Group #28745121 — subgroups

Subgroups — #28745121

Order 15104.

464 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 143 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	#21434755	C₂	1	2	yes	maximal
2	#21426261	C₂	1	4	yes	maximal, complemented
3	#21426204	C₂	1	2	yes	maximal
4	#21426187	C₂	1	2	yes	maximal
5	#21426147	C₂	1	4	yes	maximal, complemented
6	C₅₉×#2151	C₂	1	1	yes	maximal, characteristic
7	#16838014	—	2	2	no	—
8	C₂×Dic₄₇₂	C₂²	1	2	yes	—
9	C₂×Dic₄₇₂	C₂²	1	2	yes	—
10	#16837804	C₂²	1	2	yes	—
11	#16837782	C₂²	1	2	yes	—
12	#16837771	—	2	2	no	—
13	#16837660	—	2	2	no	—
14	#16837643	C₂²	1	4	yes	—
15	#16837510	—	2	2	no	—
16	C₄×Dic₂₃₆	C₂²	1	4	yes	—
17	#16837230	C₂²	1	2	yes	—
18	#16837228	C₂²	1	2	yes	—
19	#16837207	C₂²	1	4	yes	—
20	#16837207	C₂²	1	4	yes	—
21	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
22	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
23	C₅₉×#405	C₂²	1	1	yes	characteristic
24	C₅₉×C₄⋊C₁₆	C₂²	1	4	yes	—
25	C₂×Dic₂₃₆	—	2	2	no	—
26	C₂×Dic₂₃₆	C₂³	1	4	yes	—
27	#14273727	—	2	2	no	—
28	#14273720	—	4	4	no	—
29	#14273719	—	2	2	no	—
30	#14273717	—	4	4	no	—
31	#14273706	—	2	4	no	—
32	C₄×Dic₁₁₈	—	2	4	no	—
33	Dic₄₇₂	—	4	2	no	—
34	Dic₄₇₂	—	4	2	no	—
35	#14273661	C₂³	1	4	yes	—
36	#14273659	—	2	2	no	—
37	#14273652	—	2	4	no	—
38	#14273652	—	2	4	no	—
39	C₂×M₁₆×C₅₉	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
41	#117×C₅₉	D₄	1	1	yes	characteristic
42	M₃₂×C₅₉	—	4	1	no	—
43	C₂×C₁₆×C₅₉	—	4	1	no	—
44	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
45	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
46	#97×C₅₉	D₄	1	2	yes	—
47	C₄×C₈×C₅₉	C₂³	1	2	yes	—
48	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
49	C₂×Dic₁₁₈	—	2	4	no	—
50	M₁₆×C₅₉	—	4	1	no	—
51	C₂×C₈×C₅₉	—	2	1	no	—
52	C₂×C₈×C₅₉	—	2	1	no	—
53	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
54	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
56	C₄⋊C₄×C₅₉	D₈	1	2	yes	—
57	C₂²⋊C₄×C₅₉	D₈	1	2	yes	—
58	C₄²×C₅₉	C₂×D₄	1	2	yes	—
59	C₂²⋊Dic₅₉	—	4	4	no	—
60	C₄⋊Dic₅₉	—	2	4	no	—
61	C₄⋊Dic₅₉	—	4	4	no	—
62	Dic₅₉⋊C₄	—	4	4	no	—
63	Dic₅₉⋊C₄	—	4	4	no	—
64	Dic₅₉⋊C₄	—	4	4	no	—
65	C₄×Dic₅₉	—	4	4	no	—
66	Dic₂₃₆	—	2	4	no	—
67	Dic₂₃₆	—	2	4	no	—
68	Dic₂₃₆	—	4	4	no	—
69	C₁₆×C₅₉	—	2	2	no	—
70	C₁₆×C₅₉	—	2	2	no	—
71	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
72	C₂×C₄×C₅₉	—	2	1	no	—
73	C₂×C₄×C₅₉	—	2	1	no	—
74	C₂×C₄×C₅₉	C₂×D₈	1	4	yes	—
75	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
76	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
77	C₂×Dic₅₉	—	4	4	no	—
78	C₂×Dic₅₉	—	4	4	no	—
79	Dic₁₁₈	—	4	4	no	—
80	Dic₁₁₈	—	4	4	no	—
81	C₈×C₅₉	—	2	1	no	—
82	C₈×C₅₉	—	2	2	no	—
83	C₈×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
84	C₈×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
85	#33827	—	59	1	no	maximal, Sylow
86	C₂²×C₅₉	—	2	2	no	—
87	C₂²×C₅₉	—	2	1	no	—
88	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
89	C₄×C₅₉	—	2	4	no	—
