Group #13014388 — subgroups

Subgroups — #13014388

Order 1280.

166 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 64 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×D₅⋊C₃₂	C₂	1	1	yes	maximal, characteristic
3	D₅×#368	C₂	1	1	yes	maximal, characteristic
4	D₅⋊#108	—	2	1	no	—
5	C₂×D₅×C₁₆	C₂²	1	1	yes	characteristic
6	D₅×#108	—	2	1	no	—
7	C₂×#3928	—	2	1	no	—
8	#64852	—	2	1	no	—
9	#64851	C₄	1	2	yes	—
10	D₅⋊C₃₂	—	2	1	no	—
11	D₅⋊C₃₂	C₄	1	2	yes	—
12	C₅×#368	C₄	1	1	yes	characteristic
13	#64726	C₄	1	1	yes	characteristic
14	#64713	—	2	1	no	—
15	#7472	—	5	1	no	maximal, Sylow
16	D₅⋊M₁₆	—	4	1	no	—
17	D₅×M₁₆	—	4	1	no	—
18	C₂×C₈×D₅	D₄	1	1	yes	characteristic
19	#3996	—	2	2	no	—
20	#3995	—	4	1	no	—
21	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
22	C₅×#108	—	2	1	no	—
23	#3954	—	4	1	no	—
24	#3951	—	2	1	no	—
25	C₂×#672	D₄	1	1	yes	characteristic
26	D₅×C₁₆	C₂×C₄	1	1	yes	characteristic
27	D₅×C₁₆	C₂×C₄	1	1	yes	characteristic
28	D₅×C₁₆	D₄	1	1	yes	characteristic
29	D₅×C₁₆	Q₈	1	1	yes	characteristic
30	#3928	—	2	2	no	—
31	C₂²×C₃₂	—	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	#700	—	4	2	no	—
37	D₅⋊C₈	—	8	1	no	—
38	C₅×M₁₆	—	4	1	no	—
39	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
40	#681	—	4	1	no	—
41	C₂×#168	SD₁₆	1	1	yes	characteristic
42	#676	—	8	1	no	—
43	C₈×D₅	—	8	1	no	—
44	C₈×D₅	C₂²⋊C₄	1	1	yes	characteristic
45	C₈×D₅	C₄⋊C₄	1	1	yes	characteristic
46	C₈×D₅	SD₁₆	1	1	yes	characteristic
47	C₈×D₅	Q₁₆	1	1	yes	characteristic
48	C₅×C₁₆	C₄²	1	1	yes	characteristic
49	C₅×C₁₆	C₄⋊C₄	1	1	yes	characteristic
50	#672	C₂²⋊C₄	1	1	yes	characteristic
51	#672	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₁₆	—	5	1	no	—
53	C₂×#108	—	10	1	no	—
54	C₂×#108	—	10	1	no	—
55	C₂×C₃₂	—	5	2	no	—
56	C₂×C₃₂	—	10	1	no	—
57	C₂×C₃₂	—	10	1	no	—
58	#368	—	5	2	no	—
59	#368	—	5	1	no	—
60	#368	—	10	1	no	—
61	#368	—	10	1	no	—
62	#368	AGL(1,𝔽₅)	1	1	yes	characteristic
63	C₂²×D₅	Q₃₂	1	1	yes	characteristic
64	C₂×C₄×C₅	#102	1	1	yes	characteristic
65	C₂×Dic₅	SD₃₂	1	1	yes	characteristic
66	C₄×D₅	#102	1	1	yes	characteristic
67	C₄×D₅	#107	1	1	yes	characteristic
68	C₄×D₅	D₁₆	1	1	yes	characteristic
69	C₄×D₅	SD₃₂	1	1	yes	characteristic
70	#170	—	8	2	no	—
