Group #21433903 — subgroups

Subgroups — #21433903

Order 7552.

376 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 145 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	#16837959	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₁₁₈	C₂	1	1	yes	maximal, characteristic
3	#16837710	C₂	1	4	yes	maximal, complemented
4	#16837698	C₂	1	2	yes	maximal, complemented
5	#16837688	C₂	1	2	yes	maximal, complemented
6	#16837444	C₂	1	2	yes	maximal
7	#16837412	C₂	1	2	yes	maximal
8	C₅₉×#423	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₁₈	C₂²	1	1	yes	characteristic
10	#14273727	C₂²	1	1	yes	characteristic
11	#14273727	C₂²	1	1	yes	characteristic
12	#14273719	C₂²	1	2	yes	—
13	#14273706	—	2	1	no	—
14	#14273706	—	2	1	no	—
15	C₄×Dic₁₁₈	—	2	1	no	—
16	C₄×Dic₁₁₈	—	2	1	no	—
17	C₄×Dic₁₁₈	—	2	1	no	—
18	C₄×Dic₁₁₈	—	2	1	no	—
19	C₄×Dic₁₁₈	C₂²	1	2	yes	—
20	C₄×Dic₁₁₈	C₂²	1	2	yes	—
21	#14273686	C₂²	1	2	yes	—
22	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
23	C₂×Dic₅₉⋊C₄	—	2	1	no	—
24	C₂×C₄×Dic₅₉	—	2	1	no	—
25	#14273661	C₂²	1	4	yes	—
26	#14273659	C₂²	1	4	yes	—
27	#14273652	C₂²	1	4	yes	—
28	#14273652	C₂²	1	4	yes	—
29	C₂×M₁₆×C₅₉	C₂²	1	2	yes	—
30	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
31	C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
32	C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
33	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
36	C₂²×Dic₅₉	—	2	1	no	—
37	C₂²×Dic₅₉	—	2	1	no	—
38	C₂×Dic₁₁₈	—	2	1	no	—
39	C₂×Dic₁₁₈	—	2	2	no	—
40	C₂×Dic₁₁₈	—	2	1	no	—
41	C₂×Dic₁₁₈	—	2	1	no	—
42	C₂×Dic₁₁₈	Q₈	1	4	yes	complemented
43	C₂×Dic₁₁₈	C₂³	1	2	yes	—
44	M₁₆×C₅₉	—	2	4	no	—
45	C₂×C₈×C₅₉	C₂³	1	4	yes	—
46	C₄²×C₅₉	D₄	1	1	yes	characteristic
47	C₄²×C₅₉	D₄	1	1	yes	characteristic
48	C₄²×C₅₉	C₂³	1	1	yes	characteristic
49	C₄²×C₅₉	C₂³	1	1	yes	characteristic
50	C₂²⋊Dic₅₉	—	2	2	no	—
51	C₄⋊Dic₅₉	—	2	1	no	—
52	C₄⋊Dic₅₉	—	2	1	no	—
53	C₄⋊Dic₅₉	—	2	1	no	—
54	C₄⋊Dic₅₉	C₂³	1	2	yes	—
55	C₄⋊Dic₅₉	C₂³	1	2	yes	—
56	C₄⋊Dic₅₉	C₂³	1	2	yes	—
57	Dic₅₉⋊C₄	—	2	2	no	—
58	Dic₅₉⋊C₄	—	2	2	no	—
59	Dic₅₉⋊C₄	—	2	2	no	—
60	Dic₅₉⋊C₄	—	2	2	no	—
61	C₄×Dic₅₉	—	2	2	no	—
62	C₄×Dic₅₉	—	2	2	no	—
63	C₄×Dic₅₉	—	2	2	no	—
64	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₅₉	—	2	2	no	—
66	C₂×C₄×C₅₉	—	4	1	no	—
67	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
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×C₄×C₅₉	C₂×Q₈	1	2	yes	—
73	C₂×C₄×C₅₉	#41	1	2	yes	—
74	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
75	C₂×Dic₅₉	—	2	4	no	—
76	C₂×Dic₅₉	—	2	2	no	—
77	C₂×Dic₅₉	—	2	2	no	—
78	C₂×Dic₅₉	—	2	2	no	—
79	C₂×Dic₅₉	—	2	2	no	—
80	C₂×Dic₅₉	—	4	2	no	—
81	Dic₁₁₈	—	2	4	no	—
82	Dic₁₁₈	—	2	2	no	—
83	Dic₁₁₈	—	4	2	no	—
84	Dic₁₁₈	—	4	2	no	—
85	Dic₁₁₈	C₂×Q₈	1	4	yes	—
86	C₈×C₅₉	—	2	4	no	—
87	C₂²×C₅₉	—	2	1	no	—
88	C₂²×C₅₉	—	2	1	no	—
89	C₂²×C₅₉	C₂²⋊Q₈	1	2	yes	—
90	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
91	C₄×C₅₉	—	2	1	no	—
92	C₄×C₅₉	—	2	1	no	—
93	C₄×C₅₉	—	2	1	no	—
94	C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
95	C₄×C₅₉	#137	1	2	yes	—
96	C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
97	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
98	Dic₅₉	—	2	4	no	—
99	Dic₅₉	—	4	2	no	—
100	Dic₅₉	—	4	2	no	—
101	Dic₅₉	—	4	2	no	—
102	#3078	—	59	1	no	maximal, Sylow
