Group #18069289 — subgroups

Subgroups — #18069289

Order 4864.

268 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 71 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	#14561719	C₂	1	2	yes	maximal, complemented
2	C₁₉×#1955	C₂	1	1	yes	maximal, characteristic
3	C₂×#10597269	—	2	2	no	—
4	C₂×C₄×D₄×C₁₉	C₄	1	1	yes	characteristic
5	C₂×C₁₉×C₄⋊C₈	C₄	1	1	yes	characteristic
6	C₂×C₁₉×C₂²⋊C₈	—	2	2	no	—
7	C₂×C₁₉×#97	C₂²	1	1	yes	characteristic
8	C₂²×D₄×C₁₉	C₈	1	1	yes	characteristic
9	C₂³×C₄×C₁₉	—	2	1	no	—
10	C₂²×C₈×C₁₉	—	2	1	no	—
11	C₂²×C₈×C₁₉	D₄	1	2	yes	—
12	C₄×D₄×C₁₉	—	4	1	no	—
13	C₄×D₄×C₁₉	—	4	1	no	—
14	C₂×C₄⋊C₄×C₁₉	C₈	1	1	yes	characteristic
15	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
16	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
17	C₁₉×C₄⋊C₈	—	4	1	no	—
18	C₁₉×C₂²⋊C₈	—	4	2	no	—
19	C₁₉×C₂²⋊C₈	—	4	2	no	—
20	C₁₉×#97	—	4	1	no	—
21	#10597269	—	2	4	no	—
22	#10597269	—	4	2	no	—
23	C₂×C₁₉⋊C₁₆	—	4	2	no	—
24	C₂⁴×C₁₉	—	2	1	no	—
25	C₂×D₄×C₁₉	—	4	1	no	—
26	C₂×D₄×C₁₉	—	4	2	no	—
27	C₂²×C₄×C₁₉	—	2	1	no	—
28	C₂²×C₄×C₁₉	—	4	1	no	—
29	C₂²×C₄×C₁₉	—	4	1	no	—
30	C₂²×C₄×C₁₉	—	4	2	no	—
31	C₂²×C₄×C₁₉	—	4	2	no	—
32	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₉	C₂×C₈	1	1	yes	characteristic
34	C₂²×C₄×C₁₉	M₁₆	1	1	yes	characteristic
35	C₂×C₈×C₁₉	—	2	2	no	—
36	C₂×C₈×C₁₉	—	2	2	no	—
37	C₂×C₈×C₁₉	—	4	1	no	—
38	C₂×C₈×C₁₉	—	4	2	no	—
39	C₂×C₈×C₁₉	D₈	1	2	yes	—
40	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
41	C₄⋊C₄×C₁₉	—	4	1	no	—
42	C₂²⋊C₄×C₁₉	—	4	2	no	—
43	C₄²×C₁₉	—	2	1	no	—
44	C₄²×C₁₉	—	2	1	no	—
45	C₁₉⋊C₁₆	—	4	4	no	—
46	#7105	—	19	1	no	maximal, Sylow
47	C₂³×C₁₉	—	2	2	no	—
48	C₂³×C₁₉	—	4	1	no	—
49	C₂³×C₁₉	—	4	2	no	—
50	C₂³×C₁₉	—	4	1	no	—
51	C₂³×C₁₉	—	4	2	no	—
52	C₂³×C₁₉	C₂²⋊C₈	1	1	yes	characteristic
53	D₄×C₁₉	—	4	2	no	—
54	D₄×C₁₉	—	4	2	no	—
55	C₂×C₄×C₁₉	—	2	1	no	—
56	C₂×C₄×C₁₉	—	2	1	no	—
57	C₂×C₄×C₁₉	—	4	1	no	—
58	C₂×C₄×C₁₉	—	4	1	no	—
59	C₂×C₄×C₁₉	—	4	1	no	—
60	C₂×C₄×C₁₉	—	4	2	no	—
61	C₂×C₄×C₁₉	—	8	2	no	—
62	C₂×C₄×C₁₉	—	8	1	no	—
63	C₂×C₄×C₁₉	—	8	1	no	—
64	C₂×C₄×C₁₉	#99	1	1	yes	characteristic
65	C₂×C₄×C₁₉	#100	1	1	yes	characteristic
66	C₂×C₄×C₁₉	#102	1	2	yes	—
67	C₂×C₄×C₁₉	#104	1	2	yes	—
68	C₈×C₁₉	—	2	2	no	—
69	C₈×C₁₉	—	2	2	no	—
70	C₈×C₁₉	—	8	1	no	—
71	#1955	D₁₉	1	1	yes	characteristic
72	#1377	—	19	2	no	—
73	C₂²×C₁₉	—	2	1	no	—
74	C₂²×C₁₉	—	2	1	no	—
75	C₂²×C₁₉	—	4	4	no	—
76	C₂²×C₁₉	—	4	2	no	—
77	C₂²×C₁₉	—	8	2	no	—
78	C₂²×C₁₉	—	8	1	no	—
79	C₂²×C₁₉	—	8	1	no	—
80	C₂²×C₁₉	#323	1	1	yes	characteristic
81	C₂²×C₁₉	D₄⋊C₈	1	2	yes	—
82	C₄×C₁₉	—	4	1	no	—
83	C₄×C₁₉	—	4	1	no	—
