Group #28481709 — subgroups

Subgroups — #28481709

Order 15104.

336 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 109 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	#21433031	C₂	1	1	yes	maximal, characteristic
2	#21422851	C₂	1	2	yes	maximal
3	#21422834	C₂	1	1	yes	maximal, characteristic
4	C₅₉⋊#1309	C₂	1	2	yes	maximal
5	C₅₉×#1561	C₂	1	1	yes	maximal, characteristic
6	#16837533	—	2	2	no	—
7	Q₈×C₅₉⋊C₈	—	2	2	no	—
8	#16837429	C₄	1	2	yes	—
9	#16837426	C₂²	1	2	yes	—
10	#16837426	C₂²	1	1	yes	characteristic
11	#16837358	C₄	1	2	yes	—
12	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
13	#16837152	—	2	2	no	—
14	#16837152	—	2	2	no	—
15	#16837148	—	2	2	no	—
16	C₅₉×#533	C₂²	1	1	yes	characteristic
17	C₅₉×#423	C₂²	1	2	yes	—
18	C₅₉×#332	—	2	2	no	—
19	C₂×#11745664	—	2	2	no	—
20	C₂×#11745664	C₂×C₄	1	2	yes	—
21	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
23	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
24	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
25	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
26	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
27	#14273638	—	2	2	no	—
28	C₄×C₅₉⋊C₈	—	2	2	no	—
29	C₄×C₅₉⋊C₈	D₄	1	2	yes	—
30	C₄×C₅₉⋊C₈	D₄	1	2	yes	—
31	C₂×M₁₆×C₅₉	—	2	2	no	—
32	C₄⋊Q₈×C₅₉	D₄	1	2	yes	—
33	#125×C₅₉	Q₈	1	2	yes	—
34	C₂²⋊Q₈×C₅₉	—	2	2	no	—
35	C₄×Q₈×C₅₉	—	2	2	no	—
36	#117×C₅₉	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
38	C₄⋊C₈×C₅₉	—	2	2	no	—
39	C₄⋊C₈×C₅₉	—	2	2	no	—
40	C₂×Q₈×C₅₉	—	2	2	no	—
41	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₉	#41	1	2	yes	—
43	M₁₆×C₅₉	—	8	2	no	—
44	C₂×C₈×C₅₉	—	4	2	no	—
45	C₄⋊C₄×C₅₉	—	2	2	no	—
46	C₄⋊C₄×C₅₉	—	2	4	no	—
47	C₄⋊C₄×C₅₉	C₄⋊C₄	1	4	yes	—
48	C₂²⋊C₄×C₅₉	—	2	2	no	—
49	C₄²×C₅₉	—	2	2	no	—
50	C₄²×C₅₉	C₂×D₄	1	2	yes	—
51	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
52	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
53	#11745664	—	4	2	no	—
54	#11745664	—	8	2	no	—
55	C₂×C₅₉⋊C₈	—	2	2	no	—
56	C₂×C₅₉⋊C₈	—	2	2	no	—
57	C₂×C₅₉⋊C₈	—	4	1	no	—
58	C₂×C₅₉⋊C₈	—	4	2	no	—
59	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	2	yes	—
61	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
62	C₂³×C₅₉	#117	1	1	yes	characteristic
63	Q₈×C₅₉	—	4	2	no	—
64	Q₈×C₅₉	—	4	2	no	—
65	C₂×C₄×C₅₉	—	2	1	no	—
66	C₂×C₄×C₅₉	—	2	1	no	—
67	C₂×C₄×C₅₉	—	2	2	no	—
68	C₂×C₄×C₅₉	—	2	2	no	—
69	C₂×C₄×C₅₉	—	2	4	no	—
70	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₅₉	#121	1	2	yes	—
73	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
74	C₈×C₅₉	—	8	2	no	—
75	C₅₉⋊C₈	—	2	1	no	—
76	C₅₉⋊C₈	—	2	2	no	—
77	C₅₉⋊C₈	—	4	2	no	—
78	C₅₉⋊C₈	—	8	2	no	—
79	C₅₉⋊C₈	#101	1	2	yes	—
80	#8637	—	59	1	no	maximal, Sylow
81	C₂²×C₅₉	—	2	1	no	—
82	C₂²×C₅₉	—	2	2	no	—
83	C₂²×C₅₉	#380	1	1	yes	characteristic
84	C₄×C₅₉	—	2	1	no	—
85	C₄×C₅₉	—	2	2	no	—
86	C₄×C₅₉	—	4	4	no	—
87	C₄×C₅₉	C₂×#101	1	1	yes	characteristic
88	C₄×C₅₉	#413	1	1	yes	characteristic
89	C₄×C₅₉	#471	1	2	yes	—
90	C₄×C₅₉	#471	1	2	yes	—
91	#2960	—	59	1	no	—
92	#1566	—	59	2	no	—
93	#1561	—	59	1	no	—
94	#1561	D₅₉	1	1	yes	characteristic
95	#1309	—	59	2	no	—
96	C₂×C₅₉	—	4	1	no	—
97	C₂×C₅₉	#1801	1	1	yes	characteristic
98	C₂×C₅₉	#1815	1	2	yes	—
99	#533	C₂×D₅₉	1	1	yes	characteristic
100	#446	—	118	2	no	—
101	C₈×Q₈	—	118	2	no	—
102	#424	—	59	2	no	—
103	#423	—	59	1	no	—
104	#423	—	59	2	no	—
105	#423	C₂×D₅₉	1	2	yes	—
106	#405	—	59	2	no	—
107	C₂×C₄×C₈	—	59	1	no	—
108	#334	—	118	2	no	—
109	#334	—	118	2	no	—
110	#332	—	2	2	no	—
111	#332	—	118	2	no	—
112	C₅₉	#8637	1	1	yes	characteristic, complemented, Sylow
113	C₂×M₁₆	—	2	2	no	—
114	C₂×M₁₆	—	59	2	no	—
115	C₂×M₁₆	—	118	2	no	—
116	C₂²×C₈	—	59	1	no	—
117	C₂²×C₈	—	59	1	no	—
118	C₄⋊Q₈	C₂²⋊D₅₉	1	2	yes	—
119	#125	Dic₁₁₈	1	2	yes	—
120	C₂²⋊Q₈	—	2	2	no	—
121	C₄×Q₈	—	2	2	no	—
122	#117	C₄×D₅₉	1	2	yes	—
123	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	118	4	no	—
127	C₄⋊C₈	—	118	2	no	—
128	C₄⋊C₈	—	118	2	no	—
129	C₄⋊C₈	—	118	2	no	—
130	#97	—	118	2	no	—
131	C₄×C₈	—	59	2	no	—
132	C₄×C₈	—	59	2	no	—
133	C₄×C₈	—	118	2	no	—
134	C₂×Q₈	—	2	2	no	—
135	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
136	C₂²×C₄	#11745687	1	2	yes	—
137	M₁₆	—	8	2	no	—
138	M₁₆	—	472	2	no	—
139	M₁₆	—	236	2	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	59	1	no	—
142	C₂×C₈	—	59	1	no	—
143	C₂×C₈	—	59	2	no	—
144	C₂×C₈	—	118	2	no	—
145	C₂×C₈	—	118	2	no	—
146	C₂×C₈	—	236	1	no	—
147	C₂×C₈	—	236	2	no	—
148	C₄⋊C₄	—	2	2	no	—
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	Dic₅₉⋊C₄	1	4	yes	—
151	C₂²⋊C₄	—	2	2	no	—
152	C₄²	—	2	2	no	—
153	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
154	C₄²	D₄×D₅₉	1	2	yes	—
155	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
156	C₂³	#14273683	1	1	yes	characteristic
157	Q₈	—	4	2	no	—
158	Q₈	—	4	2	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
165	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
166	C₂×C₄	#14273699	1	2	yes	—
167	C₂×C₄	#14273704	1	2	yes	—
168	C₈	—	8	2	no	—
169	C₈	—	59	2	no	—
170	C₈	—	118	1	no	—
171	C₈	—	118	2	no	—
172	C₈	—	472	2	no	—
173	C₈	—	236	2	no	—
174	C₂²	—	2	2	no	—
175	C₂²	—	2	1	no	—
176	C₂²	#16837260	1	1	yes	characteristic
177	C₄	—	2	2	no	—
178	C₄	—	2	1	no	—
179	C₄	—	4	4	no	—
180	C₄	#101×D₅₉	1	1	yes	characteristic
181	C₄	#16837385	1	1	yes	characteristic
182	C₄	#16837663	1	2	yes	—
183	C₄	#16837667	1	2	yes	—
184	C₂	—	4	1	no	—
185	C₂	#21424090	1	1	yes	characteristic
186	C₂	#21424153	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21433031

