Group #24912499 — subgroups

Subgroups — #24912499

Order 10624.

452 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 181 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×#14581783	C₂	1	2	yes	maximal
2	#19127634	C₂	1	4	yes	maximal
3	#19127600	C₂	1	2	yes	maximal
4	#19127597	C₂	1	2	yes	maximal
5	#19127570	C₂	1	2	yes	maximal
6	C₄×C₄⋊Dic₈₃	C₂	1	1	yes	maximal, characteristic
7	C₄×C₄⋊Dic₈₃	C₂	1	1	yes	maximal, characteristic
8	#384×C₈₃	C₂	1	1	yes	maximal, characteristic
9	#14581783	—	2	8	no	—
10	C₂×C₄⋊Dic₈₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₈₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₈₃	C₂²	1	4	yes	—
13	C₂×C₄⋊Dic₈₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₈₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₈₃⋊C₄	C₂²	1	4	yes	—
16	C₂×Dic₈₃⋊C₄	C₂²	1	4	yes	—
17	C₂×C₄×Dic₈₃	C₂²	1	2	yes	—
18	C₂×C₄×Dic₈₃	C₂²	1	2	yes	—
19	C₈₃⋊#95	C₂²	1	2	yes	—
20	C₈₃⋊#95	C₂²	1	2	yes	—
21	C₈₃⋊#95	C₂²	1	4	yes	—
22	C₂×C₄⋊C₄×C₈₃	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₈₃	C₂²	1	2	yes	—
24	C₂×C₄⋊C₄×C₈₃	C₂²	1	2	yes	—
25	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
26	C₂²×Dic₈₃	C₂³	1	2	yes	—
27	C₂²×Dic₈₃	C₂³	1	2	yes	—
28	C₂²×Dic₈₃	C₂³	1	2	yes	—
29	C₂²×Dic₈₃	C₂³	1	2	yes	—
30	C₄⋊Dic₈₃	—	2	2	no	—
31	C₄⋊Dic₈₃	—	2	2	no	—
32	C₄⋊Dic₈₃	—	2	8	no	—
33	C₄⋊Dic₈₃	D₄	1	4	yes	—
34	C₄⋊Dic₈₃	Q₈	1	4	yes	—
35	Dic₈₃⋊C₄	—	2	8	no	—
36	Dic₈₃⋊C₄	—	4	4	no	—
37	C₄×Dic₈₃	—	2	4	no	—
38	C₄×Dic₈₃	—	2	4	no	—
39	C₂²×C₄×C₈₃	C₂³	1	4	yes	—
40	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
43	C₄⋊C₄×C₈₃	—	2	4	no	—
44	C₄⋊C₄×C₈₃	—	2	4	no	—
45	C₄⋊C₄×C₈₃	—	4	2	no	—
46	C₄²×C₈₃	Q₈	1	4	yes	—
47	C₂³×C₈₃	C₂⁴	1	1	yes	characteristic
48	C₂×C₄×C₈₃	—	2	1	no	—
49	C₂×C₄×C₈₃	—	2	1	no	—
50	C₂×C₄×C₈₃	—	2	4	no	—
51	C₂×C₄×C₈₃	—	2	4	no	—
52	C₂×C₄×C₈₃	—	2	4	no	—
53	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₈₃	C₂×Q₈	1	2	yes	—
55	C₂×C₄×C₈₃	C₂×Q₈	1	2	yes	—
56	C₂×C₄×C₈₃	C₂×Q₈	1	2	yes	—
57	C₂×C₄×C₈₃	C₂×Q₈	1	2	yes	—
58	C₂×C₄×C₈₃	#41	1	2	yes	—
59	C₂×C₄×C₈₃	#41	1	2	yes	—
60	C₂×Dic₈₃	—	2	2	no	—
61	C₂×Dic₈₃	—	2	2	no	—
62	C₂×Dic₈₃	—	2	2	no	—
63	C₂×Dic₈₃	—	2	2	no	—
64	C₂×Dic₈₃	—	2	2	no	—
65	C₂×Dic₈₃	—	2	2	no	—
66	C₂×Dic₈₃	—	2	2	no	—
67	C₂×Dic₈₃	—	2	2	no	—
68	C₂×Dic₈₃	—	2	2	no	—
69	C₂×Dic₈₃	—	2	2	no	—
70	C₂×Dic₈₃	C₂×D₄	1	4	yes	—
71	C₂×Dic₈₃	C₂×Q₈	1	4	yes	—
72	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
73	C₂²×C₈₃	C₂²×Q₈	1	1	yes	characteristic
74	C₂²×C₈₃	C₂²×Q₈	1	1	yes	characteristic
75	C₂²×C₈₃	C₂×#41	1	1	yes	characteristic
76	C₂²×C₈₃	C₂×#41	1	1	yes	characteristic
77	C₂²×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
78	C₂²×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
79	C₄×C₈₃	—	2	2	no	—
80	C₄×C₈₃	—	4	4	no	—
81	C₄×C₈₃	#125	1	4	yes	complemented
82	C₄×C₈₃	C₄⋊Q₈	1	4	yes	—
83	Dic₈₃	—	2	4	no	—
84	Dic₈₃	—	2	4	no	—
85	Dic₈₃	—	4	2	no	—
86	Dic₈₃	—	4	2	no	—
87	C₂×C₈₃	C₂×#125	1	1	yes	characteristic
