Group #13013816 — subgroups

Subgroups — #13013816

Order 1280.

192 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 62 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	#10603827	C₂	1	1	yes	maximal, characteristic
2	#10603821	C₂	1	1	yes	maximal, characteristic
3	#10599322	C₂	1	1	yes	maximal, characteristic
4	C₂×#3990	C₄	1	2	yes	—
5	#65683	—	2	1	no	—
6	#65126	—	2	1	no	—
7	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
8	#64888	—	2	1	no	—
9	#64884	C₄	1	2	yes	—
10	#64883	—	2	1	no	—
11	C₄⋊#674	—	2	1	no	—
12	C₅×#334	C₄	1	1	yes	characteristic
13	#64700	—	2	1	no	—
14	#64677	C₄	1	1	yes	characteristic
15	#7244	—	5	1	no	maximal, Sylow
16	C₂×#676	—	4	1	no	—
17	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×D₅	Q₈	1	1	yes	characteristic
19	C₄²×D₅	D₄	1	1	yes	characteristic
20	#4003	—	4	1	no	—
21	#4002	—	4	1	no	—
22	C₄⋊#170	—	2	2	no	—
23	C₂×#674	—	2	2	no	—
24	#3990	—	4	2	no	—
25	C₅×C₄⋊C₈	—	2	1	no	—
26	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	#3946	—	4	1	no	—
28	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₅	D₄	1	1	yes	characteristic
30	C₄⋊#168	—	2	1	no	—
31	C₄×#168	D₄	1	1	yes	characteristic
32	#2150	—	5	1	no	—
33	#1568	—	5	1	no	—
34	#1566	—	5	1	no	—
35	C₂×C₄×D₅	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×D₅	D₈	1	1	yes	characteristic
37	C₂×C₄×D₅	Q₁₆	1	1	yes	characteristic
38	C₂×#170	—	4	2	no	—
39	C₂×C₅×C₈	—	4	1	no	—
40	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
41	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
42	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
43	C₄×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
44	C₄×Dic₅	SD₁₆	1	1	yes	characteristic
45	C₄×Dic₅	SD₁₆	1	1	yes	characteristic
46	C₂×#168	—	4	1	no	—
47	C₂×#168	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×#168	C₄⋊C₄	1	1	yes	characteristic
49	#676	—	8	2	no	—
50	C₈×D₅	—	2	1	no	—
51	C₈×D₅	—	2	1	no	—
52	C₈×D₅	—	4	1	no	—
53	#674	—	4	2	no	—
54	C₂×M₃₂	—	5	2	no	—
55	#432	—	10	1	no	—
56	#423	—	10	1	no	—
57	C₂×C₄×C₈	—	5	1	no	—
58	C₄⋊C₁₆	—	10	1	no	—
59	C₄⋊C₁₆	—	10	1	no	—
60	#335	—	5	2	no	—
61	#334	—	5	1	no	—
62	#334	—	10	1	no	—
63	#334	—	10	1	no	—
64	#334	AGL(1,𝔽₅)	1	1	yes	characteristic
65	C₂²×D₅	#107	1	1	yes	characteristic
66	C₂×C₄×C₅	#95	1	1	yes	characteristic
67	C₂×C₄×C₅	#102	1	1	yes	characteristic
68	C₂×C₄×C₅	#103	1	1	yes	characteristic
69	C₂×Dic₅	#102	1	1	yes	characteristic
70	C₂×Dic₅	#103	1	1	yes	characteristic
71	C₂×Dic₅	#106	1	1	yes	characteristic
72	C₄×D₅	—	2	1	no	—
73	C₄×D₅	—	2	1	no	—
74	C₄×D₅	—	2	2	no	—
