Group #13018748 — subgroups

Subgroups — #13018748

Order 1280.

342 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 116 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	#10604191	C₂	1	1	yes	maximal, characteristic
2	#10604185	C₂	1	1	yes	maximal, characteristic
3	#10599389	C₂	1	1	yes	maximal, characteristic
4	#10598771	C₂	1	4	yes	maximal
5	C₂×#4004	—	2	2	no	—
6	C₂×C₄⋊#170	C₂²	1	2	yes	—
7	C₂×C₄⋊#170	C₂²	1	2	yes	—
8	C₂×C₈×Dic₅	C₂²	1	2	yes	—
9	C₄²×Dic₅	C₂²	1	1	yes	characteristic
10	#64918	—	2	2	no	—
11	C₅×#336	C₄	1	1	yes	characteristic
12	#64770	C₄	1	2	yes	—
13	#64744	C₄	1	1	yes	characteristic
14	#8789	—	5	1	no	maximal, Sylow
15	C₂²×#170	—	2	2	no	—
16	C₂²×#170	—	2	2	no	—
17	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₅	D₄	1	2	yes	—
21	C₂×C₄×Dic₅	Q₈	1	2	yes	—
22	C₂×C₄×Dic₅	C₂³	1	1	yes	characteristic
23	C₂²×#168	C₂×C₄	1	2	yes	—
24	#4004	—	4	4	no	—
25	C₄⋊#170	—	2	4	no	—
26	C₄⋊#170	—	2	4	no	—
27	C₈×Dic₅	—	2	2	no	—
28	C₈×Dic₅	—	2	2	no	—
29	#1842	—	5	1	no	—
30	#1831	—	5	1	no	—
31	#1744	—	5	1	no	—
32	#1277	—	5	4	no	—
33	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
34	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
35	C₂²×Dic₅	C₄⋊C₄	1	2	yes	—
36	C₂²×Dic₅	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₅	C₂×Q₈	1	1	yes	characteristic
38	C₂×#170	—	4	2	no	—
39	C₂×#170	—	4	2	no	—
40	C₂×#170	—	4	2	no	—
41	C₂×#170	—	4	4	no	—
42	C₂×#170	—	4	2	no	—
43	C₂×C₅×C₈	—	2	2	no	—
44	C₂×C₅×C₈	—	2	2	no	—
45	C₂×C₅×C₈	—	2	2	no	—
46	C₄²×C₅	—	2	2	no	—
47	C₄×Dic₅	—	2	1	no	—
48	C₄×Dic₅	—	2	1	no	—
49	C₄×Dic₅	—	2	4	no	—
50	C₄×Dic₅	—	2	2	no	—
51	C₄×Dic₅	—	2	2	no	—
52	C₄×Dic₅	#41	1	2	yes	—
53	C₄×Dic₅	#41	1	2	yes	—
54	C₂×#168	—	2	2	no	—
55	C₂×#168	—	2	2	no	—
56	C₂×#168	—	2	2	no	—
57	C₂×C₄⋊C₈	—	5	2	no	—
58	C₂×C₄⋊C₈	—	5	2	no	—
59	C₂×C₄⋊C₈	—	10	2	no	—
60	C₂×C₄×C₈	—	5	2	no	—
61	C₄³	—	5	1	no	—
62	#336	—	5	2	no	—
63	#336	—	5	1	no	—
64	#336	—	10	2	no	—
65	#336	AGL(1,𝔽₅)	1	1	yes	characteristic
66	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₅	—	2	2	no	—
68	C₂×C₄×C₅	—	2	2	no	—
69	C₂×C₄×C₅	—	2	2	no	—
70	C₂×C₄×C₅	#117	1	2	yes	—
71	C₂×C₄×C₅	#131	1	2	yes	—
72	C₂×C₄×C₅	#131	1	1	yes	characteristic
73	C₂×C₄×C₅	#131	1	1	yes	characteristic
74	C₂×Dic₅	—	2	2	no	—
75	C₂×Dic₅	—	2	2	no	—
76	C₂×Dic₅	—	2	2	no	—
77	C₂×Dic₅	#108	1	4	yes	—
78	C₂×Dic₅	#124	1	1	yes	characteristic
79	C₂×Dic₅	#124	1	1	yes	characteristic
80	C₂×Dic₅	#125	1	2	yes	—
81	C₂×Dic₅	AGL(1,ℤ/8)	1	2	yes	—
82	C₂×Dic₅	#137	1	2	yes	—
83	#170	—	8	2	no	—
84	#170	—	8	2	no	—
85	C₅×C₈	—	4	2	no	—
86	#168	—	4	2	no	—
