Group #28508242 — subgroups

Subgroups — #28508242

Order 15104.

378 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 91 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×Dic₄₇₂	C₂	1	1	yes	maximal, characteristic
2	#21425854	C₂	1	1	yes	maximal, characteristic
3	C₂×#16837217	C₂	1	2	yes	maximal
4	#21425009	C₂	1	2	yes	maximal
5	C₅₉×#1381	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₃₆	C₂²	1	2	yes	—
7	C₂×Dic₄₇₂	—	2	4	no	—
8	C₂×Dic₄₇₂	—	2	2	no	—
9	C₂×#14273667	C₂²	1	1	yes	characteristic
10	#16837412	—	2	2	no	—
11	C₂×#14273645	—	2	2	no	—
12	#16837357	C₂²	1	1	yes	characteristic
13	Dic₅₉⋊C₁₆	—	2	2	no	—
14	#16837217	—	2	4	no	—
15	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×#108×C₅₉	C₂²	1	2	yes	—
17	C₂²×Dic₁₁₈	—	2	2	no	—
18	C₂×Dic₂₃₆	—	2	2	no	—
19	C₂×Dic₂₃₆	—	2	2	no	—
20	C₂×Dic₂₃₆	—	2	2	no	—
21	C₂×Dic₂₃₆	—	2	2	no	—
22	C₂×Dic₂₃₆	—	2	2	no	—
23	C₂×Dic₂₃₆	D₄	1	4	yes	—
24	C₂×#11745664	D₄	1	1	yes	characteristic
25	#14273686	D₄	1	1	yes	characteristic
26	Dic₄₇₂	—	2	8	no	—
27	#14273667	—	2	2	no	—
28	C₂×C₅₉⋊C₁₆	—	2	1	no	—
29	#14273652	—	4	4	no	—
30	#14273645	—	4	4	no	—
31	#14273637	—	2	1	no	—
32	C₈×Dic₅₉	—	2	1	no	—
33	C₂×M₁₆×C₅₉	—	2	2	no	—
34	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
35	C₂×C₁₆×C₅₉	—	2	2	no	—
36	C₂×C₁₆×C₅₉	D₄	1	2	yes	—
37	#108×C₅₉	—	2	2	no	—
38	#108×C₅₉	—	2	2	no	—
39	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
40	C₂²×Dic₅₉	—	4	2	no	—
41	C₂×Dic₁₁₈	—	2	4	no	—
42	C₂×Dic₁₁₈	—	4	2	no	—
43	C₂×Dic₁₁₈	—	4	2	no	—
44	C₂×Dic₁₁₈	—	4	2	no	—
45	C₂×Dic₁₁₈	—	4	2	no	—
46	C₂×Dic₁₁₈	—	4	2	no	—
47	M₁₆×C₅₉	—	4	2	no	—
48	M₁₆×C₅₉	—	4	2	no	—
49	C₂×C₈×C₅₉	—	4	2	no	—
50	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
51	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
53	C₂×C₈×C₅₉	#41	1	2	yes	—
54	C₂²⋊Dic₅₉	—	2	1	no	—
55	C₄⋊Dic₅₉	D₈	1	2	yes	—
56	C₄×Dic₅₉	—	2	1	no	—
57	#11745664	—	2	2	no	—
58	C₂×C₅₉⋊C₈	—	2	1	no	—
59	Dic₂₃₆	—	2	4	no	—
60	Dic₂₃₆	—	2	4	no	—
61	Dic₂₃₆	—	4	2	no	—
62	Dic₂₃₆	—	4	2	no	—
63	C₁₆×C₅₉	—	2	2	no	—
64	C₅₉⋊C₁₆	—	4	1	no	—
65	C₂³×C₅₉	#135	1	1	yes	characteristic
66	C₂×C₄×C₅₉	#121	1	2	yes	—
67	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
68	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
69	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
70	C₂×Dic₅₉	—	2	2	no	—
71	C₂×Dic₅₉	—	4	4	no	—
72	C₂×Dic₅₉	—	8	2	no	—
73	C₂×Dic₅₉	—	8	2	no	—
74	Dic₁₁₈	—	4	4	no	—
75	Dic₁₁₈	—	4	4	no	—
76	Dic₁₁₈	—	8	2	no	—
77	Dic₁₁₈	—	8	2	no	—
78	C₈×C₅₉	—	8	2	no	—
79	C₈×C₅₉	#120	1	1	yes	characteristic
80	C₈×C₅₉	#123	1	2	yes	—
81	C₈×C₅₉	#124	1	1	yes	characteristic
82	C₅₉⋊C₈	—	4	1	no	—
83	#13441	—	59	1	no	maximal, Sylow
84	C₂²×C₅₉	—	2	2	no	—
85	C₂²×C₅₉	#463	1	2	yes	—
86	C₂²×C₅₉	#466	1	1	yes	characteristic
87	C₄×C₅₉	#394	1	1	yes	characteristic
88	C₄×C₅₉	#459	1	2	yes	—
