Group #13018746 — subgroups

Subgroups — #13018746

Order 1280.

322 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 112 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	#10604187	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×C₄⋊#170	C₂²	1	2	yes	—
6	C₂×C₄⋊#170	C₂²	1	2	yes	—
7	C₂×C₈×Dic₅	C₂²	1	2	yes	—
8	C₄²×Dic₅	C₂²	1	1	yes	characteristic
9	#64918	—	2	2	no	—
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	#8788	—	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	C₄⋊#170	—	2	4	no	—
25	C₄⋊#170	—	2	4	no	—
26	C₈×Dic₅	—	2	2	no	—
27	C₈×Dic₅	—	2	2	no	—
28	#1831	—	5	1	no	—
29	#1831	—	5	1	no	—
30	#1744	—	5	1	no	—
31	#1277	—	5	4	no	—
32	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
33	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
34	C₂²×Dic₅	C₄⋊C₄	1	2	yes	—
35	C₂²×Dic₅	C₂×D₄	1	1	yes	characteristic
36	C₂²×Dic₅	C₂×Q₈	1	1	yes	characteristic
37	C₂×#170	—	4	2	no	—
38	C₂×#170	—	4	2	no	—
39	C₂×#170	—	4	4	no	—
40	C₂×#170	—	4	4	no	—
41	C₂×C₅×C₈	—	2	2	no	—
42	C₂×C₅×C₈	—	2	2	no	—
43	C₂×C₅×C₈	—	2	2	no	—
44	C₄²×C₅	—	2	2	no	—
45	C₄×Dic₅	—	2	1	no	—
46	C₄×Dic₅	—	2	1	no	—
47	C₄×Dic₅	—	2	4	no	—
48	C₄×Dic₅	—	4	2	no	—
49	C₄×Dic₅	#41	1	2	yes	—
50	C₄×Dic₅	#41	1	2	yes	—
51	C₂×#168	—	2	2	no	—
52	C₂×#168	—	2	2	no	—
53	C₂×#168	—	2	2	no	—
54	C₂×C₄⋊C₈	—	5	2	no	—
55	C₂×C₄⋊C₈	—	5	2	no	—
56	C₂×C₄×C₈	—	5	2	no	—
57	C₄³	—	5	1	no	—
58	#336	—	5	2	no	—
59	#336	—	5	1	no	—
60	#336	—	10	2	no	—
61	#336	—	10	2	no	—
62	#336	AGL(1,𝔽₅)	1	1	yes	characteristic
63	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₅	—	2	2	no	—
65	C₂×C₄×C₅	—	2	2	no	—
66	C₂×C₄×C₅	—	2	2	no	—
67	C₂×C₄×C₅	#117	1	2	yes	—
68	C₂×C₄×C₅	#131	1	2	yes	—
69	C₂×C₄×C₅	#131	1	1	yes	characteristic
70	C₂×C₄×C₅	#131	1	1	yes	characteristic
71	C₂×Dic₅	—	2	2	no	—
72	C₂×Dic₅	—	2	2	no	—
73	C₂×Dic₅	—	2	2	no	—
74	C₂×Dic₅	—	2	2	no	—
75	C₂×Dic₅	#124	1	1	yes	characteristic
76	C₂×Dic₅	#124	1	1	yes	characteristic
77	C₂×Dic₅	#125	1	2	yes	—
78	C₂×Dic₅	#135	1	4	yes	—
79	#170	—	8	2	no	—
80	#170	—	8	2	no	—
81	C₅×C₈	—	4	2	no	—
82	#168	—	4	2	no	—
83	C₂²×C₈	—	5	2	no	—
84	C₂²×C₈	—	10	2	no	—
85	C₂²×C₈	—	10	2	no	—
86	C₂²×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
87	C₂×C₄²	—	5	1	no	—
88	C₂×C₄²	—	5	2	no	—
89	C₂×C₄²	—	5	1	no	—
90	C₂×C₄²	—	5	2	no	—
91	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
92	C₄⋊C₈	—	10	4	no	—
93	C₄⋊C₈	—	10	4	no	—
94	C₄×C₈	—	10	2	no	—
95	C₄×C₈	—	10	2	no	—
96	C₂²×C₅	#382	1	1	yes	characteristic
97	C₂²×C₅	#424	1	1	yes	characteristic
98	C₂²×C₅	#424	1	1	yes	characteristic
99	C₂²×C₅	#427	1	2	yes	—
100	C₂²×C₅	#430	1	2	yes	—
101	C₄×C₅	—	4	2	no	—
102	C₄×C₅	#433	1	2	yes	—
103	C₄×C₅	#433	1	2	yes	—
104	Dic₅	—	2	2	no	—
105	Dic₅	—	4	2	no	—
106	Dic₅	#488	1	2	yes	—
107	Dic₅	#488	1	2	yes	—
108	C₂²×C₄	—	5	1	no	—
109	C₂²×C₄	—	5	1	no	—
110	C₂²×C₄	—	5	2	no	—
111	C₂²×C₄	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
112	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	10	2	no	—
117	C₂×C₈	—	10	2	no	—
118	C₂×C₈	—	10	2	no	—
119	C₂×C₈	—	20	2	no	—
120	C₂×C₈	—	20	2	no	—
121	C₂×C₈	—	20	4	no	—
122	C₂×C₈	—	20	4	no	—
123	C₄²	—	2	2	no	—
124	C₄²	—	5	2	no	—
125	C₄²	—	5	2	no	—
126	C₄²	—	10	4	no	—
127	C₄²	—	10	1	no	—
128	C₄²	—	10	1	no	—
129	C₄²	—	20	2	no	—
130	C₂×C₅	#1572	1	2	yes	—
131	C₂×C₅	#1572	1	2	yes	—
132	C₂×C₅	#1832	1	1	yes	characteristic
133	C₂×C₅	#1835	1	1	yes	characteristic
134	C₂×C₅	#1838	1	1	yes	characteristic
135	C₂³	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	5	1	no	—
140	C₂×C₄	—	5	1	no	—
141	C₂×C₄	—	5	4	no	—
142	C₂×C₄	—	5	2	no	—
143	C₂×C₄	—	10	2	no	—
144	C₂×C₄	—	10	2	no	—
145	C₂×C₄	—	10	2	no	—
146	C₂×C₄	—	10	2	no	—
147	C₂×C₄	#4130	1	2	yes	—
148	C₂×C₄	#4131	1	1	yes	characteristic
149	C₂×C₄	#4131	1	2	yes	—
150	C₂×C₄	#4133	1	1	yes	characteristic
151	C₈	—	4	2	no	—
152	C₈	—	20	2	no	—
153	C₈	—	40	2	no	—
154	C₈	—	40	2	no	—
155	C₅	#8788	1	1	yes	characteristic, complemented, Sylow
156	C₂²	#65689	1	1	yes	characteristic
157	C₂²	#65696	1	1	yes	characteristic
158	C₂²	#65708	1	1	yes	characteristic
159	C₂²	#65721	1	2	yes	—
160	C₂²	#65729	1	2	yes	—
161	C₄	—	4	2	no	—
162	C₄	—	5	2	no	—
163	C₄	—	5	2	no	—
164	C₄	—	10	2	no	—
165	C₄	—	20	2	no	—
166	C₄	#65697	1	2	yes	—
167	C₄	#65706	1	2	yes	—
168	C₂	#10603782	1	1	yes	characteristic
169	C₂	#10603846	1	1	yes	characteristic
170	C₂	#10603889	1	1	yes	characteristic
171	C₂	#10603899	1	2	yes	—
172	C₂	#10604017	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10604187