87	C₂²×C₈	—	5	2	no	—
88	C₂²×C₈	—	10	2	no	—
89	C₂²×C₈	—	10	2	no	—
90	C₂²×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
91	C₂×C₄²	—	5	1	no	—
92	C₂×C₄²	—	5	2	no	—
93	C₂×C₄²	—	5	2	no	—
94	C₂×C₄²	—	5	1	no	—
95	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
96	C₄⋊C₈	—	10	4	no	—
97	C₄⋊C₈	—	10	4	no	—
98	C₄⋊C₈	—	20	4	no	—
99	C₄×C₈	—	10	2	no	—
100	C₄×C₈	—	10	2	no	—
101	C₂²×C₅	#382	1	1	yes	characteristic
102	C₂²×C₅	#424	1	1	yes	characteristic
103	C₂²×C₅	#424	1	1	yes	characteristic
104	C₂²×C₅	#428	1	2	yes	—
105	C₂²×C₅	C₂×#108	1	2	yes	—
106	C₄×C₅	—	4	2	no	—
107	C₄×C₅	#433	1	2	yes	—
108	C₄×C₅	#433	1	2	yes	—
109	Dic₅	—	2	4	no	—
110	Dic₅	—	2	2	no	—
111	Dic₅	#489	1	2	yes	—
112	Dic₅	#490	1	1	yes	characteristic
113	Dic₅	#491	1	1	yes	characteristic
114	C₂²×C₄	—	5	1	no	—
115	C₂²×C₄	—	5	2	no	—
116	C₂²×C₄	—	5	1	no	—
117	C₂²×C₄	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
118	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	10	2	no	—
123	C₂×C₈	—	10	2	no	—
124	C₂×C₈	—	10	2	no	—
125	C₂×C₈	—	20	2	no	—
126	C₂×C₈	—	20	2	no	—
127	C₂×C₈	—	20	4	no	—
128	C₂×C₈	—	20	2	no	—
129	C₂×C₈	—	20	2	no	—
130	C₄²	—	2	2	no	—
131	C₄²	—	5	2	no	—
132	C₄²	—	5	2	no	—
133	C₄²	—	10	2	no	—
134	C₄²	—	10	4	no	—
135	C₄²	—	10	2	no	—
136	C₄²	—	10	1	no	—
137	C₄²	—	10	1	no	—
138	C₂×C₅	#1573	1	2	yes	—
139	C₂×C₅	#1574	1	2	yes	—
140	C₂×C₅	#1832	1	1	yes	characteristic
141	C₂×C₅	#1836	1	1	yes	characteristic
142	C₂×C₅	#1837	1	1	yes	characteristic
143	C₂³	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	5	1	no	—
148	C₂×C₄	—	5	1	no	—
149	C₂×C₄	—	5	2	no	—
150	C₂×C₄	—	5	2	no	—
151	C₂×C₄	—	5	2	no	—
152	C₂×C₄	—	5	4	no	—
153	C₂×C₄	—	10	2	no	—
154	C₂×C₄	—	10	2	no	—
155	C₂×C₄	—	10	2	no	—
156	C₂×C₄	#4130	1	2	yes	—
157	C₂×C₄	#4131	1	1	yes	characteristic
158	C₂×C₄	#4131	1	2	yes	—
159	C₂×C₄	#4133	1	1	yes	characteristic
160	C₈	—	4	2	no	—
161	C₈	—	20	2	no	—
162	C₈	—	40	2	no	—
163	C₈	—	40	2	no	—
164	C₅	#8789	1	1	yes	characteristic, complemented, Sylow
165	C₂²	#65689	1	1	yes	characteristic
166	C₂²	#65696	1	1	yes	characteristic
167	C₂²	#65708	1	1	yes	characteristic
168	C₂²	D₅⋊#108	1	2	yes	—
169	C₂²	#65727	1	2	yes	—
170	C₄	—	4	2	no	—
171	C₄	—	5	1	no	—
172	C₄	—	5	2	no	—
173	C₄	—	5	1	no	—
174	C₄	—	10	2	no	—
175	C₄	—	10	4	no	—
176	C₄	#65697	1	2	yes	—
177	C₄	#65706	1	2	yes	—
178	C₂	#10603782	1	1	yes	characteristic
179	C₂	#10603857	1	1	yes	characteristic
180	C₂	#10603882	1	1	yes	characteristic
181	C₂	#10603902	1	2	yes	—
182	C₂	#10604018	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10604191