C₂

yes

maximal, characteristic

#21422851

C₂

yes

maximal

#21422834

C₂

yes

maximal, characteristic

C₅₉⋊#1309

C₂

yes

maximal

C₅₉×#1561

C₂

yes

maximal, characteristic

#16837533

—

Q₈×C₅₉⋊C₈

—

#16837429

C₄

yes

—

#16837426

C₂²

yes

—

#16837426

C₂²

yes

characteristic

#16837358

C₄

yes

—

C₂×C₄×C₅₉⋊C₈

C₂²

yes

—

—

—

C₅₉×#533

C₂²

yes

characteristic

C₅₉×#423

C₂²

yes

—

C₅₉×#332

—

C₂×#11745664

—

C₂×#11745664

C₂×C₄

yes

—

C₂²×C₅₉⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₅₉⋊C₈

C₂×C₄

yes

characteristic

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

#14273638

—

C₄×C₅₉⋊C₈

—

C₄×C₅₉⋊C₈

D₄

yes

—

C₄×C₅₉⋊C₈

D₄

yes

—

C₂×M₁₆×C₅₉

—

C₄⋊Q₈×C₅₉

D₄

yes

—

#125×C₅₉

Q₈

yes

—

C₂²⋊Q₈×C₅₉

—

C₄×Q₈×C₅₉

—

#117×C₅₉

C₂×C₄

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊C₈×C₅₉

—

C₄⋊C₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂²×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₉

