Group #21423197 — subgroups

Subgroups — #21423197

Order 7552.

250 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 59 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	Q₈×D₂₃₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837748	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837594	C₂	1	1	yes	maximal, characteristic
4	#16837149	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₈⋊(C₅₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837129	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#460	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₅₉	—	2	1	no	—
9	C₂×#11745669	—	2	1	no	—
10	#14273727	—	2	1	no	—
11	#14273727	C₂²	1	1	yes	characteristic
12	#14273707	—	2	1	no	—
13	#14273707	—	2	2	no	—
14	C₄×D₂₃₆	—	2	1	no	—
15	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
16	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
17	#14273648	—	2	2	no	—
18	#14273648	—	2	1	no	—
19	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×SD₁₆×C₅₉	—	2	1	no	—
21	#127×C₅₉	C₂²	1	1	yes	characteristic
22	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
24	#102×C₅₉	—	2	1	no	—
25	C₂×Q₈×C₅₉	D₄	1	1	yes	characteristic
26	C₂×D₄×C₅₉	—	2	1	no	—
27	C₂×D₄×C₅₉	—	2	1	no	—
28	C₂×D₄×C₅₉	—	2	1	no	—
29	Q₈×D₅₉	—	4	2	no	—
30	Q₈×D₅₉	—	4	2	no	—
31	C₂×D₂₃₆	D₄	1	1	yes	characteristic
32	C₂×C₄×D₅₉	—	2	2	no	—
33	C₂×C₄×D₅₉	—	2	1	no	—
34	C₂×Dic₁₁₈	—	2	2	no	—
35	C₂×Dic₁₁₈	—	2	1	no	—
36	SD₁₆×C₅₉	—	4	2	no	—
37	C₂×C₈×C₅₉	—	2	1	no	—
38	C₄⋊C₄×C₅₉	—	2	1	no	—
39	C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
40	C₄²×C₅₉	—	2	1	no	—
41	C₄²×C₅₉	C₂³	1	1	yes	characteristic
42	#11745669	—	4	2	no	—
43	(C₂×D₅₉)⋊C₄	—	2	2	no	—
44	(C₂×D₅₉)⋊C₄	—	2	1	no	—
45	C₄⋊Dic₅₉	—	2	2	no	—
46	C₄⋊Dic₅₉	—	2	1	no	—
47	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
48	C₄⋊Dic₅₉	D₄	1	2	yes	—
49	C₂×C₅₉⋊C₈	—	2	1	no	—
50	C₂×C₅₉⋊C₈	—	2	1	no	—
51	C₂³×C₅₉	—	4	1	no	—
52	C₂²×D₅₉	—	2	1	no	—
53	Q₈×C₅₉	—	2	1	no	—
54	Q₈×C₅₉	D₈	1	2	yes	complemented
55	D₄×C₅₉	—	4	2	no	—
56	D₄×C₅₉	—	4	1	no	—
57	D₄×C₅₉	—	4	1	no	—
58	D₄×C₅₉	—	4	1	no	—
59	D₄×C₅₉	—	4	1	no	—
60	C₂×C₄×C₅₉	—	2	1	no	—
61	C₂×C₄×C₅₉	—	2	1	no	—
62	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
65	C₂×Dic₅₉	—	2	1	no	—
66	C₂×Dic₅₉	—	2	2	no	—
67	D₂₃₆	—	2	1	no	—
68	D₂₃₆	SD₁₆	1	2	yes	complemented
69	C₄×D₅₉	—	4	4	no	—
70	C₄×D₅₉	—	4	1	no	—
71	C₄×D₅₉	—	4	1	no	—
72	Dic₁₁₈	—	4	4	no	—
73	Dic₁₁₈	—	4	1	no	—
74	Dic₁₁₈	—	4	1	no	—
75	C₈×C₅₉	—	4	1	no	—
76	C₅₉⋊C₈	—	4	1	no	—
77	C₅₉⋊C₈	—	4	1	no	—
78	C₂²×C₅₉	—	8	1	no	—
79	C₂²×C₅₉	—	8	1	no	—
80	C₂²×C₅₉	—	8	1	no	—
81	C₂²×C₅₉	#120	1	1	yes	characteristic
82	C₂×D₅₉	—	2	2	no	—
83	C₂×D₅₉	—	4	1	no	—
84	C₂×D₅₉	—	4	1	no	—
85	C₄×C₅₉	—	2	2	no	—
86	C₄×C₅₉	—	2	1	no	—
87	C₄×C₅₉	—	4	1	no	—
88	C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
89	C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
90	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
91	C₄×C₅₉	#137	1	1	yes	characteristic
