Group #21425616 — subgroups

Subgroups — #21425616

Order 7552.

274 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 87 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×#14273661	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14273659	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14273649	C₂	1	1	yes	maximal, characteristic
4	C₂×#14273649	C₂	1	1	yes	maximal, characteristic
5	#16837304	C₂	1	1	yes	maximal, characteristic
6	#16837268	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#336	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₂₃₆	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
12	C₂×(C₂×D₅₉)⋊C₄	—	2	2	no	—
13	#14273661	—	2	2	no	—
14	#14273659	—	2	2	no	—
15	#14273649	—	2	2	no	—
16	#14273649	—	2	2	no	—
17	C₅₉⋊#95	—	2	2	no	—
18	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
21	C₂³×D₅₉	—	2	1	no	—
22	C₂²×C₄×C₅₉	D₄	1	2	yes	—
23	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
24	C₂²×Dic₅₉	—	2	1	no	—
25	C₂²×Dic₅₉	—	2	1	no	—
26	C₂²×Dic₅₉	—	2	1	no	—
27	C₂×D₂₃₆	—	2	1	no	—
28	C₂×D₂₃₆	—	2	1	no	—
29	C₂×D₂₃₆	—	2	1	no	—
30	C₂×D₂₃₆	—	2	1	no	—
31	C₂×D₂₃₆	—	2	1	no	—
32	C₂×D₂₃₆	Q₈	1	2	yes	—
33	C₂×C₈×C₅₉	—	2	1	no	—
34	C₂×C₈×C₅₉	—	2	1	no	—
35	C₂×C₈×C₅₉	—	2	1	no	—
36	C₂×C₈×C₅₉	—	2	1	no	—
37	C₂×C₈×C₅₉	D₄	1	2	yes	—
38	C₂×C₈×C₅₉	D₄	1	2	yes	—
39	C₄²×C₅₉	—	2	1	no	—
40	C₄²×C₅₉	—	2	1	no	—
41	(C₂×D₅₉)⋊C₄	—	4	4	no	—
42	C₄⋊Dic₅₉	—	2	1	no	—
43	C₄⋊Dic₅₉	—	2	1	no	—
44	C₄⋊Dic₅₉	—	2	1	no	—
45	C₄⋊Dic₅₉	—	2	1	no	—
46	C₄⋊Dic₅₉	—	2	1	no	—
47	C₄⋊Dic₅₉	—	2	1	no	—
48	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×D₅₉	—	2	2	no	—
50	C₂²×D₅₉	—	4	1	no	—
51	C₂²×D₅₉	—	4	1	no	—
52	C₂²×D₅₉	—	4	2	no	—
53	C₂²×D₅₉	—	4	2	no	—
54	C₂×C₄×C₅₉	—	2	2	no	—
55	C₂×C₄×C₅₉	—	2	2	no	—
56	C₂×C₄×C₅₉	—	2	2	no	—
57	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
61	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
62	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
63	C₂×Dic₅₉	—	4	1	no	—
64	C₂×Dic₅₉	—	4	1	no	—
65	C₂×Dic₅₉	—	4	1	no	—
66	C₂×Dic₅₉	—	4	2	no	—
67	C₂×Dic₅₉	—	4	2	no	—
68	C₂×Dic₅₉	—	4	2	no	—
69	D₂₃₆	—	2	2	no	—
70	D₂₃₆	—	2	2	no	—
71	D₂₃₆	—	4	1	no	—
72	D₂₃₆	—	4	1	no	—
73	C₈×C₅₉	—	2	2	no	—
74	C₈×C₅₉	—	2	2	no	—
75	C₂²×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂²×C₅₉	#123	1	1	yes	characteristic
77	C₂²×C₅₉	#127	1	1	yes	characteristic
78	C₂²×C₅₉	#135	1	2	yes	—
79	C₂²×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
80	C₂×D₅₉	—	4	2	no	—
81	C₂×D₅₉	—	4	4	no	—
82	C₂×D₅₉	—	8	2	no	—
83	C₂×D₅₉	—	8	1	no	—
84	C₂×D₅₉	—	8	1	no	—
85	C₄×C₅₉	—	4	2	no	—
86	C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
87	C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
88	C₄×C₅₉	#123	1	1	yes	characteristic
89	C₄×C₅₉	#123	1	1	yes	characteristic
90	Dic₅₉	—	8	1	no	—
91	Dic₅₉	—	8	1	no	—
92	Dic₅₉	—	8	1	no	—
93	#2092	—	59	1	no	maximal, Sylow
94	C₂×C₅₉	#399	1	1	yes	characteristic
