Group #21433874 — subgroups

Subgroups — C₂×#16837710 #21433874

Order 7552.

384 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 157 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×#14273719	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₁₁₈	C₂	1	1	yes	maximal, characteristic
3	#16837710	C₂	1	8	yes	maximal, direct product factor
4	C₂×#14273661	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273659	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273652	C₂	1	1	yes	maximal, characteristic
7	C₂×#14273652	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₁₈	C₂²	1	1	yes	characteristic
10	#14273719	—	2	2	no	—
11	#14273719	C₂²	1	4	yes	—
12	C₄×Dic₁₁₈	—	2	2	no	—
13	C₄×Dic₁₁₈	C₂²	1	4	yes	—
14	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₅₉⋊C₄	—	2	1	no	—
18	C₂×Dic₅₉⋊C₄	—	2	1	no	—
19	C₂×Dic₅₉⋊C₄	—	2	1	no	—
20	C₂×C₄×Dic₅₉	—	2	1	no	—
21	#14273661	C₂²	1	4	yes	—
22	#14273659	C₂²	1	4	yes	—
23	#14273652	C₂²	1	4	yes	—
24	#14273652	C₂²	1	4	yes	—
25	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
26	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
28	C₄⋊C₈×C₅₉	C₂²	1	4	yes	—
29	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
32	C₂²×Dic₅₉	—	2	1	no	—
33	C₂²×Dic₅₉	—	2	1	no	—
34	C₂²×Dic₅₉	—	2	1	no	—
35	C₂²×Dic₅₉	—	2	1	no	—
36	C₂×Dic₁₁₈	—	2	1	no	—
37	C₂×Dic₁₁₈	—	2	2	no	—
38	C₂×Dic₁₁₈	Q₈	1	4	yes	—
39	C₂×Dic₁₁₈	C₂³	1	2	yes	—
40	C₂×C₈×C₅₉	—	2	2	no	—
41	C₂×C₈×C₅₉	—	2	2	no	—
42	C₂×C₈×C₅₉	C₂³	1	2	yes	—
43	C₂×C₈×C₅₉	C₂³	1	2	yes	—
44	C₄²×C₅₉	D₄	1	2	yes	—
45	C₄²×C₅₉	C₂³	1	2	yes	—
46	C₄⋊Dic₅₉	—	2	1	no	—
47	C₄⋊Dic₅₉	—	2	1	no	—
48	C₄⋊Dic₅₉	—	2	1	no	—
49	C₄⋊Dic₅₉	C₂³	1	2	yes	—
50	C₄⋊Dic₅₉	C₂³	1	2	yes	—
51	C₄⋊Dic₅₉	C₂³	1	2	yes	—
52	Dic₅₉⋊C₄	—	2	4	no	—
53	Dic₅₉⋊C₄	—	2	4	no	—
54	Dic₅₉⋊C₄	—	2	4	no	—
55	C₄×Dic₅₉	—	2	4	no	—
56	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₅₉	—	2	2	no	—
58	C₂×C₄×C₅₉	—	2	2	no	—
59	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₅₉	C₂×Q₈	1	2	yes	—
63	C₂×C₄×C₅₉	#41	1	2	yes	—
64	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
65	C₂×Dic₅₉	—	2	4	no	—
66	C₂×Dic₅₉	—	2	2	no	—
67	C₂×Dic₅₉	—	2	2	no	—
68	C₂×Dic₅₉	—	2	2	no	—
69	C₂×Dic₅₉	—	2	2	no	—
70	C₂×Dic₅₉	—	4	2	no	—
71	C₂×Dic₅₉	—	4	2	no	—
72	C₂×Dic₅₉	—	4	2	no	—
73	Dic₁₁₈	—	2	4	no	—
74	Dic₁₁₈	—	2	2	no	—
75	Dic₁₁₈	C₂×Q₈	1	4	yes	—
76	C₈×C₅₉	—	2	2	no	—
77	C₈×C₅₉	—	2	2	no	—
78	C₂²×C₅₉	C₂²⋊Q₈	1	2	yes	—
79	C₂²×C₅₉	#135	1	2	yes	—
80	C₂²×C₅₉	#137	1	2	yes	—
81	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
82	C₄×C₅₉	—	2	2	no	—
83	C₄×C₅₉	—	2	2	no	—
84	C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
85	C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
86	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
87	Dic₅₉	—	2	4	no	—
88	Dic₅₉	—	4	2	no	—
89	Dic₅₉	—	4	2	no	—
90	Dic₅₉	—	4	2	no	—
91	C₂×#479	—	59	1	no	maximal, Sylow
92	C₂×C₅₉	#479	1	4	yes	complemented
