Group #24905949 — subgroups

Subgroups — #24905949

Order 10624.

238 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 81 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	#19127754	C₂	1	1	yes	maximal, characteristic, complemented
2	#19127547	C₂	1	1	yes	maximal, characteristic, complemented
3	#19127438	C₂	1	2	yes	maximal, complemented
4	#19127435	C₂	1	2	yes	maximal
5	#529×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₈₃	—	2	1	no	—
7	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
8	C₂²⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
9	C₄×C₈₃⋊C₈	—	2	1	no	—
10	C₈₃⋊#95	—	2	1	no	—
11	C₈₃⋊#95	C₂²	1	2	yes	—
12	#126×C₈₃	—	2	1	no	—
13	#125×C₈₃	C₄	1	1	yes	characteristic, complemented
14	#124×C₈₃	C₄	1	1	yes	characteristic, complemented
15	#123×C₈₃	—	2	1	no	—
16	C₂²⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
17	#121×C₈₃	C₂²	1	1	yes	characteristic
18	C₄×Q₈×C₈₃	C₄	1	1	yes	characteristic, complemented
19	C₄×D₄×C₈₃	—	2	1	no	—
20	C₄×D₄×C₈₃	C₄	1	1	yes	characteristic, complemented
21	#117×C₈₃	—	2	1	no	—
22	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
23	C₂²×Dic₈₃	—	2	2	no	—
24	C₂×C₈₃⋊C₈	—	2	2	no	—
25	C₄×Dic₈₃	—	2	4	no	—
26	C₂×Q₈×C₈₃	C₂×C₄	1	1	yes	characteristic
27	C₂×D₄×C₈₃	—	2	1	no	—
28	C₂×D₄×C₈₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₈₃	—	2	1	no	—
30	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₈₃	—	2	1	no	—
34	C₄⋊C₄×C₈₃	—	2	1	no	—
35	C₄⋊C₄×C₈₃	—	2	1	no	—
36	C₄⋊C₄×C₈₃	—	2	1	no	—
37	C₄⋊C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
38	C₄⋊C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
39	C₂²⋊C₄×C₈₃	—	2	1	no	—
40	C₂²⋊C₄×C₈₃	—	2	1	no	—
41	C₂²⋊C₄×C₈₃	—	2	1	no	—
42	C₂²⋊C₄×C₈₃	—	2	1	no	—
43	C₂²⋊C₄×C₈₃	—	2	1	no	—
44	C₄²×C₈₃	—	2	1	no	—
45	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
46	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
47	C₄²×C₈₃	D₄	1	1	yes	characteristic
48	C₄²×C₈₃	D₄	1	1	yes	characteristic
49	C₂³×C₈₃	—	2	1	no	—
50	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
51	Q₈×C₈₃	—	4	1	no	—
52	D₄×C₈₃	—	4	1	no	—
53	D₄×C₈₃	—	4	2	no	—
54	C₂×C₄×C₈₃	—	2	2	no	—
55	C₂×C₄×C₈₃	—	2	2	no	—
56	C₂×C₄×C₈₃	—	2	1	no	—
57	C₂×C₄×C₈₃	—	2	1	no	—
58	C₂×C₄×C₈₃	—	2	1	no	—
59	C₂×C₄×C₈₃	—	4	1	no	—
60	C₂×C₄×C₈₃	—	4	2	no	—
61	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
67	C₂×Dic₈₃	—	2	4	no	—
68	C₂×Dic₈₃	—	4	2	no	—
69	C₂×Dic₈₃	—	4	2	no	—
70	C₈₃⋊C₈	—	4	2	no	—
71	C₂²×C₈₃	—	2	1	no	—
72	C₂²×C₈₃	—	2	1	no	—
73	C₂²×C₈₃	—	4	1	no	—
74	C₂²×C₈₃	—	4	1	no	—
75	C₂²×C₈₃	—	4	1	no	—
76	C₂²×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₈₃	#135	1	2	yes	—
78	C₄×C₈₃	—	2	1	no	—
79	C₄×C₈₃	—	2	1	no	—
80	C₄×C₈₃	—	2	1	no	—
81	C₄×C₈₃	—	2	1	no	—
82	C₄×C₈₃	—	2	1	no	—
83	C₄×C₈₃	—	4	1	no	—
84	C₄×C₈₃	—	4	1	no	—
85	C₄×C₈₃	—	4	1	no	—
86	C₄×C₈₃	#104	1	2	yes	—
87	Dic₈₃	—	4	4	no	—
88	C₂×C₈₃	—	2	2	no	—
89	C₂×C₈₃	—	8	1	no	—
90	C₂×C₈₃	#410	1	1	yes	characteristic
