Group #28493494 — subgroups

Subgroups — #28493494

Order 15104.

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

normal subgroups only

This group has 346 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	#21424237	C₂	1	1	yes	maximal, characteristic, complemented
2	#21422764	C₂	1	2	yes	maximal
3	#21421954	C₂	1	2	yes	maximal, complemented
4	#21421941	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#1961	C₂	1	1	yes	maximal, characteristic
6	#16837455	C₂²	1	2	yes	—
7	#16837426	C₂²	1	2	yes	—
8	#16837167	—	2	2	no	—
9	#16837146	—	2	4	no	—
10	C₂×#127×C₅₉	C₂²	1	1	yes	characteristic
11	C₂×#102×C₅₉	—	2	2	no	—
12	C₂×#97×C₅₉	C₂²	1	1	yes	characteristic
13	C₅₉×#382	C₂²	1	1	yes	characteristic
14	C₂×#11745664	—	2	2	no	—
15	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
16	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
17	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
18	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
19	C₄×C₅₉⋊C₈	—	2	2	no	—
20	C₂²×D₄×C₅₉	—	2	1	no	—
21	C₂²×D₄×C₅₉	—	2	2	no	—
22	C₂²×C₈×C₅₉	D₄	1	2	yes	—
23	#127×C₅₉	—	2	1	no	—
24	#127×C₅₉	—	2	2	no	—
25	#127×C₅₉	D₄	1	2	yes	—
26	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
27	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
28	#102×C₅₉	—	4	4	no	—
29	#97×C₅₉	—	2	2	no	—
30	#95×C₅₉	—	2	2	no	—
31	C₂⁴×C₅₉	—	4	1	no	—
32	C₂×D₄×C₅₉	—	2	2	no	—
33	C₂×D₄×C₅₉	—	2	4	no	—
34	C₂×D₄×C₅₉	—	4	1	no	—
35	C₂×D₄×C₅₉	—	4	2	no	—
36	C₂×D₄×C₅₉	—	4	2	no	—
37	C₂×D₄×C₅₉	—	4	2	no	—
38	C₂×D₄×C₅₉	—	8	4	no	—
39	C₂²×C₄×C₅₉	—	4	1	no	—
40	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₉	#41	1	2	yes	—
42	C₂×C₈×C₅₉	—	2	2	no	—
43	C₂×C₈×C₅₉	—	2	2	no	—
44	C₂×C₈×C₅₉	—	2	2	no	—
45	C₄⋊C₄×C₅₉	—	4	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	#11745664	—	4	2	no	—
50	#11745664	—	4	2	no	—
51	C₂×C₅₉⋊C₈	—	4	2	no	—
52	C₂×C₅₉⋊C₈	—	4	2	no	—
53	C₂³×C₅₉	—	4	2	no	—
54	C₂³×C₅₉	—	8	4	no	—
55	C₂³×C₅₉	—	8	1	no	—
56	C₂³×C₅₉	—	8	1	no	—
57	C₂³×C₅₉	#124	1	1	yes	characteristic
58	D₄×C₅₉	—	4	4	no	—
59	D₄×C₅₉	—	8	4	no	—
60	D₄×C₅₉	—	8	4	no	—
61	D₄×C₅₉	—	8	2	no	—
62	C₂×C₄×C₅₉	—	4	2	no	—
63	C₂×C₄×C₅₉	—	8	2	no	—
64	C₂×C₄×C₅₉	—	8	1	no	—
65	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
66	C₂×C₄×C₅₉	#123	1	1	yes	characteristic
67	C₂×C₄×C₅₉	#123	1	2	yes	—
68	C₂×C₄×C₅₉	#126	1	2	yes	—
69	C₂×C₄×C₅₉	#135	1	2	yes	—
70	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
71	C₈×C₅₉	—	4	2	no	—
72	C₅₉⋊C₈	—	8	2	no	—
73	C₅₉⋊C₈	—	8	2	no	—
74	#10634	—	59	1	no	maximal, Sylow
75	C₂²×C₅₉	—	2	2	no	—
76	C₂²×C₅₉	—	8	2	no	—
77	C₂²×C₅₉	—	8	4	no	—
78	C₂²×C₅₉	—	16	2	no	—
79	C₂²×C₅₉	—	16	2	no	—
80	C₂²×C₅₉	#397	1	1	yes	characteristic
81	C₂²×C₅₉	#488	1	1	yes	characteristic
