Group #28487310 — subgroups

Subgroups — #28487310

Order 15104.

294 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 123 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	C₄×#14273663	C₂	1	1	yes	maximal, characteristic
2	#21422849	C₂	1	2	yes	maximal
3	C₂×#16837153	C₂	1	1	yes	maximal, characteristic
4	#21421945	C₂	1	2	yes	maximal
5	C₅₉×#1452	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273663	C₄	1	2	yes	—
7	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
8	C₄×#11745664	C₂²	1	2	yes	—
9	#16837167	C₄	1	1	yes	characteristic
10	#16837167	C₄	1	1	yes	characteristic
11	#16837154	—	2	2	no	—
12	#16837153	—	2	2	no	—
13	#16837152	—	2	2	no	—
14	C₅₉×#431	C₂²	1	1	yes	characteristic
15	C₅₉×#431	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₅₉×#322	—	2	2	no	—
18	C₂×#11745664	C₂×C₄	1	2	yes	—
19	C₂×#11745664	C₂×C₄	1	2	yes	—
20	C₂²×C₅₉⋊C₈	—	2	2	no	—
21	#14273663	—	2	4	no	—
22	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
23	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
24	#14273638	—	2	2	no	—
25	C₄×C₅₉⋊C₈	—	2	2	no	—
26	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₅₉	—	2	1	no	—
30	C₄⋊C₈×C₅₉	D₄	1	2	yes	—
31	C₂²⋊C₈×C₅₉	—	2	1	no	—
32	C₂²⋊C₈×C₅₉	Q₈	1	2	yes	—
33	C₄×C₈×C₅₉	—	2	1	no	—
34	C₄×C₈×C₅₉	—	2	1	no	—
35	C₄×C₈×C₅₉	D₄	1	2	yes	—
36	C₄×C₈×C₅₉	Q₈	1	2	yes	—
37	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
40	C₂×C₈×C₅₉	—	2	2	no	—
41	C₂×C₈×C₅₉	—	2	2	no	—
42	C₂×C₈×C₅₉	—	2	2	no	—
43	C₂×C₈×C₅₉	—	2	2	no	—
44	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
45	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
46	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
49	C₄²×C₅₉	#41	1	1	yes	characteristic
50	#11745664	—	2	2	no	—
51	#11745664	—	2	2	no	—
52	#11745664	—	2	2	no	—
53	#11745664	M₁₆	1	4	yes	—
54	C₂×C₅₉⋊C₈	—	2	2	no	—
55	C₂×C₅₉⋊C₈	—	2	2	no	—
56	C₂×C₅₉⋊C₈	—	4	2	no	—
57	C₂×C₅₉⋊C₈	—	8	2	no	—
58	C₂³×C₅₉	C₄×Q₈	1	1	yes	characteristic
59	C₂×C₄×C₅₉	—	2	2	no	—
60	C₂×C₄×C₅₉	—	4	1	no	—
61	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
63	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
64	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₅₉	#125	1	1	yes	characteristic
66	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₅₉	C₂×M₁₆	1	2	yes	—
68	C₂×C₄×C₅₉	#131	1	2	yes	—
69	C₈×C₅₉	—	2	2	no	—
70	C₈×C₅₉	—	2	2	no	—
71	C₈×C₅₉	—	4	2	no	—
72	C₈×C₅₉	—	4	2	no	—
73	C₅₉⋊C₈	—	2	4	no	—
74	C₅₉⋊C₈	—	4	2	no	—
75	C₅₉⋊C₈	—	8	2	no	—
76	#9609	—	59	1	no	maximal, Sylow
77	C₂²×C₅₉	—	2	1	no	—
78	C₂²×C₅₉	—	2	1	no	—
79	C₂²×C₅₉	#389	1	1	yes	characteristic
80	C₂²×C₅₉	#446	1	2	yes	—
81	C₄×C₅₉	—	2	1	no	—
82	C₄×C₅₉	—	2	1	no	—
83	C₄×C₅₉	—	2	1	no	—
84	C₄×C₅₉	#423	1	1	yes	characteristic
85	C₄×C₅₉	#424	1	1	yes	characteristic
86	C₄×C₅₉	#435	1	2	yes	—
87	C₄×C₅₉	#471	1	2	yes	—
88	C₄×#108	—	59	1	no	—
89	#1565	—	59	2	no	—