90	C₄×C₅₉	—	2	1	no	—
91	C₄×C₅₉	#537	1	1	yes	characteristic
92	C₄×C₅₉	#575	1	1	yes	characteristic
93	Dic₅₉	—	8	4	no	—
94	Dic₅₉	—	8	4	no	—
95	#3177	—	59	2	no	—
96	#2229	—	59	4	no	—
97	#2219	—	59	2	no	—
98	#2216	—	59	2	no	—
99	#2209	—	59	4	no	—
100	#2151	D₅₉	1	1	yes	characteristic
101	C₂×C₅₉	—	4	1	no	—
102	C₂×C₅₉	#3245	1	1	yes	characteristic
103	C₂×C₅₉	#3417	1	2	yes	—
104	#541	—	118	2	no	—
105	C₂×Q₃₂	—	59	2	no	—
106	C₂×Q₃₂	—	59	2	no	—
107	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
108	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
109	#499	—	59	2	no	—
110	#494	—	59	2	no	—
111	#491	—	118	2	no	—
112	#470	—	118	2	no	—
113	#467	—	59	4	no	—
114	#441	—	118	2	no	—
115	C₄×Q₁₆	—	59	4	no	—
116	#405	C₂×D₅₉	1	1	yes	characteristic
117	#367	—	59	2	no	—
118	#366	—	59	2	no	—
119	C₄⋊C₁₆	C₂×D₅₉	1	4	yes	—
120	#358	—	59	4	no	—
121	#358	—	59	4	no	—
122	C₅₉	#33827	1	1	yes	characteristic, complemented, Sylow
123	C₂×Q₁₆	—	59	4	no	—
124	C₂×Q₁₆	—	118	2	no	—
125	C₂×M₁₆	D₂₃₆	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
127	C₄⋊Q₈	—	118	2	no	—
128	#126	—	236	4	no	—
129	#125	—	118	2	no	—
130	#125	—	236	4	no	—
131	C₂²⋊Q₈	—	118	4	no	—
132	C₄×Q₈	—	118	4	no	—
133	#117	D₂₃₆	1	1	yes	characteristic
134	Q₃₂	—	236	2	no	—
135	Q₃₂	—	236	2	no	—
136	M₃₂	—	4	1	no	—
137	C₂×C₁₆	—	4	1	no	—
138	C₂×C₁₆	C₂²×D₅₉	1	2	yes	—
139	C₂×C₁₆	C₂²×D₅₉	1	2	yes	—
140	#107	—	59	4	no	—
141	#106	—	118	2	no	—
142	#103	—	118	4	no	—
143	#103	—	118	4	no	—
144	#97	D₂₃₆	1	2	yes	—
145	C₄×C₈	C₂²×D₅₉	1	2	yes	—
146	C₂×Q₈	—	118	4	no	—
147	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
148	Q₁₆	—	118	4	no	—
149	Q₁₆	—	118	4	no	—
150	Q₁₆	—	236	4	no	—
151	M₁₆	—	4	1	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
155	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
156	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
157	C₄⋊C₄	—	118	4	no	—
158	C₄⋊C₄	—	236	4	no	—
159	C₄⋊C₄	—	236	4	no	—
160	C₄⋊C₄	—	236	4	no	—
161	C₄⋊C₄	—	236	4	no	—
162	C₄⋊C₄	D₄₇₂	1	2	yes	—
163	C₂²⋊C₄	—	236	4	no	—
164	C₂²⋊C₄	D₄₇₂	1	2	yes	—
165	C₄²	—	236	4	no	—
166	C₄²	C₂×D₂₃₆	1	2	yes	—
167	C₁₆	—	2	2	no	—
168	C₁₆	—	2	2	no	—
169	C₂³	C₂×D₄₇₂	1	1	yes	characteristic
170	Q₈	—	236	4	no	—
171	Q₈	—	236	4	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	236	4	no	—
175	C₂×C₄	—	236	4	no	—
176	C₂×C₄	C₂×D₄₇₂	1	4	yes	—
177	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
178	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
179	C₈	—	2	2	no	—
180	C₈	—	2	1	no	—
181	C₈	#14273772	1	1	yes	characteristic
182	C₈	#14273775	1	1	yes	characteristic
183	C₂²	—	2	1	no	—
184	C₂²	—	2	2	no	—
185	C₂²	C₂²×D₄₇₂	1	1	yes	characteristic
186	C₄	—	2	1	no	—
187	C₄	—	2	4	no	—
188	C₄	—	472	4	no	—
189	C₄	—	472	4	no	—
190	C₄	#16837983	1	1	yes	characteristic
191	C₄	#16838202	1	1	yes	characteristic
192	C₂	—	4	1	no	—
193	C₂	#21435406	1	1	yes	characteristic
194	C₂	#21437249	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21434755