91	C₂×C₈₃	#416	1	1	yes	characteristic
92	C₂×C₈₃	C₂×#104	1	1	yes	characteristic
93	#1500	—	83	1	no	maximal, Sylow
94	C₈₃	#1500	1	1	yes	characteristic, complemented, Sylow
95	#529	D₈₃	1	1	yes	characteristic
96	#431	—	83	1	no	—
97	#376	—	83	1	no	—
98	#328	—	83	2	no	—
99	#327	—	83	2	no	—
100	#126	—	2	1	no	—
101	#125	Dic₈₃	1	1	yes	characteristic, complemented
102	#124	Dic₈₃	1	1	yes	characteristic, complemented
103	#123	—	2	1	no	—
104	C₂²⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
105	#121	C₂×D₈₃	1	1	yes	characteristic
106	C₄×Q₈	Dic₈₃	1	1	yes	characteristic, complemented
107	C₄×D₄	—	2	1	no	—
108	C₄×D₄	Dic₈₃	1	1	yes	characteristic, complemented
109	#117	—	2	1	no	—
110	C₂×C₄²	—	166	1	no	—
111	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
112	C₄⋊C₈	—	166	1	no	—
113	C₂²⋊C₈	—	83	2	no	—
114	C₄×C₈	—	166	1	no	—
115	#95	—	83	2	no	—
116	#95	—	166	1	no	—
117	C₂×Q₈	C₂×Dic₈₃	1	1	yes	characteristic
118	C₂×D₄	—	2	1	no	—
119	C₂×D₄	C₂×Dic₈₃	1	1	yes	characteristic
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	—	166	2	no	—
122	C₂²×C₄	C₂²⋊D₈₃	1	1	yes	characteristic
123	C₂²×C₄	C₂²⋊D₈₃	1	1	yes	characteristic
124	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
125	C₂×C₈	—	166	2	no	—
126	C₄⋊C₄	—	2	1	no	—
127	C₄⋊C₄	—	2	1	no	—
128	C₄⋊C₄	—	2	1	no	—
129	C₄⋊C₄	—	2	1	no	—
130	C₄⋊C₄	C₂×Dic₈₃	1	1	yes	characteristic
131	C₄⋊C₄	C₂×Dic₈₃	1	1	yes	characteristic
132	C₂²⋊C₄	—	2	1	no	—
133	C₂²⋊C₄	—	2	1	no	—
134	C₂²⋊C₄	—	2	1	no	—
135	C₂²⋊C₄	—	2	1	no	—
136	C₂²⋊C₄	—	2	1	no	—
137	C₄²	—	2	1	no	—
138	C₄²	—	166	4	no	—
139	C₄²	C₂×Dic₈₃	1	1	yes	characteristic
140	C₄²	C₂×Dic₈₃	1	1	yes	characteristic
141	C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
142	C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
143	C₂³	—	2	1	no	—
144	C₂³	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
145	Q₈	—	4	1	no	—
146	D₄	—	4	1	no	—
147	D₄	—	4	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	332	2	no	—
156	C₂×C₄	—	332	2	no	—
157	C₂×C₄	—	166	4	no	—
158	C₂×C₄	C₂²⋊Dic₈₃	1	1	yes	characteristic
159	C₂×C₄	C₂²⋊Dic₈₃	1	1	yes	characteristic
160	C₂×C₄	C₂²⋊Dic₈₃	1	1	yes	characteristic
161	C₂×C₄	C₂²⋊Dic₈₃	1	1	yes	characteristic
162	C₂×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
163	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
164	C₈	—	332	2	no	—
165	C₂²	—	2	1	no	—
166	C₂²	—	2	1	no	—
167	C₂²	—	4	1	no	—
168	C₂²	—	4	1	no	—
169	C₂²	—	4	1	no	—
170	C₂²	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
171	C₂²	#14581812	1	2	yes	—
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	—	2	1	no	—
177	C₄	—	4	1	no	—
178	C₄	—	4	1	no	—
179	C₄	—	4	1	no	—
180	C₄	—	332	4	no	—
181	C₄	#14581719	1	2	yes	—
182	C₂	—	2	2	no	—
183	C₂	—	8	1	no	—
184	C₂	#19127675	1	1	yes	characteristic
185	C₂	#19127697	1	1	yes	characteristic
186	C₂	C₂×#14581719	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19127754