C₂

yes

maximal, characteristic

#10604185

C₂

yes

maximal, characteristic

#10599389

C₂

yes

maximal, characteristic

#10598771

C₂

yes

maximal

C₂×#4004

—

C₂×C₄⋊#170

C₂²

yes

—

C₂×C₄⋊#170

C₂²

yes

—

C₂×C₈×Dic₅

C₂²

yes

—

C₄²×Dic₅

C₂²

yes

characteristic

#64918

—

C₅×#336

C₄

yes

characteristic

#64770

C₄

yes

—

#64744

C₄

yes

characteristic

#8789

—

maximal, Sylow

C₂²×#170

—

C₂²×#170

—

C₂²×C₅×C₈

C₂×C₄

yes

—

C₂×C₄²×C₅

C₂×C₄

yes

characteristic

C₂×C₄×Dic₅

C₂×C₄

yes

characteristic

C₂×C₄×Dic₅

D₄

yes

—

C₂×C₄×Dic₅

Q₈

yes

—

C₂×C₄×Dic₅

C₂³

yes

characteristic

C₂²×#168

C₂×C₄

yes

—

#4004

—

C₄⋊#170

—

C₄⋊#170

—

C₈×Dic₅

—

C₈×Dic₅

—

—

—

—

—

C₂²×C₄×C₅

C₄⋊C₄

yes

—

C₂²×C₄×C₅

C₂²×C₄

yes

characteristic

C₂²×Dic₅

C₄⋊C₄

yes

—

C₂²×Dic₅

C₂×D₄

yes

characteristic

C₂²×Dic₅

C₂×Q₈

yes

—

—

—

—

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₄²×C₅

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

#41

yes

—

C₄×Dic₅

#41

yes

—

C₂×#168

—

C₂×#168

—

C₂×#168

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

C₄³

—

—

—

—

yes

C₂³×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#117

yes

—

C₂×C₄×C₅

#131

yes

—

C₂×C₄×C₅

yes

C₂×C₄×C₅

yes

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

#108

yes

—

C₂×Dic₅

yes

C₂×Dic₅

yes

C₂×Dic₅

yes

—

C₂×Dic₅

AGL(1,ℤ/8)

yes

—

C₂×Dic₅

#137

yes

—

#170

—

#170

—

C₅×C₈

—

#168

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×AGL(1,𝔽₅)

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×AGL(1,𝔽₅)

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

100

C₄×C₈

—

101

C₂²×C₅

#382

yes

characteristic

102

C₂²×C₅

#424

yes

characteristic

103

C₂²×C₅

#424

yes

characteristic

104

C₂²×C₅

#428

yes

—

105

C₂²×C₅

C₂×#108

yes

—

106

C₄×C₅

—

107

C₄×C₅

#433

yes

—

108

C₄×C₅

#433

yes

—

109

Dic₅

—

110

Dic₅

—

111

Dic₅

#489

yes

—

112

Dic₅

#490

yes

characteristic

113

Dic₅

#491

yes

characteristic

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

C₄⋊AGL(1,𝔽₅)

yes

—

118

C₂²×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₄²

—

131

C₄²

—

132

C₄²

—

133

C₄²

—

134

C₄²

—

135

C₄²

—

136

C₄²

—

137

C₄²

—

138

C₂×C₅

#1573

yes

—

139

C₂×C₅

#1574

yes

—

140

C₂×C₅

#1832

yes

characteristic

141

C₂×C₅

#1836

yes

characteristic

142

C₂×C₅

#1837

yes

characteristic

143

C₂³

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

#4130

yes

—

157

C₂×C₄

#4131

yes

characteristic

158

C₂×C₄

#4131

yes

—

159

C₂×C₄

#4133

yes

characteristic

160

C₈

—

161

C₈

—

162

C₈

—

163

C₈

—

164

C₅

#8789

yes

characteristic, complemented, Sylow

165

C₂²

#65689

yes

characteristic

166

C₂²

#65696

yes

characteristic

167

C₂²

#65708

yes

characteristic

168

C₂²

D₅⋊#108

yes

—

169

C₂²

#65727

yes

—

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

#65697

yes

—

177

C₄

#65706

yes

—

178

C₂

#10603782

yes

characteristic

179

C₂

#10603857

yes

characteristic

180

C₂

#10603882

yes

characteristic

181

C₂

#10603902

yes

—

182

C₂

#10604018

yes

—

	subgroup	G/H
center	C₂³	C₂×C₄⋊AGL(1,𝔽₅)
commutator	C₂²×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#336	C₄
socle, prime radical	C₂³×C₅	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₅	C₂³
hypercenter	#336	AGL(1,𝔽₅)
hypocenter	C₅	#8789
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×C₄⋊AGL(1,𝔽₅)

commutator

C₂²×C₄×C₅

C₂²×C₄

Frattini

C₂²×C₄

C₂²×AGL(1,𝔽₅)

Fitting

C₅×#336

C₄

socle, prime radical

C₂³×C₅

C₂×C₄⋊C₄

cosocle

C₂×C₄×Dic₅

C₂³

C₅

G ▷ C₂²×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
164. C₅	#8789	semidirect

H ≅ G/N

type

164. C₅

#8789

semidirect

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

subgroup	index
#10604191	2
#10604185	2
#10599389	2
#10598771	2
#8789	5

subgroup

index

Subgroups — #13018748

Conjugacy classes of subgroups

Complements

Maximal subgroups