C₂

yes

maximal

#21426261

C₂

yes

maximal, complemented

#21426204

C₂

yes

maximal

#21426187

C₂

yes

maximal

#21426147

C₂

yes

maximal, complemented

C₅₉×#2151

C₂

yes

maximal, characteristic

#16838014

—

C₂×Dic₄₇₂

C₂²

yes

—

C₂×Dic₄₇₂

C₂²

yes

—

#16837804

C₂²

yes

—

#16837782

C₂²

yes

—

#16837771

—

#16837660

—

#16837643

C₂²

yes

—

#16837510

—

C₄×Dic₂₃₆

C₂²

yes

—

#16837230

C₂²

yes

—

#16837228

C₂²

yes

—

#16837207

C₂²

yes

—

#16837207

C₂²

yes

—

C₂×M₃₂×C₅₉

C₂²

yes

characteristic

C₂²×C₁₆×C₅₉

C₂²

yes

characteristic

C₅₉×#405

C₂²

yes

characteristic

C₅₉×C₄⋊C₁₆

C₂²

yes

—

C₂×Dic₂₃₆

—

C₂×Dic₂₃₆

C₂³

yes

—

—

—

—

—

—

C₄×Dic₁₁₈

—

Dic₄₇₂

—

Dic₄₇₂

—

#14273661

C₂³

yes

—

#14273659

—

#14273652

—

#14273652

—

C₂×M₁₆×C₅₉

D₄

yes

characteristic

C₂²×C₈×C₅₉

C₂³

yes

characteristic

#117×C₅₉

D₄

yes

characteristic

M₃₂×C₅₉

—

C₂×C₁₆×C₅₉

—

C₂×C₁₆×C₅₉

C₂³

yes

—

C₂×C₁₆×C₅₉

C₂³

yes

—

#97×C₅₉

D₄

yes

—

C₄×C₈×C₅₉

C₂³

yes

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₁₁₈

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×D₄

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

C₂⁴

yes

characteristic

C₄⋊C₄×C₅₉

D₈

yes

—

C₂²⋊C₄×C₅₉

D₈

yes

—

C₄²×C₅₉

C₂×D₄

yes

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

Dic₂₃₆

—

Dic₂₃₆

—

Dic₂₃₆

—

C₁₆×C₅₉

—

C₁₆×C₅₉

—

C₂³×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₈

yes

—

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

C₂²×D₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

H(5,𝔽₂)

