Group #13020839 — subgroups

Subgroups — #13020839

Order 1280.

294 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 98 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	#10604739	C₂	1	2	yes	maximal
2	C₂×#64881	C₂	1	1	yes	maximal, characteristic
3	#10604689	C₂	1	1	yes	maximal, characteristic
4	#10599709	C₂	1	1	yes	maximal, characteristic
5	#10598793	C₂	1	2	yes	maximal
6	#65748	C₂²	1	1	yes	characteristic
7	#65748	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊#170	C₂²	1	2	yes	—
9	C₂×#3954	C₄	1	2	yes	—
10	M₁₆×Dic₅	C₂²	1	2	yes	—
11	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
12	#64884	—	2	2	no	—
13	#64883	—	2	2	no	—
14	#64881	—	2	2	no	—
15	#170⋊C₈	—	2	2	no	—
16	C₅×#341	C₄	1	1	yes	characteristic
17	#64772	C₄	1	1	yes	characteristic
18	#9609	—	5	1	no	maximal, Sylow
19	C₂²×#170	—	2	2	no	—
20	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
21	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₅	C₂³	1	1	yes	characteristic
23	C₂×#681	C₂×C₄	1	2	yes	—
24	C₂²×#168	C₂×C₄	1	1	yes	characteristic
25	C₂²⋊#170	—	2	1	no	—
26	C₂²⋊#170	Q₈	1	2	yes	—
27	C₄⋊#170	—	2	2	no	—
28	C₄⋊#170	—	2	2	no	—
29	C₄⋊#170	—	2	1	no	—
30	C₄⋊#170	D₄	1	2	yes	—
31	C₄×#170	—	2	1	no	—
32	C₄×#170	—	2	1	no	—
33	#3954	—	2	4	no	—
34	#3947	—	2	2	no	—
35	C₈×Dic₅	—	2	2	no	—
36	C₈×Dic₅	D₄	1	2	yes	—
37	C₈×Dic₅	Q₈	1	2	yes	—
38	C₄×#108	—	5	1	no	—
39	#1565	—	5	2	no	—
40	C₂×#334	—	5	1	no	—
41	#1452	—	5	1	no	—
42	#1283	—	5	2	no	—
43	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
44	C₂²×Dic₅	C₂×Q₈	1	1	yes	characteristic
45	C₂²×Dic₅	#41	1	1	yes	characteristic
46	C₂×#170	—	2	2	no	—
47	C₂×#170	—	2	2	no	—
48	C₂×#170	—	4	2	no	—
49	C₂×#170	—	8	2	no	—
50	C₅×M₁₆	—	2	2	no	—
51	C₅×M₁₆	—	2	2	no	—
52	C₂×C₅×C₈	—	2	2	no	—
53	C₂×C₅×C₈	—	2	2	no	—
54	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
55	C₄×Dic₅	C₂×D₄	1	1	yes	characteristic
56	C₄×Dic₅	C₂×D₄	1	1	yes	characteristic
57	C₄×Dic₅	C₂×Q₈	1	1	yes	characteristic
58	C₄×Dic₅	#41	1	1	yes	characteristic
59	#681	—	2	2	no	—
60	#681	M₁₆	1	4	yes	—
61	C₂×#168	—	2	2	no	—
62	C₂×#168	—	2	2	no	—
63	C₂×#168	C₄⋊C₄	1	2	yes	—
64	#431	—	5	1	no	—
65	#431	—	5	1	no	—
66	C₂×#108	—	5	2	no	—
67	C₂×C₄⋊C₈	—	5	2	no	—
68	C₄×M₁₆	—	5	2	no	—
69	C₂×C₄×C₈	—	5	1	no	—
70	#341	—	5	1	no	—
71	#341	AGL(1,𝔽₅)	1	1	yes	characteristic
72	#335	—	10	2	no	—
73	#334	—	10	2	no	—
74	#334	—	10	2	no	—
75	#322	—	10	2	no	—
76	C₂³×C₅	C₄×Q₈	1	1	yes	characteristic
77	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
80	C₂×C₄×C₅	#131	1	2	yes	—
81	C₂×Dic₅	—	2	2	no	—
82	C₂×Dic₅	—	4	1	no	—
83	C₂×Dic₅	#121	1	1	yes	characteristic
84	C₂×Dic₅	C₂²⋊Q₈	1	1	yes	characteristic
85	C₂×Dic₅	#125	1	1	yes	characteristic