88	C₂×C₈₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
89	C₂×C₈₃	#544	1	1	yes	characteristic
90	C₂×C₈₃	#547	1	1	yes	characteristic
91	C₂×C₈₃	#550	1	1	yes	characteristic
92	C₂×C₈₃	#554	1	1	yes	characteristic
93	C₂×C₈₃	#557	1	1	yes	characteristic
94	#2552	—	83	1	no	maximal, Sylow
95	C₈₃	#2552	1	1	yes	characteristic, complemented, Sylow
96	C₂×#125	—	83	2	no	—
97	#398	—	83	4	no	—
98	#389	—	83	2	no	—
99	#389	—	83	2	no	—
100	#384	—	83	2	no	—
101	#384	D₈₃	1	1	yes	characteristic
102	C₄×C₄⋊C₄	—	83	1	no	—
103	C₄×C₄⋊C₄	—	83	1	no	—
104	#125	—	166	8	no	—
105	C₂×C₄⋊C₄	—	83	4	no	—
106	C₂×C₄⋊C₄	—	83	1	no	—
107	C₂×C₄⋊C₄	—	83	1	no	—
108	C₂×C₄⋊C₄	—	83	4	no	—
109	C₂×C₄⋊C₄	—	83	4	no	—
110	C₂×C₄⋊C₄	—	83	1	no	—
111	C₂×C₄⋊C₄	—	83	1	no	—
112	C₂×C₄⋊C₄	C₂×D₈₃	1	2	yes	—
113	C₂×C₄⋊C₄	C₂×D₈₃	1	2	yes	—
114	C₂×C₄⋊C₄	C₂×D₈₃	1	2	yes	—
115	C₂×C₄²	—	83	2	no	—
116	C₂×C₄²	—	83	2	no	—
117	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
118	#95	—	83	2	no	—
119	#95	—	83	2	no	—
120	#95	—	83	4	no	—
121	C₂²×C₄	—	83	2	no	—
122	C₂²×C₄	—	83	2	no	—
123	C₂²×C₄	—	83	2	no	—
124	C₂²×C₄	—	83	2	no	—
125	C₂²×C₄	C₂²×D₈₃	1	4	yes	—
126	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
128	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
129	C₄⋊C₄	—	2	4	no	—
130	C₄⋊C₄	—	2	4	no	—
131	C₄⋊C₄	—	4	2	no	—
132	C₄⋊C₄	—	332	4	no	—
133	C₄⋊C₄	—	83	4	no	—
134	C₄⋊C₄	—	83	4	no	—
135	C₄⋊C₄	—	166	8	no	—
136	C₄⋊C₄	—	166	8	no	—
137	C₄⋊C₄	—	166	2	no	—
138	C₄⋊C₄	—	166	2	no	—
139	C₄²	—	166	4	no	—
140	C₄²	—	166	4	no	—
141	C₄²	Dic₁₆₆	1	4	yes	—
142	C₂³	C₂³×D₈₃	1	1	yes	characteristic
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	83	4	no	—
149	C₂×C₄	—	83	4	no	—
150	C₂×C₄	—	166	2	no	—
151	C₂×C₄	—	166	2	no	—
152	C₂×C₄	—	166	2	no	—
153	C₂×C₄	—	166	2	no	—
154	C₂×C₄	—	166	2	no	—
155	C₂×C₄	—	166	2	no	—
156	C₂×C₄	—	166	2	no	—
157	C₂×C₄	—	166	2	no	—
158	C₂×C₄	—	166	2	no	—
159	C₂×C₄	—	166	2	no	—
160	C₂×C₄	D₄×D₈₃	1	2	yes	—
161	C₂×C₄	Q₈×D₈₃	1	2	yes	—
162	C₂×C₄	C₂×Dic₁₆₆	1	2	yes	—
163	C₂×C₄	C₂×Dic₁₆₆	1	2	yes	—
164	C₂×C₄	C₂×Dic₁₆₆	1	2	yes	—
165	C₂×C₄	#13050589	1	2	yes	—
166	C₂×C₄	#13050591	1	2	yes	—
167	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
168	C₂²	C₂×Q₈×D₈₃	1	1	yes	characteristic
169	C₂²	C₂²×Dic₁₆₆	1	1	yes	characteristic
170	C₂²	C₂×#13050589	1	1	yes	characteristic
171	C₂²	C₂×#13050591	1	1	yes	characteristic
172	C₂²	#14581837	1	1	yes	characteristic
173	C₂²	#14581840	1	1	yes	characteristic
174	C₄	—	2	2	no	—
175	C₄	—	4	4	no	—
176	C₄	—	332	2	no	—
177	C₄	—	332	2	no	—
178	C₄	—	166	4	no	—
179	C₄	—	166	4	no	—
180	C₄	#14581783	1	4	yes	complemented
181	C₄	#14581791	1	4	yes	—
182	C₂	C₂×#14581783	1	1	yes	characteristic
183	C₂	C₂×#14581791	1	1	yes	characteristic
184	C₂	#19128332	1	1	yes	characteristic
185	C₂	#19128354	1	1	yes	characteristic
186	C₂	#19128371	1	1	yes	characteristic
187	C₂	#19128412	1	1	yes	characteristic
188	C₂	#19128426	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581783