71	C₅×C₈	—	8	1	no	—
72	C₅×C₈	#95	1	1	yes	characteristic
73	C₅×C₈	#103	1	1	yes	characteristic
74	#168	—	8	1	no	—
75	#168	#103	1	1	yes	characteristic
76	#168	#106	1	1	yes	characteristic
77	C₂×M₁₆	—	20	1	no	—
78	C₂×M₁₆	—	20	1	no	—
79	C₂²×C₈	—	5	1	no	—
80	C₂×C₁₆	—	5	1	no	—
81	C₂×C₁₆	—	5	1	no	—
82	C₂×C₁₆	—	5	1	no	—
83	C₂×C₁₆	—	5	1	no	—
84	C₂×C₁₆	—	5	1	no	—
85	C₂×C₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
86	#108	—	2	1	no	—
87	#108	—	10	2	no	—
88	#108	—	10	1	no	—
89	#108	—	20	1	no	—
90	#108	—	20	1	no	—
91	C₃₂	—	10	2	no	—
92	C₂²×C₅	#358	1	1	yes	characteristic
93	C₂×D₅	—	2	2	no	—
94	C₂×D₅	#358	1	1	yes	characteristic
95	C₂×D₅	#366	1	1	yes	characteristic
96	C₄×C₅	#340	1	1	yes	characteristic
97	C₄×C₅	#357	1	1	yes	characteristic
98	Dic₅	#357	1	1	yes	characteristic
99	Dic₅	#367	1	1	yes	characteristic
100	C₂²×C₄	—	5	1	no	—
101	M₁₆	—	4	1	no	—
102	M₁₆	—	20	2	no	—
103	M₁₆	—	20	1	no	—
104	M₁₆	—	40	1	no	—
105	M₁₆	—	40	1	no	—
106	C₂×C₈	—	5	1	no	—
107	C₂×C₈	—	5	1	no	—
108	C₂×C₈	—	5	1	no	—
109	C₂×C₈	—	5	1	no	—
110	C₂×C₈	—	5	1	no	—
111	C₂×C₈	—	40	1	no	—
112	C₂×C₈	—	40	1	no	—
113	C₂×C₈	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
114	C₁₆	—	5	1	no	—
115	C₁₆	—	5	1	no	—
116	C₁₆	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
117	C₁₆	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
118	C₂×C₅	—	2	1	no	—
119	C₂×C₅	#1380	1	1	yes	characteristic
120	D₅	—	2	1	no	—
121	D₅	#1425	1	2	yes	—
122	C₂³	—	5	1	no	—
123	C₂×C₄	—	5	1	no	—
124	C₂×C₄	—	5	1	no	—
125	C₂×C₄	—	5	1	no	—
126	C₂×C₄	—	5	1	no	—
127	C₂×C₄	—	5	1	no	—
128	C₂×C₄	#4007	1	1	yes	characteristic
129	C₈	—	5	1	no	—
130	C₈	—	5	1	no	—
131	C₈	—	8	1	no	—
132	C₈	—	40	2	no	—
133	C₈	—	40	1	no	—
134	C₈	(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic
135	C₈	#4009	1	1	yes	characteristic
136	C₅	#7472	1	1	yes	characteristic, complemented, Sylow
137	C₂²	—	5	1	no	—
138	C₂²	—	5	1	no	—
139	C₂²	—	10	2	no	—
140	C₂²	#64908	1	1	yes	characteristic
141	C₄	—	5	1	no	—
142	C₄	—	5	1	no	—
143	C₄	#64874	1	1	yes	characteristic
144	C₄	#64904	1	1	yes	characteristic
145	C₂	—	2	1	no	—
146	C₂	—	5	2	no	—
147	C₂	—	10	1	no	—
148	C₂	#10598619	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₅⋊#368

