Group #24987523 — subgroups

Subgroups — #24987523

Order 11008.

444 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 151 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	#19134344	C₂	1	2	yes	maximal
2	C₂×#14582641	C₂	1	2	yes	maximal
3	#19132476	C₂	1	2	yes	maximal
4	C₄×#125×C₄₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13063236	C₂²	1	4	yes	—
6	#14582654	—	2	2	no	—
7	#14582654	—	2	2	no	—
8	#14582641	C₄	1	8	yes	complemented
9	C₂×#125×C₄₃	C₂²	1	1	yes	characteristic
10	C₄₃×#389	C₄	1	2	yes	—
11	C₄₃×#387	—	2	2	no	—
12	C₄₃×#382	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₄₃	—	2	2	no	—
14	C₄×C₄⋊C₄×C₄₃	C₄	1	2	yes	—
15	C₄³×C₄₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄₃⋊C₈	—	2	4	no	—
17	#13063236	—	2	4	no	—
18	#13063236	C₂×C₄	1	8	yes	—
19	#125×C₄₃	—	2	2	no	—
20	#125×C₄₃	C₂×C₄	1	4	yes	—
21	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
22	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
23	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
24	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₄₃	—	2	2	no	—
26	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₄₃	D₄	1	2	yes	—
29	C₂×C₄²×C₄₃	D₄	1	2	yes	—
30	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
31	#95×C₄₃	—	2	2	no	—
32	#95×C₄₃	—	2	2	no	—
33	#95×C₄₃	—	2	2	no	—
34	C₂²×C₄×C₄₃	—	2	2	no	—
35	C₂²×C₄×C₄₃	—	2	2	no	—
36	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
38	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
41	C₄⋊C₄×C₄₃	—	2	2	no	—
42	C₄⋊C₄×C₄₃	—	2	8	no	—
43	C₄⋊C₄×C₄₃	—	4	2	no	—
44	C₄⋊C₄×C₄₃	—	4	4	no	—
45	C₄⋊C₄×C₄₃	C₄²	1	4	yes	—
46	C₄²×C₄₃	—	2	2	no	—
47	C₄²×C₄₃	—	4	1	no	—
48	C₄²×C₄₃	—	4	2	no	—
49	C₄²×C₄₃	—	4	2	no	—
50	C₄²×C₄₃	—	4	4	no	—
51	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
52	C₄²×C₄₃	#41	1	2	yes	—
53	C₂×C₄₃⋊C₈	—	2	8	no	—
54	C₂×C₄₃⋊C₈	—	4	4	no	—
55	C₂×C₄₃⋊C₈	—	4	4	no	—
56	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₄₃	—	2	1	no	—
58	C₂×C₄×C₄₃	—	2	1	no	—
59	C₂×C₄×C₄₃	—	2	2	no	—
60	C₂×C₄×C₄₃	—	2	2	no	—
61	C₂×C₄×C₄₃	—	2	2	no	—
62	C₂×C₄×C₄₃	—	2	4	no	—
63	C₂×C₄×C₄₃	—	2	1	no	—
64	C₂×C₄×C₄₃	—	2	2	no	—
65	C₂×C₄×C₄₃	—	2	2	no	—
66	C₂×C₄×C₄₃	—	2	2	no	—
67	C₂×C₄×C₄₃	—	2	2	no	—
68	C₂×C₄×C₄₃	—	4	2	no	—
69	C₂×C₄×C₄₃	—	4	2	no	—
70	C₂×C₄×C₄₃	—	4	2	no	—
71	C₂×C₄×C₄₃	—	4	2	no	—
72	C₂×C₄×C₄₃	—	4	2	no	—
73	C₂×C₄×C₄₃	#101	1	2	yes	—
74	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
75	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
76	C₂×C₄×C₄₃	C₄×D₄	1	4	yes	—
77	C₄₃⋊C₈	—	4	8	no	—
78	#8225	—	43	1	no	maximal, Sylow
79	C₂²×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₄₃	C₂×#101	1	1	yes	characteristic
81	C₂²×C₄₃	#413	1	1	yes	characteristic
82	C₂²×C₄₃	#421	1	2	yes	—
83	C₂²×C₄₃	#421	1	2	yes	—
84	C₄×C₄₃	—	2	1	no	—
85	C₄×C₄₃	—	2	1	no	—
86	C₄×C₄₃	—	2	4	no	—
