Group #28493469 — subgroups

Subgroups — #28493469

Order 15104.

310 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 85 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	#21424928	C₂	1	1	yes	maximal, characteristic
2	#21424216	C₂	1	1	yes	maximal, characteristic, complemented
3	#21422743	C₂	1	2	yes	maximal
4	#21421992	C₂	1	2	yes	maximal, complemented
5	C₅₉×#1274	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837952	C₂²	1	1	yes	characteristic
7	#16837426	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
9	C₂×#14273655	—	2	2	no	—
10	#16837157	—	2	2	no	—
11	#16837147	—	2	2	no	—
12	#16837145	—	2	2	no	—
13	C₅₉×#431	C₂²	1	2	yes	—
14	C₅₉×#376	C₂²	1	1	yes	characteristic
15	C₂×#11745685	—	2	1	no	—
16	C₂×#11745664	—	2	1	no	—
17	C₂²×C₅₉⋊C₈	—	2	2	no	—
18	#14273727	D₄	1	1	yes	characteristic
19	#14273724	D₄	1	1	yes	characteristic
20	#14273715	—	2	1	no	—
21	#14273701	—	2	2	no	—
22	C₄×D₂₃₆	—	2	2	no	—
23	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
24	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
25	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
26	#14273655	—	4	4	no	—
27	C₂×C₄²×C₅₉	D₄	1	2	yes	—
28	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₅₉	—	2	2	no	—
30	C₂²⋊C₈×C₅₉	—	2	2	no	—
31	C₄×C₈×C₅₉	—	2	2	no	—
32	#95×C₅₉	—	2	2	no	—
33	C₂²×C₄×C₅₉	—	2	2	no	—
34	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₅₉	#41	1	2	yes	—
36	C₂×C₂²⋊D₅₉	—	4	1	no	—
37	#11745685	—	4	2	no	—
38	#11745685	—	4	2	no	—
39	C₂×D₂₃₆	—	2	1	no	—
40	C₂×D₂₃₆	—	2	2	no	—
41	C₂×C₄×D₅₉	—	4	1	no	—
42	C₂×Dic₁₁₈	—	2	1	no	—
43	C₂×C₈×C₅₉	—	4	2	no	—
44	C₄²×C₅₉	—	2	2	no	—
45	C₄²×C₅₉	—	2	2	no	—
46	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₅₉	#41	1	1	yes	characteristic
48	C₄²×C₅₉	#41	1	2	yes	—
49	(C₂×D₅₉)⋊C₄	—	4	2	no	—
50	C₄⋊Dic₅₉	—	2	2	no	—
51	#11745664	—	4	2	no	—
52	C₂×C₅₉⋊C₈	—	4	2	no	—
53	C₂×C₅₉⋊C₈	—	4	1	no	—
54	C₂×C₅₉⋊C₈	—	4	2	no	—
55	C₂×C₅₉⋊C₈	—	4	2	no	—
56	C₂³×C₅₉	#123	1	1	yes	characteristic
57	C₂²×D₅₉	—	4	1	no	—
58	C₂×C₄×C₅₉	—	4	2	no	—
59	C₂×C₄×C₅₉	—	4	4	no	—
60	C₂×C₄×C₅₉	—	4	2	no	—
61	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
62	C₂×C₄×C₅₉	#123	1	2	yes	—
63	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
64	C₂×C₄×C₅₉	#126	1	2	yes	—
65	C₂×C₄×C₅₉	#135	1	2	yes	—
66	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
67	C₂²⋊D₅₉	—	8	2	no	—
68	C₂×Dic₅₉	—	4	1	no	—
69	D₂₃₆	—	4	2	no	—
70	D₂₃₆	—	8	2	no	—
71	C₄×D₅₉	—	8	1	no	—
72	C₄×D₅₉	—	8	1	no	—
73	Dic₁₁₈	—	4	2	no	—
74	C₈×C₅₉	—	8	2	no	—
75	C₅₉⋊C₈	—	8	2	no	—
76	C₅₉⋊C₈	—	8	1	no	—
77	#10629	—	59	1	no	maximal, Sylow
78	C₂²×C₅₉	—	2	2	no	—
79	C₂²×C₅₉	#397	1	1	yes	characteristic
80	C₂²×C₅₉	#482	1	2	yes	—
81	C₂×D₅₉	—	8	1	no	—
82	C₂×D₅₉	—	8	2	no	—
83	C₄×C₅₉	—	2	2	no	—
84	C₄×C₅₉	—	2	2	no	—
85	C₄×C₅₉	—	8	2	no	—