90	C₂×#334	—	59	1	no	—
91	#1452	D₅₉	1	1	yes	characteristic
92	#1283	—	59	2	no	—
93	C₂×C₅₉	—	2	2	no	—
94	C₂×C₅₉	#1940	1	1	yes	characteristic
95	C₂×C₅₉	#1951	1	1	yes	characteristic
96	C₂×C₅₉	#1953	1	1	yes	characteristic
97	#431	C₂×D₅₉	1	1	yes	characteristic
98	#431	C₂×D₅₉	1	1	yes	characteristic
99	C₂×#108	—	59	2	no	—
100	C₂×C₄⋊C₈	—	59	2	no	—
101	C₄×M₁₆	—	59	2	no	—
102	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
103	#341	—	59	1	no	—
104	#341	—	59	1	no	—
105	#335	—	118	2	no	—
106	#334	—	118	2	no	—
107	#334	—	118	2	no	—
108	#322	—	2	2	no	—
109	C₅₉	#9609	1	1	yes	characteristic, complemented, Sylow
110	C₂×M₁₆	—	59	2	no	—
111	C₂×M₁₆	—	59	2	no	—
112	C₂²×C₈	—	118	2	no	—
113	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
114	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
115	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
116	#108	—	118	4	no	—
117	C₄⋊C₈	—	2	1	no	—
118	C₄⋊C₈	—	118	2	no	—
119	C₄⋊C₈	—	118	2	no	—
120	C₄⋊C₈	D₂₃₆	1	2	yes	—
121	C₂²⋊C₈	—	2	1	no	—
122	C₂²⋊C₈	Dic₁₁₈	1	2	yes	—
123	#97	—	118	2	no	—
124	C₄×C₈	—	2	1	no	—
125	C₄×C₈	—	2	1	no	—
126	C₄×C₈	—	118	2	no	—
127	C₄×C₈	Dic₁₁₈	1	2	yes	—
128	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
129	C₂²×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
130	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
131	C₂²×C₄	#11745685	1	1	yes	characteristic
132	M₁₆	—	59	4	no	—
133	M₁₆	—	118	2	no	—
134	M₁₆	—	118	2	no	—
135	M₁₆	—	118	2	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	118	2	no	—
141	C₂×C₈	—	118	2	no	—
142	C₂×C₈	—	472	2	no	—
143	C₂×C₈	—	236	2	no	—
144	C₂×C₈	Dic₅₉⋊C₄	1	2	yes	—
145	C₂×C₈	Dic₅₉⋊C₄	1	2	yes	—
146	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
147	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
148	C₄²	#11745685	1	1	yes	characteristic
149	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
150	C₂³	C₄×Dic₁₁₈	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
154	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
155	C₂×C₄	#14273701	1	1	yes	characteristic
156	C₂×C₄	#14273706	1	1	yes	characteristic
157	C₂×C₄	#14273719	1	1	yes	characteristic
158	C₂×C₄	#14273727	1	1	yes	characteristic
159	C₂×C₄	M₁₆×D₅₉	1	2	yes	—
160	C₂×C₄	#14273735	1	2	yes	—
161	C₈	—	2	2	no	—
162	C₈	—	2	2	no	—
163	C₈	—	4	2	no	—
164	C₈	—	4	2	no	—
165	C₈	—	118	4	no	—
166	C₈	—	472	2	no	—
167	C₈	—	236	2	no	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	#16837300	1	1	yes	characteristic
171	C₂²	#16837530	1	2	yes	—
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	#16837425	1	1	yes	characteristic
176	C₄	#16837430	1	1	yes	characteristic
177	C₄	#16837475	1	2	yes	—
178	C₄	#16837665	1	2	yes	—
179	C₂	—	2	2	no	—
180	C₂	#21424819	1	1	yes	characteristic
181	C₂	#21424876	1	1	yes	characteristic
182	C₂	#21424886	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14273663