84	C₄×C₁₉	—	8	1	no	—
85	C₄×C₁₉	#327	1	2	yes	—
86	C₄×C₁₉	#329	1	1	yes	characteristic
87	C₄×C₁₉	#331	1	1	yes	characteristic
88	C₂×C₄×D₄	Dic₁₉	1	1	yes	characteristic
89	C₂×M₃₂	—	38	2	no	—
90	C₂×C₄⋊C₈	Dic₁₉	1	1	yes	characteristic
91	C₂×C₂²⋊C₈	—	2	2	no	—
92	C₂×#97	C₂×D₁₉	1	1	yes	characteristic
93	C₂×C₁₉	—	2	2	no	—
94	C₂×C₁₉	—	8	2	no	—
95	C₂×C₁₉	#1270	1	1	yes	characteristic
96	C₂×C₁₉	#1320	1	1	yes	characteristic
97	C₂×C₁₉	#1321	1	1	yes	characteristic
98	C₂²×D₄	C₁₉⋊C₈	1	1	yes	characteristic
99	C₂³×C₄	—	2	1	no	—
100	C₂²×C₈	—	2	1	no	—
101	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
102	C₄×D₄	—	4	1	no	—
103	C₄×D₄	—	4	1	no	—
104	C₂×C₄⋊C₄	C₁₉⋊C₈	1	1	yes	characteristic
105	C₂×C₂²⋊C₄	—	2	1	no	—
106	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
107	M₃₂	—	38	4	no	—
108	M₃₂	—	76	2	no	—
109	C₂×C₁₆	—	76	2	no	—
110	C₄⋊C₈	—	4	1	no	—
111	C₂²⋊C₈	—	4	2	no	—
112	C₂²⋊C₈	—	4	2	no	—
113	#97	—	4	1	no	—
114	C₁₉	#7105	1	1	yes	characteristic, complemented, Sylow
115	C₂⁴	—	2	1	no	—
116	C₂×D₄	—	4	1	no	—
117	C₂×D₄	—	4	2	no	—
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	—	4	2	no	—
120	C₂²×C₄	—	4	1	no	—
121	C₂²×C₄	—	4	1	no	—
122	C₂²×C₄	—	4	2	no	—
123	C₂²×C₄	C₂×C₁₉⋊C₈	1	1	yes	characteristic
124	C₂²×C₄	#64520	1	1	yes	characteristic
125	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	4	2	no	—
129	C₂×C₈	—	4	1	no	—
130	C₂×C₈	#64525	1	2	yes	—
131	C₂×C₈	#64526	1	2	yes	—
132	C₄⋊C₄	—	4	1	no	—
133	C₂²⋊C₄	—	4	2	no	—
134	C₄²	—	2	1	no	—
135	C₄²	—	2	1	no	—
136	C₁₆	—	76	4	no	—
137	C₂³	—	2	2	no	—
138	C₂³	—	4	1	no	—
139	C₂³	—	4	2	no	—
140	C₂³	—	4	2	no	—
141	C₂³	—	4	1	no	—
142	C₂³	C₂²⋊(C₁₉⋊C₈)	1	1	yes	characteristic
143	D₄	—	4	2	no	—
144	D₄	—	4	2	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	8	2	no	—
152	C₂×C₄	—	8	1	no	—
153	C₂×C₄	—	8	1	no	—
154	C₂×C₄	#10597289	1	2	yes	—
155	C₂×C₄	#10597290	1	1	yes	characteristic
156	C₂×C₄	#10597291	1	1	yes	characteristic
157	C₂×C₄	#10597294	1	2	yes	—
158	C₈	—	2	2	no	—
159	C₈	—	2	2	no	—
160	C₈	—	8	1	no	—
161	C₂²	—	2	1	no	—
162	C₂²	—	2	1	no	—
163	C₂²	—	4	4	no	—
164	C₂²	—	4	2	no	—
165	C₂²	—	8	2	no	—
166	C₂²	—	8	1	no	—
167	C₂²	—	8	1	no	—
168	C₂²	#11926268	1	1	yes	characteristic
169	C₂²	D₄⋊(C₁₉⋊C₈)	1	2	yes	—
170	C₄	—	4	1	no	—
171	C₄	—	4	1	no	—
172	C₄	—	8	1	no	—
173	C₄	#11926278	1	2	yes	—
174	C₄	#11926284	1	1	yes	characteristic
175	C₄	#11926288	1	1	yes	characteristic
176	C₂	—	2	2	no	—
177	C₂	—	8	2	no	—
178	C₂	#14561456	1	1	yes	characteristic
179	C₂	#14561571	1	1	yes	characteristic
180	C₂	#14561573	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14561719

