Group #14574177 — subgroups

Subgroups — #14574177

Order 2432.

380 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 157 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	#11927101	C₂	1	1	yes	maximal, characteristic
2	#11926950	C₂	1	4	yes	maximal, complemented
3	#11926947	C₂	1	2	yes	maximal, complemented
4	#11926940	C₂	1	2	yes	maximal, complemented
5	#11926584	C₂	1	4	yes	maximal
6	C₄×#64520	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₈×C₁₉	—	2	1	no	—
9	C₁₉×C₄⋊Q₈	—	2	1	no	—
10	C₁₉×C₄⋊Q₈	—	2	1	no	—
11	C₁₉×C₄⋊Q₈	C₂²	1	2	yes	—
12	C₁₉×C₄⋊Q₈	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
14	C₂×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
16	#10597372	C₂²	1	2	yes	—
17	#10597370	—	2	1	no	—
18	#10597370	—	2	1	no	—
19	C₂×#64520	C₂²	1	2	yes	—
20	#10597316	C₂²	1	2	yes	—
21	#10597315	C₂²	1	1	yes	characteristic
22	#10597315	C₂²	1	1	yes	characteristic
23	C₄×Dic₃₈	—	2	1	no	—
24	C₄×Dic₃₈	—	2	1	no	—
25	#10597265	C₂²	1	8	yes	—
26	#10597264	C₂²	1	8	yes	—
27	#10597260	C₂²	1	2	yes	—
28	C₄×C₁₉⋊C₈	C₂²	1	2	yes	—
29	C₂×Q₈×C₁₉	—	2	2	no	—
30	C₂×Q₈×C₁₉	—	2	2	no	—
31	C₂×Q₈×C₁₉	—	4	4	no	—
32	C₂²×C₄×C₁₉	—	2	2	no	—
33	C₂²×C₄×C₁₉	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₉	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
36	C₂×Dic₃₈	—	2	1	no	—
37	C₂×Dic₃₈	—	2	1	no	—
38	C₄⋊C₄×C₁₉	—	2	2	no	—
39	C₄⋊C₄×C₁₉	—	2	2	no	—
40	C₄⋊C₄×C₁₉	—	2	2	no	—
41	C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
42	C₄²×C₁₉	D₄	1	1	yes	characteristic
43	C₄²×C₁₉	D₄	1	1	yes	characteristic
44	C₄²×C₁₉	C₂³	1	1	yes	characteristic
45	C₄²×C₁₉	C₂³	1	1	yes	characteristic
46	C₂²⋊Dic₁₉	—	2	2	no	—
47	C₄⋊Dic₁₉	—	2	1	no	—
48	C₄⋊Dic₁₉	—	2	1	no	—
49	C₄⋊Dic₁₉	C₂³	1	2	yes	—
50	C₄⋊Dic₁₉	C₂³	1	2	yes	—
51	Dic₁₉⋊C₄	—	2	2	no	—
52	Dic₁₉⋊C₄	—	2	2	no	—
53	C₄×Dic₁₉	—	2	2	no	—
54	#64520	Q₈	1	8	yes	complemented
55	C₂×C₁₉⋊C₈	C₂³	1	4	yes	—
56	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
57	Q₈×C₁₉	—	4	4	no	—
58	Q₈×C₁₉	—	4	4	no	—
59	C₂×C₄×C₁₉	—	2	2	no	—
60	C₂×C₄×C₁₉	—	2	2	no	—
61	C₂×C₄×C₁₉	—	2	4	no	—
62	C₂×C₄×C₁₉	—	4	4	no	—
63	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄×C₁₉	C₂×Q₈	1	4	yes	—
69	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
70	C₂×Dic₁₉	—	2	2	no	—
71	C₂×Dic₁₉	—	2	2	no	—
72	Dic₃₈	—	4	2	no	—
73	Dic₃₈	—	4	2	no	—
74	C₁₉⋊C₈	C₂×Q₈	1	8	yes	complemented
75	#3165	—	19	1	no	maximal, Sylow
76	C₂²×C₁₉	—	2	1	no	—
77	C₂²×C₁₉	—	2	1	no	—
78	C₂²×C₁₉	C₄⋊Q₈	1	2	yes	—
79	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
80	C₄×C₁₉	—	2	1	no	—
81	C₄×C₁₉	—	2	1	no	—
82	C₄×C₁₉	—	4	4	no	—
83	C₄×C₁₉	C₄⋊Q₈	1	2	yes	—
84	C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
85	C₄×C₁₉	#137	1	2	yes	—
86	C₄×C₁₉	C₂²×Q₈	1	2	yes	—
87	Dic₁₉	—	4	2	no	—
88	Dic₁₉	—	4	2	no	—
89	#533	—	19	1	no	—
90	C₂×C₄⋊Q₈	D₁₉	1	1	yes	characteristic
91	#501	—	19	4	no	—