C₂

yes

maximal, characteristic, complemented

#19127547

C₂

yes

maximal, characteristic, complemented

#19127438

C₂

yes

maximal, complemented

#19127435

C₂

yes

maximal

#529×C₈₃

C₂

yes

maximal, characteristic

C₂×C₄×Dic₈₃

—

C₄⋊(C₈₃⋊C₈)

—

C₂²⋊(C₈₃⋊C₈)

C₂²

yes

—

C₄×C₈₃⋊C₈

—

C₈₃⋊#95

—

C₈₃⋊#95

C₂²

yes

—

#126×C₈₃

—

#125×C₈₃

C₄

yes

characteristic, complemented

#124×C₈₃

C₄

yes

characteristic, complemented

#123×C₈₃

—

C₂²⋊Q₈×C₈₃

C₂²

yes

characteristic

#121×C₈₃

C₂²

yes

characteristic

C₄×Q₈×C₈₃

C₄

yes

characteristic, complemented

C₄×D₄×C₈₃

—

C₄×D₄×C₈₃

C₄

yes

characteristic, complemented

#117×C₈₃

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₂²×Dic₈₃

—

C₂×C₈₃⋊C₈

—

C₄×Dic₈₃

—

C₂×Q₈×C₈₃

C₂×C₄

yes

characteristic

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

D₄

yes

characteristic

C₂²×C₄×C₈₃

D₄

yes

characteristic

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²⋊C₄×C₈₃

—

C₂²⋊C₄×C₈₃

—

C₂²⋊C₄×C₈₃

—

C₂²⋊C₄×C₈₃

—

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

C₂×C₄

yes

characteristic

C₄²×C₈₃

C₂×C₄

yes

characteristic

C₄²×C₈₃

D₄

yes

characteristic

C₄²×C₈₃

D₄

yes

characteristic

C₂³×C₈₃

—

C₂³×C₈₃

C₂×D₄

yes

characteristic

Q₈×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₈₃

#135

yes

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#104

yes

—

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

#529

D₈₃

yes

—

—

—

—

100

#126

—

101

#125

Dic₈₃

yes

characteristic, complemented

102

#124

Dic₈₃

yes

characteristic, complemented

103

#123

—

104

C₂²⋊Q₈

C₂×D₈₃

yes

characteristic

105

#121

C₂×D₈₃

yes

characteristic

106

C₄×Q₈

Dic₈₃

yes

characteristic, complemented

107

C₄×D₄

—

108

C₄×D₄

Dic₈₃

yes

characteristic, complemented

109

#117

—

110

C₂×C₄²

—

166

—

111

C₂×C₄²

C₂×D₈₃

yes

characteristic

112

C₄⋊C₈

—

166

—

113

C₂²⋊C₈

—

114

C₄×C₈

—

166

—

115

#95

—

116

#95

—

166

—

117

C₂×Q₈

C₂×Dic₈₃

yes

characteristic

118

C₂×D₄

—

119

C₂×D₄

C₂×Dic₈₃

yes

characteristic

120

C₂²×C₄

—

121

C₂²×C₄

—

166

—

122

C₂²×C₄

C₂²⋊D₈₃

yes

characteristic

123

C₂²×C₄

C₂²⋊D₈₃

yes

characteristic

124

C₂²×C₄

C₂²×D₈₃

yes

characteristic

125

C₂×C₈

—

166

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

C₂×Dic₈₃

yes

characteristic

131

C₄⋊C₄

C₂×Dic₈₃

yes

characteristic

132

C₂²⋊C₄

—

133

C₂²⋊C₄

—

134

C₂²⋊C₄

—

