Group #21423223 — subgroups

Subgroups — #21423223

Order 7552.

256 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 61 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	#16837774	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837697	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837692	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837146	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₈⋊(C₅₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄⋊(C₅₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#550	C₂	1	1	yes	maximal, characteristic
8	C₂×#11745670	—	2	2	no	—
9	#14273727	C₂²	1	1	yes	characteristic
10	#14273660	—	2	1	no	—
11	#14273658	—	2	1	no	—
12	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
13	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
14	#14273651	—	2	1	no	—
15	#14273648	—	2	1	no	—
16	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×#41×C₅₉	—	2	1	no	—
18	#127×C₅₉	—	2	1	no	—
19	#127×C₅₉	C₂²	1	1	yes	characteristic
20	#121×C₅₉	—	2	2	no	—
21	#121×C₅₉	—	2	1	no	—
22	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₅₉	—	2	1	no	—
24	C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
25	#41×C₅₉	—	4	2	no	—
26	#41×C₅₉	—	4	2	no	—
27	C₂×Q₈×C₅₉	D₄	1	1	yes	characteristic
28	C₂×D₄×C₅₉	—	2	2	no	—
29	C₂×D₄×C₅₉	—	2	2	no	—
30	C₂×D₄×C₅₉	—	2	1	no	—
31	C₂×D₄×C₅₉	—	2	1	no	—
32	C₂×D₄×C₅₉	D₄	1	1	yes	characteristic
33	C₂×D₄×C₅₉	D₄	1	2	yes	—
34	C₂²×C₄×C₅₉	—	2	2	no	—
35	C₂²×C₄×C₅₉	—	2	1	no	—
36	C₂×Dic₁₁₈	—	2	1	no	—
37	C₄⋊C₄×C₅₉	—	2	1	no	—
38	C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
39	C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
40	C₂²⋊C₄×C₅₉	—	2	2	no	—
41	C₂²⋊C₄×C₅₉	—	2	1	no	—
42	C₄²×C₅₉	—	2	1	no	—
43	C₄²×C₅₉	C₂³	1	1	yes	characteristic
44	#11745670	—	4	4	no	—
45	C₄⋊Dic₅₉	—	2	1	no	—
46	C₄⋊Dic₅₉	—	2	1	no	—
47	C₂×C₅₉⋊C₈	—	2	1	no	—
48	C₂×C₅₉⋊C₈	—	2	1	no	—
49	C₂×C₅₉⋊C₈	—	2	1	no	—
50	C₂³×C₅₉	—	2	1	no	—
51	C₂³×C₅₉	—	2	2	no	—
52	Q₈×C₅₉	—	2	1	no	—
53	Q₈×C₅₉	SD₁₆	1	2	yes	complemented
54	D₄×C₅₉	—	2	1	no	—
55	D₄×C₅₉	—	2	1	no	—
56	D₄×C₅₉	—	4	2	no	—
57	D₄×C₅₉	—	4	4	no	—
58	D₄×C₅₉	—	4	1	no	—
59	D₄×C₅₉	—	4	1	no	—
60	D₄×C₅₉	—	4	1	no	—
61	D₄×C₅₉	—	4	1	no	—
62	D₄×C₅₉	—	8	2	no	—
63	C₂×C₄×C₅₉	—	2	1	no	—
64	C₂×C₄×C₅₉	—	2	1	no	—
65	C₂×C₄×C₅₉	—	2	1	no	—
66	C₂×C₄×C₅₉	—	4	4	no	—
67	C₂×C₄×C₅₉	—	4	1	no	—
68	C₂×C₄×C₅₉	—	4	1	no	—
69	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
72	C₂×Dic₅₉	—	4	1	no	—
73	Dic₁₁₈	—	4	2	no	—
74	C₅₉⋊C₈	—	2	2	no	—
75	C₅₉⋊C₈	—	4	1	no	—
76	C₅₉⋊C₈	—	4	1	no	—
77	C₂²×C₅₉	—	4	2	no	—
78	C₂²×C₅₉	—	4	1	no	—
79	C₂²×C₅₉	—	4	1	no	—
80	C₂²×C₅₉	—	4	2	no	—
81	C₂²×C₅₉	—	4	2	no	—
82	C₂²×C₅₉	—	4	1	no	—
83	C₂²×C₅₉	#120	1	1	yes	characteristic
84	C₄×C₅₉	—	2	2	no	—
85	C₄×C₅₉	—	2	1	no	—
86	C₄×C₅₉	—	4	1	no	—
87	C₄×C₅₉	—	4	1	no	—
88	C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