93	C₂×C₅₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
94	C₂×C₅₉	C₂×#135	1	1	yes	characteristic
95	C₂×C₅₉	C₂×#137	1	1	yes	characteristic
96	C₂×#125	—	59	1	no	—
97	C₂×C₄×Q₈	—	59	1	no	—
98	#479	—	59	8	no	—
99	C₂×#107	—	59	1	no	—
100	C₂×#106	—	59	1	no	—
101	C₂×C₄⋊C₈	D₅₉	1	1	yes	characteristic
102	C₂×#103	—	59	1	no	—
103	C₂×#103	—	59	1	no	—
104	C₅₉	C₂×#479	1	1	yes	characteristic, complemented, Sylow
105	C₂²×Q₈	—	59	1	no	—
106	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
107	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
108	#125	—	59	4	no	—
109	#125	—	118	2	no	—
110	C₄×Q₈	—	59	4	no	—
111	C₄×Q₈	—	118	2	no	—
112	C₂×C₄⋊C₄	—	59	1	no	—
113	C₂×C₄⋊C₄	—	59	1	no	—
114	C₂×C₄⋊C₄	—	59	1	no	—
115	C₂×C₄⋊C₄	—	118	1	no	—
116	C₂×C₄⋊C₄	—	118	1	no	—
117	C₂×C₄⋊C₄	—	118	1	no	—
118	C₂×C₄²	—	118	1	no	—
119	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
120	#107	—	59	4	no	—
121	#106	—	59	4	no	—
122	C₄⋊C₈	C₂×D₅₉	1	4	yes	—
123	#103	—	59	4	no	—
124	#103	—	59	4	no	—
125	C₂×Q₈	—	59	4	no	—
126	C₂×Q₈	—	59	2	no	—
127	C₂×Q₈	—	118	1	no	—
128	C₂×Q₈	—	118	2	no	—
129	C₂²×C₄	—	118	1	no	—
130	C₂²×C₄	—	118	1	no	—
131	C₂²×C₄	—	118	1	no	—
132	C₂²×C₄	—	118	1	no	—
133	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
134	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
135	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	C₂²×D₅₉	1	2	yes	—
139	C₂×C₈	C₂²×D₅₉	1	2	yes	—
140	C₄⋊C₄	—	59	2	no	—
141	C₄⋊C₄	—	59	2	no	—
142	C₄⋊C₄	—	59	2	no	—
143	C₄⋊C₄	—	118	1	no	—
144	C₄⋊C₄	—	118	1	no	—
145	C₄⋊C₄	—	118	1	no	—
146	C₄⋊C₄	—	118	4	no	—
147	C₄⋊C₄	—	118	4	no	—
148	C₄⋊C₄	—	118	4	no	—
149	C₄²	—	118	4	no	—
150	C₄²	D₂₃₆	1	2	yes	—
151	C₄²	C₂²×D₅₉	1	2	yes	—
152	C₂³	C₂×D₂₃₆	1	1	yes	characteristic
153	Q₈	—	59	4	no	—
154	Q₈	—	118	4	no	—
155	Q₈	—	118	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	118	4	no	—
159	C₂×C₄	—	118	2	no	—
160	C₂×C₄	—	118	2	no	—
161	C₂×C₄	—	118	2	no	—
162	C₂×C₄	—	118	2	no	—
163	C₂×C₄	—	236	2	no	—
164	C₂×C₄	—	236	2	no	—
165	C₂×C₄	—	236	2	no	—
166	C₂×C₄	C₂×D₂₃₆	1	2	yes	—
167	C₂×C₄	C₂×D₂₃₆	1	2	yes	—
168	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
169	C₂×C₄	#11745687	1	2	yes	—
170	C₂×C₄	Q₈×D₅₉	1	2	yes	—
171	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
172	C₈	—	2	2	no	—
173	C₈	—	2	2	no	—
174	C₂²	#14273707	1	2	yes	—
175	C₂²	#14273749	1	2	yes	—
176	C₂²	#14273760	1	2	yes	—
177	C₂²	C₂²×D₂₃₆	1	1	yes	characteristic
178	C₄	—	2	2	no	—
179	C₄	—	2	2	no	—
180	C₄	—	118	4	no	—
181	C₄	—	236	2	no	—
182	C₄	—	236	2	no	—
183	C₄	—	236	2	no	—
184	C₄	#14273707	1	2	yes	—
185	C₄	C₂×Q₈×D₅₉	1	1	yes	characteristic
186	C₄	C₂×#11745687	1	1	yes	characteristic
187	C₂	#16837710	1	4	yes	direct product factor
188	C₂	C₂×#14273707	1	1	yes	characteristic
189	C₂	C₂×#14273749	1	1	yes	characteristic
190	C₂	C₂×#14273760	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273719