87	C₄×C₄₃	—	2	2	no	—
88	C₄×C₄₃	—	4	1	no	—
89	C₄×C₄₃	—	4	4	no	—
90	C₄×C₄₃	—	4	2	no	—
91	C₄×C₄₃	—	8	2	no	—
92	C₄×#125	D₄₃	1	1	yes	characteristic
93	#1745	—	43	2	no	—
94	C₂×#330	—	43	2	no	—
95	#1295	—	43	2	no	—
96	C₂×C₄₃	#1534	1	1	yes	characteristic
97	C₂×C₄₃	#1535	1	2	yes	—
98	C₂×C₄₃	#1537	1	1	yes	characteristic
99	C₂×C₄₃	C₄×#101	1	1	yes	characteristic
100	C₂×C₄₃	#1759	1	2	yes	—
101	C₂×#125	C₂×D₄₃	1	1	yes	characteristic
102	C₂×#97	—	43	4	no	—
103	#389	Dic₄₃	1	2	yes	—
104	#387	—	2	2	no	—
105	#382	C₂×D₄₃	1	1	yes	characteristic
106	C₄×C₄⋊C₄	—	2	2	no	—
107	C₄×C₄⋊C₄	Dic₄₃	1	2	yes	—
108	C₄³	C₂×D₄₃	1	1	yes	characteristic
109	#336	—	86	2	no	—
110	#336	—	86	2	no	—
111	#330	—	43	8	no	—
112	C₄₃	#8225	1	1	yes	characteristic, complemented, Sylow
113	C₂²×C₈	—	86	4	no	—
114	#125	—	2	2	no	—
115	#125	C₄×D₄₃	1	4	yes	—
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
119	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
120	C₂×C₄²	—	2	2	no	—
121	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
122	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
123	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
124	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
125	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
126	#97	—	43	8	no	—
127	#97	—	86	4	no	—
128	#95	—	2	2	no	—
129	#95	—	2	2	no	—
130	#95	—	2	2	no	—
131	C₂²×C₄	—	2	2	no	—
132	C₂²×C₄	—	2	2	no	—
133	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
134	C₂²×C₄	C₂²⋊Dic₄₃	1	2	yes	—
135	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
136	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
137	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
138	C₂×C₈	—	86	8	no	—
139	C₂×C₈	—	172	4	no	—
140	C₂×C₈	—	172	4	no	—
141	C₄⋊C₄	—	2	2	no	—
142	C₄⋊C₄	—	2	8	no	—
143	C₄⋊C₄	—	4	2	no	—
144	C₄⋊C₄	—	4	4	no	—
145	C₄⋊C₄	C₄×Dic₄₃	1	4	yes	—
146	C₄²	—	2	2	no	—
147	C₄²	—	4	1	no	—
148	C₄²	—	4	2	no	—
149	C₄²	—	4	2	no	—
150	C₄²	—	4	4	no	—
151	C₄²	C₂×C₄×D₄₃	1	2	yes	—
152	C₄²	#10621898	1	2	yes	—
153	C₂³	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	#13063244	1	2	yes	—
171	C₂×C₄	C₂×C₄×Dic₄₃	1	1	yes	characteristic
172	C₂×C₄	#13063284	1	1	yes	characteristic
173	C₂×C₄	C₄×C₂²⋊D₄₃	1	4	yes	—
174	C₈	—	172	8	no	—
175	C₂²	C₄×C₂²⋊Dic₄₃	1	1	yes	characteristic
176	C₂²	C₂×#13063244	1	1	yes	characteristic
177	C₂²	#14582883	1	1	yes	characteristic
178	C₂²	#14582916	1	2	yes	—
179	C₂²	#14582916	1	2	yes	—
180	C₄	—	2	2	no	—
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	4	no	—
184	C₄	—	4	2	no	—
185	C₄	—	4	1	no	—
186	C₄	—	4	4	no	—
187	C₄	—	8	2	no	—
188	C₂	#19133222	1	1	yes	characteristic
189	C₂	#19133226	1	2	yes	—
190	C₂	#19133233	1	1	yes	characteristic
191	C₂	C₄×#13063244	1	1	yes	characteristic
192	C₂	#19134405	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19134344