C₂

yes

maximal, characteristic

#10604185

C₂

yes

maximal, characteristic

#10599389

C₂

yes

maximal, characteristic

#10598771

C₂

yes

maximal

C₂×C₄⋊#170

C₂²

yes

—

C₂×C₄⋊#170

C₂²

yes

—

C₂×C₈×Dic₅

C₂²

yes

—

C₄²×Dic₅

C₂²

yes

—

—

C₄

yes

characteristic

#64770

C₄

yes

—

#64744

C₄

yes

characteristic

#8788

—

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

—

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

#41

yes

—

C₄×Dic₅

#41

yes

—

C₂×#168

—

C₂×#168

—

C₂×#168

—

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

—

C₂×Dic₅

yes

C₂×Dic₅

yes

C₂×Dic₅

yes

—

C₂×Dic₅

#135

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

—

C₂²×C₅

yes

C₂²×C₅

yes

C₂²×C₅

yes

C₂²×C₅

yes

—

100

C₂²×C₅

#430

yes

—

101

C₄×C₅

—

102

C₄×C₅

#433

yes

—

103

C₄×C₅

#433

yes

—

104

Dic₅

—

105

Dic₅

—

106

Dic₅

#488

yes

—

107

Dic₅

#488

yes

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

C₄⋊AGL(1,𝔽₅)

yes

—

112

C₂²×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₄²

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₄²

—

129

C₄²

—

130

C₂×C₅

#1572

yes

—

131

C₂×C₅

#1572

yes

—

132

C₂×C₅

#1832

yes

characteristic

133

C₂×C₅

#1835

yes

characteristic

134

C₂×C₅

#1838

yes

characteristic

135

C₂³

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

#4130

yes

—

148

C₂×C₄

#4131

yes

characteristic

149

C₂×C₄

#4131

yes

—

150

C₂×C₄

#4133

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

—

155

C₅

#8788

yes

characteristic, complemented, Sylow

156

C₂²

#65689

yes

characteristic

157

C₂²

#65696

yes

characteristic

158

C₂²

#65708

yes

characteristic

159

C₂²

#65721

yes

—

160

C₂²

#65729

yes

—

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

#65697

yes

—

167

C₄

#65706

yes

—

168

C₂

#10603782

yes

characteristic

169

C₂

#10603846

yes

characteristic

170

C₂

#10603889

yes

characteristic

171

C₂

#10603899

yes

—

172

C₂

#10604017

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	#8788
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
155. C₅	#8788	semidirect

H ≅ G/N

type

155. C₅

#8788

semidirect

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

subgroup	index
#10604187	2
#10604185	2
#10599389	2
#10598771	2
#8788	5

subgroup

index

Subgroups — #13018746

Conjugacy classes of subgroups

Complements

Maximal subgroups