Group #13027833 — subgroups

Subgroups — #13027833

Order 1280.

164 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 68 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	#10599844	C₂	1	1	yes	maximal, characteristic
2	#10598658	C₂	1	2	yes	maximal, complemented
3	#10598657	C₂	1	2	yes	maximal
4	#10598537	C₂	1	2	yes	maximal, complemented
5	C₂×#3930	C₂²	1	1	yes	characteristic
6	C₂×D₅×C₁₆	C₄	1	1	yes	characteristic
7	#64978	C₄	1	1	yes	characteristic
8	#64877	C₂²	1	4	yes	—
9	C₄×C₅×C₁₆	C₄	1	1	yes	characteristic
10	#64722	C₄	1	1	yes	characteristic
11	C₁₆×Dic₅	C₂²	1	1	yes	characteristic
12	#64681	C₂²	1	1	yes	characteristic
13	#13643	—	5	1	no	maximal, Sylow
14	C₂×#676	C₈	1	1	yes	characteristic
15	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
16	#4018	C₈	1	1	yes	characteristic
17	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
18	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
19	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
20	#3947	C₈	1	1	yes	characteristic
21	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
22	C₂×#672	C₂×C₄	1	1	yes	characteristic
23	C₂×#672	C₂³	1	1	yes	characteristic
24	#3935	C₈	1	1	yes	characteristic
25	#3930	—	2	2	no	—
26	D₅×C₁₆	—	2	2	no	—
27	#3928	—	2	4	no	—
28	#2108	—	5	1	no	—
29	#1422	—	5	2	no	—
30	#1400	—	5	2	no	—
31	#1398	—	5	2	no	—
32	C₂×C₄×D₅	C₂×C₈	1	1	yes	characteristic
33	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
34	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
35	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
36	#685	—	2	1	no	—
37	#683	—	2	1	no	—
38	C₄×Dic₅	C₂×C₈	1	1	yes	characteristic
39	C₂×#168	C₂×C₈	1	1	yes	characteristic
40	C₂×#168	C₂×C₈	1	1	yes	characteristic
41	#676	—	2	2	no	—
42	C₈×D₅	—	2	1	no	—
43	C₈×D₅	—	2	1	no	—
44	C₅×C₁₆	—	2	1	no	—
45	C₅×C₁₆	—	2	1	no	—
46	#672	M₁₆	1	2	yes	—
47	#672	#41	1	2	yes	—
48	C₂×M₃₂	—	5	1	no	—
49	C₂²×C₁₆	—	5	1	no	—
50	#405	—	5	1	no	—
51	M₆₄	—	5	4	no	—
52	#346	—	5	1	no	—
53	#346	—	5	1	no	—
54	C₄×C₁₆	—	5	1	no	—
55	C₄×C₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic
56	C₂²×D₅	M₃₂	1	1	yes	characteristic
57	C₂×C₄×C₅	—	2	1	no	—