103	C₂×C₅₉	—	2	2	no	—
104	C₂×C₅₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
105	C₂×C₅₉	C₂×#137	1	1	yes	characteristic
106	C₂×C₅₉	#575	1	1	yes	characteristic
107	#533	—	59	1	no	—
108	C₂×C₄×Q₈	—	59	1	no	—
109	#479	—	59	4	no	—
110	#477	—	59	2	no	—
111	#475	—	59	2	no	—
112	#428	—	59	2	no	—
113	#423	D₅₉	1	1	yes	characteristic
114	#419	—	59	2	no	—
115	C₅₉	#3078	1	1	yes	characteristic, complemented, Sylow
116	C₂²×Q₈	—	59	1	no	—
117	C₂×M₁₆	C₂×D₅₉	1	2	yes	—
118	C₄⋊Q₈	—	59	1	no	—
119	C₄⋊Q₈	—	59	1	no	—
120	#125	—	59	2	no	—
121	C₂²⋊Q₈	—	118	1	no	—
122	C₂²⋊Q₈	—	118	1	no	—
123	C₄×Q₈	—	59	2	no	—
124	C₄×Q₈	—	59	2	no	—
125	C₄×Q₈	—	118	1	no	—
126	C₄×Q₈	—	118	1	no	—
127	C₄×Q₈	—	118	1	no	—
128	C₄×Q₈	—	118	1	no	—
129	#117	—	59	2	no	—
130	C₂×C₄⋊C₄	—	59	1	no	—
131	C₂×C₄⋊C₄	—	118	1	no	—
132	C₂×C₄²	—	118	1	no	—
133	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
134	#107	—	59	4	no	—
135	#106	—	59	4	no	—
136	C₄⋊C₈	C₂×D₅₉	1	2	yes	—
137	C₄⋊C₈	C₂×D₅₉	1	2	yes	—
138	#103	—	59	4	no	—
139	#103	—	59	4	no	—
140	C₂×Q₈	—	59	4	no	—
141	C₂×Q₈	—	59	2	no	—
142	C₂×Q₈	—	118	1	no	—
143	C₂×Q₈	—	118	1	no	—
144	C₂×Q₈	—	118	1	no	—
145	C₂×Q₈	—	118	2	no	—
146	C₂²×C₄	—	118	1	no	—
147	C₂²×C₄	—	118	1	no	—
148	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
149	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
150	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
151	M₁₆	—	2	4	no	—
152	C₂×C₈	C₂²×D₅₉	1	4	yes	—
153	C₄⋊C₄	—	59	2	no	—
154	C₄⋊C₄	—	59	2	no	—
155	C₄⋊C₄	—	59	2	no	—
156	C₄⋊C₄	—	118	1	no	—
157	C₄⋊C₄	—	118	2	no	—
158	C₄⋊C₄	—	118	2	no	—
159	C₄⋊C₄	—	118	2	no	—
160	C₄⋊C₄	—	118	2	no	—
161	C₄⋊C₄	—	118	1	no	—
162	C₄⋊C₄	—	118	1	no	—
163	C₂²⋊C₄	—	118	2	no	—
164	C₄²	—	118	2	no	—
165	C₄²	—	118	2	no	—
166	C₄²	—	118	2	no	—
167	C₄²	D₂₃₆	1	1	yes	characteristic
168	C₄²	D₂₃₆	1	1	yes	characteristic
169	C₄²	C₂²×D₅₉	1	1	yes	characteristic
170	C₄²	C₂²×D₅₉	1	1	yes	characteristic
171	C₂³	C₂×D₂₃₆	1	1	yes	characteristic
172	Q₈	—	59	4	no	—
173	Q₈	—	118	4	no	—
174	Q₈	—	118	2	no	—
175	Q₈	—	236	2	no	—
176	Q₈	—	236	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	4	1	no	—
179	C₂×C₄	—	118	4	no	—
180	C₂×C₄	—	118	2	no	—
181	C₂×C₄	—	118	2	no	—
182	C₂×C₄	—	118	2	no	—
183	C₂×C₄	—	118	2	no	—
184	C₂×C₄	—	236	2	no	—
185	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
186	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
187	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
188	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
189	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
190	C₂×C₄	#11745687	1	2	yes	—
191	C₂×C₄	Q₈×D₅₉	1	2	yes	—
192	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
193	C₈	—	2	4	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	2	1	no	—
196	C₂²	#14273707	1	2	yes	—
197	C₂²	C₂²×D₂₃₆	1	1	yes	characteristic
198	C₄	—	2	1	no	—
199	C₄	—	2	1	no	—
200	C₄	—	2	1	no	—
201	C₄	—	118	4	no	—
202	C₄	—	236	2	no	—
203	C₄	—	236	2	no	—
204	C₄	—	236	2	no	—
205	C₄	#14273707	1	2	yes	—
206	C₄	#14273760	1	2	yes	—
207	C₄	C₂×Q₈×D₅₉	1	1	yes	characteristic
208	C₄	C₂×#11745687	1	1	yes	characteristic
209	C₂	—	2	2	no	—
210	C₂	C₂×#14273707	1	1	yes	characteristic
211	C₂	C₂×#14273760	1	1	yes	characteristic
212	C₂	#16838202	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837959

