Group #28576396 — subgroups

Subgroups — #28576396

Order 15104.

706 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 706 conjugacy classes of subgroups — too many to draw. Only its 187 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	#21434681	C₂	1	4	yes	maximal
2	#21434607	C₂	1	2	yes	maximal
3	#21423713	C₂	1	2	yes	maximal, complemented
4	#21423706	C₂	1	1	yes	maximal, characteristic, complemented
5	#21423692	C₂	1	4	yes	maximal, complemented
6	C₈²×D₅₉	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#1710	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837957	C₂²	1	4	yes	—
9	#16837807	C₂²	1	2	yes	—
10	#16837805	C₂²	1	4	yes	—
11	#16837803	—	2	4	no	—
12	#16837798	C₂²	1	4	yes	—
13	#16837781	C₂²	1	4	yes	—
14	#16837753	C₂²	1	4	yes	—
15	#16837642	—	2	8	no	—
16	#16837494	—	2	8	no	—
17	#16837439	—	2	4	no	—
18	C₄×C₈×D₅₉	C₂²	1	2	yes	—
19	C₄×C₈×D₅₉	C₂²	1	1	yes	characteristic
20	C₈×C₅₉⋊C₈	C₂²	1	2	yes	—
21	C₈×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
22	C₅₉×#500	C₂²	1	2	yes	—
23	C₅₉×#487	C₂²	1	4	yes	—
24	C₈²×C₅₉	C₂²	1	1	yes	characteristic
25	C₂×#11745672	—	2	8	no	—
26	C₂×C₈×D₅₉	D₄	1	4	yes	—
27	C₂×C₈×D₅₉	D₄	1	2	yes	—
28	#14273727	C₂³	1	4	yes	—
29	#14273718	—	2	4	no	—
30	#14273707	—	2	4	no	—
31	#14273703	—	2	4	no	—
32	Q₈×Dic₅₉	—	2	4	no	—
33	#14273692	—	2	4	no	—
34	#14273685	—	2	4	no	—
35	C₄²×D₅₉	C₂³	1	1	yes	characteristic
36	#14273661	—	2	4	no	—
37	#14273660	—	2	8	no	—
38	#14273658	—	2	8	no	—
39	#14273652	—	2	8	no	—
40	C₄×C₅₉⋊C₈	C₂³	1	2	yes	—
41	C₄×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
42	C₈×Dic₅₉	D₄	1	4	yes	—
43	C₈×Dic₅₉	D₄	1	2	yes	—
44	C₄⋊Q₈×C₅₉	C₂³	1	4	yes	—
45	#107×C₅₉	—	2	4	no	—
46	#103×C₅₉	—	2	8	no	—
47	C₄×C₈×C₅₉	C₂³	1	2	yes	—
48	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
49	C₂×Q₈×C₅₉	—	2	4	no	—
50	C₂×C₄×D₅₉	C₂×D₄	1	2	yes	—
51	C₂×C₄×D₅₉	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₁₁₈	—	2	4	no	—
53	C₂×C₈×C₅₉	C₂×D₄	1	4	yes	—
54	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
55	C₄⋊C₄×C₅₉	—	2	4	no	—
56	C₄⋊C₄×C₅₉	—	2	4	no	—
57	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
58	#11745672	—	4	16	no	—
59	(C₂×D₅₉)⋊C₄	—	4	4	no	—
60	C₄⋊Dic₅₉	—	2	4	no	—
61	C₄⋊Dic₅₉	—	2	4	no	—
62	C₄⋊Dic₅₉	—	4	4	no	—
63	Dic₅₉⋊C₄	—	4	4	no	—
64	C₄×Dic₅₉	—	4	4	no	—
65	C₄×Dic₅₉	C₂×D₄	1	2	yes	—
66	C₄×Dic₅₉	C₂×D₄	1	1	yes	characteristic
67	C₂×C₅₉⋊C₈	C₂×D₄	1	2	yes	—
68	C₂×C₅₉⋊C₈	C₂×D₄	1	4	yes	—
69	C₈×D₅₉	—	2	8	no	—
70	C₈×D₅₉	Q₁₆	1	8	yes	complemented
71	C₂²×D₅₉	#127	1	1	yes	characteristic
72	Q₈×C₅₉	—	4	8	no	—
73	C₂×C₄×C₅₉	—	4	4	no	—
74	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
75	C₂×C₄×C₅₉	C₂²×D₄	1	2	yes	—
76	C₂×Dic₅₉	—	4	4	no	—
77	C₂×Dic₅₉	#127	1	2	yes	—
78	C₂×Dic₅₉	#127	1	1	yes	characteristic
79	C₄×D₅₉	—	2	4	no	—
80	C₄×D₅₉	C₂×Q₁₆	1	4	yes	—
81	Dic₁₁₈	—	4	8	no	—
82	C₈×C₅₉	—	2	4	no	—
83	C₈×C₅₉	C₂×Q₁₆	1	4	yes	—
84	C₅₉⋊C₈	C₂×Q₁₆	1	4	yes	—
85	C₅₉⋊C₈	#135	1	8	yes	—