82	C₂²×C₅₉	#490	1	1	yes	characteristic
83	C₄×C₅₉	—	2	2	no	—
84	C₄×C₅₉	—	2	2	no	—
85	C₄×C₅₉	—	16	1	no	—
86	C₄×C₅₉	#463	1	2	yes	—
87	C₄×C₅₉	#482	1	2	yes	—
88	#1961	D₅₉	1	1	yes	characteristic
89	#1838	—	59	1	no	—
90	#1545	—	59	2	no	—
91	#1286	—	59	2	no	—
92	#1281	—	59	1	no	—
93	C₂×C₅₉	—	2	1	no	—
94	C₂×C₅₉	—	2	1	no	—
95	C₂×C₅₉	—	16	2	no	—
96	C₂×C₅₉	#2073	1	1	yes	characteristic
97	C₂×C₅₉	#2081	1	2	yes	—
98	C₂×#127	C₂×D₅₉	1	1	yes	characteristic
99	#431	—	59	2	no	—
100	#423	—	59	2	no	—
101	C₂×#102	—	2	2	no	—
102	C₂×#97	C₂×D₅₉	1	1	yes	characteristic
103	#382	C₂×D₅₉	1	1	yes	characteristic
104	#341	—	118	2	no	—
105	#331	—	118	4	no	—
106	C₅₉	#10634	1	1	yes	characteristic, complemented, Sylow
107	C₂²×D₄	—	2	2	no	—
108	C₂²×D₄	—	2	1	no	—
109	C₂×M₁₆	—	118	2	no	—
110	C₂²×C₈	C₂²⋊D₅₉	1	2	yes	—
111	#127	—	2	1	no	—
112	#127	—	2	2	no	—
113	#127	D₂₃₆	1	2	yes	—
114	C₂×C₄⋊C₄	—	2	1	no	—
115	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
116	C₄⋊C₈	—	118	2	no	—
117	C₄⋊C₈	—	118	2	no	—
118	C₄⋊C₈	—	118	2	no	—
119	#102	—	4	4	no	—
120	C₂²⋊C₈	—	118	2	no	—
121	#97	—	2	2	no	—
122	C₄×C₈	—	118	2	no	—
123	#95	—	2	2	no	—
124	C₂⁴	—	4	1	no	—
125	C₂×D₄	—	2	2	no	—
126	C₂×D₄	—	2	4	no	—
127	C₂×D₄	—	4	1	no	—
128	C₂×D₄	—	4	2	no	—
129	C₂×D₄	—	4	2	no	—
130	C₂×D₄	—	4	2	no	—
131	C₂×D₄	—	8	4	no	—
132	C₂²×C₄	—	4	1	no	—
133	C₂²×C₄	#11745687	1	2	yes	—
134	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
135	M₁₆	—	236	2	no	—
136	M₁₆	—	236	2	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	236	2	no	—
141	C₂×C₈	—	236	2	no	—
142	C₄⋊C₄	—	4	2	no	—
143	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
144	C₄²	#11745685	1	1	yes	characteristic
145	C₄²	#11745687	1	2	yes	—
146	C₂³	—	4	2	no	—
147	C₂³	—	8	4	no	—
148	C₂³	—	8	1	no	—
149	C₂³	—	8	1	no	—
150	C₂³	#14273714	1	1	yes	characteristic
151	D₄	—	4	4	no	—
152	D₄	—	8	4	no	—
153	D₄	—	8	4	no	—
154	D₄	—	8	2	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	—	8	1	no	—
158	C₂×C₄	#14273701	1	1	yes	characteristic
159	C₂×C₄	#14273708	1	1	yes	characteristic
160	C₂×C₄	#14273711	1	2	yes	—
161	C₂×C₄	#14273720	1	2	yes	—
162	C₂×C₄	#14273747	1	2	yes	—
163	C₂×C₄	#14273753	1	2	yes	—
164	C₈	—	4	2	no	—
165	C₈	—	472	2	no	—
166	C₈	—	472	2	no	—
167	C₂²	—	2	2	no	—
168	C₂²	—	8	2	no	—
169	C₂²	—	8	4	no	—
170	C₂²	—	16	2	no	—
171	C₂²	—	16	2	no	—
172	C₂²	#16837327	1	1	yes	characteristic
173	C₂²	#16837756	1	1	yes	characteristic
174	C₂²	#16837766	1	1	yes	characteristic
175	C₄	—	2	2	no	—
176	C₄	—	2	2	no	—
177	C₄	—	16	1	no	—
178	C₄	#16837617	1	2	yes	—
179	C₄	#16837723	1	2	yes	—
180	C₂	—	2	1	no	—
181	C₂	—	2	1	no	—
182	C₂	—	16	2	no	—
183	C₂	#21425509	1	1	yes	characteristic
184	C₂	#21425553	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21424237

