Group #28422738 — subgroups

Subgroups — C₅₉×#10629 #28422738

Order 15104.

310 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 98 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	C₅₉×#1963	C₂	1	1	yes	maximal, characteristic
2	C₅₉×#1832	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₉×#1540	C₂	1	2	yes	maximal
4	C₅₉×#1301	C₂	1	2	yes	maximal, complemented
5	C₅₉×#1274	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₉×#532	C₂²	1	1	yes	characteristic
7	C₅₉×#431	C₂²	1	2	yes	—
8	C₅₉×#423	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
10	C₂×#104×C₅₉	—	2	2	no	—
11	C₅₉×#376	C₂²	1	1	yes	characteristic
12	C₅₉×#336	—	2	2	no	—
13	C₅₉×#332	—	2	2	no	—
14	C₅₉×#331	—	2	2	no	—
15	C₂×#41×C₅₉	—	2	1	no	—
16	C₂×M₁₆×C₅₉	—	2	1	no	—
17	C₂²×C₈×C₅₉	—	2	2	no	—
18	C₄⋊Q₈×C₅₉	D₄	1	1	yes	characteristic
19	#127×C₅₉	D₄	1	1	yes	characteristic
20	#124×C₅₉	—	2	1	no	—
21	#121×C₅₉	—	2	2	no	—
22	C₄×D₄×C₅₉	—	2	2	no	—
23	C₂×C₄²×C₅₉	D₄	1	2	yes	—
24	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
25	C₄⋊C₈×C₅₉	—	2	2	no	—
26	C₄⋊C₈×C₅₉	—	2	2	no	—
27	C₄⋊C₈×C₅₉	—	2	2	no	—
28	C₄⋊C₈×C₅₉	—	2	2	no	—
29	#104×C₅₉	—	4	4	no	—
30	C₂²⋊C₈×C₅₉	—	2	2	no	—
31	C₄×C₈×C₅₉	—	2	2	no	—
32	#95×C₅₉	—	2	2	no	—
33	#41×C₅₉	—	4	2	no	—
34	#41×C₅₉	—	4	2	no	—
35	C₂×Q₈×C₅₉	—	2	1	no	—
36	C₂×D₄×C₅₉	—	2	1	no	—
37	C₂×D₄×C₅₉	—	2	2	no	—
38	C₂×D₄×C₅₉	—	4	1	no	—
39	C₂²×C₄×C₅₉	—	2	2	no	—
40	C₂²×C₄×C₅₉	—	4	1	no	—
41	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₉	#41	1	2	yes	—
43	M₁₆×C₅₉	—	4	2	no	—
44	C₂×C₈×C₅₉	—	4	2	no	—
45	C₂×C₈×C₅₉	—	4	2	no	—
46	C₂×C₈×C₅₉	—	4	1	no	—
47	C₂×C₈×C₅₉	—	4	2	no	—
48	C₂×C₈×C₅₉	—	4	2	no	—
49	C₄⋊C₄×C₅₉	—	2	2	no	—
50	C₂²⋊C₄×C₅₉	—	4	2	no	—
51	C₄²×C₅₉	—	2	2	no	—
52	C₄²×C₅₉	—	2	2	no	—
53	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₅₉	#41	1	1	yes	characteristic
55	C₄²×C₅₉	#41	1	2	yes	—
56	C₂³×C₅₉	—	4	1	no	—
57	C₂³×C₅₉	#123	1	1	yes	characteristic
58	Q₈×C₅₉	—	4	2	no	—
59	D₄×C₅₉	—	4	2	no	—
60	D₄×C₅₉	—	8	2	no	—
61	D₄×C₅₉	—	8	2	no	—
62	C₂×C₄×C₅₉	—	4	2	no	—
63	C₂×C₄×C₅₉	—	4	4	no	—
64	C₂×C₄×C₅₉	—	4	1	no	—
65	C₂×C₄×C₅₉	—	4	2	no	—
66	C₂×C₄×C₅₉	—	8	1	no	—
67	C₂×C₄×C₅₉	—	8	1	no	—
68	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
69	C₂×C₄×C₅₉	#123	1	2	yes	—
70	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
71	C₂×C₄×C₅₉	#126	1	2	yes	—
72	C₂×C₄×C₅₉	#135	1	2	yes	—
73	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
74	C₈×C₅₉	—	8	2	no	—
75	C₈×C₅₉	—	8	2	no	—
76	C₈×C₅₉	—	8	1	no	—
77	#10629	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂²×C₅₉	—	2	2	no	—
79	C₂²×C₅₉	—	8	1	no	—
80	C₂²×C₅₉	—	8	2	no	—
81	C₂²×C₅₉	#397	1	1	yes	characteristic
82	C₂²×C₅₉	#482	1	2	yes	—
83	C₄×C₅₉	—	2	2	no	—
84	C₄×C₅₉	—	2	2	no	—
85	C₄×C₅₉	—	8	2	no	—
86	C₄×C₅₉	—	8	1	no	—
87	C₄×C₅₉	#463	1	2	yes	—
88	C₄×C₅₉	#488	1	1	yes	characteristic