86	C₄×C₅₉	#463	1	2	yes	—
87	C₄×C₅₉	#488	1	1	yes	characteristic
88	C₄×C₅₉	#490	1	1	yes	characteristic
89	Dic₅₉	—	8	1	no	—
90	#1963	—	59	1	no	—
91	#1832	—	59	1	no	—
92	#1540	—	59	2	no	—
93	#1301	—	59	2	no	—
94	#1274	D₅₉	1	1	yes	characteristic, complemented
95	C₂×C₅₉	—	2	2	no	—
96	C₂×C₅₉	#2073	1	1	yes	characteristic
97	C₂×C₅₉	#2080	1	2	yes	—
98	D₅₉	—	16	1	no	—
99	#532	—	59	1	no	—
100	#431	C₂×D₅₉	1	2	yes	—
101	#423	—	59	1	no	—
102	C₂×C₄⋊C₈	—	59	2	no	—
103	C₂×#104	—	118	2	no	—
104	#376	C₂×D₅₉	1	1	yes	characteristic
105	#336	—	118	2	no	—
106	#332	—	118	2	no	—
107	#331	—	118	2	no	—
108	C₅₉	#10629	1	1	yes	characteristic, complemented, Sylow
109	C₂×#41	—	118	1	no	—
110	C₂×M₁₆	—	118	1	no	—
111	C₂²×C₈	—	118	2	no	—
112	C₄⋊Q₈	—	59	1	no	—
113	#127	—	59	1	no	—
114	#124	—	118	1	no	—
115	#121	—	118	2	no	—
116	C₄×D₄	—	118	2	no	—
117	C₂×C₄²	D₂₃₆	1	2	yes	—
118	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	118	2	no	—
121	C₄⋊C₈	—	118	2	no	—
122	C₄⋊C₈	—	118	2	no	—
123	#104	—	236	4	no	—
124	C₂²⋊C₈	—	2	2	no	—
125	C₄×C₈	—	2	2	no	—
126	#95	—	2	2	no	—
127	#41	—	236	2	no	—
128	#41	—	236	2	no	—
129	C₂×Q₈	—	118	1	no	—
130	C₂×D₄	—	118	1	no	—
131	C₂×D₄	—	118	2	no	—
132	C₂×D₄	—	236	1	no	—
133	C₂²×C₄	—	2	2	no	—
134	C₂²×C₄	—	236	1	no	—
135	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
136	C₂²×C₄	#11745687	1	2	yes	—
137	M₁₆	—	236	2	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	236	2	no	—
140	C₂×C₈	—	236	1	no	—
141	C₂×C₈	—	236	2	no	—
142	C₂×C₈	—	236	2	no	—
143	C₄⋊C₄	—	118	2	no	—
144	C₂²⋊C₄	—	236	2	no	—
145	C₄²	—	2	2	no	—
146	C₄²	—	2	2	no	—
147	C₄²	#11745685	1	2	yes	—
148	C₄²	D₄×D₅₉	1	1	yes	characteristic
149	C₄²	#11745689	1	1	yes	characteristic
150	C₂³	—	236	1	no	—
151	C₂³	#14273711	1	1	yes	characteristic
152	Q₈	—	236	2	no	—
153	D₄	—	472	2	no	—
154	D₄	—	472	2	no	—
155	D₄	—	236	2	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	—	4	4	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	472	1	no	—
160	C₂×C₄	—	472	1	no	—
161	C₂×C₄	—	236	1	no	—
162	C₂×C₄	#14273702	1	1	yes	characteristic
163	C₂×C₄	#14273709	1	2	yes	—
164	C₂×C₄	#14273715	1	1	yes	characteristic
165	C₂×C₄	#14273722	1	2	yes	—
166	C₂×C₄	#14273749	1	2	yes	—
167	C₂×C₄	#14273755	1	2	yes	—
168	C₈	—	8	2	no	—
169	C₈	—	472	1	no	—
170	C₈	—	472	2	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	472	1	no	—
173	C₂²	—	472	2	no	—
174	C₂²	#16837329	1	1	yes	characteristic
175	C₂²	#16837727	1	2	yes	—
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	8	2	no	—
179	C₄	—	472	1	no	—
180	C₄	#16837618	1	2	yes	—
181	C₄	#16837757	1	1	yes	characteristic
182	C₄	#16837767	1	1	yes	characteristic
183	C₂	—	2	2	no	—
184	C₂	—	944	1	no	—
185	C₂	#21425511	1	1	yes	characteristic
186	C₂	#21425548	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21424928

