Group #13014407 — subgroups

Subgroups — #13014407

Order 1280.

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

normal subgroups only

This group has 162 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	D₅⋊#368	C₂	1	1	yes	maximal, characteristic
2	C₂×#64852	C₂	1	1	yes	maximal, characteristic
3	D₅×M₆₄	C₂	1	1	yes	maximal, characteristic
4	D₅⋊#108	—	2	1	no	—
5	C₂×#3996	—	2	1	no	—
6	C₂×D₅×C₁₆	C₂²	1	1	yes	characteristic
7	#64852	—	2	1	no	—
8	#64852	—	2	1	no	—
9	#64852	C₄	1	2	yes	—
10	#64851	C₄	1	2	yes	—
11	C₅×M₆₄	C₄	1	1	yes	characteristic
12	#64719	C₄	1	1	yes	characteristic
13	#64667	—	2	1	no	—
14	D₅×C₃₂	—	2	1	no	—
15	#7479	—	5	1	no	maximal, Sylow
16	D₅⋊M₁₆	—	4	1	no	—
17	C₂×#700	—	4	1	no	—
18	C₂×C₈×D₅	D₄	1	1	yes	characteristic
19	#3996	—	2	2	no	—
20	#3996	—	2	2	no	—
21	#3996	—	4	1	no	—
22	#3995	—	4	1	no	—
23	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
24	C₂×#672	Q₈	1	1	yes	characteristic
25	D₅×C₁₆	C₂×C₄	1	1	yes	characteristic
26	D₅×C₁₆	C₂×C₄	1	1	yes	characteristic
27	D₅×C₁₆	D₄	1	1	yes	characteristic
28	D₅×C₁₆	D₄	1	1	yes	characteristic
29	C₅×C₃₂	—	2	1	no	—
30	#3926	—	2	1	no	—
31	C₂×M₆₄	—	5	1	no	—
32	C₂×#368	—	5	1	no	—
33	C₂×#368	—	5	1	no	—
34	C₂×C₄×D₅	D₈	1	1	yes	characteristic
35	#704	—	8	1	no	—
36	C₂×#170	—	8	1	no	—
37	#700	—	4	2	no	—
38	#700	—	4	2	no	—
39	#700	—	8	1	no	—
40	D₅⋊C₈	—	8	1	no	—
41	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
42	C₂×#168	Q₁₆	1	1	yes	characteristic
43	C₈×D₅	C₂²⋊C₄	1	1	yes	characteristic
44	C₈×D₅	C₂²⋊C₄	1	1	yes	characteristic
45	C₈×D₅	SD₁₆	1	1	yes	characteristic
46	C₈×D₅	SD₁₆	1	1	yes	characteristic
47	C₅×C₁₆	C₄²	1	1	yes	characteristic
48	C₅×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
49	#672	C₄⋊C₄	1	1	yes	characteristic
50	#672	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₁₆	—	5	1	no	—
52	C₂×#108	—	10	1	no	—
53	C₂×#108	—	10	1	no	—
54	M₆₄	—	5	1	no	—
55	M₆₄	—	10	1	no	—
56	M₆₄	AGL(1,𝔽₅)	1	1	yes	characteristic
57	C₂×C₃₂	—	10	1	no	—
58	#368	—	5	2	no	—
59	#368	—	5	2	no	—
60	#368	—	10	1	no	—
61	#368	—	10	1	no	—
62	C₂²×D₅	Q₃₂	1	1	yes	characteristic
63	C₂×C₄×C₅	#107	1	1	yes	characteristic
64	C₂×Dic₅	D₁₆	1	1	yes	characteristic
65	C₄×D₅	#102	1	1	yes	characteristic
66	C₄×D₅	#102	1	1	yes	characteristic