89	C₄×C₅₉	#490	1	1	yes	characteristic
90	#1963	C₂×C₅₉	1	1	yes	characteristic
91	#1832	C₂×C₅₉	1	1	yes	characteristic, complemented
92	#1540	C₂×C₅₉	1	2	yes	—
93	#1301	C₂×C₅₉	1	2	yes	complemented
94	#1274	C₂×C₅₉	1	1	yes	characteristic, complemented
95	C₂×C₅₉	—	2	2	no	—
96	C₂×C₅₉	—	16	1	no	—
97	C₂×C₅₉	#2073	1	1	yes	characteristic
98	C₂×C₅₉	#2080	1	2	yes	—
99	#532	C₂²×C₅₉	1	1	yes	characteristic
100	#431	C₂²×C₅₉	1	2	yes	—
101	#423	C₂²×C₅₉	1	1	yes	characteristic
102	C₂×C₄⋊C₈	C₂²×C₅₉	1	2	yes	—
103	C₂×#104	—	2	2	no	—
104	#376	C₂²×C₅₉	1	1	yes	characteristic
105	#336	—	2	2	no	—
106	#332	—	2	2	no	—
107	#331	—	2	2	no	—
108	C₅₉	#10629	1	1	yes	characteristic, direct product factor, Sylow
109	C₂×#41	—	2	1	no	—
110	C₂×M₁₆	—	2	1	no	—
111	C₂²×C₈	—	2	2	no	—
112	C₄⋊Q₈	D₄×C₅₉	1	1	yes	characteristic
113	#127	D₄×C₅₉	1	1	yes	characteristic
114	#124	—	2	1	no	—
115	#121	—	2	2	no	—
116	C₄×D₄	—	2	2	no	—
117	C₂×C₄²	D₄×C₅₉	1	2	yes	—
118	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	2	2	no	—
121	C₄⋊C₈	—	2	2	no	—
122	C₄⋊C₈	—	2	2	no	—
123	#104	—	4	4	no	—
124	C₂²⋊C₈	—	2	2	no	—
125	C₄×C₈	—	2	2	no	—
126	#95	—	2	2	no	—
127	#41	—	4	2	no	—
128	#41	—	4	2	no	—
129	C₂×Q₈	—	2	1	no	—
130	C₂×D₄	—	2	1	no	—
131	C₂×D₄	—	2	2	no	—
132	C₂×D₄	—	4	1	no	—
133	C₂²×C₄	—	2	2	no	—
134	C₂²×C₄	—	4	1	no	—
135	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
136	C₂²×C₄	#41×C₅₉	1	2	yes	—
137	M₁₆	—	4	2	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	—	4	1	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	—	4	2	no	—
143	C₄⋊C₄	—	2	2	no	—
144	C₂²⋊C₄	—	4	2	no	—
145	C₄²	—	2	2	no	—
146	C₄²	—	2	2	no	—
147	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
148	C₄²	#41×C₅₉	1	1	yes	characteristic
149	C₄²	#41×C₅₉	1	2	yes	—
150	C₂³	—	4	1	no	—
151	C₂³	#123×C₅₉	1	1	yes	characteristic
152	Q₈	—	4	2	no	—
153	D₄	—	4	2	no	—
154	D₄	—	8	2	no	—
155	D₄	—	8	2	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	—	4	4	no	—
158	C₂×C₄	—	4	1	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	—	8	1	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	#121×C₅₉	1	1	yes	characteristic
163	C₂×C₄	#123×C₅₉	1	2	yes	—
164	C₂×C₄	#124×C₅₉	1	1	yes	characteristic
165	C₂×C₄	#126×C₅₉	1	2	yes	—
166	C₂×C₄	#135×C₅₉	1	2	yes	—
167	C₂×C₄	AGL(1,ℤ/8)×C₅₉	1	2	yes	—
168	C₈	—	8	2	no	—
169	C₈	—	8	2	no	—
170	C₈	—	8	1	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	8	1	no	—
173	C₂²	—	8	2	no	—
174	C₂²	C₅₉×#397	1	1	yes	characteristic
175	C₂²	C₅₉×#482	1	2	yes	—
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	8	2	no	—
179	C₄	—	8	1	no	—
180	C₄	C₅₉×#463	1	2	yes	—
181	C₄	C₅₉×#488	1	1	yes	characteristic
182	C₄	C₅₉×#490	1	1	yes	characteristic
183	C₂	—	2	2	no	—
184	C₂	—	16	1	no	—
185	C₂	C₅₉×#2073	1	1	yes	characteristic
186	C₂	C₅₉×#2080	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₉×#1963