86	#19604	—	59	1	no	maximal, Sylow
87	C₂²×C₅₉	C₂×#127	1	1	yes	characteristic
88	C₂×D₅₉	—	2	2	no	—
89	C₂×D₅₉	#494	1	2	yes	—
90	C₄×C₅₉	—	8	4	no	—
91	C₄×C₅₉	C₂²×Q₁₆	1	2	yes	—
92	C₄×C₅₉	C₂×#135	1	4	yes	—
93	Dic₅₉	—	2	2	no	—
94	Dic₅₉	—	8	4	no	—
95	Dic₅₉	#494	1	2	yes	—
96	#3170	—	59	4	no	—
97	#3160	—	59	2	no	—
98	#1713	—	59	2	no	—
99	#1710	—	59	1	no	—
100	#1710	D₅₉	1	1	yes	characteristic, complemented
101	#1707	—	59	4	no	—
102	C₂×C₈²	—	59	1	no	—
103	C₂×C₅₉	C₂×#494	1	1	yes	characteristic
104	C₂×C₅₉	#3147	1	2	yes	—
105	D₅₉	—	2	2	no	—
106	#533	—	59	4	no	—
107	#500	—	59	4	no	—
108	#500	—	59	2	no	—
109	#500	C₂×D₅₉	1	2	yes	—
110	#499	—	118	4	no	—
111	#498	—	59	4	no	—
112	#494	—	59	4	no	—
113	#487	—	59	4	no	—
114	#487	C₂×D₅₉	1	4	yes	—
115	#467	—	118	8	no	—
116	C₄×Q₁₆	—	118	8	no	—
117	#427	—	118	4	no	—
118	C₂×C₄×C₈	—	59	1	no	—
119	C₂×C₄×C₈	—	59	2	no	—
120	C₈²	—	59	2	no	—
121	C₈²	—	59	1	no	—
122	C₈²	C₂×D₅₉	1	1	yes	characteristic
123	C₅₉	#19604	1	1	yes	characteristic, complemented, Sylow
124	C₂×Q₁₆	—	118	8	no	—
125	C₂²×C₈	—	59	2	no	—
126	C₂²×C₈	—	59	4	no	—
127	C₄⋊Q₈	—	59	4	no	—
128	C₄⋊Q₈	C₂²×D₅₉	1	4	yes	—
129	#125	—	118	4	no	—
130	C₂²⋊Q₈	—	118	4	no	—
131	C₂²⋊Q₈	—	118	4	no	—
132	C₄×Q₈	—	118	4	no	—
133	C₄×Q₈	—	118	4	no	—
134	#117	—	118	4	no	—
135	C₂×C₄²	—	59	1	no	—
136	#107	—	2	4	no	—
137	#107	—	118	8	no	—
138	#107	—	118	4	no	—
139	#106	—	118	8	no	—
140	#103	—	2	8	no	—
141	#103	—	118	8	no	—
142	C₄×C₈	—	59	4	no	—
143	C₄×C₈	—	59	2	no	—
144	C₄×C₈	—	59	2	no	—
145	C₄×C₈	—	59	1	no	—
146	C₄×C₈	C₂²×D₅₉	1	2	yes	—
147	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
148	C₂×Q₈	—	2	4	no	—
149	C₂×Q₈	—	118	4	no	—
150	C₂²×C₄	—	59	1	no	—
151	C₂²×C₄	—	59	2	no	—
152	Q₁₆	—	236	16	no	—
153	C₂×C₈	—	59	2	no	—
154	C₂×C₈	—	59	4	no	—
155	C₂×C₈	—	59	8	no	—
156	C₂×C₈	—	118	8	no	—
157	C₂×C₈	D₄×D₅₉	1	4	yes	—
158	C₂×C₈	D₄×D₅₉	1	2	yes	—
159	C₄⋊C₄	—	2	4	no	—
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	118	4	no	—
162	C₄⋊C₄	—	118	4	no	—
163	C₄⋊C₄	—	236	4	no	—
164	C₄⋊C₄	—	236	4	no	—
165	C₂²⋊C₄	—	236	4	no	—
166	C₄²	—	59	2	no	—
167	C₄²	—	59	1	no	—
168	C₄²	—	236	4	no	—
169	C₄²	C₂³×D₅₉	1	1	yes	characteristic
170	C₂³	—	59	1	no	—
171	Q₈	—	4	8	no	—
172	Q₈	—	236	8	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	—	59	1	no	—
175	C₂×C₄	—	59	4	no	—
176	C₂×C₄	—	59	2	no	—
177	C₂×C₄	—	118	4	no	—
178	C₂×C₄	—	236	4	no	—
179	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
180	C₂×C₄	C₂×D₄×D₅₉	1	2	yes	—
181	C₈	—	2	4	no	—
182	C₈	—	59	4	no	—
183	C₈	—	59	8	no	—
184	C₈	Q₁₆×D₅₉	1	4	yes	—
185	C₂²	—	59	2	no	—
186	C₂²	—	118	2	no	—
187	C₂²	#127×D₅₉	1	1	yes	characteristic
188	C₄	—	8	4	no	—
189	C₄	—	59	2	no	—
190	C₄	—	118	2	no	—
191	C₄	—	472	4	no	—
192	C₄	C₂×Q₁₆×D₅₉	1	2	yes	—
193	C₄	#135×D₅₉	1	4	yes	—
194	C₂	—	118	2	no	—
195	C₂	#494×D₅₉	1	1	yes	characteristic
196	C₂	#21434521	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21434681

