Group #28493467 — subgroups

Subgroups — #28493467

Order 15104.

310 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 87 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	#21424216	C₂	1	1	yes	maximal, characteristic, complemented
2	#21422745	C₂	1	2	yes	maximal
3	#21421991	C₂	1	2	yes	maximal, complemented
4	#21421925	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#1963	C₂	1	1	yes	maximal, characteristic
6	#16837455	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
8	#16837157	—	2	2	no	—
9	#16837148	—	2	2	no	—
10	#16837146	—	2	2	no	—
11	C₅₉×#532	C₂²	1	1	yes	characteristic
12	C₅₉×#423	C₂²	1	1	yes	characteristic
13	C₂×#104×C₅₉	—	2	2	no	—
14	C₅₉×#376	C₂²	1	1	yes	characteristic
15	C₂²×C₅₉⋊C₈	—	2	2	no	—
16	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
17	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
18	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
19	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
20	C₄×C₅₉⋊C₈	—	2	2	no	—
21	C₂×#41×C₅₉	—	2	1	no	—
22	C₂×M₁₆×C₅₉	—	2	1	no	—
23	C₄⋊Q₈×C₅₉	D₄	1	1	yes	characteristic
24	#127×C₅₉	D₄	1	1	yes	characteristic
25	#124×C₅₉	—	2	1	no	—
26	#121×C₅₉	—	2	2	no	—
27	C₄×D₄×C₅₉	—	2	2	no	—
28	C₂×C₄²×C₅₉	D₄	1	2	yes	—
29	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₅₉	—	2	2	no	—
31	#104×C₅₉	—	4	4	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	1	no	—
45	C₄⋊C₄×C₅₉	—	2	2	no	—
46	C₂²⋊C₄×C₅₉	—	4	2	no	—
47	C₄²×C₅₉	—	2	2	no	—
48	C₄²×C₅₉	—	2	2	no	—
49	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
50	C₄²×C₅₉	#41	1	1	yes	characteristic
51	C₄²×C₅₉	#41	1	2	yes	—
52	C₂×C₅₉⋊C₈	—	4	2	no	—
53	C₂×C₅₉⋊C₈	—	4	2	no	—
54	C₂×C₅₉⋊C₈	—	4	2	no	—
55	C₂×C₅₉⋊C₈	—	4	2	no	—
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	1	no	—
75	C₅₉⋊C₈	—	8	2	no	—
76	C₅₉⋊C₈	—	8	2	no	—
77	#10629	—	59	1	no	maximal, 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	D₅₉	1	1	yes	characteristic
91	#1832	—	59	1	no	—
92	#1540	—	59	2	no	—
93	#1301	—	59	2	no	—
94	#1274	—	59	1	no	—
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	C₂×D₅₉	1	1	yes	characteristic
100	#431	—	59	2	no	—
101	#423	C₂×D₅₉	1	1	yes	characteristic
102	C₂×C₄⋊C₈	—	59	2	no	—
103	C₂×#104	—	2	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	—	2	1	no	—
110	C₂×M₁₆	—	2	1	no	—
111	C₂²×C₈	—	118	2	no	—
112	C₄⋊Q₈	D₂₃₆	1	1	yes	characteristic
113	#127	D₂₃₆	1	1	yes	characteristic
114	#124	—	2	1	no	—
115	#121	—	2	2	no	—
116	C₄×D₄	—	2	2	no	—
117	C₂×C₄²	C₂²⋊D₅₉	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	—	4	4	no	—
124	C₂²⋊C₈	—	118	2	no	—
125	C₄×C₈	—	118	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	#11745687	1	2	yes	—
136	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
137	M₁₆	—	4	2	no	—
138	C₂×C₈	—	4	1	no	—
139	C₂×C₈	—	236	2	no	—
140	C₂×C₈	—	236	2	no	—
141	C₂×C₈	—	236	2	no	—
142	C₂×C₈	—	236	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×D₂₃₆	1	1	yes	characteristic
148	C₄²	#11745685	1	1	yes	characteristic
149	C₄²	#11745687	1	2	yes	—
150	C₂³	—	4	1	no	—
151	C₂³	#14273708	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	2	no	—
159	C₂×C₄	—	4	1	no	—
160	C₂×C₄	—	8	1	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	#14273701	1	1	yes	characteristic
163	C₂×C₄	#14273711	1	2	yes	—
164	C₂×C₄	#14273714	1	1	yes	characteristic
165	C₂×C₄	#14273720	1	2	yes	—
166	C₂×C₄	#14273747	1	2	yes	—
167	C₂×C₄	#14273753	1	2	yes	—
168	C₈	—	8	1	no	—
169	C₈	—	472	2	no	—
170	C₈	—	472	2	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	8	1	no	—
173	C₂²	—	8	2	no	—
174	C₂²	#16837327	1	1	yes	characteristic
175	C₂²	#16837723	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	#16837617	1	2	yes	—
181	C₄	#16837756	1	1	yes	characteristic
182	C₄	#16837766	1	1	yes	characteristic
183	C₂	—	2	2	no	—
184	C₂	—	16	1	no	—
185	C₂	#21425509	1	1	yes	characteristic
186	C₂	#21425546	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21424216