C₂

yes

maximal, characteristic

C₅₉×#1832

C₂

yes

maximal, characteristic, complemented

C₅₉×#1540

C₂

yes

maximal

C₅₉×#1301

C₂

yes

maximal, complemented

C₅₉×#1274

C₂

yes

maximal, characteristic, complemented

C₅₉×#532

C₂²

yes

characteristic

C₅₉×#431

C₂²

yes

—

C₅₉×#423

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₅₉

C₂²

yes

—

C₂×#104×C₅₉

—

C₅₉×#376

C₂²

yes

characteristic

C₅₉×#336

—

C₅₉×#332

—

C₅₉×#331

—

C₂×#41×C₅₉

—

C₂×M₁₆×C₅₉

—

C₂²×C₈×C₅₉

—

C₄⋊Q₈×C₅₉

D₄

yes

characteristic

#127×C₅₉

D₄

yes

characteristic

#124×C₅₉

—

#121×C₅₉

—

C₄×D₄×C₅₉

—

C₂×C₄²×C₅₉

D₄

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊C₈×C₅₉

—

C₄⋊C₈×C₅₉

—

C₄⋊C₈×C₅₉

—

C₄⋊C₈×C₅₉

—

#104×C₅₉

—

C₂²⋊C₈×C₅₉

—

C₄×C₈×C₅₉

—

#95×C₅₉

—

#41×C₅₉

—

#41×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

#41

yes

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₄⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

C₂×D₄

yes

C₄²×C₅₉

yes

C₄²×C₅₉

yes

—

C₂³×C₅₉

—

C₂³×C₅₉

#123

yes

characteristic

Q₈×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#121

yes

characteristic

C₂×C₄×C₅₉

#123

yes

—

C₂×C₄×C₅₉

#124

yes

characteristic

C₂×C₄×C₅₉

#126

yes

—

C₂×C₄×C₅₉

#135

yes

—

C₂×C₄×C₅₉

AGL(1,ℤ/8)