86	C₂×Dic₅	C₄⋊Q₈	1	1	yes	characteristic
87	#170	—	4	2	no	—
88	#170	—	4	2	no	—
89	#170	—	8	2	no	—
90	C₅×C₈	—	2	2	no	—
91	C₅×C₈	—	4	2	no	—
92	#168	—	2	2	no	—
93	#168	—	2	4	no	—
94	C₂×M₁₆	—	5	2	no	—
95	C₂×M₁₆	C₂×AGL(1,𝔽₅)	1	2	yes	—
96	C₂²×C₈	—	5	1	no	—
97	C₂²×C₈	—	10	2	no	—
98	C₂²×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
99	C₂×C₄²	—	5	1	no	—
100	#108	—	10	4	no	—
101	C₄⋊C₈	—	5	2	no	—
102	C₄⋊C₈	—	10	1	no	—
103	C₄⋊C₈	—	10	2	no	—
104	C₄⋊C₈	—	10	2	no	—
105	C₂²⋊C₈	—	5	2	no	—
106	C₂²⋊C₈	—	10	1	no	—
107	#97	—	10	2	no	—
108	C₄×C₈	—	5	2	no	—
109	C₄×C₈	—	5	2	no	—
110	C₄×C₈	—	10	2	no	—
111	C₄×C₈	—	10	1	no	—
112	C₄×C₈	—	10	1	no	—
113	C₂²×C₅	—	2	1	no	—
114	C₂²×C₅	—	2	1	no	—
115	C₂²×C₅	#389	1	1	yes	characteristic
116	C₂²×C₅	#446	1	2	yes	—
117	C₄×C₅	#423	1	1	yes	characteristic
118	C₄×C₅	#424	1	1	yes	characteristic
119	C₄×C₅	#435	1	2	yes	—
120	Dic₅	—	2	1	no	—
121	Dic₅	—	2	1	no	—
122	Dic₅	—	2	1	no	—
123	Dic₅	#471	1	2	yes	—
124	C₂²×C₄	—	5	1	no	—
125	C₂²×C₄	—	5	1	no	—
126	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
127	M₁₆	—	2	2	no	—
128	M₁₆	—	2	2	no	—
129	M₁₆	—	5	4	no	—
130	M₁₆	—	10	2	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	5	2	no	—
134	C₂×C₈	—	10	2	no	—
135	C₂×C₈	—	10	2	no	—
136	C₂×C₈	—	10	2	no	—
137	C₂×C₈	—	10	2	no	—
138	C₂×C₈	—	20	2	no	—
139	C₂×C₈	—	40	2	no	—
140	C₂×C₈	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
141	C₄²	—	5	1	no	—
142	C₄²	—	5	1	no	—
143	C₄²	—	5	1	no	—
144	C₄²	—	5	1	no	—
145	C₂×C₅	—	2	2	no	—
146	C₂×C₅	#1940	1	1	yes	characteristic
147	C₂×C₅	#1951	1	1	yes	characteristic
148	C₂×C₅	#1953	1	1	yes	characteristic
149	C₂³	Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
150	C₂×C₄	—	5	1	no	—
151	C₂×C₄	—	5	1	no	—
152	C₂×C₄	—	5	1	no	—
153	C₂×C₄	—	5	1	no	—
154	C₂×C₄	—	10	2	no	—
155	C₂×C₄	—	20	1	no	—
156	C₂×C₄	D₅⋊M₁₆	1	2	yes	—
157	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
158	C₂×C₄	#4131	1	2	yes	—
159	C₂×C₄	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
160	C₈	—	2	2	no	—
161	C₈	—	4	2	no	—
162	C₈	—	10	2	no	—
163	C₈	—	10	4	no	—
164	C₈	—	20	2	no	—
165	C₈	—	20	2	no	—
166	C₈	—	40	2	no	—
167	C₅	#9609	1	1	yes	characteristic, complemented, Sylow
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	#65707	1	2	yes	—
171	C₂²	#65712	1	1	yes	characteristic
172	C₄	—	5	2	no	—
173	C₄	—	10	1	no	—
174	C₄	—	10	1	no	—
175	C₄	—	10	1	no	—
176	C₄	#65682	1	1	yes	characteristic
177	C₄	#65694	1	2	yes	—
178	C₄	#65696	1	1	yes	characteristic
179	C₂	—	2	2	no	—
180	C₂	#10603797	1	1	yes	characteristic
181	C₂	#10604087	1	1	yes	characteristic
182	C₂	#10604091	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10604739