C₂

yes

maximal, characteristic, complemented

#21422764

C₂

yes

maximal

#21421954

C₂

yes

maximal, complemented

#21421941

C₂

yes

maximal, characteristic, complemented

C₅₉×#1961

C₂

yes

maximal, characteristic

#16837455

C₂²

yes

—

#16837426

C₂²

yes

—

#16837167

—

#16837146

—

C₂×#127×C₅₉

C₂²

yes

characteristic

C₂×#102×C₅₉

—

C₂×#97×C₅₉

C₂²

yes

characteristic

C₅₉×#382

C₂²

yes

characteristic

C₂×#11745664

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₂²⋊(C₅₉⋊C₈)

—

C₄×C₅₉⋊C₈

—

C₂²×D₄×C₅₉

—

C₂²×D₄×C₅₉

—

C₂²×C₈×C₅₉

D₄

yes

—

#127×C₅₉

—

#127×C₅₉

—

#127×C₅₉

D₄

yes

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

#102×C₅₉

—

#97×C₅₉

—

#95×C₅₉

—

C₂⁴×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

#41

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂×D₄

yes

C₄²×C₅₉

yes

C₄²×C₅₉

yes

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

#124

yes

characteristic

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

—

C₂×C₄×C₅₉

#126

yes

—

C₂×C₄×C₅₉

#135

yes

—

C₂×C₄×C₅₉

AGL(1,ℤ/8)

yes

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

#10634

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#463

yes

—

C₄×C₅₉

#482

yes

—

#1961

D₅₉

yes

—

—

—

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#2073

yes

characteristic

C₂×C₅₉

#2081

yes

—

C₂×#127

C₂×D₅₉

yes

characteristic

#431

—

100

#423

—

101

C₂×#102

—

102

C₂×#97

C₂×D₅₉

yes

characteristic

103

#382

C₂×D₅₉

yes

characteristic

104

#341

—

118

—

105

#331

—

118

—

106

C₅₉

#10634

yes

characteristic, complemented, Sylow

107

C₂²×D₄

—

108

C₂²×D₄

—

109

C₂×M₁₆

—

118

—

110

C₂²×C₈

C₂²⋊D₅₉

yes

—

111

#127

—

112

#127

—

113

#127

D₂₃₆

yes

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄²

C₂²×D₅₉

yes

characteristic

116

C₄⋊C₈

—

118

—

117

C₄⋊C₈

—

118

—

118

C₄⋊C₈

—

118

—

119

#102

—

120

C₂²⋊C₈

—

118

—

121

#97

—

122

C₄×C₈

—

118

—

123

#95

—

124

C₂⁴

—

125

C₂×D₄

—

126

C₂×D₄

—

127

C₂×D₄

—

128

C₂×D₄

—

129

C₂×D₄

—

130

C₂×D₄

—

131

C₂×D₄

—

132

C₂²×C₄

—

133

C₂²×C₄

#11745687

yes

—

134

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

135

M₁₆

—

236

—

136

M₁₆

—

236

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

236

—

141

C₂×C₈

—

236

—

142

C₄⋊C₄

—

143

C₄²

C₂×D₂₃₆

yes

characteristic

144

C₄²

#11745685

yes

characteristic

145

C₄²

#11745687

yes

—

146

C₂³

—

147

C₂³

—

148

C₂³

—

149

C₂³

—

150

C₂³

#14273714

yes

characteristic

151

D₄

—

152

D₄

—

153

D₄

—

154

D₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

#14273701

yes

characteristic

159

C₂×C₄

#14273708

yes

characteristic

160

C₂×C₄

#14273711

yes

—

161

C₂×C₄

#14273720

yes

—

162

C₂×C₄

#14273747

yes

—

163

C₂×C₄

#14273753

yes

—

164

C₈

—

165

C₈

—

472

—

166

C₈

—

472

—

167

C₂²

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

#16837327

yes

characteristic

173

C₂²

#16837756

yes

characteristic

174

C₂²

#16837766

yes

characteristic

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

#16837617

yes

—

179

C₄

#16837723

yes

—

180

C₂

—

181

C₂

—

182

C₂

—

183

C₂

#21425509

yes

characteristic

184

C₂

#21425553

yes

—

	subgroup	G/H
center	C₂²	#16837327
commutator, cosocle	C₂×C₄²×C₅₉	C₂³
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1961	C₂
socle	C₂²×C₅₉	#397
prime radical	C₂×#127×C₅₉	C₂²
hypercenter	#1961	D₅₉
hypocenter	C₅₉	#10634
LCS	G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1961

subgroup

G/H

center

C₂²

#16837327

commutator, cosocle

C₂×C₄²×C₅₉

C₂³

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#1961

C₂

socle

C₂²×C₅₉

#397

prime radical

C₂×#127×C₅₉

C₂²

D₅₉

C₅₉

G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1961

N	H ≅ G/N	type
1. #21424237	C₂	semidirect
3. #21421954	C₂	semidirect
4. #21421941	C₂	semidirect
106. C₅₉	#10634	semidirect

H ≅ G/N

type

1. #21424237

C₂

semidirect

3. #21421954

C₂

semidirect

4. #21421941

C₂

semidirect

106. C₅₉

#10634

semidirect

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

subgroup	index
#21424237	2
#21422764	2
#21421954	2
#21421941	2
C₅₉×#1961	2
#10634	59

subgroup

index

C₅₉×#1961

Subgroups — #28493494

Conjugacy classes of subgroups

Complements

Maximal subgroups