89	C₄×C₅₉	#135	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	—	8	1	no	—
93	#1631	—	59	1	no	maximal, Sylow
94	C₂×C₅₉	—	8	1	no	—
95	C₂×C₅₉	—	8	2	no	—
96	C₂×C₅₉	#450	1	1	yes	characteristic
97	C₂×C₅₉	#452	1	1	yes	characteristic
98	C₂×C₅₉	#453	1	1	yes	characteristic
99	#550	D₅₉	1	1	yes	characteristic
100	#492	—	59	1	no	—
101	#477	—	59	1	no	—
102	#476	—	59	1	no	—
103	#331	—	59	1	no	—
104	Q₈⋊C₈	—	59	1	no	—
105	D₄⋊C₈	—	59	1	no	—
106	C₅₉	#1631	1	1	yes	characteristic, complemented, Sylow
107	C₂×#41	—	2	1	no	—
108	C₂×SD₁₆	—	118	2	no	—
109	C₄⋊Q₈	—	59	1	no	—
110	#127	—	2	1	no	—
111	#127	C₂×D₅₉	1	1	yes	characteristic
112	#121	—	2	2	no	—
113	#121	—	2	1	no	—
114	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
115	C₄×D₄	—	2	1	no	—
116	C₄×D₄	C₂×D₅₉	1	1	yes	characteristic
117	#107	—	118	1	no	—
118	#106	—	118	1	no	—
119	C₄⋊C₈	—	59	1	no	—
120	C₄⋊C₈	—	59	1	no	—
121	#103	—	118	1	no	—
122	#102	—	118	1	no	—
123	C₄×C₈	—	59	1	no	—
124	#41	—	4	2	no	—
125	#41	—	4	2	no	—
126	C₂×Q₈	—	118	1	no	—
127	C₂×Q₈	C₂²⋊D₅₉	1	1	yes	characteristic
128	C₂×D₄	—	2	2	no	—
129	C₂×D₄	—	2	2	no	—
130	C₂×D₄	—	2	1	no	—
131	C₂×D₄	—	2	1	no	—
132	C₂×D₄	C₂²⋊D₅₉	1	1	yes	characteristic
133	C₂×D₄	C₂²⋊D₅₉	1	2	yes	—
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	—	2	1	no	—
136	SD₁₆	—	236	4	no	—
137	C₂×C₈	—	118	1	no	—
138	C₂×C₈	—	118	1	no	—
139	C₂×C₈	—	118	1	no	—
140	C₄⋊C₄	—	2	1	no	—
141	C₄⋊C₄	—	118	1	no	—
142	C₄⋊C₄	—	118	1	no	—
143	C₄⋊C₄	C₂²⋊D₅₉	1	1	yes	characteristic
144	C₄⋊C₄	C₂²⋊D₅₉	1	1	yes	characteristic
145	C₂²⋊C₄	—	2	2	no	—
146	C₂²⋊C₄	—	2	1	no	—
147	C₄²	—	2	1	no	—
148	C₄²	C₂²×D₅₉	1	1	yes	characteristic
149	C₂³	—	2	1	no	—
150	C₂³	—	2	2	no	—
151	Q₈	—	2	1	no	—
152	Q₈	—	236	2	no	—
153	Q₈	#11745670	1	2	yes	complemented
154	D₄	—	2	1	no	—
155	D₄	—	2	1	no	—
156	D₄	—	4	2	no	—
157	D₄	—	4	4	no	—
158	D₄	—	4	1	no	—
159	D₄	—	4	1	no	—
160	D₄	—	4	1	no	—
161	D₄	—	4	1	no	—
162	D₄	—	8	2	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	—	4	1	no	—
168	C₂×C₄	—	4	1	no	—
169	C₂×C₄	—	236	1	no	—
170	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
171	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
172	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
173	C₈	—	118	2	no	—
174	C₈	—	236	1	no	—
175	C₈	—	236	1	no	—
176	C₂²	—	4	2	no	—
177	C₂²	—	4	1	no	—
178	C₂²	—	4	1	no	—
179	C₂²	—	4	2	no	—
180	C₂²	—	4	2	no	—
181	C₂²	—	4	1	no	—
182	C₂²	#14273696	1	1	yes	characteristic
183	C₄	—	2	2	no	—
184	C₄	—	2	1	no	—
185	C₄	—	4	1	no	—
186	C₄	—	4	1	no	—
187	C₄	—	472	1	no	—
188	C₄	C₂×#11745670	1	1	yes	characteristic
189	C₄	#14273747	1	1	yes	characteristic
190	C₄	#14273753	1	1	yes	characteristic
191	C₄	#14273757	1	1	yes	characteristic
192	C₂	—	8	1	no	—
193	C₂	—	8	2	no	—
194	C₂	#16837553	1	1	yes	characteristic
195	C₂	#16837564	1	1	yes	characteristic
196	C₂	#16837570	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837774

