Group #18069364 — subgroups

Subgroups — #18069364

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	#14564600	C₂	1	1	yes	maximal, characteristic
2	#14561717	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1376	C₂	1	1	yes	maximal, characteristic
4	C₂×#10597316	C₄	1	1	yes	characteristic
5	C₂×#10597269	—	2	1	no	—
6	#11926410	C₄	1	1	yes	characteristic
7	#11926307	—	2	1	no	—
8	#11926307	—	2	1	no	—
9	C₂×C₁₉×M₃₂	—	2	1	no	—
10	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
12	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
13	C₂×C₄²×C₁₉	C₂×C₄	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	#10597316	—	2	1	no	—
18	#10597316	—	4	1	no	—
19	#10597316	C₈	1	2	yes	—
20	C₁₉×M₃₂	—	2	2	no	—
21	C₁₉×M₃₂	—	4	1	no	—
22	C₂×C₁₆×C₁₉	—	4	1	no	—
23	C₄×C₈×C₁₉	—	4	1	no	—
24	C₁₉⋊#95	—	4	1	no	—
25	#10597269	—	2	2	no	—
26	#10597269	—	4	1	no	—
27	C₂×C₁₉⋊C₁₆	—	4	1	no	—
28	C₂²×C₄×C₁₉	—	2	1	no	—
29	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×Dic₁₉	—	4	1	no	—
31	C₂²×Dic₁₉	—	4	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	—	2	1	no	—
36	C₂×C₈×C₁₉	SD₁₆	1	1	yes	characteristic
37	C₂×C₈×C₁₉	SD₁₆	1	1	yes	characteristic
38	C₂×C₈×C₁₉	Q₁₆	1	1	yes	characteristic
39	C₂×C₈×C₁₉	Q₁₆	1	1	yes	characteristic
40	C₄²×C₁₉	—	2	1	no	—
41	C₄²×C₁₉	C₂×C₈	1	1	yes	characteristic
42	C₄²×C₁₉	M₁₆	1	1	yes	characteristic
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	—	4	2	no	—
48	Dic₁₉⋊C₄	—	4	2	no	—
49	C₁₆×C₁₉	—	4	2	no	—
50	C₁₉⋊C₁₆	—	4	2	no	—
51	#7135	—	19	1	no	maximal, Sylow
52	C₂³×C₁₉	#100	1	1	yes	characteristic
53	C₂×C₄×C₁₉	—	2	1	no	—
54	C₂×C₄×C₁₉	—	2	1	no	—
55	C₂×C₄×C₁₉	—	4	1	no	—
56	C₂×C₄×C₁₉	—	4	1	no	—
57	C₂×C₄×C₁₉	C₂²⋊C₈	1	1	yes	characteristic
58	C₂×C₄×C₁₉	#99	1	1	yes	characteristic
59	C₂×C₄×C₁₉	#103	1	1	yes	characteristic
60	C₂×C₄×C₁₉	#103	1	1	yes	characteristic
61	C₂×C₄×C₁₉	#104	1	1	yes	characteristic
62	C₂×C₄×C₁₉	#104	1	1	yes	characteristic
63	C₂×Dic₁₉	—	4	2	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	C₂×Dic₁₉	—	8	1	no	—
69	C₈×C₁₉	—	2	1	no	—
70	C₈×C₁₉	—	2	1	no	—
71	C₈×C₁₉	—	2	1	no	—
72	C₈×C₁₉	—	2	1	no	—
73	#1991	—	19	1	no	—
74	#1376	—	19	1	no	—
75	#1376	D₁₉	1	1	yes	characteristic
76	C₂²×C₁₉	—	2	1	no	—
77	C₂²×C₁₉	—	2	1	no	—
78	C₂²×C₁₉	#323	1	1	yes	characteristic
79	C₂²×C₁₉	#329	1	1	yes	characteristic
80	C₂²×C₁₉	#333	1	1	yes	characteristic
81	C₄×C₁₉	—	4	1	no	—
82	C₄×C₁₉	—	4	1	no	—
83	C₄×C₁₉	Q₈⋊C₈	1	1	yes	characteristic
84	C₄×C₁₉	Q₈⋊C₈	1	1	yes	characteristic
85	C₄×C₁₉	#328	1	1	yes	characteristic
86	C₄×C₁₉	#328	1	1	yes	characteristic
