Group #24917921 — subgroups

Subgroups — #24917921

Order 10624.

552 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 145 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	C₂×#14581824	C₂	1	4	yes	maximal, complemented
2	#19128252	C₂	1	1	yes	maximal, characteristic
3	C₄⋊Q₈×D₈₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128094	C₂	1	4	yes	maximal, complemented
5	#19128080	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128075	C₂	1	1	yes	maximal, characteristic, complemented
7	#19128074	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#13050573	C₂	1	1	yes	maximal, characteristic, complemented
9	#494×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13050591	C₂²	1	2	yes	—
11	C₂×Q₈×D₈₃	C₂²	1	2	yes	—
12	#14581824	—	2	16	no	—
13	C₂×#13050580	C₂²	1	4	yes	—
14	C₂×#13050579	C₂²	1	4	yes	—
15	C₂×#13050577	C₂²	1	4	yes	—
16	C₂×#13050573	C₂²	1	2	yes	—
17	#14581792	C₂²	1	1	yes	characteristic
18	#14581791	—	2	2	no	—
19	#14581790	C₂²	1	2	yes	—
20	#14581789	C₂²	1	1	yes	characteristic
21	#14581781	C₂²	1	2	yes	—
22	#14581767	—	2	2	no	—
23	#14581754	—	2	2	no	—
24	C₄⋊C₄×D₈₃	—	2	2	no	—
25	C₄²×D₈₃	C₂²	1	1	yes	characteristic
26	#14581702	C₂²	1	2	yes	—
27	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×Q₁₆×C₈₃	C₂²	1	4	yes	—
29	C₄⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
30	C₄⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
31	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
32	#13050591	—	2	8	no	—
33	C₂×Dic₁₆₆	—	2	2	no	—
34	C₂×Dic₁₆₆	C₂³	1	2	yes	—
35	Q₈×D₈₃	—	2	8	no	—
36	C₂×D₃₃₂	—	2	2	no	—
37	C₂×D₃₃₂	—	2	2	no	—
38	C₂×D₃₃₂	C₂³	1	2	yes	—
39	C₂×C₄×D₈₃	—	2	2	no	—
40	C₂×C₄×D₈₃	—	2	2	no	—
41	C₂×C₄×D₈₃	D₄	1	2	yes	—
42	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
43	#13050580	—	2	8	no	—
44	#13050579	—	2	8	no	—
45	#13050577	—	2	8	no	—
46	#13050573	D₄	1	8	yes	complemented
47	C₂×C₈₃⋊C₈	C₂³	1	2	yes	—
48	C₄⋊Dic₈₃	—	2	2	no	—
49	Dic₈₃⋊C₄	—	2	4	no	—
50	(C₂×D₈₃)⋊C₄	—	2	4	no	—
51	C₄×Dic₈₃	D₄	1	2	yes	—
52	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
53	C₂×Q₈×C₈₃	C₂³	1	2	yes	—
54	C₂×Q₈×C₈₃	C₂³	1	2	yes	—
55	Q₁₆×C₈₃	—	2	8	no	—
56	C₂×C₈×C₈₃	C₂³	1	2	yes	—
57	C₄⋊C₄×C₈₃	—	2	2	no	—
58	C₄⋊C₄×C₈₃	—	2	2	no	—
59	C₄²×C₈₃	C₂³	1	1	yes	characteristic
60	C₂²×D₈₃	—	2	2	no	—
61	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
62	Q₈×C₈₃	—	2	4	no	—
63	Q₈×C₈₃	—	2	4	no	—
64	C₂×C₄×C₈₃	—	2	2	no	—
65	C₂×C₄×C₈₃	—	2	2	no	—
66	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
67	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
68	C₂×Dic₈₃	—	2	2	no	—
69	C₂×Dic₈₃	C₂×D₄	1	2	yes	—
70	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
71	D₃₃₂	—	2	4	no	—
72	D₃₃₂	—	4	4	no	—
73	D₃₃₂	—	4	4	no	—
74	C₄×D₈₃	—	2	4	no	—
75	C₄×D₈₃	—	4	4	no	—
76	C₄×D₈₃	—	4	4	no	—
77	C₄×D₈₃	C₂×D₄	1	4	yes	—
78	Dic₁₆₆	—	2	4	no	—
79	Dic₁₆₆	—	4	4	no	—
80	C₈×C₈₃	C₂×D₄	1	4	yes	—
81	C₈₃⋊C₈	C₂×D₄	1	4	yes	—
82	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
83	C₂×D₈₃	—	2	2	no	—