67	C₄×D₅	SD₃₂	1	1	yes	characteristic
68	C₄×D₅	SD₃₂	1	1	yes	characteristic
69	#170	—	8	2	no	—
70	#170	—	8	2	no	—
71	C₅×C₈	#95	1	1	yes	characteristic
72	C₅×C₈	#106	1	1	yes	characteristic
73	#168	#103	1	1	yes	characteristic
74	#168	#103	1	1	yes	characteristic
75	C₂×M₁₆	—	20	1	no	—
76	C₂×M₁₆	—	20	1	no	—
77	C₂²×C₈	—	5	1	no	—
78	C₂×C₁₆	—	5	1	no	—
79	C₂×C₁₆	—	5	1	no	—
80	C₂×C₁₆	—	5	1	no	—
81	C₂×C₁₆	—	5	1	no	—
82	C₂×C₁₆	—	5	1	no	—
83	C₂×C₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
84	#108	—	10	2	no	—
85	#108	—	10	2	no	—
86	#108	—	20	1	no	—
87	#108	—	20	1	no	—
88	C₃₂	—	2	1	no	—
89	C₃₂	—	10	1	no	—
90	C₂²×C₅	#366	1	1	yes	characteristic
91	C₂×D₅	—	4	1	no	—
92	C₂×D₅	#358	1	1	yes	characteristic
93	C₂×D₅	#358	1	1	yes	characteristic
94	C₄×C₅	#340	1	1	yes	characteristic
95	C₄×C₅	#367	1	1	yes	characteristic
96	Dic₅	#357	1	1	yes	characteristic
97	Dic₅	#357	1	1	yes	characteristic
98	C₂²×C₄	—	5	1	no	—
99	M₁₆	—	20	2	no	—
100	M₁₆	—	20	2	no	—
101	M₁₆	—	40	1	no	—
102	M₁₆	—	40	1	no	—
103	C₂×C₈	—	5	1	no	—
104	C₂×C₈	—	5	1	no	—
105	C₂×C₈	—	5	1	no	—
106	C₂×C₈	—	5	1	no	—
107	C₂×C₈	—	5	1	no	—
108	C₂×C₈	—	40	1	no	—
109	C₂×C₈	—	40	1	no	—
110	C₂×C₈	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
111	C₁₆	—	5	1	no	—
112	C₁₆	—	5	1	no	—
113	C₁₆	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
114	C₁₆	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
115	C₂×C₅	—	2	1	no	—
116	C₂×C₅	#1380	1	1	yes	characteristic
117	D₅	—	2	1	no	—
118	D₅	—	2	1	no	—
119	C₂³	—	5	1	no	—
120	C₂×C₄	—	5	1	no	—
121	C₂×C₄	—	5	1	no	—
122	C₂×C₄	—	5	1	no	—
123	C₂×C₄	—	5	1	no	—
124	C₂×C₄	—	5	1	no	—
125	C₂×C₄	#3994	1	1	yes	characteristic
126	C₈	—	5	1	no	—
127	C₈	—	5	1	no	—
128	C₈	—	40	2	no	—
129	C₈	—	40	2	no	—
130	C₈	#3993	1	1	yes	characteristic
131	C₈	(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic
132	C₅	#7479	1	1	yes	characteristic, complemented, Sylow
133	C₂²	—	5	1	no	—
134	C₂²	—	5	1	no	—
135	C₂²	—	20	1	no	—
136	C₂²	#64849	1	1	yes	characteristic
137	C₄	—	5	1	no	—
138	C₄	—	5	1	no	—
139	C₄	#64850	1	1	yes	characteristic
140	C₄	#64893	1	1	yes	characteristic
141	C₂	—	2	1	no	—
142	C₂	—	10	1	no	—
143	C₂	—	10	1	no	—
144	C₂	#10598664	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₅⋊#368