#41

yes

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

C₄⋊C₄

yes

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

C₂×D₄

yes

—

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄²×C₅₉

C₂×Q₈

yes

characteristic

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂²⋊C₄

yes

characteristic

C₂×C₅₉⋊C₈

C₂²⋊C₄

yes

—

C₂×C₅₉⋊C₈

C₂²⋊C₄

yes

characteristic

C₂³×C₅₉

#117

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

#121

yes

—

C₂×C₄×C₅₉

C₂²⋊Q₈

yes

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

#101

yes

—

#8637

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#380

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

C₄×C₅₉

#471

yes

—

—

—

—

D₅₉

yes

characteristic

#1309

—

C₂×C₅₉

—

C₂×C₅₉

#1801

yes

characteristic

C₂×C₅₉

#1815

yes

—

#533

C₂×D₅₉

yes

characteristic

100

#446

—

118

—

101

C₈×Q₈

—

118

—

102

#424

—

103

#423

—

104

#423

—

105

#423

C₂×D₅₉

yes

—

106

#405

—

107

C₂×C₄×C₈

—

108

#334

—

118

—

109

#334

—

118

—

110

#332

—

111

#332

—

118

—

112

C₅₉

#8637

yes

characteristic, complemented, Sylow

113

C₂×M₁₆

—

114

C₂×M₁₆

—

115

C₂×M₁₆

—

118

—

116

C₂²×C₈

—

117

C₂²×C₈

—

118

C₄⋊Q₈

C₂²⋊D₅₉

yes

—

119

#125

Dic₁₁₈

yes

—

120

C₂²⋊Q₈

—

121

C₄×Q₈

—

122

#117

C₄×D₅₉

yes

—

123

C₂×C₄²

C₂²×D₅₉

yes

characteristic

124

C₄⋊C₈

—

125

C₄⋊C₈

—

126

C₄⋊C₈

—

118

—

127

C₄⋊C₈

—

118

—

128

C₄⋊C₈

—

118

—

129

C₄⋊C₈

—

118

—

130

#97

—

118

—

131

C₄×C₈

—

132

C₄×C₈

—

133

C₄×C₈

—

118

—

134

C₂×Q₈

—

135

C₂²×C₄

C₂×C₄×D₅₉

yes

characteristic

136

C₂²×C₄

#11745687

yes

—

137

M₁₆

—

138

M₁₆

—

472

—

139

M₁₆

—

236

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

118

—

145

C₂×C₈

—

118

—

146

C₂×C₈

—

236

—

147

C₂×C₈

—

236

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

Dic₅₉⋊C₄

yes

—

151

C₂²⋊C₄

—

152

C₄²

—

153

C₄²

C₂×Dic₁₁₈

yes

characteristic

154

C₄²

D₄×D₅₉

yes

—

155

C₄²

C₂×C₂²⋊D₅₉

yes

characteristic

156

C₂³

#14273683

yes

characteristic

157

Q₈

—

158

Q₈

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

C₂²⋊C₄×D₅₉

yes

characteristic

165

C₂×C₄

C₂×Dic₅₉⋊C₄

yes

characteristic

166

C₂×C₄

#14273699

yes

—

167

C₂×C₄

#14273704

yes

—

168

C₈

—

169

C₈

—

170

C₈

—

118

—

171

C₈

—

118

—

172

C₈

—

472

—

173

C₈

—

236

—

174

C₂²

—

175

C₂²

—

176

C₂²

#16837260

yes

characteristic

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

#101×D₅₉

yes

characteristic

181

C₄

#16837385

yes

characteristic

182

C₄

#16837663

yes

—

183

C₄

#16837667

yes

—

184

C₂

—

185

C₂

#21424090

yes

characteristic

186

C₂

#21424153

yes

—

	subgroup	G/H
center	C₂²	#16837260
commutator	C₂²×C₄×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1561	C₂
socle	C₂²×C₅₉	#380
prime radical	C₂³×C₅₉	#117
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1561	D₅₉
hypocenter	C₅₉	#8637
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1561

subgroup

G/H

center

C₂²

#16837260

commutator

C₂²×C₄×C₅₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#1561

C₂

C₂²×C₅₉

C₂³×C₅₉

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1561

N	H ≅ G/N	type
112. C₅₉	#8637	semidirect

H ≅ G/N

type

112. C₅₉

#8637

semidirect

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

subgroup	index
#21433031	2
#21422851	2
#21422834	2
C₅₉⋊#1309	2
C₅₉×#1561	2
#8637	59

subgroup

index

#21433031

#21422851

#21422834

C₅₉⋊#1309

C₅₉×#1561

#8637

Subgroups — #28481709

Conjugacy classes of subgroups

Complements

Maximal subgroups