92	Dic₅₉	—	4	1	no	—
93	Dic₅₉	—	4	2	no	—
94	#1627	—	59	1	no	maximal, Sylow
95	C₂×C₅₉	—	16	1	no	—
96	C₂×C₅₉	#447	1	1	yes	characteristic
97	C₂×C₅₉	#448	1	1	yes	characteristic
98	C₂×C₅₉	#454	1	1	yes	characteristic
99	D₅₉	—	4	2	no	—
100	D₄×Q₈	—	59	1	no	—
101	#486	—	59	1	no	—
102	#460	D₅₉	1	1	yes	characteristic, complemented
103	#459	—	59	1	no	—
104	#332	—	59	1	no	—
105	Q₈⋊C₈	—	59	1	no	—
106	D₄⋊C₈	—	59	1	no	—
107	C₅₉	#1627	1	1	yes	characteristic, complemented, Sylow
108	C₂²×Q₈	—	118	1	no	—
109	C₂×SD₁₆	—	2	1	no	—
110	C₂×D₈	—	118	1	no	—
111	C₄⋊Q₈	—	59	1	no	—
112	C₄⋊Q₈	—	118	1	no	—
113	#127	C₂×D₅₉	1	1	yes	characteristic
114	C₂²⋊Q₈	—	118	2	no	—
115	C₂²⋊Q₈	—	118	1	no	—
116	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
117	C₄×D₄	—	59	1	no	—
118	C₄×D₄	—	118	1	no	—
119	C₄⋊C₈	—	59	1	no	—
120	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
121	#102	—	2	1	no	—
122	#102	—	118	1	no	—
123	#102	—	118	2	no	—
124	C₄×C₈	—	59	1	no	—
125	C₂×Q₈	—	118	2	no	—
126	C₂×Q₈	—	118	1	no	—
127	C₂×Q₈	—	236	2	no	—
128	C₂×Q₈	—	236	2	no	—
129	C₂×Q₈	C₂²⋊D₅₉	1	1	yes	characteristic
130	C₂×D₄	—	2	1	no	—
131	C₂×D₄	—	2	1	no	—
132	C₂×D₄	—	2	1	no	—
133	C₂×D₄	—	59	1	no	—
134	C₂²×C₄	—	118	2	no	—
135	C₂²×C₄	—	118	1	no	—
136	SD₁₆	—	4	2	no	—
137	D₈	—	236	2	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	118	1	no	—
140	C₂×C₈	—	118	1	no	—
141	C₄⋊C₄	—	2	1	no	—
142	C₄⋊C₄	—	59	1	no	—
143	C₄⋊C₄	—	59	2	no	—
144	C₄⋊C₄	—	118	2	no	—
145	C₄⋊C₄	—	118	1	no	—
146	C₄⋊C₄	C₂²⋊D₅₉	1	1	yes	characteristic
147	C₂²⋊C₄	—	118	2	no	—
148	C₂²⋊C₄	—	118	1	no	—
149	C₄²	—	2	1	no	—
150	C₄²	C₂²×D₅₉	1	1	yes	characteristic
151	C₂³	—	4	1	no	—
152	C₂³	—	118	1	no	—
153	Q₈	—	2	1	no	—
154	Q₈	—	236	4	no	—
155	Q₈	—	236	1	no	—
156	Q₈	—	236	1	no	—
157	Q₈	#11745669	1	2	yes	complemented
158	D₄	—	4	2	no	—
159	D₄	—	4	1	no	—
160	D₄	—	4	1	no	—
161	D₄	—	4	1	no	—
162	D₄	—	4	1	no	—
163	D₄	—	59	2	no	—
164	D₄	—	118	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	118	2	no	—
168	C₂×C₄	—	118	1	no	—
169	C₂×C₄	—	236	4	no	—
170	C₂×C₄	—	236	1	no	—
171	C₂×C₄	—	236	1	no	—
172	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
173	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
174	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
175	C₈	—	4	1	no	—
176	C₈	—	236	1	no	—
177	C₈	—	236	1	no	—
178	C₂²	—	8	1	no	—
179	C₂²	—	8	1	no	—
180	C₂²	—	8	1	no	—
181	C₂²	—	118	2	no	—
182	C₂²	—	236	1	no	—
183	C₂²	—	236	1	no	—
184	C₂²	#14273695	1	1	yes	characteristic
185	C₄	—	2	2	no	—
186	C₄	—	2	1	no	—
187	C₄	—	4	1	no	—
188	C₄	—	236	2	no	—
189	C₄	—	236	1	no	—
190	C₄	C₂×#11745669	1	1	yes	characteristic
191	C₄	SD₁₆×D₅₉	1	1	yes	characteristic
192	C₄	#14273752	1	1	yes	characteristic
193	C₄	#14273756	1	1	yes	characteristic
194	C₂	—	16	1	no	—
195	C₂	—	236	2	no	—
196	C₂	#16837535	1	1	yes	characteristic
197	C₂	#16837539	1	1	yes	characteristic
198	C₂	#16837573	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×D₂₃₆