89	C₄×C₅₉	#469	1	1	yes	characteristic
90	Dic₅₉	—	4	2	no	—
91	Dic₅₉	—	8	4	no	—
92	C₂²×Q₃₂	—	59	1	no	—
93	#2151	—	59	1	no	—
94	C₂×#362	—	59	2	no	—
95	#1979	—	59	2	no	—
96	#1381	D₅₉	1	1	yes	characteristic
97	C₂×C₅₉	—	2	2	no	—
98	C₂×C₅₉	#2054	1	1	yes	characteristic
99	C₂×C₅₉	#2222	1	2	yes	—
100	C₂²×Q₁₆	—	59	2	no	—
101	C₂×Q₃₂	—	118	4	no	—
102	C₂×Q₃₂	—	118	2	no	—
103	C₂×M₃₂	—	59	1	no	—
104	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
105	C₂×#108	C₂×D₅₉	1	2	yes	—
106	#419	—	118	2	no	—
107	C₂×#101	—	118	2	no	—
108	#405	—	59	1	no	—
109	C₄⋊C₁₆	—	118	2	no	—
110	#362	—	118	4	no	—
111	C₅₉	#13441	1	1	yes	characteristic, complemented, Sylow
112	C₂²×Q₈	—	118	2	no	—
113	C₂×Q₁₆	—	59	4	no	—
114	C₂×Q₁₆	—	118	2	no	—
115	C₂×Q₁₆	—	118	2	no	—
116	C₂×Q₁₆	—	118	2	no	—
117	C₂×Q₁₆	—	118	2	no	—
118	C₂×Q₁₆	—	118	2	no	—
119	C₂×M₁₆	—	2	2	no	—
120	C₂×M₁₆	—	59	1	no	—
121	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
122	#117	—	59	1	no	—
123	Q₃₂	—	118	8	no	—
124	M₃₂	—	118	2	no	—
125	C₂×C₁₆	—	2	2	no	—
126	C₂×C₁₆	—	118	1	no	—
127	C₂×C₁₆	D₂₃₆	1	2	yes	—
128	#108	—	2	2	no	—
129	#108	—	2	2	no	—
130	#103	—	236	4	no	—
131	#101	—	236	4	no	—
132	#97	—	118	1	no	—
133	C₄×C₈	—	118	1	no	—
134	C₂×Q₈	—	118	4	no	—
135	C₂×Q₈	—	236	2	no	—
136	C₂×Q₈	—	236	2	no	—
137	C₂×Q₈	—	236	2	no	—
138	C₂×Q₈	—	236	2	no	—
139	C₂×Q₈	—	236	2	no	—
140	C₂²×C₄	—	236	2	no	—
141	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
142	Q₁₆	—	118	4	no	—
143	Q₁₆	—	118	4	no	—
144	Q₁₆	—	236	2	no	—
145	Q₁₆	—	236	2	no	—
146	M₁₆	—	4	2	no	—
147	M₁₆	—	4	2	no	—
148	M₁₆	—	118	2	no	—
149	C₂×C₈	—	4	2	no	—
150	C₂×C₈	—	118	1	no	—
151	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
152	C₂×C₈	#11745685	1	2	yes	—
153	C₂×C₈	D₄×D₅₉	1	2	yes	—
154	C₂×C₈	#11745689	1	1	yes	characteristic
155	C₄⋊C₄	—	59	2	no	—
156	C₂²⋊C₄	—	118	1	no	—
157	C₄²	—	118	1	no	—
158	C₁₆	—	2	2	no	—
159	C₁₆	—	236	1	no	—
160	C₂³	#14273750	1	1	yes	characteristic
161	Q₈	—	472	2	no	—
162	Q₈	—	472	2	no	—
163	Q₈	—	236	4	no	—
164	Q₈	—	236	4	no	—
165	C₂×C₄	—	118	2	no	—
166	C₂×C₄	—	472	2	no	—
167	C₂×C₄	—	472	2	no	—
168	C₂×C₄	—	236	4	no	—
169	C₂×C₄	#14273700	1	2	yes	—
170	C₂×C₄	#14273702	1	1	yes	characteristic
171	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
172	C₂×C₄	#14273754	1	2	yes	—
173	C₈	—	8	2	no	—
174	C₈	—	236	1	no	—
175	C₈	#14273697	1	1	yes	characteristic
176	C₈	#14273712	1	2	yes	—
177	C₈	#14273715	1	1	yes	characteristic
178	C₂²	—	2	2	no	—
179	C₂²	#16837620	1	2	yes	—
180	C₂²	#16837639	1	1	yes	characteristic
181	C₄	—	472	4	no	—
182	C₄	—	236	2	no	—
183	C₄	#16837320	1	1	yes	characteristic
184	C₄	#16837596	1	2	yes	—
185	C₄	#16837656	1	1	yes	characteristic
186	C₂	—	2	2	no	—
187	C₂	#21425408	1	1	yes	characteristic
188	C₂	#21426222	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₄₇₂

