Group #28504869 — subgroups

Subgroups — #28504869

Order 15104.

394 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 119 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	C₂²×C₁₆×D₅₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837231	C₂	1	2	yes	maximal, complemented
3	#21424131	C₂	1	2	yes	maximal
4	#21422190	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#1381	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₅₉	C₂²	1	1	yes	characteristic
7	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₁₆×D₅₉	—	2	4	no	—
9	C₂×C₁₆×D₅₉	C₄	1	4	yes	—
10	C₂×#14273663	C₂²	1	2	yes	—
11	#16837365	—	2	2	no	—
12	#16837231	—	2	4	no	—
13	#16837166	—	2	2	no	—
14	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×#108×C₅₉	C₂²	1	2	yes	—
16	C₂²×C₄×D₅₉	D₄	1	1	yes	characteristic
17	C₂×#11745664	—	2	2	no	—
18	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
19	C₂×C₈×D₅₉	—	2	2	no	—
20	C₂×C₈×D₅₉	—	2	2	no	—
21	C₂×C₈×D₅₉	C₂×C₄	1	2	yes	—
22	C₂×C₈×D₅₉	Q₈	1	2	yes	—
23	C₂×C₅₉⋊C₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₅₉⋊C₁₆	D₄	1	4	yes	—
25	C₁₆×D₅₉	—	2	8	no	—
26	#14273663	—	2	2	no	—
27	#14273663	—	2	2	no	—
28	(C₂×D₅₉)⋊C₈	—	4	4	no	—
29	C₂×M₁₆×C₅₉	—	2	2	no	—
30	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
31	C₂×C₁₆×C₅₉	—	2	2	no	—
32	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
33	#108×C₅₉	—	2	2	no	—
34	#108×C₅₉	—	2	2	no	—
35	C₂³×D₅₉	Q₁₆	1	1	yes	characteristic
36	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₅₉	Q₁₆	1	1	yes	characteristic
38	C₂×C₄×D₅₉	—	2	2	no	—
39	C₂×C₄×D₅₉	—	2	2	no	—
40	C₂×C₄×D₅₉	C₄⋊C₄	1	2	yes	—
41	C₂×C₄×D₅₉	D₈	1	2	yes	—
42	M₁₆×C₅₉	—	4	2	no	—
43	M₁₆×C₅₉	—	4	2	no	—
44	C₂×C₈×C₅₉	—	4	2	no	—
45	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
47	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₅₉	#41	1	2	yes	—
49	#11745664	—	4	2	no	—
50	#11745664	—	4	2	no	—
51	C₂×C₅₉⋊C₈	—	2	2	no	—
52	C₂×C₅₉⋊C₈	—	4	2	no	—
53	C₂×C₅₉⋊C₈	C₄⋊C₄	1	2	yes	—
54	C₈×D₅₉	—	2	2	no	—
55	C₈×D₅₉	—	2	2	no	—
56	C₈×D₅₉	—	2	4	no	—
57	C₁₆×C₅₉	—	2	2	no	—
58	C₅₉⋊C₁₆	C₂²⋊C₄	1	4	yes	—
59	C₂³×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
60	C₂²×D₅₉	—	2	2	no	—
61	C₂²×D₅₉	—	2	4	no	—
62	C₂²×D₅₉	#107	1	2	yes	—
63	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₉	#121	1	2	yes	—
65	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₅₉	#135	1	2	yes	—
67	C₂×Dic₅₉	—	2	2	no	—
68	C₂×Dic₅₉	#107	1	2	yes	—
69	C₄×D₅₉	—	2	2	no	—
70	C₄×D₅₉	—	2	2	no	—
71	C₄×D₅₉	—	2	4	no	—
72	C₈×C₅₉	—	8	2	no	—
73	C₈×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₈×C₅₉	#117	1	1	yes	characteristic
75	C₈×C₅₉	C₂²⋊Q₈	1	2	yes	—
76	C₅₉⋊C₈	—	2	2	no	—
77	C₅₉⋊C₈	—	8	2	no	—
78	#12837	—	59	1	no	maximal, Sylow
79	C₂²×C₅₉	—	2	2	no	—
80	C₂²×C₅₉	C₂×#107	1	1	yes	characteristic
81	C₂²×C₅₉	#467	1	2	yes	—
82	C₂×D₅₉	—	2	8	no	—
83	C₂×D₅₉	—	2	2	no	—
84	C₂×D₅₉	—	2	2	no	—
85	C₂×D₅₉	#368	1	4	yes	complemented