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₁₈

C₂

yes

maximal, characteristic

#16837710

C₂

yes

maximal, complemented

#16837698

C₂

yes

maximal, complemented

#16837688

C₂

yes

maximal, complemented

#16837444

C₂

yes

maximal

#16837412

C₂

yes

maximal

C₅₉×#423

C₂

yes

maximal, characteristic

C₂²×Dic₁₁₈

C₂²

yes

characteristic

#14273727

C₂²

yes

characteristic

#14273727

C₂²

yes

characteristic

#14273719

C₂²

yes

—

#14273706

—

#14273706

—

C₄×Dic₁₁₈

—

C₄×Dic₁₁₈

—

C₄×Dic₁₁₈

—

C₄×Dic₁₁₈

—

C₄×Dic₁₁₈

C₂²

yes

—

C₄×Dic₁₁₈

C₂²

yes

—

#14273686

C₂²

yes

—

C₂×C₄⋊Dic₅₉

C₂²

yes

characteristic

C₂×Dic₅₉⋊C₄

—

C₂×C₄×Dic₅₉

—

#14273661

C₂²

yes

—

#14273659

C₂²

yes

—

#14273652

C₂²

yes

—

#14273652

C₂²

yes

—

C₂×M₁₆×C₅₉

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

—

C₄⋊C₈×C₅₉

C₂²

yes

—

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

Q₈

yes

complemented

C₂×Dic₁₁₈

C₂³

yes

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

C₂³

yes

—

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂³

yes

—

C₄⋊Dic₅₉

C₂³

yes

—

C₄⋊Dic₅₉

C₂³

yes

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

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×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