C₂

yes

maximal, complemented

C₁₉×#1955

C₂

yes

maximal, characteristic

C₂×#10597269

—

C₂×C₄×D₄×C₁₉

C₄

yes

characteristic

C₂×C₁₉×C₄⋊C₈

C₄

yes

characteristic

C₂×C₁₉×C₂²⋊C₈

—

C₂×C₁₉×#97

C₂²

yes

characteristic

C₂²×D₄×C₁₉

C₈

yes

characteristic

C₂³×C₄×C₁₉

—

C₂²×C₈×C₁₉

—

C₂²×C₈×C₁₉

D₄

yes

—

C₄×D₄×C₁₉

—

C₄×D₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

C₈

yes

characteristic

C₂×C₂²⋊C₄×C₁₉

—

C₂×C₄²×C₁₉

C₂×C₄

yes

characteristic

C₁₉×C₄⋊C₈

—

C₁₉×C₂²⋊C₈

—

C₁₉×C₂²⋊C₈

—

C₁₉×#97

—

#10597269

—

#10597269

—

C₂×C₁₉⋊C₁₆

—

C₂⁴×C₁₉

—

C₂×D₄×C₁₉

—

C₂×D₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₉

C₂×C₈

yes

characteristic

C₂²×C₄×C₁₉

M₁₆

yes

characteristic

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

—

C₂×C₈×C₁₉

D₈

yes

—

C₂×C₈×C₁₉

SD₁₆

yes

—

C₄⋊C₄×C₁₉

—

C₂²⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

—

C₁₉⋊C₁₆

—

#7105

—

maximal, Sylow

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

—

C₂³×C₁₉

C₂²⋊C₈

yes

characteristic

D₄×C₁₉

—

D₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

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×C₄×C₁₉

yes

—

C₂×C₄×C₁₉

#104

yes

—

C₈×C₁₉

—

C₈×C₁₉

—

C₈×C₁₉

—

#1955

D₁₉

yes

characteristic

#1377

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

#323

yes

characteristic

C₂²×C₁₉

D₄⋊C₈

yes

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

#327

yes

—

C₄×C₁₉

yes

C₄×C₁₉

yes

C₂×C₄×D₄

Dic₁₉

yes

characteristic

C₂×M₃₂

—

C₂×C₄⋊C₈

Dic₁₉

yes

characteristic

C₂×C₂²⋊C₈

—

C₂×#97

C₂×D₁₉

yes

characteristic

C₂×C₁₉

—

C₂×C₁₉

—

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂²×D₄

C₁₉⋊C₈

yes

characteristic

C₂³×C₄

—

100

C₂²×C₈

—

101

C₂²×C₈

C₂²⋊D₁₉

yes

—

102

C₄×D₄

—

103

C₄×D₄

—

104

C₂×C₄⋊C₄

C₁₉⋊C₈

yes

characteristic

105

C₂×C₂²⋊C₄

—

106

C₂×C₄²

C₂×Dic₁₉

yes

characteristic

107

M₃₂

—

108

M₃₂

—

109

C₂×C₁₆

—

110

C₄⋊C₈

—

111

C₂²⋊C₈

—

112

C₂²⋊C₈

—

113

#97

—

114

C₁₉

#7105

yes

characteristic, complemented, Sylow

115

C₂⁴

—

116

C₂×D₄

—

117