75	C₄×D₅	—	4	1	no	—
76	#170	—	8	2	no	—
77	C₅×C₈	—	2	1	no	—
78	C₅×C₈	—	2	1	no	—
79	C₅×C₈	—	8	1	no	—
80	#168	—	2	1	no	—
81	#168	—	8	1	no	—
82	#168	#100	1	1	yes	characteristic
83	#168	#101	1	1	yes	characteristic
84	C₂×M₁₆	—	20	1	no	—
85	C₂²×C₈	—	5	1	no	—
86	C₂²×C₈	—	5	1	no	—
87	C₂×C₄²	—	5	1	no	—
88	M₃₂	—	20	2	no	—
89	C₂×C₁₆	—	10	2	no	—
90	C₄⋊C₈	—	2	1	no	—
91	C₄⋊C₈	—	10	2	no	—
92	C₄⋊C₈	—	10	1	no	—
93	C₄⋊C₈	—	20	1	no	—
94	C₂²⋊C₈	—	20	1	no	—
95	#97	—	20	1	no	—
96	C₄×C₈	—	5	1	no	—
97	C₄×C₈	—	5	1	no	—
98	C₄×C₈	—	5	1	no	—
99	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
100	C₂²×C₅	#340	1	1	yes	characteristic
101	C₂×D₅	—	2	1	no	—
102	C₂×D₅	—	2	1	no	—
103	C₂×D₅	—	2	1	no	—
104	C₄×C₅	—	2	1	no	—
105	C₄×C₅	—	2	1	no	—
106	C₄×C₅	#338	1	1	yes	characteristic
107	C₄×C₅	#343	1	1	yes	characteristic
108	Dic₅	—	2	1	no	—
109	Dic₅	—	2	1	no	—
110	Dic₅	#360	1	2	yes	—
111	C₂²×C₄	—	5	1	no	—
112	C₂²×C₄	—	5	1	no	—
113	C₂²×C₄	—	5	1	no	—
114	M₁₆	—	40	2	no	—
115	C₂×C₈	—	4	1	no	—
116	C₂×C₈	—	5	1	no	—
117	C₂×C₈	—	5	1	no	—
118	C₂×C₈	—	10	1	no	—
119	C₂×C₈	—	10	1	no	—
120	C₂×C₈	—	20	1	no	—
121	C₂×C₈	—	20	2	no	—
122	C₂×C₈	—	20	1	no	—
123	C₂×C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
124	C₂×C₈	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
125	C₄²	—	5	1	no	—
126	C₄²	—	5	1	no	—
127	C₄²	—	5	1	no	—
128	C₄²	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
129	C₁₆	—	20	2	no	—
130	C₂×C₅	#1292	1	1	yes	characteristic
131	C₂×C₅	#1386	1	1	yes	characteristic
132	C₂×C₅	#1388	1	1	yes	characteristic
133	D₅	—	4	1	no	—
134	C₂³	—	5	1	no	—
135	C₂×C₄	—	5	1	no	—
136	C₂×C₄	—	5	1	no	—
137	C₂×C₄	—	5	1	no	—
138	C₂×C₄	—	10	2	no	—
139	C₂×C₄	—	10	1	no	—
140	C₂×C₄	—	10	1	no	—
141	C₂×C₄	—	20	1	no	—
142	C₂×C₄	(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic
143	C₂×C₄	#4007	1	1	yes	characteristic
144	C₂×C₄	#4009	1	1	yes	characteristic
145	C₈	—	2	1	no	—
146	C₈	—	2	1	no	—
147	C₈	—	5	1	no	—
148	C₈	—	5	1	no	—
149	C₈	—	8	1	no	—
150	C₈	—	10	1	no	—
151	C₈	—	40	1	no	—
152	C₈	—	40	2	no	—
153	C₅	#7244	1	1	yes	characteristic, complemented, Sylow
154	C₂²	—	10	1	no	—
155	C₂²	—	10	1	no	—
156	C₂²	—	10	1	no	—
157	C₂²	#64874	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	—	5	2	no	—
161	C₄	—	10	1	no	—
162	C₄	—	10	1	no	—
163	C₄	#64897	1	1	yes	characteristic
164	C₄	#64899	1	1	yes	characteristic
165	C₂	—	20	1	no	—
166	C₂	#10598593	1	1	yes	characteristic
167	C₂	#10598605	1	1	yes	characteristic
168	C₂	#10598630	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10603827