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₁₈

C₂

yes

maximal, characteristic

#16837710

C₂

yes

maximal, direct product factor

C₂×#14273661

C₂

yes

maximal, characteristic

C₂×#14273659

C₂

yes

maximal, characteristic

C₂×#14273652

C₂

yes

maximal, characteristic

C₂×#14273652

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₅₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₁₈

C₂²

yes

characteristic

#14273719

—

#14273719

C₂²

yes

—

C₄×Dic₁₁₈

—

C₄×Dic₁₁₈

C₂²

yes

—

C₂×C₄⋊Dic₅₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₉

C₂²

yes

characteristic

C₂×Dic₅₉⋊C₄

—

C₂×Dic₅₉⋊C₄

—

C₂×Dic₅₉⋊C₄

—

C₂×C₄×Dic₅₉

—

#14273661

C₂²

yes

—

#14273659

C₂²

yes

—

#14273652

C₂²

yes

—

#14273652

C₂²

yes

—

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂×C₄²×C₅₉

C₂²

yes

characteristic

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

—

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

Q₈

yes

—

C₂×Dic₁₁₈

C₂³

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂³

yes

—

C₂×C₈×C₅₉

C₂³

yes

—

C₄²×C₅₉

D₄

yes

—

C₄²×C₅₉

C₂³

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂³

yes

—

C₄⋊Dic₅₉

C₂³

yes

—

C₄⋊Dic₅₉

C₂³

yes

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

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

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

C₂×Q₈

yes

—

C₈×C₅₉

—

C₈×C₅₉

—

C₂²×C₅₉

C₂²⋊Q₈

yes

—

C₂²×C₅₉

#135

yes

—

C₂²×C₅₉

#137

yes

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂²⋊Q₈

yes

—

C₄×C₅₉

C₂²×Q₈

yes

characteristic

C₄×C₅₉

C₂×#41

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

C₂×C₂²⋊Q₈

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×C₄×Q₈

—

#479

—

C₂×#107

—

100

C₂×#106

—

101

C₂×C₄⋊C₈

D₅₉

yes

characteristic

102

C₂×#103

—

103

C₂×#103

—

104

C₅₉

C₂×#479

yes

characteristic, complemented, Sylow

105

C₂²×Q₈

—

106

C₂²×C₈

C₂×D₅₉

yes

characteristic

107

C₂²×C₈

C₂×D₅₉

yes

characteristic

108

#125

—

109

#125

—

118

—

110

C₄×Q₈

—

111

C₄×Q₈

—

118

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

118

—

116

C₂×C₄⋊C₄

—

118

—

117

C₂×C₄⋊C₄

—

118

—

118

C₂×C₄²

—

118

—

119

C₂×C₄²

C₂×D₅₉

yes

characteristic

120

#107

—

121

#106

—

122

C₄⋊C₈

C₂×D₅₉

yes

—

123

#103

—

124

#103

—

125

C₂×Q₈

—

126

C₂×Q₈

—

127

C₂×Q₈

—

118

—

128

C₂×Q₈

—

118

—

129

C₂²×C₄

—

118

—

130

C₂²×C₄