92	#500	—	19	2	no	—
93	#498	—	19	2	no	—
94	#428	—	19	4	no	—
95	C₄×M₁₆	—	19	1	no	—
96	C₂×C₁₉	—	2	2	no	—
97	C₂×C₁₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
98	C₂×C₁₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
99	C₂×C₁₉	C₂×#137	1	1	yes	characteristic
100	C₂²×Q₈	—	2	1	no	—
101	C₂×M₁₆	—	19	2	no	—
102	C₄⋊Q₈	—	2	1	no	—
103	C₄⋊Q₈	—	2	1	no	—
104	C₄⋊Q₈	—	19	1	no	—
105	C₄⋊Q₈	—	19	1	no	—
106	C₄⋊Q₈	C₂×D₁₉	1	2	yes	—
107	C₄⋊Q₈	C₂×D₁₉	1	2	yes	—
108	#125	—	19	2	no	—
109	C₂²⋊Q₈	—	38	1	no	—
110	C₂²⋊Q₈	—	38	1	no	—
111	C₄×Q₈	—	38	1	no	—
112	C₄×Q₈	—	38	1	no	—
113	#117	—	19	2	no	—
114	C₂×C₄⋊C₄	—	2	1	no	—
115	C₂×C₄⋊C₄	C₂×D₁₉	1	2	yes	—
116	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
117	#107	—	19	8	no	—
118	#106	—	19	8	no	—
119	#97	—	19	2	no	—
120	C₄×C₈	—	19	2	no	—
121	C₁₉	#3165	1	1	yes	characteristic, complemented, Sylow
122	C₂×Q₈	—	2	2	no	—
123	C₂×Q₈	—	2	2	no	—
124	C₂×Q₈	—	4	4	no	—
125	C₂×Q₈	—	38	1	no	—
126	C₂×Q₈	—	38	1	no	—
127	C₂²×C₄	—	2	2	no	—
128	C₂²×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
129	C₂²×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
130	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
131	M₁₆	—	19	8	no	—
132	C₂×C₈	—	19	4	no	—
133	C₄⋊C₄	—	2	2	no	—
134	C₄⋊C₄	—	2	2	no	—
135	C₄⋊C₄	—	2	2	no	—
136	C₄⋊C₄	—	19	2	no	—
137	C₄⋊C₄	—	19	2	no	—
138	C₄⋊C₄	—	38	2	no	—
139	C₄⋊C₄	—	38	2	no	—
140	C₄⋊C₄	—	38	1	no	—
141	C₄⋊C₄	—	38	1	no	—
142	C₄⋊C₄	C₂²×D₁₉	1	4	yes	—
143	C₂²⋊C₄	—	38	2	no	—
144	C₄²	—	38	2	no	—
145	C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
146	C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
147	C₄²	C₂²×D₁₉	1	1	yes	characteristic
148	C₄²	C₂²×D₁₉	1	1	yes	characteristic
149	C₂³	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
150	Q₈	—	4	4	no	—
151	Q₈	—	4	4	no	—
152	Q₈	—	76	2	no	—
153	Q₈	—	76	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	4	4	no	—
158	C₂×C₄	—	38	2	no	—
159	C₂×C₄	—	38	2	no	—
160	C₂×C₄	Q₈×D₁₉	1	4	yes	—
161	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
162	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
163	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
164	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
165	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
166	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
167	C₈	—	19	8	no	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	#10597390	1	2	yes	—
171	C₂²	C₂²×C₂²⋊D₁₉	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	4	4	no	—
175	C₄	—	76	2	no	—
176	C₄	—	76	2	no	—
177	C₄	#10597378	1	2	yes	—
178	C₄	#10597388	1	2	yes	—
179	C₄	#10597390	1	2	yes	—
180	C₄	C₂×Q₈×D₁₉	1	2	yes	—
181	C₂	—	2	2	no	—
182	C₂	C₂×#10597390	1	1	yes	characteristic
183	C₂	C₂×#10597378	1	1	yes	characteristic
184	C₂	C₂×#10597388	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11927101