C₂

yes

maximal, characteristic

#21425854

C₂

yes

maximal, characteristic

C₂×#16837217

C₂

yes

maximal

#21425009

C₂

yes

maximal

C₅₉×#1381

C₂

yes

maximal, characteristic

C₂²×Dic₂₃₆

C₂²

yes

—

C₂×Dic₄₇₂

—

C₂×Dic₄₇₂

—

C₂×#14273667

C₂²

yes

—

—

C₂²

yes

characteristic

Dic₅₉⋊C₁₆

—

#16837217

—

C₂²×C₁₆×C₅₉

C₂²

yes

characteristic

C₂×#108×C₅₉

C₂²

yes

—

C₂²×Dic₁₁₈

—

C₂×Dic₂₃₆

—

C₂×Dic₂₃₆

—

C₂×Dic₂₃₆

—

C₂×Dic₂₃₆

—

C₂×Dic₂₃₆

—

C₂×Dic₂₃₆

D₄

yes

—

C₂×#11745664

D₄

yes

characteristic

#14273686

D₄

yes

characteristic

Dic₄₇₂

—

#14273667

—

C₂×C₅₉⋊C₁₆

—

#14273652

—

#14273645

—

#14273637

—

C₈×Dic₅₉

—

C₂×M₁₆×C₅₉

—

C₂²×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₁₆×C₅₉

—

C₂×C₁₆×C₅₉

D₄

yes

—

#108×C₅₉

—

#108×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×Dic₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×D₄

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

C₂×C₈×C₅₉

yes

C₂×C₈×C₅₉

yes

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

D₈

yes

—

C₄×Dic₅₉

—

#11745664

—

C₂×C₅₉⋊C₈

—

Dic₂₃₆

—

Dic₂₃₆

—

Dic₂₃₆

—

Dic₂₃₆

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

#135

yes

characteristic

C₂×C₄×C₅₉

#121

yes

—

C₂×C₄×C₅₉

#121

yes

characteristic

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

AGL(1,ℤ/8)