87	Dic₁₉	—	8	2	no	—
88	Dic₁₉	—	16	1	no	—
89	C₂×#125	—	19	1	no	—
90	C₂×M₃₂	—	2	1	no	—
91	C₂×M₃₂	—	38	1	no	—
92	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
93	#382	—	19	1	no	—
94	#340	—	38	1	no	—
95	#340	—	38	1	no	—
96	C₂×C₁₉	—	2	1	no	—
97	C₂×C₁₉	—	2	1	no	—
98	C₂×C₁₉	#1272	1	1	yes	characteristic
99	C₂×C₁₉	#1366	1	1	yes	characteristic
100	C₂×C₁₉	#1366	1	1	yes	characteristic
101	C₂²×C₈	D₇₆	1	1	yes	characteristic
102	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
103	#125	—	19	2	no	—
104	#125	—	38	1	no	—
105	#125	—	76	1	no	—
106	C₂×C₄⋊C₄	—	38	1	no	—
107	C₂×C₄⋊C₄	—	38	1	no	—
108	C₂×C₄⋊C₄	—	76	1	no	—
109	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
110	M₃₂	—	2	2	no	—
111	M₃₂	—	4	1	no	—
112	M₃₂	—	38	2	no	—
113	M₃₂	—	76	1	no	—
114	C₂×C₁₆	—	4	1	no	—
115	C₂×C₁₆	—	76	1	no	—
116	C₄×C₈	—	4	1	no	—
117	#95	—	76	1	no	—
118	C₁₉	#7135	1	1	yes	characteristic, complemented, Sylow
119	C₂²×C₄	—	2	1	no	—
120	C₂²×C₄	—	76	1	no	—
121	C₂²×C₄	—	76	1	no	—
122	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	#64516	1	1	yes	characteristic
128	C₂×C₈	Dic₇₆	1	1	yes	characteristic
129	C₂×C₈	#64526	1	1	yes	characteristic
130	C₂×C₈	#64528	1	1	yes	characteristic
131	C₄⋊C₄	—	38	2	no	—
132	C₄⋊C₄	—	76	2	no	—
133	C₄⋊C₄	—	76	2	no	—
134	C₄⋊C₄	—	76	1	no	—
135	C₄⋊C₄	—	76	1	no	—
136	C₄⋊C₄	—	76	1	no	—
137	C₄²	—	2	1	no	—
138	C₄²	C₈×D₁₉	1	1	yes	characteristic
139	C₄²	#64515	1	1	yes	characteristic
140	C₁₆	—	4	2	no	—
141	C₁₆	—	76	2	no	—
142	C₂³	#10597280	1	1	yes	characteristic
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	76	2	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	—	152	1	no	—
153	C₂×C₄	#10597262	1	1	yes	characteristic
154	C₂×C₄	#10597263	1	1	yes	characteristic
155	C₂×C₄	#10597267	1	1	yes	characteristic
156	C₂×C₄	#10597273	1	1	yes	characteristic
157	C₂×C₄	(C₂×D₁₉)⋊C₈	1	1	yes	characteristic
158	C₂×C₄	#10597282	1	1	yes	characteristic
159	C₈	—	2	1	no	—
160	C₈	—	2	1	no	—
161	C₈	—	2	1	no	—
162	C₈	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	#11926267	1	1	yes	characteristic
166	C₂²	#11926283	1	1	yes	characteristic
167	C₂²	#11926292	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	#11926274	1	1	yes	characteristic
173	C₄	#11926276	1	1	yes	characteristic
174	C₄	#11926280	1	1	yes	characteristic
175	C₄	#11926282	1	1	yes	characteristic
176	C₂	—	2	1	no	—
177	C₂	—	2	1	no	—
178	C₂	#14561460	1	1	yes	characteristic
179	C₂	#14561695	1	1	yes	characteristic
180	C₂	#14561697	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14564600