84	C₂×D₈₃	—	4	2	no	—
85	C₂×D₈₃	—	4	4	no	—
86	C₂×D₈₃	#127	1	2	yes	—
87	C₄×C₈₃	—	4	2	no	—
88	C₄×C₈₃	—	4	2	no	—
89	C₄×C₈₃	#137	1	4	yes	complemented
90	C₄×C₈₃	C₂²×D₄	1	2	yes	—
91	Dic₈₃	—	2	2	no	—
92	Dic₈₃	—	4	2	no	—
93	Dic₈₃	#127	1	2	yes	—
94	C₂×C₈₃	C₂×#127	1	1	yes	characteristic
95	C₂×C₈₃	C₂×#137	1	2	yes	—
96	D₈₃	—	2	2	no	—
97	D₈₃	—	8	2	no	—
98	#3152	—	83	1	no	maximal, Sylow
99	C₈₃	#3152	1	1	yes	characteristic, complemented, Sylow
100	C₂×#137	—	83	4	no	—
101	#532	—	83	1	no	—
102	C₂×C₄⋊Q₈	—	83	1	no	—
103	#497	—	83	4	no	—
104	#494	—	83	1	no	—
105	#494	D₈₃	1	1	yes	characteristic, complemented
106	#492	—	83	1	no	—
107	#492	—	83	1	no	—
108	C₄×M₁₆	—	83	1	no	—
109	C₂×#41	—	83	2	no	—
110	C₂²×Q₈	—	83	2	no	—
111	#137	—	166	16	no	—
112	C₂×Q₁₆	—	83	4	no	—
113	C₂×Q₁₆	C₂×D₈₃	1	4	yes	—
114	C₂×SD₁₆	—	83	4	no	—
115	C₂×SD₁₆	—	83	4	no	—
116	C₂×M₁₆	—	83	2	no	—
117	C₄⋊Q₈	—	83	2	no	—
118	C₄⋊Q₈	—	83	1	no	—
119	C₄⋊Q₈	—	166	2	no	—
120	C₄⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
121	C₄⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
122	#127	—	83	1	no	—
123	#124	—	83	2	no	—
124	#121	—	166	2	no	—
125	C₄×D₄	—	166	2	no	—
126	C₂×C₄⋊C₄	—	166	2	no	—
127	C₂×C₄²	—	83	1	no	—
128	#97	—	83	2	no	—
129	C₄×C₈	—	83	1	no	—
130	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
131	#41	—	166	8	no	—
132	C₂×Q₈	—	83	2	no	—
133	C₂×Q₈	—	166	2	no	—
134	C₂×Q₈	—	166	8	no	—
135	C₂×Q₈	C₂²×D₈₃	1	2	yes	—
136	C₂×Q₈	C₂²×D₈₃	1	2	yes	—
137	C₂×D₄	—	83	2	no	—
138	C₂×D₄	—	166	2	no	—
139	C₂×D₄	—	166	2	no	—
140	C₂²×C₄	—	83	1	no	—
141	C₂²×C₄	—	83	2	no	—
142	C₂²×C₄	—	166	2	no	—
143	C₂²×C₄	—	166	2	no	—
144	Q₁₆	—	2	8	no	—
145	Q₁₆	—	166	8	no	—
146	SD₁₆	—	166	8	no	—
147	SD₁₆	—	166	8	no	—
148	M₁₆	—	83	8	no	—
149	C₂×C₈	—	83	2	no	—
150	C₂×C₈	C₂²×D₈₃	1	2	yes	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	2	2	no	—
153	C₄⋊C₄	—	166	4	no	—
154	C₄⋊C₄	—	166	2	no	—
155	C₂²⋊C₄	—	166	4	no	—
156	C₄²	—	83	2	no	—
157	C₄²	—	83	1	no	—
158	C₄²	C₂²×D₈₃	1	1	yes	characteristic
159	C₂³	—	83	1	no	—
160	C₂³	—	166	2	no	—
161	Q₈	—	2	4	no	—
162	Q₈	—	2	4	no	—
163	Q₈	—	332	4	no	—
164	Q₈	—	166	4	no	—
165	D₄	—	332	4	no	—
166	D₄	—	332	4	no	—
167	D₄	—	166	4	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	332	4	no	—
171	C₂×C₄	—	332	4	no	—
172	C₂×C₄	—	83	1	no	—
173	C₂×C₄	—	83	4	no	—
174	C₂×C₄	—	83	2	no	—
175	C₂×C₄	—	166	2	no	—
176	C₂×C₄	—	166	4	no	—
177	C₂×C₄	D₄×D₈₃	1	2	yes	—
178	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
179	C₈	—	83	4	no	—
180	C₈	D₄×D₈₃	1	4	yes	—
181	C₂²	—	332	2	no	—
182	C₂²	—	332	4	no	—
183	C₂²	—	83	2	no	—
184	C₂²	—	166	2	no	—
185	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
186	C₄	—	4	2	no	—
187	C₄	—	4	2	no	—
188	C₄	—	332	2	no	—
189	C₄	—	83	2	no	—
190	C₄	—	166	2	no	—
191	C₄	#14581824	1	4	yes	complemented
192	C₄	C₂×D₄×D₈₃	1	2	yes	—
193	C₂	—	664	2	no	—
194	C₂	—	166	2	no	—
195	C₂	#127×D₈₃	1	1	yes	characteristic
196	C₂	C₂×#14581824	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581824