C₂

yes

maximal

#21434607

C₂

yes

maximal

#21423713

C₂

yes

maximal, complemented

#21423706

C₂

yes

maximal, characteristic, complemented

#21423692

C₂

yes

maximal, complemented

C₈²×D₅₉

C₂

yes

maximal, characteristic

C₅₉×#1710

C₂

yes

maximal, characteristic, complemented

#16837957

C₂²

yes

—

#16837807

C₂²

yes

—

#16837805

C₂²

yes

—

#16837803

—

#16837798

C₂²

yes

—

#16837781

C₂²

yes

—

#16837753

C₂²

yes

—

#16837642

—

#16837494

—

#16837439

—

C₄×C₈×D₅₉

C₂²

yes

—

C₄×C₈×D₅₉

C₂²

yes

characteristic

C₈×C₅₉⋊C₈

C₂²

yes

—

C₈×C₅₉⋊C₈

C₂²

yes

characteristic

C₅₉×#500

C₂²

yes

—

C₅₉×#487

C₂²

yes

—

C₈²×C₅₉

C₂²

yes

characteristic

C₂×#11745672

—

C₂×C₈×D₅₉

D₄

yes

—

C₂×C₈×D₅₉

D₄

yes

—

#14273727

C₂³

yes

—

#14273718

—

#14273707

—

#14273703

—

Q₈×Dic₅₉

—

#14273692

—

#14273685

—

C₄²×D₅₉

C₂³

yes

—

—

—

—

C₄×C₅₉⋊C₈

C₂³

yes

—

C₄×C₅₉⋊C₈

C₂³

yes

characteristic

C₈×Dic₅₉

D₄

yes

—

C₈×Dic₅₉

D₄

yes

—

C₄⋊Q₈×C₅₉

C₂³

yes

—

#107×C₅₉

—

#103×C₅₉

—

C₄×C₈×C₅₉

C₂³

yes

—

C₄×C₈×C₅₉

C₂³

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×C₄×D₅₉

C₂×D₄

yes

—

C₂×C₄×D₅₉

C₂×D₄

yes

characteristic

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

C₂×D₄

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂⁴

yes

characteristic

#11745672

—

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₄×Dic₅₉

C₂×D₄

yes

—

C₄×Dic₅₉

C₂×D₄

yes

characteristic

C₂×C₅₉⋊C₈

C₂×D₄

yes

—

C₂×C₅₉⋊C₈

C₂×D₄

yes

—

C₈×D₅₉

—

C₈×D₅₉

Q₁₆

yes

complemented

C₂²×D₅₉

#127

yes

characteristic

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂²×D₄

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

#127

yes

—

C₂×Dic₅₉

#127

yes

characteristic

C₄×D₅₉

—

C₄×D₅₉

C₂×Q₁₆

yes

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

C₂×Q₁₆

yes

—

C₅₉⋊C₈

C₂×Q₁₆

yes

—

C₅₉⋊C₈

#135

yes

—

—

C₂²×C₅₉

yes

C₂×D₅₉

—

C₂×D₅₉

#494

yes

—

C₄×C₅₉

—

C₄×C₅₉

C₂²×Q₁₆

yes

—

C₄×C₅₉

C₂×#135

yes

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

#494

yes

—

—

—

—

—

100

#1710

D₅₉

yes

characteristic, complemented

101

#1707

—

102

C₂×C₈²

—

103

C₂×C₅₉

C₂×#494

yes

characteristic

104

C₂×C₅₉

#3147

yes

—

105

D₅₉

—

106

#533

—

107

#500

—

108

#500

—

109

#500

C₂×D₅₉

yes

—

110

#499

—

118

—

111

#498

—

112

#494

—

113

#487

—

114

#487

C₂×D₅₉

yes

—

115

#467

—

118

—

116

C₄×Q₁₆

—

118

—

117

#427

—

118

—

118

C₂×C₄×C₈

—

119

C₂×C₄×C₈

—

120

C₈²

—

121

C₈²

—

122

C₈²

C₂×D₅₉

yes

characteristic

123

C₅₉

#19604

yes

characteristic, complemented, Sylow

124

C₂×Q₁₆

—