C₂

yes

maximal

C₂×#14582641

C₂

yes

maximal

#19132476

C₂

yes

maximal

C₄×#125×C₄₃

C₂

yes

maximal, characteristic

C₂×#13063236

C₂²

yes

—

#14582654

—

#14582654

—

#14582641

C₄

yes

complemented

C₂×#125×C₄₃

C₂²

yes

characteristic

C₄₃×#389

C₄

yes

—

C₄₃×#387

—

C₄₃×#382

C₂²

yes

characteristic

C₄×C₄⋊C₄×C₄₃

—

C₄×C₄⋊C₄×C₄₃

C₄

yes

—

C₄³×C₄₃

C₂²

yes

characteristic

C₂²×C₄₃⋊C₈

—

#13063236

—

#13063236

C₂×C₄

yes

—

#125×C₄₃

—

#125×C₄₃

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₄₃

C₂×C₄

yes

—

C₂×C₄²×C₄₃

—

C₂×C₄²×C₄₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

#95×C₄₃

—

#95×C₄₃

—

#95×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×D₄

yes

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

C₄²

yes

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

C₂²×C₄

yes

—

C₄²×C₄₃

#41

yes

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#101

yes

—

C₂×C₄×C₄₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₄₃

#117

yes

characteristic

C₂×C₄×C₄₃

C₄×D₄

yes

—

C₄₃⋊C₈

—

#8225

—

maximal, Sylow

C₂²×C₄₃

C₄×C₂²⋊C₄

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₂²×C₄₃

#421

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×#125

D₄₃

yes

—

—

—

C₂×C₄₃

#1534

yes

characteristic

C₂×C₄₃

#1535

yes

—

C₂×C₄₃

yes

C₂×C₄₃

yes

100

C₂×C₄₃

#1759

yes

—

101

C₂×#125

C₂×D₄₃

yes

characteristic

102

C₂×#97

—

103

#389

Dic₄₃

yes

—

104

#387

—

105

#382

C₂×D₄₃

yes

characteristic

106

C₄×C₄⋊C₄

—

107

C₄×C₄⋊C₄

Dic₄₃

yes

—

108

C₄³

C₂×D₄₃

yes

characteristic

109

#336

—

110

#336

—

111

#330

—

112

C₄₃

#8225

yes

characteristic, complemented, Sylow

113

C₂²×C₈

—

114

#125

—

115

#125

C₄×D₄₃

yes

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

C₂×Dic₄₃

yes

—

119

C₂×C₄⋊C₄

C₂×Dic₄₃

yes

—

120

C₂×C₄²

—

121

C₂×C₄²

C₂×Dic₄₃

yes

characteristic

122

C₂×C₄²

C₂×Dic₄₃

yes

characteristic

123

C₂×C₄²

C₂²⋊D₄₃

yes

—

124

C₂×C₄²

C₂²⋊D₄₃

yes

—

125

C₂×C₄²

C₂²×D₄₃

yes

characteristic

126

#97

—

127

#97

—

128

#95

—

129

#95

—

130

#95

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

C₂²⋊Dic₄₃

yes

characteristic

134

C₂²×C₄

C₂²⋊Dic₄₃

yes

—

135

C₂²×C₄

C₂²⋊Dic₄₃

yes

characteristic

136

C₂²×C₄

C₂²×Dic₄₃

yes

characteristic

137

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

—

138

C₂×C₈

—

139

C₂×C₈

—

172

—

140

C₂×C₈

—

172

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

C₄×Dic₄₃

yes

—

146

C₄²

—

147

C₄²

—

148

C₄²

—

149

C₄²

—

150

C₄²

—

151

C₄²

C₂×C₄×D₄₃

yes

—

152

C₄²

#10621898

yes

—

153

C₂³

C₂×C₂²⋊Dic₄₃

yes

characteristic

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

#13063244

yes

—

171

C₂×C₄

C₂×C₄×Dic₄₃

yes

characteristic

172

C₂×C₄

#13063284

yes

characteristic

173

C₂×C₄

C₄×C₂²⋊D₄₃

yes

—

174

C₈

—

172

—

175

C₂²

C₄×C₂²⋊Dic₄₃

yes

characteristic

176

C₂²

C₂×#13063244

yes

characteristic

177

C₂²

#14582883

yes

characteristic

178

C₂²

#14582916

yes

—

179

C₂²

#14582916

yes

—

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

—

188

C₂

#19133222

yes

characteristic

189

C₂

#19133226

yes

—

190

C₂

#19133233

yes

characteristic

191

C₂

C₄×#13063244

yes

characteristic

192

C₂

#19134405

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₃
commutator	C₂×C₄×C₄₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄×#125×C₄₃	C₂
socle, prime radical	C₂³×C₄₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₄×#125	D₄₃
hypocenter	C₄₃	#8225
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₄₃

commutator

C₂×C₄×C₄₃

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄×#125×C₄₃

C₂

socle, prime radical

C₂³×C₄₃

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

N	H ≅ G/N	type
8. #14582641	C₄	semidirect
112. C₄₃	#8225	semidirect

H ≅ G/N

type

8. #14582641

C₄

semidirect

112. C₄₃

#8225

semidirect

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

subgroup	index
#19134344	2
C₂×#14582641	2
#19132476	2
C₄×#125×C₄₃	2
#8225	43

subgroup

index

C₄×#125×C₄₃

Subgroups — #24987523

Conjugacy classes of subgroups

Complements

Maximal subgroups