C₂

yes

maximal, complemented

#19128252

C₂

yes

maximal, characteristic

C₄⋊Q₈×D₈₃

C₂

yes

maximal, characteristic, complemented

#19128094

C₂

yes

maximal, complemented

#19128080

C₂

yes

maximal, characteristic, complemented

#19128075

C₂

yes

maximal, characteristic, complemented

#19128074

C₂

yes

maximal, characteristic, complemented

C₄×#13050573

C₂

yes

maximal, characteristic, complemented

#494×C₈₃

C₂

yes

maximal, characteristic, complemented

C₂×#13050591

C₂²

yes

—

C₂×Q₈×D₈₃

C₂²

yes

—

#14581824

—

C₂×#13050580

C₂²

yes

—

C₂×#13050579

C₂²

yes

—

C₂×#13050577

C₂²

yes

—

C₂×#13050573

C₂²

yes

—

#14581792

C₂²

yes

characteristic

#14581791

—

#14581790

C₂²

yes

—

#14581789

C₂²

yes

characteristic

#14581781

C₂²

yes

—

#14581767

—

#14581754

—

C₄⋊C₄×D₈₃

—

C₄²×D₈₃

C₂²

yes

characteristic

#14581702

C₂²

yes

—

C₄×C₈₃⋊C₈

C₂²

yes

characteristic

C₂×Q₁₆×C₈₃

C₂²

yes

—

C₄⋊Q₈×C₈₃

C₂²

yes

characteristic

C₄⋊Q₈×C₈₃

C₂²

yes

characteristic

C₄×C₈×C₈₃

C₂²

yes

characteristic

#13050591

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

C₂³

yes

—

Q₈×D₈₃

—

C₂×D₃₃₂

—

C₂×D₃₃₂

—

C₂×D₃₃₂

C₂³

yes

—

C₂×C₄×D₈₃

—

C₂×C₄×D₈₃

—

C₂×C₄×D₈₃

D₄

yes

—

C₂×C₄×D₈₃

C₂³

yes

—

—

—

D₄

yes

complemented

C₂×C₈₃⋊C₈

C₂³

yes

—

C₄⋊Dic₈₃

—

Dic₈₃⋊C₄

—

(C₂×D₈₃)⋊C₄

—

C₄×Dic₈₃

D₄

yes

—

C₄×Dic₈₃

C₂³

yes

characteristic

C₂×Q₈×C₈₃

C₂³

yes

—

C₂×Q₈×C₈₃

C₂³

yes

—

Q₁₆×C₈₃

—

C₂×C₈×C₈₃

C₂³

yes

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

C₂³

yes

characteristic

C₂²×D₈₃

—

C₂²×D₈₃

C₂×D₄

yes

characteristic

Q₈×C₈₃

—

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂⁴

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

C₂×D₄

yes

—

C₂×Dic₈₃

C₂×D₄

yes

characteristic

D₃₃₂

—

D₃₃₂

—

D₃₃₂

—

C₄×D₈₃

—

C₄×D₈₃

—

C₄×D₈₃

—

C₄×D₈₃

C₂×D₄

yes

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

C₂×D₄

yes

—

C₈₃⋊C₈

C₂×D₄

yes

—

C₂²×C₈₃

C₂²×D₄

yes

characteristic

C₂×D₈₃

—

C₂×D₈₃

—

C₂×D₈₃

—

C₂×D₈₃

#127

yes

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#137

yes

complemented

C₄×C₈₃

C₂²×D₄

yes

—

Dic₈₃

—

Dic₈₃

—

Dic₈₃

#127

yes

—

C₂×C₈₃

C₂×#127

yes

characteristic

C₂×C₈₃

C₂×#137

yes

—

D₈₃

—

D₈₃

—

#3152

—

maximal, Sylow

C₈₃

#3152

yes

characteristic, complemented, Sylow

100

C₂×#137

—

101

#532

—

102

C₂×C₄⋊Q₈

—

103

#497

—

104

#494

—

105

#494

D₈₃

yes

characteristic, complemented

106

#492

—

107

#492

—

108

C₄×M₁₆

—

109

C₂×#41

—

110

C₂²×Q₈

—

111

#137

—

166

—

112

C₂×Q₁₆

—

113

C₂×Q₁₆

C₂×D₈₃

yes

—

114

C₂×SD₁₆

—

115

C₂×SD₁₆

—

116

C₂×M₁₆

—

117

C₄⋊Q₈

—

118

C₄⋊Q₈

—

119

C₄⋊Q₈

—

166

—

120

C₄⋊Q₈

C₂×D₈₃

yes

characteristic

121

C₄⋊Q₈

C₂×D₈₃

yes

characteristic

122

#127

—

123

#124

—

124

#121

—

166

—

125

C₄×D₄

—

166

—

126

C₂×C₄⋊C₄

—

166

—

127

C₂×C₄²

—

128