C₂×D₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

C₂×C₁₉⋊C₈

yes

characteristic

124

C₂²×C₄

#64520

yes

characteristic

125

C₂²×C₄

C₂²⋊Dic₁₉

yes

characteristic

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

#64525

yes

—

131

C₂×C₈

#64526

yes

—

132

C₄⋊C₄

—

133

C₂²⋊C₄

—

134

C₄²

—

135

C₄²

—

136

C₁₆

—

137

C₂³

—

138

C₂³

—

139

C₂³

—

140

C₂³

—

141

C₂³

—

142

C₂³

C₂²⋊(C₁₉⋊C₈)

yes

characteristic

143

D₄

—

144

D₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

#10597289

yes

—

155

C₂×C₄

#10597290

yes

characteristic

156

C₂×C₄

#10597291

yes

characteristic

157

C₂×C₄

#10597294

yes

—

158

C₈

—

159

C₈

—

160

C₈

—

161

C₂²

—

162

C₂²

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

#11926268

yes

characteristic

169

C₂²

D₄⋊(C₁₉⋊C₈)

yes

—

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

#11926278

yes

—

174

C₄

#11926284

yes

characteristic

175

C₄

#11926288

yes

characteristic

176

C₂

—

177

C₂

—

178

C₂

#14561456

yes

characteristic

179

C₂

#14561571

yes

characteristic

180

C₂

#14561573

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#10597291
commutator	C₂²×C₄×C₁₉	C₂×C₈
Frattini	C₂×#97	C₂×D₁₉
Fitting	C₁₉×#1955	C₂
socle	C₂²×C₁₉	#323
prime radical	C₂²×D₄×C₁₉	C₈
cosocle	C₂×C₁₉×#97	C₂²
hypercenter	#1955	D₁₉
hypocenter	C₁₉	#7105
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂³×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1955

subgroup

G/H

center

C₂×C₄

#10597291

commutator

C₂²×C₄×C₁₉

C₂×C₈

Frattini

C₂×#97

C₂×D₁₉

Fitting

C₁₉×#1955

C₂

socle

C₂²×C₁₉

#323

prime radical

C₂²×D₄×C₁₉

C₈

cosocle

C₂×C₁₉×#97

C₂²

D₁₉

C₁₉

G ▷ C₂²×C₄×C₁₉ ▷ C₂³×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1955

N	H ≅ G/N	type
1. #14561719	C₂	semidirect
114. C₁₉	#7105	semidirect

H ≅ G/N

type

1. #14561719

C₂

semidirect

114. C₁₉

#7105

semidirect

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

subgroup	index
#14561719	2
C₁₉×#1955	2
#7105	19

subgroup

index

#14561719

C₁₉×#1955

#7105

Subgroups — #18069289

Conjugacy classes of subgroups

Complements

Maximal subgroups