86	C₄×C₅₉	#380	1	1	yes	characteristic
87	C₄×C₅₉	#427	1	1	yes	characteristic
88	C₄×C₅₉	#466	1	2	yes	—
89	Dic₅₉	—	2	2	no	—
90	C₂³×C₁₆	—	59	1	no	—
91	C₂×#368	—	59	2	no	—
92	#1810	—	59	2	no	—
93	#1381	—	59	1	no	—
94	#1381	D₅₉	1	1	yes	characteristic, complemented
95	C₂×C₅₉	—	2	2	no	—
96	C₂×C₅₉	#1808	1	1	yes	characteristic
97	C₂×C₅₉	C₂×#368	1	2	yes	—
98	D₅₉	—	2	4	no	—
99	C₂³×C₈	—	59	1	no	—
100	C₂²×C₁₆	—	59	1	no	—
101	C₂²×C₁₆	—	59	4	no	—
102	C₂²×C₁₆	—	118	4	no	—
103	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
104	C₂×#108	—	59	2	no	—
105	C₂×#108	C₂×D₅₉	1	2	yes	—
106	#407	—	118	2	no	—
107	#368	—	118	4	no	—
108	#341	—	118	2	no	—
109	C₅₉	#12837	1	1	yes	characteristic, complemented, Sylow
110	C₂³×C₄	—	59	1	no	—
111	C₂×M₁₆	—	2	2	no	—
112	C₂×M₁₆	—	118	2	no	—
113	C₂²×C₈	—	59	1	no	—
114	C₂²×C₈	—	59	2	no	—
115	C₂²×C₈	—	59	2	no	—
116	C₂²×C₈	—	118	2	no	—
117	C₂²×C₈	—	118	2	no	—
118	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
119	C₂×C₁₆	—	2	2	no	—
120	C₂×C₁₆	—	59	2	no	—
121	C₂×C₁₆	—	59	4	no	—
122	C₂×C₁₆	—	118	8	no	—
123	C₂×C₁₆	C₄×D₅₉	1	2	yes	—
124	#108	—	2	2	no	—
125	#108	—	2	2	no	—
126	#108	—	118	2	no	—
127	#108	—	118	2	no	—
128	C₂²⋊C₈	—	236	4	no	—
129	C₂⁴	—	59	1	no	—
130	C₂²×C₄	—	59	1	no	—
131	C₂²×C₄	—	59	2	no	—
132	C₂²×C₄	—	59	2	no	—
133	C₂²×C₄	—	118	2	no	—
134	C₂²×C₄	—	118	2	no	—
135	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
136	M₁₆	—	4	2	no	—
137	M₁₆	—	4	2	no	—
138	M₁₆	—	236	2	no	—
139	M₁₆	—	236	2	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	59	2	no	—
142	C₂×C₈	—	118	2	no	—
143	C₂×C₈	—	118	2	no	—
144	C₂×C₈	—	118	2	no	—
145	C₂×C₈	—	118	4	no	—
146	C₂×C₈	—	236	2	no	—
147	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
148	C₂×C₈	#11745685	1	2	yes	—
149	C₂×C₈	D₄×D₅₉	1	2	yes	—
150	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
151	C₁₆	—	2	2	no	—
152	C₁₆	—	59	4	no	—
153	C₂³	—	59	2	no	—
154	C₂³	—	118	4	no	—
155	C₂³	—	118	2	no	—
156	C₂³	Q₁₆×D₅₉	1	1	yes	characteristic
157	C₂×C₄	—	59	2	no	—
158	C₂×C₄	—	118	2	no	—
159	C₂×C₄	—	118	4	no	—
160	C₂×C₄	—	118	2	no	—
161	C₂×C₄	—	118	2	no	—
162	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
163	C₂×C₄	#14273700	1	2	yes	—
164	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
165	C₂×C₄	#14273748	1	2	yes	—
166	C₈	—	8	2	no	—
167	C₈	—	118	2	no	—
168	C₈	—	472	2	no	—
169	C₈	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
170	C₈	#14273684	1	1	yes	characteristic
171	C₈	#14273705	1	2	yes	—
172	C₂²	—	2	2	no	—
173	C₂²	—	59	4	no	—
174	C₂²	—	118	8	no	—
175	C₂²	—	118	2	no	—
176	C₂²	—	118	2	no	—
177	C₂²	#107×D₅₉	1	1	yes	characteristic
178	C₂²	#16837642	1	2	yes	—
179	C₄	—	118	2	no	—
180	C₄	#16837261	1	1	yes	characteristic
181	C₄	#16837437	1	1	yes	characteristic
182	C₄	#16837637	1	2	yes	—
183	C₂	—	2	2	no	—
184	C₂	—	118	4	no	—
185	C₂	#21424124	1	1	yes	characteristic
186	C₂	#368×D₅₉	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₁₆×D₅₉