#97

—

129

C₄×C₈

—

130

C₄×C₈

C₂×D₈₃

yes

characteristic

131

#41

—

166

—

132

C₂×Q₈

—

133

C₂×Q₈

—

166

—

134

C₂×Q₈

—

166

—

135

C₂×Q₈

C₂²×D₈₃

yes

—

136

C₂×Q₈

C₂²×D₈₃

yes

—

137

C₂×D₄

—

138

C₂×D₄

—

166

—

139

C₂×D₄

—

166

—

140

C₂²×C₄

—

141

C₂²×C₄

—

142

C₂²×C₄

—

166

—

143

C₂²×C₄

—

166

—

144

Q₁₆

—

145

Q₁₆

—

166

—

146

SD₁₆

—

166

—

147

SD₁₆

—

166

—

148

M₁₆

—

149

C₂×C₈

—

150

C₂×C₈

C₂²×D₈₃

yes

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

166

—

154

C₄⋊C₄

—

166

—

155

C₂²⋊C₄

—

166

—

156

C₄²

—

157

C₄²

—

158

C₄²

C₂²×D₈₃

yes

characteristic

159

C₂³

—

160

C₂³

—

166

—

161

Q₈

—

162

Q₈

—

163

Q₈

—

332

—

164

Q₈

—

166

—

165

D₄

—

332

—

166

D₄

—

332

—

167

D₄

—

166

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

332

—

171

C₂×C₄

—

332

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

166

—

176

C₂×C₄

—

166

—

177

C₂×C₄

D₄×D₈₃

yes

—

178

C₂×C₄

C₂³×D₈₃

yes

characteristic

179

C₈

—

180

C₈

D₄×D₈₃

yes

—

181

C₂²

—

332

—

182

C₂²

—

332

—

183

C₂²

—

184

C₂²

—

166

—

185

C₂²

C₂×D₄×D₈₃

yes

characteristic

186

C₄

—

187

C₄

—

188

C₄

—

332

—

189

C₄

—

190

C₄

—

166

—

191

C₄

#14581824

yes

complemented

192

C₄

C₂×D₄×D₈₃

yes

—

193

C₂

—

664

—

194

C₂

—

166

—

195

C₂

#127×D₈₃

yes

characteristic

196

C₂

C₂×#14581824

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#494×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#19128252	C₂
hypercenter	#494	D₈₃
hypocenter	C₈₃	#3152
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

C₂×D₄×D₈₃

commutator, cosocle

C₂×C₄×C₈₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₃

Fitting

#494×C₈₃

C₂

socle

C₂²×C₈₃

C₂²×D₄

C₂

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
1. C₂×#14581824	C₂	semidirect
3. C₄⋊Q₈×D₈₃	C₂	semidirect
4. #19128094	C₂	semidirect
5. #19128080	C₂	semidirect
6. #19128075	C₂	semidirect
7. #19128074	C₂	semidirect
8. C₄×#13050573	C₂	semidirect
9. #494×C₈₃	C₂	semidirect
46. #13050573	D₄	semidirect
89. C₄×C₈₃	#137	semidirect
99. C₈₃	#3152	semidirect
105. #494	D₈₃	semidirect
191. C₄	#14581824	semidirect

H ≅ G/N

type

1. C₂×#14581824

C₂

semidirect

3. C₄⋊Q₈×D₈₃

C₂

semidirect

4. #19128094

C₂

semidirect

5. #19128080

C₂

semidirect

6. #19128075

C₂

semidirect

7. #19128074

C₂

semidirect

8. C₄×#13050573

C₂

semidirect

9. #494×C₈₃

C₂

semidirect

46. #13050573

D₄

semidirect

89. C₄×C₈₃

#137

semidirect

99. C₈₃

#3152

semidirect

105. #494

D₈₃

semidirect

191. C₄

#14581824

semidirect

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

subgroup	index
C₂×#14581824	2
#19128252	2
C₄⋊Q₈×D₈₃	2
#19128094	2
#19128080	2
#19128075	2
#19128074	2
C₄×#13050573	2
#494×C₈₃	2
#3152	83

subgroup

index

C₄⋊Q₈×D₈₃

#494×C₈₃

Subgroups — #24917921

Conjugacy classes of subgroups

Complements

Maximal subgroups