—

C₂×C₄×C₅₉

#41

yes

—

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

C₂×Q₈

yes

—

C₈×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂²⋊Q₈

yes

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂²⋊Q₈

yes

—

C₄×C₅₉

#137

yes

—

C₄×C₅₉

C₂²×Q₈

yes

characteristic

C₄×C₅₉

C₂×#41

yes

characteristic

Dic₅₉

—

Dic₅₉

—

100

Dic₅₉

—

101

Dic₅₉

—

102

#3078

—

maximal, Sylow

103

C₂×C₅₉

—

104

C₂×C₅₉

C₂×C₂²⋊Q₈

yes

characteristic

105

C₂×C₅₉

C₂×#137

yes

characteristic

106

C₂×C₅₉

#575

yes

characteristic

107

#533

—

108

C₂×C₄×Q₈

—

109

#479

—

110

#477

—

111

#475

—

112

#428

—

113

#423

D₅₉

yes

characteristic

114

#419

—

115

C₅₉

#3078

yes

characteristic, complemented, Sylow

116

C₂²×Q₈

—

117

C₂×M₁₆

C₂×D₅₉

yes

—

118

C₄⋊Q₈

—

119

C₄⋊Q₈

—

120

#125

—

121

C₂²⋊Q₈

—

118

—

122

C₂²⋊Q₈

—

118

—

123

C₄×Q₈

—

124

C₄×Q₈

—

125

C₄×Q₈

—

118

—

126

C₄×Q₈

—

118

—

127

C₄×Q₈

—

118

—

128

C₄×Q₈

—

118

—

129

#117

—

130

C₂×C₄⋊C₄

—

131

C₂×C₄⋊C₄

—

118

—

132

C₂×C₄²

—

118

—

133

C₂×C₄²

C₂×D₅₉

yes

characteristic

134

#107

—

135

#106

—

136

C₄⋊C₈

C₂×D₅₉

yes

—

137

C₄⋊C₈

C₂×D₅₉

yes

—

138

#103

—

139

#103

—

140

C₂×Q₈

—

141

C₂×Q₈

—

142

C₂×Q₈

—

118

—

143

C₂×Q₈

—

118

—

144

C₂×Q₈

—

118

—

145

C₂×Q₈

—

118

—

146

C₂²×C₄

—

118

—

147

C₂²×C₄

—

118

—

148

C₂²×C₄

D₂₃₆

yes

characteristic

149

C₂²×C₄

D₂₃₆

yes

characteristic

150

C₂²×C₄

C₂²×D₅₉

yes

characteristic

151

M₁₆

—

152

C₂×C₈

C₂²×D₅₉

yes

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

118

—

157

C₄⋊C₄

—

118

—

158

C₄⋊C₄

—

118

—

159

C₄⋊C₄

—

118

—

160

C₄⋊C₄

—

118

—

161

C₄⋊C₄

—

118

—

162

C₄⋊C₄

—

118

—

163

C₂²⋊C₄

—

118

—

164

C₄²

—

118

—

165

C₄²

—

118

—

166

C₄²

—

118

—

167

C₄²

D₂₃₆

yes

characteristic

168

C₄²

D₂₃₆

yes

characteristic

169

C₄²

C₂²×D₅₉

yes

characteristic

170

C₄²

C₂²×D₅₉

yes

characteristic

171

C₂³

C₂×D₂₃₆

yes

characteristic

172

Q₈

—

173

Q₈

—

118

—

174

Q₈

—

118

—

175

Q₈

—

236

—

176

Q₈

—

236

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

118

—

180

C₂×C₄

—

118

—

181

C₂×C₄

—

118

—

182

C₂×C₄

—

118

—

183

C₂×C₄

—

118

—

184

C₂×C₄

—

236

—

185

C₂×C₄

C₂×D₂₃₆

yes

characteristic

186

C₂×C₄

C₂×D₂₃₆

yes

characteristic

187

C₂×C₄

C₂×D₂₃₆

yes

characteristic

188

C₂×C₄

C₂×D₂₃₆

yes

characteristic

189

C₂×C₄

C₂×D₂₃₆

yes

characteristic

190

C₂×C₄

#11745687

yes

—

191

C₂×C₄

Q₈×D₅₉

yes

—

192

C₂×C₄

C₂³×D₅₉

yes

characteristic

193

C₈

—

194

C₂²

—

195

C₂²

—

196

C₂²

#14273707

yes

—

197

C₂²

C₂²×D₂₃₆

yes

characteristic

198

C₄

—

199

C₄

—

200

C₄

—

201

C₄

—

118

—

202

C₄

—

236

—

203

C₄

—

236

—

204

C₄

—

236

—

205

C₄

#14273707

yes

—

206

C₄

#14273760

yes

—

207

C₄

C₂×Q₈×D₅₉

yes

characteristic

208

C₄

C₂×#11745687

yes

characteristic

209

C₂

—

210

C₂

C₂×#14273707

yes

characteristic

211

C₂

C₂×#14273760

yes

characteristic

212

C₂

#16838202

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#423	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#423	D₅₉
hypocenter	C₅₉	#3078
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂²×D₂₃₆

commutator

C₂×C₄×C₅₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₉

Fitting

C₅₉×#423

C₂

socle

C₂²×C₅₉

C₂²×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
3. #16837710	C₂	semidirect
4. #16837698	C₂	semidirect
5. #16837688	C₂	semidirect
42. C₂×Dic₁₁₈	Q₈	semidirect
115. C₅₉	#3078	semidirect

H ≅ G/N

type

3. #16837710

C₂

semidirect

4. #16837698

C₂

semidirect

5. #16837688

C₂

semidirect

42. C₂×Dic₁₁₈

Q₈

semidirect

115. C₅₉

#3078

semidirect

Subgroups — #21433903

Conjugacy classes of subgroups

Complements