C₂

yes

maximal, characteristic

#21422849

C₂

yes

maximal

C₂×#16837153

C₂

yes

maximal, characteristic

#21421945

C₂

yes

maximal

C₅₉×#1452

C₂

yes

maximal, characteristic

C₂×#14273663

C₄

yes

—

C₂×C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

C₄×#11745664

C₂²

yes

—

#16837167

C₄

yes

characteristic

#16837167

C₄

yes

—

—

—

C₅₉×#431

C₂²

yes

characteristic

C₅₉×#431

C₂²

yes

characteristic

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×#322

—

C₂×#11745664

C₂×C₄

yes

—

C₂×#11745664

C₂×C₄

yes

—

C₂²×C₅₉⋊C₈

—

#14273663

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

#14273638

—

C₄×C₅₉⋊C₈

—

C₂²×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊C₈×C₅₉

—

C₄⋊C₈×C₅₉

D₄

yes

—

C₂²⋊C₈×C₅₉

—

C₂²⋊C₈×C₅₉

Q₈

yes

—

C₄×C₈×C₅₉

—

C₄×C₈×C₅₉

—

C₄×C₈×C₅₉

D₄

yes

—

C₄×C₈×C₅₉

Q₈

yes

—

C₂²×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₂²×C₄×C₅₉

#41

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄²×C₅₉

C₂×Q₈

yes

C₄²×C₅₉

yes

—

—

—

M₁₆

yes

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₄×Q₈

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄×D₄

yes

characteristic

C₂×C₄×C₅₉

#121

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅₉

#125

yes

characteristic

C₂×C₄×C₅₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₉

C₂×M₁₆

yes

—

C₂×C₄×C₅₉

#131

yes

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

#9609

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#389

yes

characteristic

C₂²×C₅₉

#446

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

C₄×C₅₉

#471

yes

—

—

—

—

D₅₉

yes

characteristic

#1283

—

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×D₅₉

yes

characteristic

#431

C₂×D₅₉

yes

characteristic

C₂×#108

—

100

C₂×C₄⋊C₈

—

101

C₄×M₁₆

—

102

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

103

#341

—

104

#341

—

105

#335

—

118

—

106

#334

—

118

—

107

#334

—

118

—

108

#322

—

109

C₅₉

#9609

yes

characteristic, complemented, Sylow

110

C₂×M₁₆

—

111

C₂×M₁₆

—

112

C₂²×C₈

—

118

—

113

C₂²×C₈

C₄×D₅₉

yes

characteristic

114

C₂²×C₈

C₄×D₅₉

yes

characteristic

115

C₂×C₄²

C₂²×D₅₉

yes

characteristic

116

#108

—

118

—

117

C₄⋊C₈

—

118

C₄⋊C₈

—

118

—

119

C₄⋊C₈

—

118

—

120

C₄⋊C₈

D₂₃₆

yes

—

121

C₂²⋊C₈

—

122

C₂²⋊C₈

Dic₁₁₈

yes

—

123

#97

—

118

—

124

C₄×C₈

—

125

C₄×C₈

—

126

C₄×C₈

—

118

—

127

C₄×C₈

Dic₁₁₈

yes

—

128

C₄×C₈

C₂²⋊D₅₉

yes

—

129

C₂²×C₄

C₂×Dic₁₁₈

yes

characteristic

130

C₂²×C₄

C₂×C₄×D₅₉

yes

characteristic

131

C₂²×C₄

#11745685

yes

characteristic

132

M₁₆

—

133

M₁₆

—

118

—

134

M₁₆

—

118

—

135

M₁₆

—

118

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

118

—

141

C₂×C₈

—

118

—

142

C₂×C₈

—

472

—

143

C₂×C₈

—

236

—

144

C₂×C₈

Dic₅₉⋊C₄

yes

—

145

C₂×C₈

Dic₅₉⋊C₄

yes

—

146

C₄²

C₂×Dic₁₁₈

yes

characteristic

147

C₄²

C₂×D₂₃₆

yes

characteristic

148

C₄²

#11745685

yes

characteristic

149

C₄²

C₂×C₂²⋊D₅₉

yes

characteristic

150

C₂³

C₄×Dic₁₁₈

yes

characteristic

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

C₂×Dic₅₉⋊C₄

yes

characteristic

154

C₂×C₄

C₄×D₂₃₆

yes

characteristic

155

C₂×C₄

#14273701

yes

characteristic

156

C₂×C₄

#14273706

yes

characteristic

157

C₂×C₄

#14273719

yes

characteristic

158

C₂×C₄

#14273727

yes

characteristic

159

C₂×C₄

M₁₆×D₅₉

yes

—

160

C₂×C₄

#14273735

yes

—

161

C₈

—

162

C₈

—

163

C₈

—

164

C₈

—

165

C₈

—

118

—

166

C₈

—

472

—

167

C₈

—

236

—

168

C₂²

—

169

C₂²

—

170

C₂²

#16837300

yes

characteristic

171

C₂²

#16837530

yes

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

#16837425

yes

characteristic

176

C₄

#16837430

yes

characteristic

177

C₄

#16837475

yes

—

178

C₄

#16837665

yes

—

179

C₂

—

180

C₂

#21424819

yes

characteristic

181

C₂

#21424876

yes

characteristic

182

C₂

#21424886

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14273701
commutator	C₂²×C₄×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1452	C₂
socle	C₂²×C₅₉	#389
prime radical	C₂³×C₅₉	C₄×Q₈
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1452	D₅₉
hypocenter	C₅₉	#9609
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1452

subgroup

G/H

center

C₂×C₄

#14273701

commutator

C₂²×C₄×C₅₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#1452

C₂

socle

C₂²×C₅₉

#389

prime radical

C₂³×C₅₉

C₄×Q₈

cosocle

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1452

N	H ≅ G/N	type
109. C₅₉	#9609	semidirect

H ≅ G/N

type

109. C₅₉

#9609

semidirect

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

subgroup	index
C₄×#14273663	2
#21422849	2
C₂×#16837153	2
#21421945	2
C₅₉×#1452	2
#9609	59

subgroup

index

C₅₉×#1452

Subgroups — #28487310

Conjugacy classes of subgroups

Complements

Maximal subgroups