C₂

yes

maximal, characteristic

#10603821

C₂

yes

maximal, characteristic

#10599322

C₂

yes

maximal, characteristic

C₂×#3990

C₄

yes

—

#65683

—

#65126

—

C₄×C₈×D₅

C₂²

yes

characteristic

#64888

—

#64884

C₄

yes

—

#64883

—

C₄⋊#674

—

C₅×#334

C₄

yes

characteristic

#64700

—

#64677

C₄

yes

—

—

C₂×C₈×D₅

C₂×C₄

yes

characteristic

C₂×C₈×D₅

Q₈

yes

characteristic

C₄²×D₅

D₄

yes

—

—

—

—

—

C₅×C₄⋊C₈

—

C₄×C₅×C₈

C₂×C₄

yes

characteristic

#3946

—

C₈×Dic₅

C₂×C₄

yes

characteristic

C₈×Dic₅

D₄

yes

characteristic

C₄⋊#168

—

C₄×#168

D₄

yes

—

—

—

C₂×C₄×D₅

C₄⋊C₄

yes

characteristic

C₂×C₄×D₅

D₈

yes

characteristic

C₂×C₄×D₅

Q₁₆

yes

characteristic

C₂×#170

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₄²

yes

characteristic

C₂×C₅×C₈

C₄⋊C₄

yes

characteristic

C₄²×C₅

C₂²⋊C₄

yes

characteristic

C₄×Dic₅

C₂²⋊C₄

yes

characteristic

C₄×Dic₅

SD₁₆

yes

characteristic

C₄×Dic₅

SD₁₆

yes

characteristic

C₂×#168

—

C₂×#168

C₂²⋊C₄

yes

characteristic

C₂×#168

C₄⋊C₄

yes

characteristic

#676

—

C₈×D₅

—

C₈×D₅

—

C₈×D₅

—

#674

—

C₂×M₃₂

—

#432

—

#423

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

—

—

—

—

yes

C₂²×D₅

yes

C₂×C₄×C₅

yes

C₂×C₄×C₅

yes

C₂×C₄×C₅

yes

C₂×Dic₅

yes

C₂×Dic₅

yes

C₂×Dic₅

yes

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

#170

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

—

—

—

yes

yes

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×AGL(1,𝔽₅)

yes

characteristic

100

C₂²×C₅

#340

yes

characteristic

101

C₂×D₅

—

102

C₂×D₅

—

103

C₂×D₅

—

104

C₄×C₅

—

105

C₄×C₅

—

106

C₄×C₅

#338

yes

characteristic

107

C₄×C₅

#343

yes

characteristic

108

Dic₅

—

109

Dic₅

—

110

Dic₅

#360

yes

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

M₁₆

—

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

C₂×C₈

C₄×AGL(1,𝔽₅)

yes

characteristic

124

C₂×C₈

C₄⋊AGL(1,𝔽₅)

yes

characteristic

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₄²

C₂²⋊AGL(1,𝔽₅)

yes

characteristic

129

C₁₆

—

130

C₂×C₅

#1292

yes

characteristic

131

C₂×C₅

#1386

yes

characteristic

132

C₂×C₅

#1388

yes

characteristic

133

D₅

—

134

C₂³

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

(C₂×AGL(1,𝔽₅))⋊C₄

yes

characteristic

143

C₂×C₄

#4007

yes

characteristic

144

C₂×C₄

#4009

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

—

152

C₈

—

153

C₅

#7244

yes

characteristic, complemented, Sylow

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

#64874

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

#64897

yes

characteristic

164

C₄

#64899

yes

characteristic

165

C₂

—

166

C₂

#10598593

yes

characteristic

167

C₂

#10598605

yes

characteristic

168

C₂

#10598630

yes

characteristic

	subgroup	G/H
center	C₂²	#64874
commutator	C₂×C₅×C₈	C₄²
Frattini	C₄×C₈	C₂×AGL(1,𝔽₅)
Fitting	C₅×#334	C₄
socle	C₂²×C₅	#340
prime radical	C₂²×D₅	#107
cosocle	C₄×C₈×D₅	C₂²
hypercenter	#334	AGL(1,𝔽₅)
hypocenter	C₅	#7244
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

subgroup

G/H

center

C₂²

#64874

commutator

C₂×C₅×C₈

C₄²

C₄×C₈

C₄

C₂²×C₅

C₂²×D₅

C₄×C₈×D₅

C₂²

C₅

G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

N	H ≅ G/N	type
153. C₅	#7244	semidirect

H ≅ G/N

type

153. C₅

#7244

semidirect

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

subgroup	index
#10603827	2
#10603821	2
#10599322	2
#7244	5

subgroup

index

Subgroups — #13013816

Conjugacy classes of subgroups

Complements

Maximal subgroups