95	C₂×C₅₉	#478	1	2	yes	—
96	C₂×C₅₉	#482	1	2	yes	—
97	C₂×C₅₉	#495	1	1	yes	characteristic
98	C₂×C₅₉	#496	1	1	yes	characteristic
99	D₅₉	—	8	2	no	—
100	C₂×#107	—	59	1	no	—
101	C₂×#106	—	59	1	no	—
102	C₂×#102	—	59	1	no	—
103	C₂×#102	—	59	1	no	—
104	#390	—	59	1	no	—
105	#382	—	59	1	no	—
106	#336	D₅₉	1	1	yes	characteristic, complemented
107	C₅₉	#2092	1	1	yes	characteristic, complemented, Sylow
108	C₂²×D₄	—	59	1	no	—
109	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
110	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
111	C₂×C₄⋊C₄	—	59	1	no	—
112	C₂×C₄⋊C₄	—	59	1	no	—
113	C₂×C₄⋊C₄	—	59	1	no	—
114	C₂×C₂²⋊C₄	—	118	2	no	—
115	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
116	#107	—	118	2	no	—
117	#106	—	118	2	no	—
118	#102	—	118	2	no	—
119	#102	—	118	2	no	—
120	#95	—	118	2	no	—
121	C₂⁴	—	118	1	no	—
122	C₂×D₄	—	59	2	no	—
123	C₂×D₄	—	118	1	no	—
124	C₂×D₄	—	118	1	no	—
125	C₂×D₄	—	118	1	no	—
126	C₂×D₄	—	118	1	no	—
127	C₂×D₄	—	118	1	no	—
128	C₂²×C₄	—	118	1	no	—
129	C₂²×C₄	—	118	1	no	—
130	C₂²×C₄	—	118	1	no	—
131	C₂²×C₄	D₂₃₆	1	2	yes	—
132	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	D₂₃₆	1	2	yes	—
138	C₂×C₈	D₂₃₆	1	2	yes	—
139	C₄⋊C₄	—	118	1	no	—
140	C₄⋊C₄	—	118	1	no	—
141	C₄⋊C₄	—	118	1	no	—
142	C₄⋊C₄	—	118	1	no	—
143	C₄⋊C₄	—	118	1	no	—
144	C₄⋊C₄	—	118	1	no	—
145	C₂²⋊C₄	—	236	4	no	—
146	C₄²	—	2	1	no	—
147	C₄²	—	2	1	no	—
148	C₂³	—	118	2	no	—
149	C₂³	—	236	1	no	—
150	C₂³	—	236	1	no	—
151	C₂³	—	236	2	no	—
152	C₂³	—	236	2	no	—
153	C₂³	C₂×D₂₃₆	1	1	yes	characteristic
154	D₄	—	118	2	no	—
155	D₄	—	118	2	no	—
156	D₄	—	236	1	no	—
157	D₄	—	236	1	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	236	1	no	—
162	C₂×C₄	—	236	1	no	—
163	C₂×C₄	—	236	1	no	—
164	C₂×C₄	—	236	2	no	—
165	C₂×C₄	—	236	2	no	—
166	C₂×C₄	—	236	2	no	—
167	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
168	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
169	C₂×C₄	#11745687	1	1	yes	characteristic
170	C₂×C₄	#11745687	1	1	yes	characteristic
171	C₂×C₄	#11745687	1	1	yes	characteristic
172	C₂×C₄	Q₈×D₅₉	1	1	yes	characteristic
173	C₈	—	2	2	no	—
174	C₈	—	2	2	no	—
175	C₂²	—	472	2	no	—
176	C₂²	—	472	1	no	—
177	C₂²	—	472	1	no	—
178	C₂²	—	236	4	no	—
179	C₂²	—	236	2	no	—
180	C₂²	#14273707	1	1	yes	characteristic
181	C₂²	#14273711	1	1	yes	characteristic
182	C₂²	#14273724	1	1	yes	characteristic
183	C₂²	#14273749	1	2	yes	—
184	C₂²	#14273755	1	2	yes	—
185	C₄	—	4	2	no	—
186	C₄	—	472	1	no	—
187	C₄	—	472	1	no	—
188	C₄	—	472	1	no	—
189	C₄	#14273707	1	1	yes	characteristic
190	C₄	#14273707	1	1	yes	characteristic
191	C₄	#14273711	1	1	yes	characteristic
192	C₄	#14273711	1	1	yes	characteristic
193	C₂	—	472	2	no	—
194	C₂	#16837339	1	1	yes	characteristic
195	C₂	#16837703	1	2	yes	—
196	C₂	#16837727	1	2	yes	—
197	C₂	#16837786	1	1	yes	characteristic
198	C₂	#16837791	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273661