C₂

yes

maximal, characteristic

C₂×#16837231

C₂

yes

maximal, complemented

#21424131

C₂

yes

maximal

#21422190

C₂

yes

maximal, characteristic, complemented

C₅₉×#1381

C₂

yes

maximal, characteristic, complemented

C₂²×C₈×D₅₉

C₂²

yes

characteristic

C₂²×C₅₉⋊C₁₆

C₂²

yes

characteristic

C₂×C₁₆×D₅₉

—

C₂×C₁₆×D₅₉

C₄

yes

—

C₂×#14273663

C₂²

yes

—

#16837365

—

#16837231

—

#16837166

—

C₂²×C₁₆×C₅₉

C₂²

yes

characteristic

C₂×#108×C₅₉

C₂²

yes

—

C₂²×C₄×D₅₉

D₄

yes

characteristic

C₂×#11745664

—

C₂²×C₅₉⋊C₈

D₄

yes

characteristic

C₂×C₈×D₅₉

—

C₂×C₈×D₅₉

—

C₂×C₈×D₅₉

C₂×C₄

yes

—

C₂×C₈×D₅₉

Q₈

yes

—

C₂×C₅₉⋊C₁₆

C₂×C₄

yes

—

C₂×C₅₉⋊C₁₆

D₄

yes

—

C₁₆×D₅₉

—

#14273663

—

#14273663

—

(C₂×D₅₉)⋊C₈

—

C₂×M₁₆×C₅₉

—

C₂²×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₁₆×C₅₉

—

C₂×C₁₆×C₅₉

C₂×C₄

yes

—

#108×C₅₉

—

#108×C₅₉

—

C₂³×D₅₉

Q₁₆

yes

characteristic

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×Dic₅₉

Q₁₆

yes

characteristic

C₂×C₄×D₅₉

—

C₂×C₄×D₅₉

—

C₂×C₄×D₅₉

C₄⋊C₄

yes

—

C₂×C₄×D₅₉

D₈

yes

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×D₄

yes

—

C₂×C₈×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₉

#41

yes

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₄⋊C₄

yes

—

C₈×D₅₉

—

C₈×D₅₉

—

C₈×D₅₉

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

C₂²⋊C₄

yes

—

C₂³×C₅₉

C₂×Q₁₆

yes

characteristic

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂²×D₅₉

#107

yes

—

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

#121

yes

—

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

#135

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

#107

yes

—

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

—

C₈×C₅₉

—

C₈×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₅₉

#117

yes

characteristic

C₈×C₅₉

C₂²⋊Q₈

yes

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

#12837

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

C₂×#107

yes

characteristic

C₂²×C₅₉

#467

yes

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

Dic₅₉

—

C₂³×C₁₆

—

—

—

—

D₅₉

yes

characteristic, complemented

C₂×C₅₉

—

C₂×C₅₉

#1808

yes

characteristic

C₂×C₅₉

C₂×#368

yes

—

D₅₉

—

C₂³×C₈

—

100

C₂²×C₁₆

—

101

C₂²×C₁₆

—

102

C₂²×C₁₆

—

118

—

103

C₂²×C₁₆

C₂×D₅₉

yes

characteristic

104

C₂×#108

—

105

C₂×#108

C₂×D₅₉

yes

—

106

#407

—

118

—

107

#368

—

118

—

108

#341

—

118

—

109

C₅₉

#12837

yes

characteristic, complemented, Sylow

110

C₂³×C₄

—

111

C₂×M₁₆

—

112

C₂×M₁₆

—

118

—

113

C₂²×C₈

—

114

C₂²×C₈

—

115

C₂²×C₈

—

116

C₂²×C₈

—

118

—

117

C₂²×C₈

—

118

—

118

C₂²×C₈

C₂²×D₅₉

yes

characteristic

119

C₂×C₁₆

—

120

C₂×C₁₆

—

121

C₂×C₁₆

—