C₂

yes

maximal

#19127634

C₂

yes

maximal

#19127600

C₂

yes

maximal

#19127597

C₂

yes

maximal

#19127570

C₂

yes

maximal

C₄×C₄⋊Dic₈₃

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₈₃

C₂

yes

maximal, characteristic

#384×C₈₃

C₂

yes

maximal, characteristic

#14581783

—

C₂×C₄⋊Dic₈₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₃

C₂²

yes

—

C₂×C₄⋊Dic₈₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₃

C₂²

yes

characteristic

C₂×Dic₈₃⋊C₄

C₂²

yes

—

C₂×Dic₈₃⋊C₄

C₂²

yes

—

C₂×C₄×Dic₈₃

C₂²

yes

—

C₂×C₄×Dic₈₃

C₂²

yes

—

C₈₃⋊#95

C₂²

yes

—

C₈₃⋊#95

C₂²

yes

—

C₈₃⋊#95

C₂²

yes

—

C₂×C₄⋊C₄×C₈₃

C₂²

yes

—

C₂×C₄⋊C₄×C₈₃

C₂²

yes

—

C₂×C₄⋊C₄×C₈₃

C₂²

yes

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₂²×Dic₈₃

C₂³

yes

—

C₂²×Dic₈₃

C₂³

yes

—

C₂²×Dic₈₃

C₂³

yes

—

C₂²×Dic₈₃

C₂³

yes

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

D₄

yes

—

C₄⋊Dic₈₃

Q₈

yes

—

Dic₈₃⋊C₄

—

Dic₈₃⋊C₄

—

C₄×Dic₈₃

—

C₄×Dic₈₃

—

C₂²×C₄×C₈₃

C₂³

yes

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

Q₈

yes

—

C₂³×C₈₃

C₂⁴

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×Q₈

yes

—

C₂×C₄×C₈₃

C₂×Q₈

yes

—

C₂×C₄×C₈₃

C₂×Q₈

yes

—

C₂×C₄×C₈₃

C₂×Q₈

yes

—

C₂×C₄×C₈₃

#41

yes

—

C₂×C₄×C₈₃

#41

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

C₂×D₄

yes

—

C₂×Dic₈₃

C₂×Q₈

yes

—

C₂²×C₈₃

C₂²×D₄

yes

characteristic

C₂²×C₈₃

C₂²×Q₈

yes

characteristic

C₂²×C₈₃

C₂²×Q₈

yes

characteristic

C₂²×C₈₃

C₂×#41

yes

characteristic

C₂²×C₈₃

C₂×#41

yes

characteristic

C₂²×C₈₃

H(5,𝔽₂)