C₂

yes

maximal, characteristic

C₂×D₅⋊C₃₂

C₂

yes

maximal, characteristic

D₅×#368

C₂

yes

maximal, characteristic

D₅⋊#108

—

C₂×D₅×C₁₆

C₂²

yes

—

—

—

C₄

yes

—

D₅⋊C₃₂

—

D₅⋊C₃₂

C₄

yes

—

C₅×#368

C₄

yes

characteristic

#64726

C₄

yes

—

—

D₅⋊M₁₆

—

D₅×M₁₆

—

C₂×C₈×D₅

D₄

yes

characteristic

#3996

—

#3995

—

C₂×C₅×C₁₆

C₂×C₄

yes

—

—

—

D₄

yes

characteristic

D₅×C₁₆

C₂×C₄

yes

characteristic

D₅×C₁₆

C₂×C₄

yes

characteristic

D₅×C₁₆

D₄

yes

characteristic

D₅×C₁₆

Q₈

yes

characteristic

#3928

—

C₂²×C₃₂

—

C₂×#368

—

C₂×#368

—

C₂×C₄×D₅

D₈

yes

characteristic

#704

—

#700

—

D₅⋊C₈

—

C₅×M₁₆

—

C₂×C₅×C₈

C₂²⋊C₄

yes

characteristic

#681

—

C₂×#168

SD₁₆

yes

characteristic

#676

—

C₈×D₅

—

C₈×D₅

C₂²⋊C₄

yes

characteristic

C₈×D₅

C₄⋊C₄

yes

characteristic

C₈×D₅

SD₁₆

yes

characteristic

C₈×D₅

Q₁₆

yes

characteristic

C₅×C₁₆

C₄²

yes

characteristic

C₅×C₁₆

C₄⋊C₄

yes

characteristic

#672

C₂²⋊C₄

yes

characteristic

#672

C₄⋊C₄

yes

characteristic

C₂²×C₁₆

—

C₂×#108

—

C₂×#108

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

—

—

—

—

yes

C₂²×D₅

Q₃₂

yes

characteristic

C₂×C₄×C₅

#102

yes

characteristic

C₂×Dic₅

SD₃₂

yes

C₄×D₅

yes

C₄×D₅

yes

C₄×D₅

D₁₆

yes

characteristic

C₄×D₅

SD₃₂

yes

characteristic

#170

—

C₅×C₈

—

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

—

C₂²×C₅

#358

yes

characteristic

C₂×D₅

—

C₂×D₅

yes

C₂×D₅

yes

C₄×C₅

yes

C₄×C₅

yes

Dic₅

yes

Dic₅

yes

100

C₂²×C₄

—

101

M₁₆

—

102

M₁₆

—

103

M₁₆

—

104

M₁₆

—

105

M₁₆

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

C₂²⋊AGL(1,𝔽₅)

yes

characteristic

114

C₁₆

—

115

C₁₆

—

116

C₁₆

C₄×AGL(1,𝔽₅)

yes

characteristic

117

C₁₆

C₄⋊AGL(1,𝔽₅)

yes

characteristic

118

C₂×C₅

—

119

C₂×C₅

#1380

yes

characteristic

120

D₅

—

121

D₅

#1425

yes

—

122

C₂³

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

#4007

yes

characteristic

129

C₈

—

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

(C₂×AGL(1,𝔽₅))⋊C₄

yes

characteristic

135

C₈

#4009

yes

characteristic

136

C₅

#7472

yes

characteristic, complemented, Sylow

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

#64908

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

#64874

yes

characteristic

144

C₄

#64904

yes

characteristic

145

C₂

—

146

C₂

—

147

C₂

—

148

C₂

#10598619

yes

characteristic

	subgroup	G/H
center	C₄	#64904
commutator	C₅×C₁₆	C₄²
Frattini	C₂×C₁₆	C₂×AGL(1,𝔽₅)
Fitting	C₅×#368	C₄
socle	C₂×C₅	#1380
prime radical	C₂²×D₅	Q₃₂
cosocle	C₂×D₅×C₁₆	C₂²
hypercenter	#368	AGL(1,𝔽₅)
hypocenter	C₅	#7472
LCS	G ▷ C₅×C₁₆ ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #368

subgroup

G/H

center

C₄

#64904

commutator

C₅×C₁₆

C₄²

C₂×C₁₆

C₄

C₂×C₅

C₂²×D₅

Q₃₂

cosocle

C₂×D₅×C₁₆

C₂²

C₅

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

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #368

N	H ≅ G/N	type
136. C₅	#7472	semidirect

H ≅ G/N

type

136. C₅

#7472

semidirect

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

subgroup	index
D₅⋊#368	2
C₂×D₅⋊C₃₂	2
D₅×#368	2
#7472	5

subgroup

index

D₅⋊#368

C₂×D₅⋊C₃₂

D₅×#368

#7472

Subgroups — #13014388

Conjugacy classes of subgroups

Complements

Maximal subgroups