122

C₂×C₁₆

—

118

—

123

C₂×C₁₆

C₄×D₅₉

yes

—

124

#108

—

125

#108

—

126

#108

—

118

—

127

#108

—

118

—

128

C₂²⋊C₈

—

236

—

129

C₂⁴

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

—

118

—

134

C₂²×C₄

—

118

—

135

C₂²×C₄

D₄×D₅₉

yes

characteristic

136

M₁₆

—

137

M₁₆

—

138

M₁₆

—

236

—

139

M₁₆

—

236

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

118

—

143

C₂×C₈

—

118

—

144

C₂×C₈

—

118

—

145

C₂×C₈

—

118

—

146

C₂×C₈

—

236

—

147

C₂×C₈

C₂×C₄×D₅₉

yes

characteristic

148

C₂×C₈

#11745685

yes

—

149

C₂×C₈

D₄×D₅₉

yes

—

150

C₂×C₈

Q₈×D₅₉

yes

characteristic

151

C₁₆

—

152

C₁₆

—

153

C₂³

—

154

C₂³

—

118

—

155

C₂³

—

118

—

156

C₂³

Q₁₆×D₅₉

yes

characteristic

157

C₂×C₄

—

158

C₂×C₄

—

118

—

159

C₂×C₄

—

118

—

160

C₂×C₄

—

118

—

161

C₂×C₄

—

118

—

162

C₂×C₄

C₄⋊C₄×D₅₉

yes

characteristic

163

C₂×C₄

#14273700

yes

—

164

C₂×C₄

D₈×D₅₉

yes

characteristic

165

C₂×C₄

#14273748

yes

—

166

C₈

—

167

C₈

—

118

—

168

C₈

—

472

—

169

C₈

C₂²⋊C₄×D₅₉

yes

characteristic

170

C₈

#14273684

yes

characteristic

171

C₈

#14273705

yes

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

118

—

175

C₂²

—

118

—

176

C₂²

—

118

—

177

C₂²

#107×D₅₉

yes

characteristic

178

C₂²

#16837642

yes

—

179

C₄

—

118

—

180

C₄

#16837261

yes

characteristic

181

C₄

#16837437

yes

characteristic

182

C₄

#16837637

yes

—

183

C₂

—

184

C₂

—

118

—

185

C₂

#21424124

yes

characteristic

186

C₂

#368×D₅₉

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₅₉
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1381	C₂
socle	C₂²×C₅₉	C₂×#107
prime radical	C₂³×D₅₉	Q₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1381	D₅₉
hypocenter	C₅₉	#12837
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

subgroup

G/H

center

C₂×C₄

D₈×D₅₉

commutator

C₂×C₈×C₅₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₅₉×#1381

C₂

socle

C₂²×C₅₉

C₂×#107

prime radical

C₂³×D₅₉

Q₁₆

cosocle

C₂²×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

N	H ≅ G/N	type
2. C₂×#16837231	C₂	semidirect
4. #21422190	C₂	semidirect
5. C₅₉×#1381	C₂	semidirect
85. C₂×D₅₉	#368	semidirect
94. #1381	D₅₉	semidirect
109. C₅₉	#12837	semidirect

H ≅ G/N

type

2. C₂×#16837231

C₂

semidirect

4. #21422190

C₂

semidirect

5. C₅₉×#1381

C₂

semidirect

85. C₂×D₅₉

#368

semidirect

94. #1381

D₅₉

semidirect

109. C₅₉

#12837

semidirect

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

subgroup	index
C₂²×C₁₆×D₅₉	2
C₂×#16837231	2
#21424131	2
#21422190	2
C₅₉×#1381	2
#12837	59

subgroup

index

C₂²×C₁₆×D₅₉

C₅₉×#1381

Subgroups — #28504869

Conjugacy classes of subgroups

Complements

Maximal subgroups