C₂

yes

maximal, characteristic

#11926950

C₂

yes

maximal, complemented

#11926947

C₂

yes

maximal, complemented

#11926940

C₂

yes

maximal, complemented

#11926584

C₂

yes

maximal

C₄×#64520

C₂

yes

maximal, characteristic

C₂×C₁₉×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₉

—

C₁₉×C₄⋊Q₈

—

C₁₉×C₄⋊Q₈

—

C₁₉×C₄⋊Q₈

C₂²

yes

—

C₁₉×C₄⋊Q₈

C₂²

yes

—

C₂×C₄⋊C₄×C₁₉

—

C₂×C₄⋊C₄×C₁₉

C₂²

yes

—

C₂×C₄²×C₁₉

C₂²

yes

characteristic

#10597372

C₂²

yes

—

#10597370

—

#10597370

—

C₂×#64520

C₂²

yes

—

#10597316

C₂²

yes

—

#10597315

C₂²

yes

characteristic

#10597315

C₂²

yes

characteristic

C₄×Dic₃₈

—

C₄×Dic₃₈

—

#10597265

C₂²

yes

—

#10597264

C₂²

yes

—

#10597260

C₂²

yes

—

C₄×C₁₉⋊C₈

C₂²

yes

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂²×C₄×C₁₉

—

C₂²×C₄×C₁₉

D₄

yes

characteristic

C₂²×C₄×C₁₉

D₄

yes

characteristic

C₂²×C₄×C₁₉

C₂³

yes

characteristic

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

C₂³

yes

—

C₄²×C₁₉

D₄

yes

characteristic

C₄²×C₁₉

D₄

yes

characteristic

C₄²×C₁₉

C₂³

yes

characteristic

C₄²×C₁₉

C₂³

yes

characteristic

C₂²⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

C₂³

yes

—

C₄⋊Dic₁₉

C₂³

yes

—

Dic₁₉⋊C₄

—

Dic₁₉⋊C₄

—

C₄×Dic₁₉

—

#64520

Q₈

yes

complemented

C₂×C₁₉⋊C₈

C₂³

yes

—

C₂³×C₁₉

C₂×D₄

yes

characteristic

Q₈×C₁₉

—

Q₈×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉

C₂×Q₈

yes

—

C₂×C₄×C₁₉

C₂⁴

yes

characteristic

C₂×Dic₁₉

—

C₂×Dic₁₉

—

Dic₃₈

—

Dic₃₈

—

C₁₉⋊C₈

C₂×Q₈

yes

complemented

#3165

—

maximal, Sylow

C₂²×C₁₉

—

C₂²×C₁₉

—

C₂²×C₁₉

C₄⋊Q₈

yes

—

C₂²×C₁₉

C₂²×D₄

yes

characteristic

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

C₄⋊Q₈

yes

—

C₄×C₁₉

AGL(1,ℤ/8)