C₂

yes

maximal

C₂×#64881

C₂

yes

maximal, characteristic

#10604689

C₂

yes

maximal, characteristic

#10599709

C₂

yes

maximal, characteristic

#10598793

C₂

yes

maximal

#65748

C₂²

yes

characteristic

#65748

C₂²

yes

characteristic

C₂×C₄⋊#170

C₂²

yes

—

C₂×#3954

C₄

yes

—

M₁₆×Dic₅

C₂²

yes

—

C₂×C₈×Dic₅

C₂²

yes

—

—

—

—

C₄

yes

characteristic

#64772

C₄

yes

characteristic

#9609

—

maximal, Sylow

C₂²×#170

—

C₂×C₅×M₁₆

C₂×C₄

yes

—

C₂²×C₅×C₈

C₂×C₄

yes

characteristic

C₂×C₄×Dic₅

C₂³

yes

characteristic

C₂×#681

C₂×C₄

yes

—

C₂²×#168

C₂×C₄

yes

characteristic

C₂²⋊#170

—

C₂²⋊#170

Q₈

yes

—

—

—

—

D₄

yes

—

—

—

—

—

C₈×Dic₅

—

C₈×Dic₅

D₄

yes

—

C₈×Dic₅

Q₈

yes

—

—

—

—

—

—

C₂²×C₄×C₅

C₂²×C₄

yes

characteristic

C₂²×Dic₅

C₂×Q₈

yes

C₂²×Dic₅

yes

—

—

—

—

C₅×M₁₆

—

C₅×M₁₆

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₄⋊C₄

yes

—

C₄×Dic₅

C₂×D₄

yes

characteristic

C₄×Dic₅

C₂×D₄

yes

characteristic

C₄×Dic₅

C₂×Q₈

yes

C₄×Dic₅

yes

—

M₁₆

yes

—

C₂×#168

—

C₂×#168

—

C₂×#168

C₄⋊C₄

yes

—

#431

—

#431

—

C₂×#108

—

C₂×C₄⋊C₈

—

C₄×M₁₆

—

C₂×C₄×C₈

—

—

yes

—

—

—

—

C₂³×C₅

C₄×Q₈

yes

characteristic

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅

C₄×D₄

yes

characteristic

C₂×C₄×C₅

C₂×M₁₆

yes

—

C₂×C₄×C₅

#131

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

#121

yes

characteristic

C₂×Dic₅

C₂²⋊Q₈

yes

characteristic

C₂×Dic₅

#125

yes

characteristic

C₂×Dic₅

C₄⋊Q₈

yes

—

—

—

C₅×C₈

—

C₅×C₈

—

#168

—

#168

—

C₂×M₁₆

—

C₂×M₁₆

C₂×AGL(1,𝔽₅)

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×AGL(1,𝔽₅)

yes

characteristic

C₂×C₄²

—

100

#108

—

101

C₄⋊C₈

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

C₂²⋊C₈

—

106

C₂²⋊C₈

—

107

#97

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₂²×C₅

—

114

C₂²×C₅

—

115

C₂²×C₅

#389

yes

characteristic

116

C₂²×C₅

#446

yes

—

117

C₄×C₅

#423

yes

characteristic

118

C₄×C₅

#424

yes

characteristic

119

C₄×C₅

#435

yes

—

120

Dic₅

—

121

Dic₅

—

122

Dic₅

—

123

Dic₅

#471

yes

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

127

M₁₆

—

128

M₁₆

—

129

M₁₆

—

130

M₁₆

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

C₄⋊AGL(1,𝔽₅)

yes

—

141

C₄²

—

142

C₄²

—

143

C₄²

—

144

C₄²

—

145

C₂×C₅

—

146

C₂×C₅

#1940

yes

characteristic

147

C₂×C₅

#1951

yes

characteristic

148

C₂×C₅

#1953

yes

characteristic

149

C₂³

Q₈×AGL(1,𝔽₅)

yes

characteristic

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

D₅⋊M₁₆

yes

—

157

C₂×C₄

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

158

C₂×C₄

#4131

yes

—

159

C₂×C₄

D₄×AGL(1,𝔽₅)

yes

characteristic

160

C₈

—

161

C₈

—

162

C₈

—

163

C₈

—

164

C₈

—

165

C₈

—

166

C₈

—

167

C₅

#9609

yes

characteristic, complemented, Sylow

168

C₂²

—

169

C₂²

—

170

C₂²

#65707

yes

—

171

C₂²

#65712

yes

characteristic

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

#65682

yes

characteristic

177

C₄

#65694

yes

—

178

C₄

#65696

yes

characteristic

179

C₂

—

180

C₂

#10603797

yes

characteristic

181

C₂

#10604087

yes

characteristic

182

C₂

#10604091

yes

characteristic

	subgroup	G/H
center	C₂²	#65712
commutator	C₂²×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#341	C₄
socle	C₂²×C₅	#389
prime radical	C₂³×C₅	C₄×Q₈
cosocle	C₂×C₄×Dic₅	C₂³
hypercenter	#341	AGL(1,𝔽₅)
hypocenter	C₅	#9609
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂²

#65712

commutator

C₂²×C₄×C₅

C₂²×C₄

Frattini

C₂²×C₄

C₂²×AGL(1,𝔽₅)

C₄

C₂²×C₅

C₂³×C₅

C₄×Q₈

cosocle

C₂×C₄×Dic₅

C₂³

C₅

G ▷ C₂²×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
167. C₅	#9609	semidirect

H ≅ G/N

type

167. C₅

#9609

semidirect

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

subgroup	index
#10604739	2
C₂×#64881	2
#10604689	2
#10599709	2
#10598793	2
#9609	5

subgroup

index

Subgroups — #13020839

Conjugacy classes of subgroups

Complements

Maximal subgroups