yes

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

#10629

C₅₉

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#397

yes

characteristic

C₂²×C₅₉

#482

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#463

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₂×C₅₉

yes

characteristic

#1832

C₂×C₅₉

yes

characteristic, complemented

#1540

C₂×C₅₉

yes

—

#1301

C₂×C₅₉

yes

complemented

#1274

C₂×C₅₉

yes

characteristic, complemented

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#2073

yes

characteristic

C₂×C₅₉

#2080

yes

—

#532

C₂²×C₅₉

yes

characteristic

100

#431

C₂²×C₅₉

yes

—

101

#423

C₂²×C₅₉

yes

characteristic

102

C₂×C₄⋊C₈

C₂²×C₅₉

yes

—

103

C₂×#104

—

104

#376

C₂²×C₅₉

yes

characteristic

105

#336

—

106

#332

—

107

#331

—

108

C₅₉

#10629

yes

characteristic, direct product factor, Sylow

109

C₂×#41

—

110

C₂×M₁₆

—

111

C₂²×C₈

—

112

C₄⋊Q₈

D₄×C₅₉

yes

characteristic

113

#127

D₄×C₅₉

yes

characteristic

114

#124

—

115

#121

—

116

C₄×D₄

—

117

C₂×C₄²

D₄×C₅₉

yes

—

118

C₂×C₄²

C₂³×C₅₉

yes

characteristic

119

C₄⋊C₈

—

120

C₄⋊C₈

—

121

C₄⋊C₈

—

122

C₄⋊C₈

—

123

#104

—

124

C₂²⋊C₈

—

125

C₄×C₈

—

126

#95

—

127

#41

—

128

#41

—

129

C₂×Q₈

—

130

C₂×D₄

—

131

C₂×D₄

—

132

C₂×D₄

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

C₂×D₄×C₅₉

yes

characteristic

136

C₂²×C₄

#41×C₅₉

yes

—

137

M₁₆

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₄⋊C₄

—

144

C₂²⋊C₄

—

145

C₄²

—

146

C₄²

—

147

C₄²

C₂×D₄×C₅₉

yes

characteristic

148

C₄²

#41×C₅₉

yes

characteristic

149

C₄²

#41×C₅₉

yes

—

150

C₂³

—

151

C₂³

#123×C₅₉

yes

characteristic

152

Q₈

—

153

D₄

—

154

D₄

—

155

D₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

#121×C₅₉

yes

characteristic

163

C₂×C₄

#123×C₅₉

yes

—

164

C₂×C₄

#124×C₅₉

yes

characteristic

165

C₂×C₄

#126×C₅₉

yes

—

166

C₂×C₄

#135×C₅₉

yes

—

167

C₂×C₄

AGL(1,ℤ/8)×C₅₉

yes

—

168

C₈

—

169

C₈

—

170

C₈

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

C₅₉×#397

yes

characteristic

175

C₂²

C₅₉×#482

yes

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

C₅₉×#463

yes

—

181

C₄

C₅₉×#488

yes

characteristic

182

C₄

C₅₉×#490

yes

characteristic

183

C₂

—

184

C₂

—

185

C₂

C₅₉×#2073

yes

characteristic

186

C₂

C₅₉×#2080

yes

—

	subgroup	G/H
center	C₂×C₄×C₅₉	#124
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅₉
socle	C₂²×C₅₉	#397
prime radical	C₅₉×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₉ ◁ C₂×C₄²×C₅₉ ◁ G

subgroup

G/H

center

C₂×C₄×C₅₉

#124

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₅₉

socle

C₂²×C₅₉

#397

prime radical

C₅₉×#532

C₂²

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₅₉ ◁ C₂×C₄²×C₅₉ ◁ G

N	H ≅ G/N	type
2. C₅₉×#1832	C₂	semidirect
4. C₅₉×#1301	C₂	semidirect
5. C₅₉×#1274	C₂	semidirect
77. #10629	C₅₉	direct (N × H)
91. #1832	C₂×C₅₉	semidirect
93. #1301	C₂×C₅₉	semidirect
94. #1274	C₂×C₅₉	semidirect
108. C₅₉	#10629	direct (N × H)

H ≅ G/N

type

2. C₅₉×#1832

C₂

semidirect

4. C₅₉×#1301

C₂

semidirect

5. C₅₉×#1274

C₂

semidirect

77. #10629

C₅₉

direct (N × H)

91. #1832

C₂×C₅₉

semidirect

93. #1301

C₂×C₅₉

semidirect

94. #1274

C₂×C₅₉

semidirect

108. C₅₉

#10629

direct (N × H)

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

subgroup	index
C₅₉×#1963	2
C₅₉×#1832	2
C₅₉×#1540	2
C₅₉×#1301	2
C₅₉×#1274	2
#10629	59

subgroup

index

C₅₉×#1963

C₅₉×#1832

C₅₉×#1540

C₅₉×#1301

C₅₉×#1274

#10629

Subgroups — C₅₉×#10629 #28422738

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C59×#10629 #28422738

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#10629 #28422738