58	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
59	C₂×Dic₅	—	2	1	no	—
60	C₂×Dic₅	M₃₂	1	1	yes	characteristic
61	C₄×D₅	—	2	1	no	—
62	C₄×D₅	—	2	1	no	—
63	C₅×C₈	—	2	1	no	—
64	C₅×C₈	#117	1	1	yes	characteristic
65	C₅×C₈	C₂×M₁₆	1	1	yes	characteristic
66	#168	—	2	1	no	—
67	#168	M₃₂	1	2	yes	—
68	C₂×M₁₆	—	5	1	no	—
69	C₂²×C₈	—	5	1	no	—
70	#117	—	5	1	no	—
71	M₃₂	—	10	2	no	—
72	C₂×C₁₆	—	5	1	no	—
73	C₂×C₁₆	—	5	1	no	—
74	C₂×C₁₆	—	10	2	no	—
75	C₂×C₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
76	C₂×C₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
77	#97	—	5	1	no	—
78	#97	—	5	1	no	—
79	C₄×C₈	—	5	1	no	—
80	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
81	C₃₂	—	10	4	no	—
82	C₂²×C₅	C₂×M₃₂	1	1	yes	characteristic
83	C₂×D₅	—	2	1	no	—
84	C₂×D₅	—	4	1	no	—
85	C₄×C₅	—	4	1	no	—
86	C₄×C₅	#431	1	1	yes	characteristic
87	C₄×C₅	C₂×M₃₂	1	1	yes	characteristic
88	Dic₅	—	2	1	no	—
89	Dic₅	—	4	1	no	—
90	C₂²×C₄	—	5	1	no	—
91	M₁₆	—	10	2	no	—
92	C₂×C₈	—	5	1	no	—
93	C₂×C₈	—	5	1	no	—
94	C₂×C₈	—	10	1	no	—
95	C₂×C₈	—	10	1	no	—
96	C₂×C₈	D₅⋊C₈	1	1	yes	characteristic
97	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
98	C₄⋊C₄	—	10	1	no	—
99	C₂²⋊C₄	—	10	1	no	—
100	C₄²	—	5	1	no	—
101	C₄²	D₅⋊C₈	1	1	yes	characteristic
102	C₁₆	—	2	1	no	—
103	C₁₆	—	2	1	no	—
104	C₁₆	—	5	2	no	—
105	C₁₆	—	5	2	no	—
106	C₂×C₅	—	2	1	no	—
107	C₂×C₅	#2163	1	1	yes	characteristic
108	D₅	—	4	1	no	—
109	C₂³	—	5	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	5	1	no	—
112	C₂×C₄	—	10	1	no	—
113	C₂×C₄	—	10	1	no	—
114	C₂×C₄	—	10	1	no	—
115	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
116	C₈	—	2	1	no	—
117	C₈	—	5	2	no	—
118	C₈	—	10	1	no	—
119	C₈	D₅⋊M₁₆	1	1	yes	characteristic
120	C₈	#4130	1	1	yes	characteristic
121	C₅	#13643	1	1	yes	characteristic, complemented, Sylow
122	C₂²	—	10	1	no	—
123	C₂²	—	20	1	no	—
124	C₂²	D₅⋊M₃₂	1	1	yes	characteristic
125	C₄	—	4	1	no	—
126	C₄	—	10	1	no	—
127	C₄	—	20	1	no	—
128	C₄	#65678	1	1	yes	characteristic
129	C₄	D₅⋊M₃₂	1	1	yes	characteristic
130	C₂	—	2	1	no	—
131	C₂	—	20	1	no	—
132	C₂	#10603830	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10599844