C₂

yes

maximal, characteristic, complemented

C₂×#14273659

C₂

yes

maximal, characteristic, complemented

C₂×#14273649

C₂

yes

maximal, characteristic

C₂×#14273649

C₂

yes

maximal, characteristic

#16837304

C₂

yes

maximal, characteristic

#16837268

C₂

yes

maximal, characteristic, complemented

C₅₉×#336

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₃₆

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₉

C₂²

yes

characteristic

C₂×(C₂×D₅₉)⋊C₄

—

—

—

—

—

C₅₉⋊#95

—

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₂³×D₅₉

—

C₂²×C₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

Q₈

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

D₄

yes

—

C₂×C₈×C₅₉

D₄

yes

—

C₄²×C₅₉

—

C₄²×C₅₉

—

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

C₈×C₅₉

—

C₈×C₅₉

—

C₂²×C₅₉

C₂²⋊Q₈

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₂²×C₅₉

AGL(1,ℤ/8)

yes

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂²⋊Q₈

yes

characteristic

C₄×C₅₉

C₂²⋊Q₈

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×C₅₉

#482

yes

—

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

100

C₂×#107

—

101

C₂×#106

—

102

C₂×#102

—

103

C₂×#102

—

104

#390

—

105

#382

—

106

#336

D₅₉

yes

characteristic, complemented

107

C₅₉

#2092

yes

characteristic, complemented, Sylow

108

C₂²×D₄

—

109

C₂²×C₈

C₂×D₅₉

yes

characteristic

110

C₂²×C₈

C₂×D₅₉

yes

characteristic

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

—

114

C₂×C₂²⋊C₄

—

118

—

115

C₂×C₄²

C₂×D₅₉

yes

characteristic

116

#107

—

118

—

117

#106

—

118

—

118

#102

—

118

—

119

#102

—

118

—

120

#95

—

118

—

121

C₂⁴

—

118

—

122

C₂×D₄

—

123

C₂×D₄

—

118

—

124

C₂×D₄

—

118

—

125

C₂×D₄

—

118

—

126

C₂×D₄

—

118

—

127

C₂×D₄

—

118

—

128

C₂²×C₄

—

118

—

129

C₂²×C₄

—

118

—

130

C₂²×C₄

—

118

—

131

C₂²×C₄

D₂₃₆

yes

—

132

C₂²×C₄

C₂²×D₅₉

yes

characteristic

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

D₂₃₆

yes

—

138

C₂×C₈

D₂₃₆

yes

—

139

C₄⋊C₄

—

118

—

140

C₄⋊C₄

—

118

—

141

C₄⋊C₄

—

118

—

142

C₄⋊C₄

—

118

—

143

C₄⋊C₄

—

118

—

144

C₄⋊C₄

—

118

—

145

C₂²⋊C₄

—

236

—

146

C₄²

—

147

C₄²

—

148

C₂³

—

118

—

149

C₂³

—

236

—

150

C₂³

—

236

—

151

C₂³

—

236

—

152

C₂³

—

236

—

153

C₂³

C₂×D₂₃₆

yes

characteristic

154

D₄

—

118

—

155

D₄

—

118

—

156

D₄

—

236

—

157

D₄

—

236

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

236

—

162

C₂×C₄

—

236

—

163

C₂×C₄

—

236

—

164

C₂×C₄

—

236

—

165

C₂×C₄

—

236

—

166

C₂×C₄

—

236

—

167

C₂×C₄

C₂×D₂₃₆

yes

characteristic

168

C₂×C₄

C₂×D₂₃₆

yes

characteristic

169

C₂×C₄

#11745687

yes

characteristic

170

C₂×C₄

#11745687

yes

characteristic

171

C₂×C₄

#11745687

yes

characteristic

172

C₂×C₄

Q₈×D₅₉

yes

characteristic

173

C₈

—

174

C₈

—

175

C₂²

—

472

—

176

C₂²

—

472

—

177

C₂²

—

472

—

178

C₂²

—

236

—

179

C₂²

—

236

—

180

C₂²

#14273707

yes

characteristic

181

C₂²

#14273711

yes

characteristic

182

C₂²

#14273724

yes

characteristic

183

C₂²

#14273749

yes

—

184

C₂²

#14273755

yes

—

185

C₄

—

186

C₄

—

472

—

187

C₄

—

472

—

188

C₄

—

472

—

189

C₄

#14273707

yes

characteristic

190

C₄

#14273707

yes

characteristic

191

C₄

#14273711

yes

characteristic

192

C₄

#14273711

yes

characteristic

193

C₂

—

472

—

194

C₂

#16837339

yes

characteristic

195

C₂

#16837703

yes

—

196

C₂

#16837727

yes

—

197

C₂

#16837786

yes

characteristic

198

C₂

#16837791

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₂₃₆
commutator	C₂²×C₄×C₅₉	C₂³
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#336	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	#336	D₅₉
hypocenter	C₅₉	#2092
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₂₃₆

commutator

C₂²×C₄×C₅₉

C₂³

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#336

C₂

socle

C₂³×C₅₉

C₂×D₄

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
1. C₂×#14273661	C₂	semidirect
2. C₂×#14273659	C₂	semidirect
6. #16837268	C₂	semidirect
7. C₅₉×#336	C₂	semidirect
106. #336	D₅₉	semidirect
107. C₅₉	#2092	semidirect

H ≅ G/N

type

1. C₂×#14273661

C₂

semidirect

2. C₂×#14273659

C₂

semidirect

6. #16837268

C₂

semidirect

7. C₅₉×#336

C₂

semidirect

106. #336

D₅₉

semidirect

107. C₅₉

#2092

semidirect

Subgroups — #21425616

Conjugacy classes of subgroups

Complements