118

—

125

C₂²×C₈

—

126

C₂²×C₈

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

C₂²×D₅₉

yes

—

129

#125

—

118

—

130

C₂²⋊Q₈

—

118

—

131

C₂²⋊Q₈

—

118

—

132

C₄×Q₈

—

118

—

133

C₄×Q₈

—

118

—

134

#117

—

118

—

135

C₂×C₄²

—

136

#107

—

137

#107

—

118

—

138

#107

—

118

—

139

#106

—

118

—

140

#103

—

141

#103

—

118

—

142

C₄×C₈

—

143

C₄×C₈

—

144

C₄×C₈

—

145

C₄×C₈

—

146

C₄×C₈

C₂²×D₅₉

yes

—

147

C₄×C₈

C₂²×D₅₉

yes

characteristic

148

C₂×Q₈

—

149

C₂×Q₈

—

118

—

150

C₂²×C₄

—

151

C₂²×C₄

—

152

Q₁₆

—

236

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

118

—

157

C₂×C₈

D₄×D₅₉

yes

—

158

C₂×C₈

D₄×D₅₉

yes

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

118

—

162

C₄⋊C₄

—

118

—

163

C₄⋊C₄

—

236

—

164

C₄⋊C₄

—

236

—

165

C₂²⋊C₄

—

236

—

166

C₄²

—

167

C₄²

—

168

C₄²

—

236

—

169

C₄²

C₂³×D₅₉

yes

characteristic

170

C₂³

—

171

Q₈

—

172

Q₈

—

236

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

118

—

178

C₂×C₄

—

236

—

179

C₂×C₄

C₂×D₄×D₅₉

yes

characteristic

180

C₂×C₄

C₂×D₄×D₅₉

yes

—

181

C₈

—

182

C₈

—

183

C₈

—

184

C₈

Q₁₆×D₅₉

yes

—

185

C₂²

—

186

C₂²

—

118

—

187

C₂²

#127×D₅₉

yes

characteristic

188

C₄

—

189

C₄

—

190

C₄

—

118

—

191

C₄

—

472

—

192

C₄

C₂×Q₁₆×D₅₉

yes

—

193

C₄

#135×D₅₉

yes

—

194

C₂

—

118

—

195

C₂

#494×D₅₉

yes

characteristic

196

C₂

#21434521

yes

—

	subgroup	G/H
center	C₂²	#127×D₅₉
commutator, cosocle	C₄²×C₅₉	C₂⁴
Frattini	C₄²	C₂³×D₅₉
Fitting	C₅₉×#1710	C₂
socle	C₂²×C₅₉	C₂×#127
prime radical	C₂²×D₅₉	#127
hypercenter	#1710	D₅₉
hypocenter	C₅₉	#19604
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1710

subgroup

G/H

center

C₂²

#127×D₅₉

commutator, cosocle

C₄²×C₅₉

C₂⁴

Frattini

C₄²

C₂³×D₅₉

Fitting

C₅₉×#1710

C₂

C₂²×C₅₉

C₂²×D₅₉

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #1710

N	H ≅ G/N	type
3. #21423713	C₂	semidirect
4. #21423706	C₂	semidirect
5. #21423692	C₂	semidirect
7. C₅₉×#1710	C₂	semidirect
70. C₈×D₅₉	Q₁₆	semidirect
100. #1710	D₅₉	semidirect
123. C₅₉	#19604	semidirect

H ≅ G/N

type

3. #21423713

C₂

semidirect

4. #21423706

C₂

semidirect

5. #21423692

C₂

semidirect

7. C₅₉×#1710

C₂

semidirect

70. C₈×D₅₉

Q₁₆

semidirect

100. #1710

D₅₉

semidirect

123. C₅₉

#19604

semidirect

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

subgroup	index
#21434681	2
#21434607	2
#21423713	2
#21423706	2
#21423692	2
C₈²×D₅₉	2
C₅₉×#1710	2
#19604	59

subgroup

index

C₈²×D₅₉

C₅₉×#1710

#19604

Subgroups — #28576396

Conjugacy classes of subgroups

Complements

Maximal subgroups