C₂

yes

maximal, characteristic

C₂×#64852

C₂

yes

maximal, characteristic

D₅×M₆₄

C₂

yes

maximal, characteristic

D₅⋊#108

—

C₂×#3996

—

C₂×D₅×C₁₆

C₂²

yes

—

—

C₄

yes

—

#64851

C₄

yes

—

C₅×M₆₄

C₄

yes

characteristic

#64719

C₄

yes

characteristic

#64667

—

D₅×C₃₂

—

#7479

—

maximal, Sylow

D₅⋊M₁₆

—

C₂×#700

—

C₂×C₈×D₅

D₄

yes

—

—

—

—

C₂×C₅×C₁₆

C₂×C₄

yes

characteristic

C₂×#672

Q₈

yes

characteristic

D₅×C₁₆

C₂×C₄

yes

characteristic

D₅×C₁₆

C₂×C₄

yes

characteristic

D₅×C₁₆

D₄

yes

characteristic

D₅×C₁₆

D₄

yes

characteristic

C₅×C₃₂

—

#3926

—

C₂×M₆₄

—

C₂×#368

—

C₂×#368

—

C₂×C₄×D₅

D₈

yes

—

—

—

—

—

D₅⋊C₈

—

C₂×C₅×C₈

C₄⋊C₄

yes

characteristic

C₂×#168

Q₁₆

yes

characteristic

C₈×D₅

C₂²⋊C₄

yes

characteristic

C₈×D₅

C₂²⋊C₄

yes

characteristic

C₈×D₅

SD₁₆

yes

characteristic

C₈×D₅

SD₁₆

yes

characteristic

C₅×C₁₆

C₄²

yes

characteristic

C₅×C₁₆

C₂²⋊C₄

yes

characteristic

#672

C₄⋊C₄

yes

characteristic

#672

C₄⋊C₄

yes

characteristic

C₂²×C₁₆

—

C₂×#108

—

C₂×#108

—

M₆₄

—

M₆₄

—

M₆₄

AGL(1,𝔽₅)

yes

characteristic

C₂×C₃₂

—

—

—

—

—

C₂²×D₅

Q₃₂

yes

characteristic

C₂×C₄×C₅

#107

yes

characteristic

C₂×Dic₅

D₁₆

yes

C₄×D₅

yes

C₄×D₅

yes

C₄×D₅

SD₃₂

yes

characteristic

C₄×D₅

SD₃₂

yes

characteristic

#170

—

#170

—

C₅×C₈

yes

C₅×C₈

yes

yes

yes

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

yes

—

—

—

—

C₃₂

—

C₃₂

—

C₂²×C₅

#366

yes

characteristic

C₂×D₅

—

C₂×D₅

yes

C₂×D₅

yes

C₄×C₅

yes

C₄×C₅

yes

Dic₅

yes

Dic₅

yes

C₂²×C₄

—

M₁₆

—

100

M₁₆

—

101

M₁₆

—

102

M₁₆

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

C₄⋊AGL(1,𝔽₅)

yes

characteristic

111

C₁₆

—

112

C₁₆

—

113

C₁₆

C₄×AGL(1,𝔽₅)

yes

characteristic

114

C₁₆

C₂²⋊AGL(1,𝔽₅)

yes

characteristic

115

C₂×C₅

—

116

C₂×C₅

#1380

yes

characteristic

117

D₅

—

118

D₅

—

119

C₂³

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

#3994

yes

characteristic

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

#3993

yes

characteristic

131

C₈

(C₂×AGL(1,𝔽₅))⋊C₄

yes

characteristic

132

C₅

#7479

yes

characteristic, complemented, Sylow

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

#64849

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

#64850

yes

characteristic

140

C₄

#64893

yes

characteristic

141

C₂

—

142

C₂

—

143

C₂

—

144

C₂

#10598664

yes

characteristic

	subgroup	G/H
center	C₂	#10598664
commutator	C₅×C₁₆	C₄²
Frattini	C₂×C₁₆	C₂×AGL(1,𝔽₅)
Fitting	C₅×M₆₄	C₄
socle	C₂×C₅	#1380
prime radical	C₂²×D₅	Q₃₂
cosocle	C₂×D₅×C₁₆	C₂²
hypercenter	M₆₄	AGL(1,𝔽₅)
hypocenter	C₅	#7479
LCS	G ▷ C₅×C₁₆ ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

subgroup

G/H

center

C₂

#10598664

commutator

C₅×C₁₆

C₄²

Frattini

C₂×C₁₆

C₂×AGL(1,𝔽₅)

Fitting

C₅×M₆₄

C₄

socle

C₂×C₅

#1380

prime radical

C₂²×D₅

Q₃₂

cosocle

C₂×D₅×C₁₆

C₂²

M₆₄

C₅

G ▷ C₅×C₁₆ ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

N	H ≅ G/N	type
132. C₅	#7479	semidirect

H ≅ G/N

type

132. C₅

#7479

semidirect

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

subgroup	index
D₅⋊#368	2
C₂×#64852	2
D₅×M₆₄	2
#7479	5

subgroup

index

D₅⋊#368

C₂×#64852

D₅×M₆₄

#7479

Subgroups — #13014407

Conjugacy classes of subgroups

Complements

Maximal subgroups