉⋊#1273

C₁₉×#1376

#7119

Subgroups — #18069325

Conjugacy classes of subgroups

Complements

Maximal subgroups