C₂

yes

maximal, characteristic, complemented

#16837748

C₂

yes

maximal, characteristic, complemented

#16837594

C₂

yes

maximal, characteristic

#16837149

C₂

yes

maximal, characteristic, complemented

Q₈⋊(C₅₉⋊C₈)

C₂

yes

maximal, characteristic, complemented

#16837129

C₂

yes

maximal, characteristic, complemented

C₅₉×#460

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₅₉

—

C₂×#11745669

—

#14273727

—

#14273727

C₂²

yes

characteristic

#14273707

—

#14273707

—

C₄×D₂₃₆

—

C₄×D₂₃₆

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

#14273648

—

#14273648

—

C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₂×SD₁₆×C₅₉

—

#127×C₅₉

C₂²

yes

characteristic

C₄×Q₈×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

#102×C₅₉

—

C₂×Q₈×C₅₉

D₄

yes

characteristic

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

Q₈×D₅₉

—

Q₈×D₅₉

—

C₂×D₂₃₆

D₄

yes

characteristic

C₂×C₄×D₅₉

—

C₂×C₄×D₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

SD₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

—

C₄²×C₅₉

C₂³

yes

characteristic

#11745669

—

(C₂×D₅₉)⋊C₄

—

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

D₄

yes

characteristic

C₄⋊Dic₅₉

D₄

yes

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

—

C₂²×D₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

D₈

yes

complemented

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×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×D₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