C₂

yes

maximal, characteristic, complemented

#21422745

C₂

yes

maximal

#21421991

C₂

yes

maximal, complemented

#21421925

C₂

yes

maximal, characteristic, complemented

C₅₉×#1963

C₂

yes

maximal, characteristic

#16837455

C₂²

yes

—

C₂×C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

#16837157

—

#16837148

—

#16837146

—

C₅₉×#532

C₂²

yes

characteristic

C₅₉×#423

C₂²

yes

characteristic

C₂×#104×C₅₉

—

C₅₉×#376

C₂²

yes

characteristic

C₂²×C₅₉⋊C₈

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₂²⋊(C₅₉⋊C₈)

—

C₄×C₅₉⋊C₈

—

C₂×#41×C₅₉

—

C₂×M₁₆×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

—

#104×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

—

C₄⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

C₂×D₄

yes

C₄²×C₅₉

yes

C₄²×C₅₉

yes

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

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

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

#10629

—

maximal, 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

D₅₉

yes

—

—

—

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#2073

yes

characteristic

C₂×C₅₉

#2080

yes

—

#532

C₂×D₅₉

yes

characteristic

100

#431

—

101

#423

C₂×D₅₉

yes

characteristic

102

C₂×C₄⋊C₈

—

103

C₂×#104

—

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

—

110

C₂×M₁₆

—

111

C₂²×C₈

—

118

—

112

C₄⋊Q₈

D₂₃₆

yes

characteristic

113

#127

D₂₃₆

yes

characteristic

114

#124

—

115

#121

—

116

C₄×D₄

—

117

C₂×C₄²

C₂²⋊D₅₉

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

—

124

C₂²⋊C₈

—

118

—

125

C₄×C₈

—

118

—

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

#11745687

yes

—

136

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

137

M₁₆

—

138

C₂×C₈

—

139

C₂×C₈

—

236

—

140

C₂×C₈

—

236

—

141

C₂×C₈

—

236

—

142

C₂×C₈

—

236

—

143

C₄⋊C₄

—

144

C₂²⋊C₄

—

145

C₄²

—

146

C₄²

—

147

C₄²

C₂×D₂₃₆

yes

characteristic

148

C₄²

#11745685

yes

characteristic

149

C₄²

#11745687

yes

—

150

C₂³

—

151

C₂³

#14273708

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

#14273701

yes

characteristic

163

C₂×C₄

#14273711

yes

—

164

C₂×C₄

#14273714

yes

characteristic

165

C₂×C₄

#14273720

yes

—

166

C₂×C₄

#14273747

yes

—

167

C₂×C₄

#14273753

yes

—

168

C₈

—

169

C₈

—

472

—

170

C₈

—

472

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

#16837327

yes

characteristic

175

C₂²

#16837723

yes

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

#16837617

yes

—

181

C₄

#16837756

yes

characteristic

182

C₄

#16837766

yes

characteristic

183

C₂

—

184

C₂

—

185

C₂

#21425509

yes

characteristic

186

C₂

#21425546

yes

—

	subgroup	G/H
center	C₂×C₄	#14273714
commutator, cosocle	C₂×C₄²×C₅₉	C₂³
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1963	C₂
socle	C₂²×C₅₉	#397
prime radical	C₅₉×#532	C₂²
hypercenter	#1963	D₅₉
hypocenter	C₅₉	#10629
LCS	G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1963

subgroup

G/H

center

C₂×C₄

#14273714

commutator, cosocle

C₂×C₄²×C₅₉

C₂³

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#1963

C₂

socle

C₂²×C₅₉

#397

prime radical

C₅₉×#532

C₂²

D₅₉

C₅₉

G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1963

N	H ≅ G/N	type
1. #21424216	C₂	semidirect
3. #21421991	C₂	semidirect
4. #21421925	C₂	semidirect
108. C₅₉	#10629	semidirect

H ≅ G/N

type

1. #21424216

C₂

semidirect

3. #21421991

C₂

semidirect

4. #21421925

C₂

semidirect

108. C₅₉

#10629

semidirect

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

subgroup	index
#21424216	2
#21422745	2
#21421991	2
#21421925	2
C₅₉×#1963	2
#10629	59

subgroup

index

C₅₉×#1963

Subgroups — #28493467

Conjugacy classes of subgroups

Complements

Maximal subgroups