C₂

yes

maximal, characteristic

#10598658

C₂

yes

maximal, complemented

#10598657

C₂

yes

maximal

#10598537

C₂

yes

maximal, complemented

C₂×#3930

C₂²

yes

characteristic

C₂×D₅×C₁₆

C₄

yes

characteristic

#64978

C₄

yes

characteristic

#64877

C₂²

yes

—

C₄×C₅×C₁₆

C₄

yes

characteristic

#64722

C₄

yes

characteristic

C₁₆×Dic₅

C₂²

yes

characteristic

#64681

C₂²

yes

—

C₈

yes

characteristic

C₂×C₈×D₅

C₂×C₄

yes

characteristic

#4018

C₈

yes

characteristic

C₂×C₅×C₁₆

C₂×C₄

yes

characteristic

C₂×C₅×C₁₆

C₂×C₄

yes

characteristic

C₄×C₅×C₈

C₂×C₄

yes

characteristic

#3947

C₈

yes

characteristic

C₈×Dic₅

C₂×C₄

yes

characteristic

C₂×#672

C₂×C₄

yes

characteristic

C₂×#672

C₂³

yes

characteristic

#3935

C₈

yes

characteristic

#3930

—

D₅×C₁₆

—

—

—

—

—

—

C₂×C₄×D₅

C₂×C₈

yes

characteristic

C₂×C₅×C₈

C₂×C₈

yes

characteristic

C₂×C₅×C₈

C₂²×C₄

yes

characteristic

C₄²×C₅

C₂×C₈

yes

characteristic

#685

—

#683

—

C₄×Dic₅

C₂×C₈

yes

characteristic

C₂×#168

C₂×C₈

yes

characteristic

C₂×#168

C₂×C₈

yes

characteristic

#676

—

C₈×D₅

—

C₈×D₅

—

C₅×C₁₆

—

C₅×C₁₆

—

#672

M₁₆

yes

—

#672

#41

yes

—

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

M₆₄

—

#346

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

AGL(1,𝔽₅)

yes

characteristic

C₂²×D₅

M₃₂

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²×C₈

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

M₃₂

yes

characteristic

C₄×D₅

—

C₄×D₅

—

C₅×C₈

—

C₅×C₈

#117

yes

characteristic

C₅×C₈

C₂×M₁₆

yes

characteristic

#168

—

#168

M₃₂

yes

—

C₂×M₁₆

—

C₂²×C₈

—

#117

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

yes

C₂×C₁₆

yes

—

—

C₄×C₈

—

C₄×C₈

C₂×AGL(1,𝔽₅)

yes

characteristic

C₃₂

—

C₂²×C₅

C₂×M₃₂

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

#431

yes

characteristic

C₄×C₅

C₂×M₃₂

yes

characteristic

Dic₅

—

Dic₅

—

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₅⋊C₈

yes

characteristic

C₂×C₈

C₂²×AGL(1,𝔽₅)

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

100

C₄²

—

101

C₄²

D₅⋊C₈

yes

characteristic

102

C₁₆

—

103

C₁₆

—

104

C₁₆

—

105

C₁₆

—

106

C₂×C₅

—

107

C₂×C₅

#2163

yes

characteristic

108

D₅

—

109

C₂³

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

C₂×D₅⋊C₈

yes

characteristic

116

C₈

—

117

C₈

—

118

C₈

—

119

C₈

D₅⋊M₁₆

yes

characteristic

120

C₈

#4130

yes

characteristic

121

C₅

#13643

yes

characteristic, complemented, Sylow

122

C₂²

—

123

C₂²

—

124

C₂²

D₅⋊M₃₂

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

#65678

yes

characteristic

129

C₄

D₅⋊M₃₂

yes

characteristic

130

C₂

—

131

C₂

—

132

C₂

#10603830

yes

characteristic

	subgroup	G/H
center	C₈	#4130
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₂×C₈	C₂²×AGL(1,𝔽₅)
Fitting	C₄×C₅×C₁₆	C₄
socle	C₂×C₅	#2163
prime radical	C₂²×D₅	M₃₂
cosocle	C₂×#672	C₂³
hypercenter	C₄×C₁₆	AGL(1,𝔽₅)
hypocenter	C₅	#13643
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₈ ◁ C₄×C₈ ◁ C₄×C₁₆

subgroup

G/H

center

C₈

#4130

commutator

C₂×C₄×C₅

C₂²×C₈

Frattini

C₂×C₈

C₂²×AGL(1,𝔽₅)

Fitting

C₄×C₅×C₁₆

C₄

socle

C₂×C₅

#2163

prime radical

C₂²×D₅

M₃₂

C₂³

C₄×C₁₆

C₅

G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₈ ◁ C₄×C₈ ◁ C₄×C₁₆

N	H ≅ G/N	type
2. #10598658	C₂	semidirect
4. #10598537	C₂	semidirect
121. C₅	#13643	semidirect

H ≅ G/N

type

2. #10598658

C₂

semidirect

4. #10598537

C₂

semidirect

121. C₅

#13643

semidirect

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

subgroup	index
#10599844	2
#10598658	2
#10598657	2
#10598537	2
#13643	5

subgroup

index

Subgroups — #13027833

Conjugacy classes of subgroups

Complements

Maximal subgroups