yes

—

C₄×C₁₉

#137

yes

—

C₄×C₁₉

C₂²×Q₈

yes

—

Dic₁₉

—

Dic₁₉

—

#533

—

C₂×C₄⋊Q₈

D₁₉

yes

—

—

—

—

C₄×M₁₆

—

C₂×C₁₉

—

C₂×C₁₉

C₂×C₄⋊Q₈

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

100

C₂²×Q₈

—

101

C₂×M₁₆

—

102

C₄⋊Q₈

—

103

C₄⋊Q₈

—

104

C₄⋊Q₈

—

105

C₄⋊Q₈

—

106

C₄⋊Q₈

C₂×D₁₉

yes

—

107

C₄⋊Q₈

C₂×D₁₉

yes

—

108

#125

—

109

C₂²⋊Q₈

—

110

C₂²⋊Q₈

—

111

C₄×Q₈

—

112

C₄×Q₈

—

113

#117

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

C₂×D₁₉

yes

—

116

C₂×C₄²

C₂×D₁₉

yes

characteristic

117

#107

—

118

#106

—

119

#97

—

120

C₄×C₈

—

121

C₁₉

#3165

yes

characteristic, complemented, Sylow

122

C₂×Q₈

—

123

C₂×Q₈

—

124

C₂×Q₈

—

125

C₂×Q₈

—

126

C₂×Q₈

—

127

C₂²×C₄

—

128

C₂²×C₄

C₂²⋊D₁₉

yes

characteristic

129

C₂²×C₄

C₂²⋊D₁₉

yes

characteristic

130

C₂²×C₄

C₂²×D₁₉

yes

characteristic

131

M₁₆

—

132

C₂×C₈

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄⋊C₄

C₂²×D₁₉

yes

—

143

C₂²⋊C₄

—

144

C₄²

—

145

C₄²

C₂²⋊D₁₉

yes

characteristic

146

C₄²

C₂²⋊D₁₉

yes

characteristic

147

C₄²

C₂²×D₁₉

yes

characteristic

148

C₄²

C₂²×D₁₉

yes

characteristic

149

C₂³

C₂×C₂²⋊D₁₉

yes

characteristic

150

Q₈

—

151

Q₈

—

152

Q₈

—

153

Q₈

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

Q₈×D₁₉

yes

—

161

C₂×C₄

C₂×C₂²⋊D₁₉

yes

characteristic

162

C₂×C₄

C₂×C₂²⋊D₁₉

yes

characteristic

163

C₂×C₄

C₂×C₂²⋊D₁₉

yes

characteristic

164

C₂×C₄

C₂×C₂²⋊D₁₉

yes

characteristic

165

C₂×C₄

C₂×C₂²⋊D₁₉

yes

characteristic

166

C₂×C₄

C₂³×D₁₉

yes

characteristic

167

C₈

—

168

C₂²

—

169

C₂²

—

170

C₂²

#10597390

yes

—

171

C₂²

C₂²×C₂²⋊D₁₉

yes

characteristic

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

#10597378

yes

—

178

C₄

#10597388

yes

—

179

C₄

#10597390

yes

—

180

C₄

C₂×Q₈×D₁₉

yes

—

181

C₂

—

182

C₂

C₂×#10597390

yes

characteristic

183

C₂

C₂×#10597378

yes

characteristic

184

C₂

C₂×#10597388

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₉
commutator, cosocle	C₂×C₄×C₁₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉
Fitting	C₂×C₁₉×C₄⋊Q₈	C₂
socle	C₂²×C₁₉	C₂²×D₄
prime radical	C₂³×C₁₉	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₁₉
hypocenter	C₁₉	#3165
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₉

commutator, cosocle

C₂×C₄×C₁₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₉

Fitting

C₂×C₁₉×C₄⋊Q₈

C₂

socle

C₂²×C₁₉

C₂²×D₄

prime radical

C₂³×C₁₉

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₁₉

hypocenter

C₁₉

#3165

LCS

G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #11926950	C₂	semidirect
3. #11926947	C₂	semidirect
4. #11926940	C₂	semidirect
54. #64520	Q₈	semidirect
74. C₁₉⋊C₈	C₂×Q₈	semidirect
121. C₁₉	#3165	semidirect

H ≅ G/N

type

2. #11926950

C₂

semidirect

3. #11926947

C₂

semidirect

4. #11926940

C₂

semidirect

54. #64520

Q₈

semidirect

74. C₁₉⋊C₈

C₂×Q₈

semidirect

121. C₁₉

#3165

semidirect

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

subgroup	index
#11927101	2
#11926950	2
#11926947	2
#11926940	2
#11926584	2
C₄×#64520	2
C₂×C₁₉×C₄⋊Q₈	2
#3165	19

subgroup

index

C₂×C₁₉×C₄⋊Q₈

Subgroups — #14574177

Conjugacy classes of subgroups

Complements

Maximal subgroups