135

C₂²⋊C₄

—

136

C₂²⋊C₄

—

137

C₄²

—

138

C₄²

—

166

—

139

C₄²

C₂×Dic₈₃

yes

characteristic

140

C₄²

C₂×Dic₈₃

yes

characteristic

141

C₄²

C₂²⋊D₈₃

yes

characteristic

142

C₄²

C₂²⋊D₈₃

yes

characteristic

143

C₂³

—

144

C₂³

C₂×C₂²⋊D₈₃

yes

characteristic

145

Q₈

—

146

D₄

—

147

D₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

332

—

156

C₂×C₄

—

332

—

157

C₂×C₄

—

166

—

158

C₂×C₄

C₂²⋊Dic₈₃

yes

characteristic

159

C₂×C₄

C₂²⋊Dic₈₃

yes

characteristic

160

C₂×C₄

C₂²⋊Dic₈₃

yes

characteristic

161

C₂×C₄

C₂²⋊Dic₈₃

yes

characteristic

162

C₂×C₄

C₂²×Dic₈₃

yes

characteristic

163

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

164

C₈

—

332

—

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

—

169

C₂²

—

170

C₂²

C₂×C₂²⋊Dic₈₃

yes

characteristic

171

C₂²

#14581812

yes

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

—

332

—

181

C₄

#14581719

yes

—

182

C₂

—

183

C₂

—

184

C₂

#19127675

yes

characteristic

185

C₂

#19127697

yes

characteristic

186

C₂

C₂×#14581719

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₈₃
commutator	C₂×C₄×C₈₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	#529×C₈₃	C₂
socle	C₂²×C₈₃	C₂×C₂²⋊C₄
prime radical	#121×C₈₃	C₂²
cosocle	C₂²×C₄×C₈₃	C₂³
hypercenter	#529	D₈₃
hypocenter	C₈₃	#1500
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #529

subgroup

G/H

center

C₂×C₄

C₂²⋊Dic₈₃

commutator

C₂×C₄×C₈₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

#529×C₈₃

C₂

socle

C₂²×C₈₃

C₂×C₂²⋊C₄

prime radical

#121×C₈₃

C₂²

cosocle

C₂²×C₄×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #529

N	H ≅ G/N	type
1. #19127754	C₂	semidirect
2. #19127547	C₂	semidirect
3. #19127438	C₂	semidirect
13. #125×C₈₃	C₄	semidirect
14. #124×C₈₃	C₄	semidirect
18. C₄×Q₈×C₈₃	C₄	semidirect
20. C₄×D₄×C₈₃	C₄	semidirect
94. C₈₃	#1500	semidirect
101. #125	Dic₈₃	semidirect
102. #124	Dic₈₃	semidirect
106. C₄×Q₈	Dic₈₃	semidirect
108. C₄×D₄	Dic₈₃	semidirect

H ≅ G/N

type

1. #19127754

C₂

semidirect

2. #19127547

C₂

semidirect

3. #19127438

C₂

semidirect

13. #125×C₈₃

C₄

semidirect

14. #124×C₈₃

C₄

semidirect

18. C₄×Q₈×C₈₃

C₄

semidirect

20. C₄×D₄×C₈₃

C₄

semidirect

94. C₈₃

#1500

semidirect

101. #125

Dic₈₃

semidirect

102. #124

Dic₈₃

semidirect

106. C₄×Q₈

Dic₈₃

semidirect

108. C₄×D₄

Dic₈₃

semidirect

Subgroups — #24905949

Conjugacy classes of subgroups

Complements