C₂

yes

maximal, characteristic

#14561717

C₂

yes

maximal, characteristic

C₁₉×#1376

C₂

yes

maximal, characteristic

C₂×#10597316

C₄

yes

characteristic

C₂×#10597269

—

#11926410

C₄

yes

characteristic

#11926307

—

#11926307

—

C₂×C₁₉×M₃₂

—

C₂×C₄×C₈×C₁₉

C₂²

yes

characteristic

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₂²×C₈×C₁₉

D₄

yes

characteristic

C₂×C₄²×C₁₉

C₂×C₄

yes

characteristic

C₂×C₄⋊Dic₁₉

—

C₂×C₄⋊Dic₁₉

—

C₂×Dic₁₉⋊C₄

—

#10597316

—

#10597316

—

#10597316

C₈

yes

—

C₁₉×M₃₂

—

C₁₉×M₃₂

—

C₂×C₁₆×C₁₉

—

C₄×C₈×C₁₉

—

C₁₉⋊#95

—

#10597269

—

#10597269

—

C₂×C₁₉⋊C₁₆

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×Dic₁₉

—

C₂²×Dic₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

SD₁₆

yes

characteristic

C₂×C₈×C₁₉

SD₁₆

yes

characteristic

C₂×C₈×C₁₉

Q₁₆

yes

characteristic

C₂×C₈×C₁₉

Q₁₆

yes

characteristic

C₄²×C₁₉

—

C₄²×C₁₉

C₂×C₈

yes

characteristic

C₄²×C₁₉

M₁₆

yes

characteristic

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

Dic₁₉⋊C₄

—

Dic₁₉⋊C₄

—

C₁₆×C₁₉

—

C₁₉⋊C₁₆

—

—

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

—

#1991

—

#1376

—

#1376

D₁₉

yes

characteristic

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₂²×C₁₉

yes

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

Q₈⋊C₈

yes

characteristic

C₄×C₁₉

Q₈⋊C₈

yes

C₄×C₁₉

yes

C₄×C₁₉

yes

Dic₁₉

—

Dic₁₉

—

C₂×#125

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂×C₄×C₈

C₂×D₁₉

yes

—

—

—

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

yes

C₂×C₁₉

yes

100

C₂×C₁₉

#1366

yes

characteristic

101

C₂²×C₈

D₇₆

yes

characteristic

102

C₂²×C₈

C₂²⋊D₁₉

yes

characteristic

103

#125

—

104

#125

—

105

#125

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄²

C₄×D₁₉

yes

characteristic

110

M₃₂

—

111

M₃₂

—

112

M₃₂

—

113

M₃₂

—

114

C₂×C₁₆

—

115

C₂×C₁₆

—

116

C₄×C₈

—

117

#95

—

118

C₁₉

#7135

yes

characteristic, complemented, Sylow

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

(C₂×D₁₉)⋊C₄

yes

characteristic

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

#64516

yes

characteristic

128

C₂×C₈

Dic₇₆

yes

characteristic

129

C₂×C₈

#64526

yes

characteristic

130

C₂×C₈

#64528

yes

characteristic

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄²

—

138

C₄²

C₈×D₁₉

yes

characteristic

139

C₄²

#64515

yes

characteristic

140

C₁₆

—

141

C₁₆

—

142

C₂³

#10597280

yes

characteristic

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

152

—

149

C₂×C₄

—

152

—

150

C₂×C₄

—

152

—

151

C₂×C₄

—

152

—

152

C₂×C₄

—

152

—

153

C₂×C₄

#10597262

yes

characteristic

154

C₂×C₄

#10597263

yes

characteristic

155

C₂×C₄

#10597267

yes

characteristic

156

C₂×C₄

#10597273

yes

characteristic

157

C₂×C₄

(C₂×D₁₉)⋊C₈

yes

characteristic

158

C₂×C₄

#10597282

yes

characteristic

159

C₈

—

160

C₈

—

161

C₈

—

162

C₈

—

163

C₂²

—

164

C₂²

—

165

C₂²

#11926267

yes

characteristic

166

C₂²

#11926283

yes

characteristic

167

C₂²

#11926292

yes

characteristic

168

C₄

—

169

C₄

—

170

C₄

—

304

—

171

C₄

—

152

—

172

C₄

#11926274

yes

characteristic

173

C₄

#11926276

yes

characteristic

174

C₄

#11926280

yes

characteristic

175

C₄

#11926282

yes

characteristic

176

C₂

—

177

C₂

—

178

C₂

#14561460

yes

characteristic

179

C₂

#14561695

yes

characteristic

180

C₂

#14561697

yes

characteristic

	subgroup	G/H
center	C₂²	#11926292
commutator	C₄²×C₁₉	C₂×C₈
Frattini	C₂×C₄×C₈	C₂×D₁₉
Fitting	C₁₉×#1376	C₂
socle	C₂²×C₁₉	#333
prime radical	C₂³×C₁₉	#100
cosocle	C₂×C₄×C₈×C₁₉	C₂²
hypercenter	#1376	D₁₉
hypocenter	C₁₉	#7135
LCS	G ▷ C₄²×C₁₉ ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1376

subgroup

G/H

center

C₂²

#11926292

commutator

C₄²×C₁₉

C₂×C₈

Frattini

C₂×C₄×C₈

C₂×D₁₉

Fitting

C₁₉×#1376

C₂

C₂²×C₁₉

C₂³×C₁₉

C₂×C₄×C₈×C₁₉

C₂²

D₁₉

C₁₉

G ▷ C₄²×C₁₉ ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1376

N	H ≅ G/N	type
118. C₁₉	#7135	semidirect

H ≅ G/N

type

118. C₁₉

#7135

semidirect

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

subgroup	index
#14564600	2
#14561717	2
C₁₉×#1376	2
#7135	19

subgroup

index

#14564600

#14561717

C₁₉×#1376

#7135

Subgroups — #18069364

Conjugacy classes of subgroups

Complements

Maximal subgroups