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

#120

yes

characteristic

C₈×C₅₉

#123

yes

—

C₈×C₅₉

#124

yes

characteristic

C₅₉⋊C₈

—

#13441

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

#463

yes

—

C₂²×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

C₄×C₅₉

#469

yes

characteristic

Dic₅₉

—

Dic₅₉

—

C₂²×Q₃₂

—

—

—

—

D₅₉

yes

characteristic

C₂×C₅₉

—

C₂×C₅₉

#2054

yes

characteristic

C₂×C₅₉

#2222

yes

—

100

C₂²×Q₁₆

—

101

C₂×Q₃₂

—

118

—

102

C₂×Q₃₂

—

118

—

103

C₂×M₃₂

—

104

C₂²×C₁₆

C₂×D₅₉

yes

characteristic

105

C₂×#108

C₂×D₅₉

yes

—

106

#419

—

118

—

107

C₂×#101

—

118

—

108

#405

—

109

C₄⋊C₁₆

—

118

—

110

#362

—

118

—

111

C₅₉

#13441

yes

characteristic, complemented, Sylow

112

C₂²×Q₈

—

118

—

113

C₂×Q₁₆

—

114

C₂×Q₁₆

—

118

—

115

C₂×Q₁₆

—

118

—

116

C₂×Q₁₆

—

118

—

117

C₂×Q₁₆

—

118

—

118

C₂×Q₁₆

—

118

—

119

C₂×M₁₆

—

120

C₂×M₁₆

—

121

C₂²×C₈

C₂²×D₅₉

yes

characteristic

122

#117

—

123

Q₃₂

—

118

—

124

M₃₂

—

118

—

125

C₂×C₁₆

—

126

C₂×C₁₆

—

118

—

127

C₂×C₁₆

D₂₃₆

yes

—

128

#108

—

129

#108

—

130

#103

—

236

—

131

#101

—

236

—

132

#97

—

118

—

133

C₄×C₈

—

118

—

134

C₂×Q₈

—

118

—

135

C₂×Q₈

—

236

—

136

C₂×Q₈

—

236

—

137

C₂×Q₈

—

236

—

138

C₂×Q₈

—

236

—

139

C₂×Q₈

—

236

—

140

C₂²×C₄

—

236

—

141

C₂²×C₄

D₄×D₅₉

yes

characteristic

142

Q₁₆

—

118

—

143

Q₁₆

—

118

—

144

Q₁₆

—

236

—

145

Q₁₆

—

236

—

146

M₁₆

—

147

M₁₆

—

148

M₁₆

—

118

—

149

C₂×C₈

—

150

C₂×C₈

—

118

—

151

C₂×C₈

C₂×D₂₃₆

yes

characteristic

152

C₂×C₈

#11745685

yes

—

153

C₂×C₈

D₄×D₅₉

yes

—

154

C₂×C₈

#11745689

yes

characteristic

155

C₄⋊C₄

—

156

C₂²⋊C₄

—

118

—

157

C₄²

—

118

—

158

C₁₆

—

159

C₁₆

—

236

—

160

C₂³

#14273750

yes

characteristic

161

Q₈

—

472

—

162

Q₈

—

472

—

163

Q₈

—

236

—

164

Q₈

—

236

—

165

C₂×C₄

—

118

—

166

C₂×C₄

—

472

—

167

C₂×C₄

—

472

—

168

C₂×C₄

—

236

—

169

C₂×C₄

#14273700

yes

—

170

C₂×C₄

#14273702

yes

characteristic

171

C₂×C₄

D₈×D₅₉

yes

characteristic

172

C₂×C₄

#14273754

yes

—

173

C₈

—

174

C₈

—

236

—

175

C₈

#14273697

yes

characteristic

176

C₈

#14273712

yes

—

177

C₈

#14273715

yes

characteristic

178

C₂²

—

179

C₂²

#16837620

yes

—

180

C₂²

#16837639

yes

characteristic

181

C₄

—

472

—

182

C₄

—

236

—

183

C₄

#16837320

yes

characteristic

184

C₄

#16837596

yes

—

185

C₄

#16837656

yes

characteristic

186

C₂

—

187

C₂

#21425408

yes

characteristic

188

C₂

#21426222

yes

—

	subgroup	G/H
center	C₂²	#16837639
commutator, cosocle	C₂²×C₈×C₅₉	C₂³
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1381	C₂
socle	C₂²×C₅₉	#466
prime radical	C₂³×C₅₉	#135
hypercenter	#1381	D₅₉
hypocenter	C₅₉	#13441
LCS	G ▷ C₂²×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

subgroup

G/H

center

C₂²

#16837639

commutator, cosocle

C₂²×C₈×C₅₉

C₂³

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₅₉×#1381

C₂

C₂²×C₅₉

C₂³×C₅₉

D₅₉

C₅₉

G ▷ C₂²×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

N	H ≅ G/N	type
111. C₅₉	#13441	semidirect

H ≅ G/N

type

111. C₅₉

#13441

semidirect

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

subgroup	index
C₂²×Dic₄₇₂	2
#21425854	2
C₂×#16837217	2
#21425009	2
C₅₉×#1381	2
#13441	59

subgroup

index

C₂²×Dic₄₇₂

C₅₉×#1381

Subgroups — #28508242

Conjugacy classes of subgroups

Complements

Maximal subgroups