C₂

yes

maximal, characteristic

#21424216

C₂

yes

maximal, characteristic, complemented

#21422743

C₂

yes

maximal

#21421992

C₂

yes

maximal, complemented

C₅₉×#1274

C₂

yes

maximal, characteristic, complemented

#16837952

C₂²

yes

characteristic

#16837426

C₂²

yes

characteristic

C₂×C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

—

—

—

—

C₅₉×#431

C₂²

yes

—

C₅₉×#376

C₂²

yes

characteristic

C₂×#11745685

—

C₂×#11745664

—

C₂²×C₅₉⋊C₈

—

#14273727

D₄

yes

characteristic

#14273724

D₄

yes

characteristic

#14273715

—

#14273701

—

C₄×D₂₃₆

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

#14273655

—

C₂×C₄²×C₅₉

D₄

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊C₈×C₅₉

—

C₂²⋊C₈×C₅₉

—

C₄×C₈×C₅₉

—

#95×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

#41

yes

—

C₂×C₂²⋊D₅₉

—

#11745685

—

#11745685

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×C₄×D₅₉

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

C₂×D₄

yes

C₄²×C₅₉

yes

C₄²×C₅₉

yes

—

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

#123

yes

characteristic

C₂²×D₅₉

—

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

—

C₂²⋊D₅₉

—

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

—

C₄×D₅₉

—

C₄×D₅₉

—

Dic₁₁₈

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

#10629

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

#397

yes

characteristic

C₂²×C₅₉

#482

yes

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#463

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

—

—

—

—

D₅₉

yes

characteristic, complemented

C₂×C₅₉

—

C₂×C₅₉

#2073

yes

characteristic

C₂×C₅₉

#2080

yes

—

D₅₉

—

#532

—

100

#431

C₂×D₅₉

yes

—

101

#423

—

102

C₂×C₄⋊C₈

—

103

C₂×#104

—

118

—

104

#376

C₂×D₅₉

yes

characteristic

105

#336

—

118

—

106

#332

—

118

—

107

#331

—

118

—

108

C₅₉

#10629

yes

characteristic, complemented, Sylow

109

C₂×#41

—

118

—

110

C₂×M₁₆

—

118

—

111

C₂²×C₈

—

118

—

112

C₄⋊Q₈

—

113

#127

—

114

#124

—

118

—

115

#121

—

118

—

116

C₄×D₄

—

118

—

117

C₂×C₄²

D₂₃₆

yes

—

118

C₂×C₄²

C₂²×D₅₉

yes

characteristic

119

C₄⋊C₈

—

120

C₄⋊C₈

—

118

—

121

C₄⋊C₈

—

118

—

122

C₄⋊C₈

—

118

—

123

#104

—

236

—

124

C₂²⋊C₈

—

125

C₄×C₈

—

126

#95

—

127

#41

—

236

—

128

#41

—

236

—

129

C₂×Q₈

—

118

—

130

C₂×D₄

—

118

—

131

C₂×D₄

—

118

—

132

C₂×D₄

—

236

—

133

C₂²×C₄

—

134

C₂²×C₄

—

236

—

135

C₂²×C₄

C₂×D₂₃₆

yes

characteristic

136

C₂²×C₄

#11745687

yes

—

137

M₁₆

—

236

—

138

C₂×C₈

—

139

C₂×C₈

—

236

—

140

C₂×C₈

—

236

—

141

C₂×C₈

—

236

—

142

C₂×C₈

—

236

—

143

C₄⋊C₄

—

118

—

144

C₂²⋊C₄

—

236

—

145

C₄²

—

146

C₄²

—

147

C₄²

#11745685

yes

—

148

C₄²

D₄×D₅₉

yes

characteristic

149

C₄²

#11745689

yes

characteristic

150

C₂³

—

236

—

151

C₂³

#14273711

yes

characteristic

152

Q₈

—

236

—

153

D₄

—

472

—

154

D₄

—

472

—

155

D₄

—

236

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

472

—

160

C₂×C₄

—

472

—

161

C₂×C₄

—

236

—

162

C₂×C₄

#14273702

yes

characteristic

163

C₂×C₄

#14273709

yes

—

164

C₂×C₄

#14273715

yes

characteristic

165

C₂×C₄

#14273722

yes

—

166

C₂×C₄

#14273749

yes

—

167

C₂×C₄

#14273755

yes

—

168

C₈

—

169

C₈

—

472

—

170

C₈

—

472

—

171

C₂²

—

172

C₂²

—

472

—

173

C₂²

—

472

—

174

C₂²

#16837329

yes

characteristic

175

C₂²

#16837727

yes

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

472

—

180

C₄

#16837618

yes

—

181

C₄

#16837757

yes

characteristic

182

C₄

#16837767

yes

characteristic

183

C₂

—

184

C₂

—

944

—

185

C₂

#21425511

yes

characteristic

186

C₂

#21425548

yes

—

	subgroup	G/H
center	C₂×C₄	#14273715
commutator, cosocle	C₂×C₄²×C₅₉	C₂³
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1274	C₂
socle	C₂²×C₅₉	#397
prime radical	#16837952	C₂²
hypercenter	#1274	D₅₉
hypocenter	C₅₉	#10629
LCS	G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

subgroup

G/H

center

C₂×C₄

#14273715

commutator, cosocle

C₂×C₄²×C₅₉

C₂³

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#1274

C₂

C₂²×C₅₉

C₂²

D₅₉

C₅₉

G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

N	H ≅ G/N	type
2. #21424216	C₂	semidirect
4. #21421992	C₂	semidirect
5. C₅₉×#1274	C₂	semidirect
94. #1274	D₅₉	semidirect
108. C₅₉	#10629	semidirect

H ≅ G/N

type

2. #21424216

C₂

semidirect

4. #21421992

C₂

semidirect

5. C₅₉×#1274

C₂

semidirect

94. #1274

D₅₉

semidirect

108. C₅₉

#10629

semidirect

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

subgroup	index
#21424928	2
#21424216	2
#21422743	2
#21421992	2
C₅₉×#1274	2
#10629	59

subgroup

index

C₅₉×#1274

Subgroups — #28493469

Conjugacy classes of subgroups

Complements

Maximal subgroups