yes

characteristic

C₈×C₅₉

2₋¹⁺⁴

yes

characteristic

#33827

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂²×D₈

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

—

—

—

—

—

100

#2151

D₅₉

yes

characteristic

101

C₂×C₅₉

—

102

C₂×C₅₉

#3245

yes

characteristic

103

C₂×C₅₉

#3417

yes

—

104

#541

—

118

—

105

C₂×Q₃₂

—

106

C₂×Q₃₂

—

107

C₂×M₃₂

C₂×D₅₉

yes

characteristic

108

C₂²×C₁₆

C₂×D₅₉

yes

characteristic

109

#499

—

110

#494

—

111

#491

—

118

—

112

#470

—

118

—

113

#467

—

114

#441

—

118

—

115

C₄×Q₁₆

—

116

#405

C₂×D₅₉

yes

characteristic

117

#367

—

118

#366

—

119

C₄⋊C₁₆

C₂×D₅₉

yes

—

120

#358

—

121

#358

—

122

C₅₉

#33827

yes

characteristic, complemented, Sylow

123

C₂×Q₁₆

—

124

C₂×Q₁₆

—

118

—

125

C₂×M₁₆

D₂₃₆

yes

characteristic

126

C₂²×C₈

C₂²×D₅₉

yes

characteristic

127

C₄⋊Q₈

—

118

—

128

#126

—

236

—

129

#125

—

118

—

130

#125

—

236

—

131

C₂²⋊Q₈

—

118

—

132

C₄×Q₈

—

118

—

133

#117

D₂₃₆

yes

characteristic

134

Q₃₂

—

236

—

135

Q₃₂

—

236

—

136

M₃₂

—

137

C₂×C₁₆

—

138

C₂×C₁₆

C₂²×D₅₉

yes

—

139

C₂×C₁₆

C₂²×D₅₉

yes

—

140

#107

—

141

#106

—

118

—

142

#103

—

118

—

143

#103

—

118

—

144

#97

D₂₃₆

yes

—

145

C₄×C₈

C₂²×D₅₉

yes

—

146

C₂×Q₈

—

118

—

147

C₂²×C₄

C₂×D₂₃₆

yes

characteristic

148

Q₁₆

—

118

—

149

Q₁₆

—

118

—

150

Q₁₆

—

236

—

151

M₁₆

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

C₂×D₂₃₆

yes

—

155

C₂×C₈

C₂×D₂₃₆

yes

characteristic

156

C₂×C₈

C₂³×D₅₉

yes

characteristic

157

C₄⋊C₄

—

118

—

158

C₄⋊C₄

—

236

—

159

C₄⋊C₄

—

236

—

160

C₄⋊C₄

—

236

—

161

C₄⋊C₄

—

236

—

162

C₄⋊C₄

D₄₇₂

yes

—

163

C₂²⋊C₄

—

236

—

164

C₂²⋊C₄

D₄₇₂

yes

—

165

C₄²

—

236

—

166

C₄²

C₂×D₂₃₆

yes

—

167

C₁₆

—

168

C₁₆

—

169

C₂³

C₂×D₄₇₂

yes

characteristic

170

Q₈

—

236

—

171

Q₈

—

236

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

236

—

175

C₂×C₄

—

236

—

176

C₂×C₄

C₂×D₄₇₂

yes

—

177

C₂×C₄

C₂×D₄₇₂

yes

characteristic

178

C₂×C₄

C₂²×D₂₃₆

yes

characteristic

179

C₈

—

180

C₈

—

181

C₈

#14273772

yes

characteristic

182

C₈

#14273775

yes

characteristic

183

C₂²

—

184

C₂²

—

185

C₂²

C₂²×D₄₇₂

yes

characteristic

186

C₄

—

187

C₄

—

188

C₄

—

472

—

189

C₄

—

472

—

190

C₄

#16837983

yes

characteristic

191

C₄

#16838202

yes

characteristic

192

C₂

—

193

C₂

#21435406

yes

characteristic

194

C₂

#21437249

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₄₇₂
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₅₉×#2151	C₂
socle	C₂²×C₅₉	C₂²×D₈
prime radical	C₂³×C₅₉	C₂×D₈
hypercenter	#2151	D₅₉
hypocenter	C₅₉	#33827
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2151

subgroup

G/H

center

C₂²

C₂²×D₄₇₂

commutator, cosocle

C₂×C₈×C₅₉

C₂⁴

Frattini

C₂×C₈

C₂³×D₅₉

Fitting

C₅₉×#2151

C₂

socle

C₂²×C₅₉

C₂²×D₈

prime radical

C₂³×C₅₉

C₂×D₈

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2151

N	H ≅ G/N	type
2. #21426261	C₂	semidirect
5. #21426147	C₂	semidirect
122. C₅₉	#33827	semidirect

H ≅ G/N

type

2. #21426261

C₂

semidirect

5. #21426147

C₂

semidirect

122. C₅₉

#33827

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#21434755	2
#21426261	2
#21426204	2
#21426187	2
#21426147	2
C₅₉×#2151	2
#33827	59

subgroup

index

C₅₉×#2151

Subgroups — #28745121

Conjugacy classes of subgroups

Complements

Maximal subgroups