SD₁₆

yes

complemented

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#120

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×D₈

yes

characteristic

C₄×C₅₉

C₂×SD₁₆

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

#1627

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

100

D₄×Q₈

—

101

#486

—

102

#460

D₅₉

yes

characteristic, complemented

103

#459

—

104

#332

—

105

Q₈⋊C₈

—

106

D₄⋊C₈

—

107

C₅₉

#1627

yes

characteristic, complemented, Sylow

108

C₂²×Q₈

—

118

—

109

C₂×SD₁₆

—

110

C₂×D₈

—

118

—

111

C₄⋊Q₈

—

112

C₄⋊Q₈

—

118

—

113

#127

C₂×D₅₉

yes

characteristic

114

C₂²⋊Q₈

—

118

—

115

C₂²⋊Q₈

—

118

—

116

C₄×Q₈

C₂×D₅₉

yes

characteristic

117

C₄×D₄

—

118

C₄×D₄

—

118

—

119

C₄⋊C₈

—

120

C₄⋊C₈

C₂×D₅₉

yes

characteristic

121

#102

—

122

#102

—

118

—

123

#102

—

118

—

124

C₄×C₈

—

125

C₂×Q₈

—

118

—

126

C₂×Q₈

—

118

—

127

C₂×Q₈

—

236

—

128

C₂×Q₈

—

236

—

129

C₂×Q₈

C₂²⋊D₅₉

yes

characteristic

130

C₂×D₄

—

131

C₂×D₄

—

132

C₂×D₄

—

133

C₂×D₄

—

134

C₂²×C₄

—

118

—

135

C₂²×C₄

—

118

—

136

SD₁₆

—

137

D₈

—

236

—

138

C₂×C₈

—

139

C₂×C₈

—

118

—

140

C₂×C₈

—

118

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

118

—

145

C₄⋊C₄

—

118

—

146

C₄⋊C₄

C₂²⋊D₅₉

yes

characteristic

147

C₂²⋊C₄

—

118

—

148

C₂²⋊C₄

—

118

—

149

C₄²

—

150

C₄²

C₂²×D₅₉

yes

characteristic

151

C₂³

—

152

C₂³

—

118

—

153

Q₈

—

154

Q₈

—

236

—

155

Q₈

—

236

—

156

Q₈

—

236

—

157

Q₈

#11745669

yes

complemented

158

D₄

—

159

D₄

—

160

D₄

—

161

D₄

—

162

D₄

—

163

D₄

—

164

D₄

—

118

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

118

—

168

C₂×C₄

—

118

—

169

C₂×C₄

—

236

—

170

C₂×C₄

—

236

—

171

C₂×C₄

—

236

—

172

C₂×C₄

D₄×D₅₉

yes

characteristic

173

C₂×C₄

D₄×D₅₉

yes

characteristic

174

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

175

C₈

—

176

C₈

—

236

—

177

C₈

—

236

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

181

C₂²

—

118

—

182

C₂²

—

236

—

183

C₂²

—

236

—

184

C₂²

#14273695

yes

characteristic

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

—

236

—

189

C₄

—

236

—

190

C₄

C₂×#11745669

yes

characteristic

191

C₄

SD₁₆×D₅₉

yes

characteristic

192

C₄

#14273752

yes

characteristic

193

C₄

#14273756

yes

characteristic

194

C₂

—

195

C₂

—

236

—

196

C₂

#16837535

yes

characteristic

197

C₂

#16837539

yes

characteristic

198

C₂

#16837573

yes

characteristic

	subgroup	G/H
center	C₂²	#14273695
commutator	C₄²×C₅₉	C₂³
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#460	C₂
socle	C₂²×C₅₉	#120
hypercenter	#460	D₅₉
hypocenter	C₅₉	#1627
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #460

subgroup

G/H

center

C₂²

#14273695

commutator

C₄²×C₅₉

C₂³

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#460

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #460

N	H ≅ G/N	type
1. Q₈×D₂₃₆	C₂	semidirect
2. #16837748	C₂	semidirect
4. #16837149	C₂	semidirect
5. Q₈⋊(C₅₉⋊C₈)	C₂	semidirect
6. #16837129	C₂	semidirect
7. C₅₉×#460	C₂	semidirect
54. Q₈×C₅₉	D₈	semidirect
68. D₂₃₆	SD₁₆	semidirect
102. #460	D₅₉	semidirect
107. C₅₉	#1627	semidirect
157. Q₈	#11745669	semidirect

H ≅ G/N

type

1. Q₈×D₂₃₆

C₂

semidirect

2. #16837748

C₂

semidirect

4. #16837149

C₂

semidirect

5. Q₈⋊(C₅₉⋊C₈)

C₂

semidirect

6. #16837129

C₂

semidirect

7. C₅₉×#460

C₂

semidirect

54. Q₈×C₅₉

D₈

semidirect

68. D₂₃₆

SD₁₆

semidirect

102. #460

D₅₉

semidirect

107. C₅₉

#1627

semidirect

157. Q₈

#11745669

semidirect

Subgroups — #21423197

Conjugacy classes of subgroups

Complements