—

118

—

131

C₂²×C₄

—

118

—

132

C₂²×C₄

—

118

—

133

C₂²×C₄

D₂₃₆

yes

characteristic

134

C₂²×C₄

D₂₃₆

yes

characteristic

135

C₂²×C₄

C₂²×D₅₉

yes

characteristic

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

C₂²×D₅₉

yes

—

139

C₂×C₈

C₂²×D₅₉

yes

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

118

—

144

C₄⋊C₄

—

118

—

145

C₄⋊C₄

—

118

—

146

C₄⋊C₄

—

118

—

147

C₄⋊C₄

—

118

—

148

C₄⋊C₄

—

118

—

149

C₄²

—

118

—

150

C₄²

D₂₃₆

yes

—

151

C₄²

C₂²×D₅₉

yes

—

152

C₂³

C₂×D₂₃₆

yes

characteristic

153

Q₈

—

154

Q₈

—

118

—

155

Q₈

—

118

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

118

—

159

C₂×C₄

—

118

—

160

C₂×C₄

—

118

—

161

C₂×C₄

—

118

—

162

C₂×C₄

—

118

—

163

C₂×C₄

—

236

—

164

C₂×C₄

—

236

—

165

C₂×C₄

—

236

—

166

C₂×C₄

C₂×D₂₃₆

yes

—

167

C₂×C₄

C₂×D₂₃₆

yes

—

168

C₂×C₄

C₂×D₂₃₆

yes

characteristic

169

C₂×C₄

#11745687

yes

—

170

C₂×C₄

Q₈×D₅₉

yes

—

171

C₂×C₄

C₂³×D₅₉

yes

characteristic

172

C₈

—

173

C₈

—

174

C₂²

#14273707

yes

—

175

C₂²

#14273749

yes

—

176

C₂²

#14273760

yes

—

177

C₂²

C₂²×D₂₃₆

yes

characteristic

178

C₄

—

179

C₄

—

180

C₄

—

118

—

181

C₄

—

236

—

182

C₄

—

236

—

183

C₄

—

236

—

184

C₄

#14273707

yes

—

185

C₄

C₂×Q₈×D₅₉

yes

characteristic

186

C₄

C₂×#11745687

yes

characteristic

187

C₂

#16837710

yes

direct product factor

188

C₂

C₂×#14273707

yes

characteristic

189

C₂

C₂×#14273749

yes

characteristic

190

C₂

C₂×#14273760

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₂×C₄⋊C₈×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₅₉
hypocenter	C₅₉	C₂×#479
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₂₃₆

commutator

C₂×C₄×C₅₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₉

Fitting

C₂×C₄⋊C₈×C₅₉

C₂

socle

C₂³×C₅₉

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₅₉

hypocenter

C₅₉

C₂×#479

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
3. #16837710	C₂	direct (N × H)
92. C₂×C₅₉	#479	semidirect
104. C₅₉	C₂×#479	semidirect
187. C₂	#16837710	direct (N × H)

H ≅ G/N

type

3. #16837710

C₂

direct (N × H)

92. C₂×C₅₉

#479

semidirect

104. C₅₉

C₂×#479

semidirect

187. C₂

#16837710

direct (N × H)

Subgroups — C₂×#16837710 #21433874

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16837710 #21433874

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16837710 #21433874