yes

characteristic

C₂²×C₈₃

2₋¹⁺⁴

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#125

yes

complemented

C₄×C₈₃

C₄⋊Q₈

yes

—

Dic₈₃

—

Dic₈₃

—

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

C₂×#125

yes

characteristic

C₂×C₈₃

C₂×C₄⋊Q₈

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

—

—

—

—

100

#384

—

101

#384

D₈₃

yes

characteristic

102

C₄×C₄⋊C₄

—

103

C₄×C₄⋊C₄

—

104

#125

—

166

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

C₂×D₈₃

yes

—

113

C₂×C₄⋊C₄

C₂×D₈₃

yes

—

114

C₂×C₄⋊C₄

C₂×D₈₃

yes

—

115

C₂×C₄²

—

116

C₂×C₄²

—

117

C₂×C₄²

C₂×D₈₃

yes

characteristic

118

#95

—

119

#95

—

120

#95

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

C₂²×D₈₃

yes

—

126

C₂²×C₄

C₂²×D₈₃

yes

characteristic

127

C₂²×C₄

C₂²×D₈₃

yes

characteristic

128

C₂²×C₄

C₂²×D₈₃

yes

characteristic

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

332

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

166

—

136

C₄⋊C₄

—

166

—

137

C₄⋊C₄

—

166

—

138

C₄⋊C₄

—

166

—

139

C₄²

—

166

—

140

C₄²

—

166

—

141

C₄²

Dic₁₆₆

yes

—

142

C₂³

C₂³×D₈₃

yes

characteristic

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

166

—

151

C₂×C₄

—

166

—

152

C₂×C₄

—

166

—

153

C₂×C₄

—

166

—

154

C₂×C₄

—

166

—

155

C₂×C₄

—

166

—

156

C₂×C₄

—

166

—

157

C₂×C₄

—

166

—

158

C₂×C₄

—

166

—

159

C₂×C₄

—

166

—

160

C₂×C₄

D₄×D₈₃

yes

—

161

C₂×C₄

Q₈×D₈₃

yes

—

162

C₂×C₄

C₂×Dic₁₆₆

yes

—

163

C₂×C₄

C₂×Dic₁₆₆

yes

—

164

C₂×C₄

C₂×Dic₁₆₆

yes

—

165

C₂×C₄

#13050589

yes

—

166

C₂×C₄

#13050591

yes

—

167

C₂²

C₂×D₄×D₈₃

yes

characteristic

168

C₂²

C₂×Q₈×D₈₃

yes

characteristic

169

C₂²

C₂²×Dic₁₆₆

yes

characteristic

170

C₂²

C₂×#13050589

yes

characteristic

171

C₂²

C₂×#13050591

yes

characteristic

172

C₂²

#14581837

yes

characteristic

173

C₂²

#14581840

yes

characteristic

174

C₄

—

175

C₄

—

176

C₄

—

332

—

177

C₄

—

332

—

178

C₄

—

166

—

179

C₄

—

166

—

180

C₄

#14581783

yes

complemented

181

C₄

#14581791

yes

—

182

C₂

C₂×#14581783

yes

characteristic

183

C₂

C₂×#14581791

yes

characteristic

184

C₂

#19128332

yes

characteristic

185

C₂

#19128354

yes

characteristic

186

C₂

#19128371

yes

characteristic

187

C₂

#19128412

yes

characteristic

188

C₂

#19128426

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₃
commutator, socle, prime radical, cosocle	C₂³×C₈₃	C₂⁴
Fitting	#384×C₈₃	C₂
hypercenter	#384	D₈₃
hypocenter	C₈₃	#2552
LCS	G ▷ C₂³×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₃

commutator, socle, prime radical, cosocle

C₂³×C₈₃

C₂⁴

Fitting

#384×C₈₃

C₂

D₈₃

C₈₃

G ▷ C₂³×C₈₃ ▷ C₈₃

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
81. C₄×C₈₃	#125	semidirect
95. C₈₃	#2552	semidirect
180. C₄	#14581783	semidirect

H ≅ G/N

type

81. C₄×C₈₃

#125

semidirect

95. C₈₃

#2552

semidirect

180. C₄

#14581783

semidirect

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

subgroup	index
C₂×#14581783	2
#19127634	2
#19127600	2
#19127597	2
#19127570	2
C₄×C₄⋊Dic₈₃	2
#384×C₈₃	2
#2552	83

subgroup

index

C₄×C₄⋊Dic₈₃

#384×C₈₃

#2552

Subgroups — #24912499

Conjugacy classes of subgroups

Complements

Maximal subgroups