C₂

yes

maximal, characteristic, complemented

#16837697

C₂

yes

maximal, characteristic, complemented

#16837692

C₂

yes

maximal, characteristic, complemented

#16837146

C₂

yes

maximal, characteristic, complemented

Q₈⋊(C₅₉⋊C₈)

C₂

yes

maximal, characteristic, complemented

D₄⋊(C₅₉⋊C₈)

C₂

yes

maximal, characteristic, complemented

C₅₉×#550

C₂

yes

maximal, characteristic

C₂×#11745670

—

#14273727

C₂²

yes

characteristic

#14273660

—

#14273658

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

#14273651

—

#14273648

—

C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₂×#41×C₅₉

—

#127×C₅₉

—

#127×C₅₉

C₂²

yes

characteristic

#121×C₅₉

—

#121×C₅₉

—

C₄×Q₈×C₅₉

C₂²

yes

characteristic

C₄×D₄×C₅₉

—

C₄×D₄×C₅₉

C₂²

yes

characteristic

#41×C₅₉

—

#41×C₅₉

—

C₂×Q₈×C₅₉

D₄

yes

characteristic

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

D₄

yes

characteristic

C₂×D₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂×Dic₁₁₈

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

D₄

yes

characteristic

C₄⋊C₄×C₅₉

D₄

yes

characteristic

C₂²⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

C₂³

yes

characteristic

#11745670

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

—

C₂³×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

SD₁₆

yes

complemented

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

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×D₄

yes

characteristic

C₂×Dic₅₉

—

Dic₁₁₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#120

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×SD₁₆

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

#1631

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

yes

characteristic

100

#492

—

101

#477

—

102

#476

—

103

#331

—

104

Q₈⋊C₈

—

105

D₄⋊C₈

—

106

C₅₉

#1631

yes

characteristic, complemented, Sylow

107

C₂×#41

—

108

C₂×SD₁₆

—

118

—

109

C₄⋊Q₈

—

110

#127

—

111

#127

C₂×D₅₉

yes

characteristic

112

#121

—

113

#121

—

114

C₄×Q₈

C₂×D₅₉

yes

characteristic

115

C₄×D₄

—

116

C₄×D₄

C₂×D₅₉

yes

characteristic

117

#107

—

118

—

118

#106

—

118

—

119

C₄⋊C₈

—

120

C₄⋊C₈

—

121

#103

—

118

—

122

#102

—

118

—

123

C₄×C₈

—

124

#41

—

125

#41

—

126

C₂×Q₈

—

118

—

127

C₂×Q₈

C₂²⋊D₅₉

yes

characteristic

128

C₂×D₄

—

129

C₂×D₄

—

130

C₂×D₄

—

131

C₂×D₄

—

132

C₂×D₄

C₂²⋊D₅₉

yes

characteristic

133

C₂×D₄

C₂²⋊D₅₉

yes

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

SD₁₆

—

236

—

137

C₂×C₈

—

118

—

138

C₂×C₈

—

118

—

139

C₂×C₈

—

118

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

118

—

142

C₄⋊C₄

—

118

—

143

C₄⋊C₄

C₂²⋊D₅₉

yes

characteristic

144

C₄⋊C₄

C₂²⋊D₅₉

yes

characteristic

145

C₂²⋊C₄

—

146

C₂²⋊C₄

—

147

C₄²

—

148

C₄²

C₂²×D₅₉

yes

characteristic

149

C₂³

—

150

C₂³

—

151

Q₈

—

152

Q₈

—

236

—

153

Q₈

#11745670

yes

complemented

154

D₄

—

155

D₄

—

156

D₄

—

157

D₄

—

158

D₄

—

159

D₄

—

160

D₄

—

161

D₄

—

162

D₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

236

—

170

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

171

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

172

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

173

C₈

—

118

—

174

C₈

—

236

—

175

C₈

—

236

—

176

C₂²

—

177

C₂²

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

181

C₂²

—

182

C₂²

#14273696

yes

characteristic

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

—

472

—

188

C₄

C₂×#11745670

yes

characteristic

189

C₄

#14273747

yes

characteristic

190

C₄

#14273753

yes

characteristic

191

C₄

#14273757

yes

characteristic

192

C₂

—

193

C₂

—

194

C₂

#16837553

yes

characteristic

195

C₂

#16837564

yes

characteristic

196

C₂

#16837570

yes

characteristic

	subgroup	G/H
center	C₂²	#14273696
commutator	C₄²×C₅₉	C₂³
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#550	C₂
socle	C₂²×C₅₉	#120
hypercenter	#550	D₅₉
hypocenter	C₅₉	#1631
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #550

subgroup

G/H

center

C₂²

#14273696

commutator

C₄²×C₅₉

C₂³

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#550

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #550

N	H ≅ G/N	type
1. #16837774	C₂	semidirect
2. #16837697	C₂	semidirect
3. #16837692	C₂	semidirect
4. #16837146	C₂	semidirect
5. Q₈⋊(C₅₉⋊C₈)	C₂	semidirect
6. D₄⋊(C₅₉⋊C₈)	C₂	semidirect
53. Q₈×C₅₉	SD₁₆	semidirect
106. C₅₉	#1631	semidirect
153. Q₈	#11745670	semidirect

H ≅ G/N

type

1. #16837774

C₂

semidirect

2. #16837697

C₂

semidirect

3. #16837692

C₂

semidirect

4. #16837146

C₂

semidirect

5. Q₈⋊(C₅₉⋊C₈)

C₂

semidirect

6. D₄⋊(C₅₉⋊C₈)

C₂

semidirect

53. Q₈×C₅₉

SD₁₆

semidirect

106. C₅₉

#1631

semidirect

153. Q₈

#11745670

semidirect

Subgroups — #21423223

Conjugacy classes of subgroups

Complements