Group #13027801 — subgroups

Subgroups — #13027801

Order 1280.

198 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 108 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	C₂×#64877	C₂	1	2	yes	maximal
2	#10602212	C₂	1	1	yes	maximal, characteristic
3	#10598658	C₂	1	4	yes	maximal, complemented
4	C₂×C₅×M₃₂	C₄	1	1	yes	characteristic
5	#65390	C₄	1	1	yes	characteristic
6	C₂²×#672	C₂²	1	1	yes	characteristic
7	#64877	—	2	2	no	—
8	#64877	C₂²	1	4	yes	—
9	C₂×#3928	—	2	2	no	—
10	#64722	C₄	1	2	yes	—
11	C₁₆×Dic₅	C₂²	1	2	yes	—
12	#13630	—	5	1	no	maximal, Sylow
13	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
14	#4072	C₈	1	1	yes	characteristic
15	C₂×#681	C₈	1	1	yes	characteristic
16	C₅×M₃₂	—	2	2	no	—
17	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
18	#3947	C₈	1	2	yes	—
19	C₈×Dic₅	C₂×C₄	1	2	yes	—
20	C₂×#672	C₂×C₄	1	1	yes	characteristic
21	C₂×#672	D₄	1	2	yes	—
22	C₂×#672	Q₈	1	2	yes	—
23	C₂×#672	C₂³	1	1	yes	characteristic
24	#3928	—	2	4	no	—
25	C₂×M₆₄	—	5	2	no	—
26	#2108	—	5	1	no	—
27	#1422	—	5	4	no	—
28	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
29	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
30	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
31	C₂×C₅×C₈	M₁₆	1	2	yes	—
32	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
33	C₂²⋊Dic₅	—	2	1	no	—
34	C₄⋊Dic₅	—	2	1	no	—
35	C₄×Dic₅	C₂×C₈	1	2	yes	—
36	#681	C₁₆	1	4	yes	—
37	C₂×#168	C₂×C₈	1	2	yes	—
38	C₅×C₁₆	—	2	2	no	—
39	#672	C₄⋊C₄	1	2	yes	—
40	#672	C₂×D₄	1	1	yes	characteristic
41	#672	C₂×Q₈	1	1	yes	characteristic
42	C₂×M₃₂	AGL(1,𝔽₅)	1	1	yes	characteristic
43	C₂²×C₁₆	—	5	1	no	—
44	#405	—	5	1	no	—
45	M₆₄	—	5	4	no	—
46	M₆₄	—	10	2	no	—
47	C₂×C₃₂	—	10	2	no	—
48	#346	—	5	2	no	—
49	C₄×C₁₆	—	5	2	no	—
50	C₂³×C₅	M₃₂	1	1	yes	characteristic
51	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
52	C₂×C₄×C₅	C₂×C₁₆	1	2	yes	—
53	C₂×C₄×C₅	M₃₂	1	1	yes	characteristic
54	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
55	C₂×Dic₅	—	2	2	no	—
56	C₅×C₈	C₄⋊C₈	1	2	yes	—
57	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₅×C₈	C₂×M₁₆	1	1	yes	characteristic
59	#168	C₂×C₁₆	1	4	yes	—
60	C₂×M₁₆	—	5	1	no	—
61	C₂²×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
62	#117	—	5	1	no	—
63	M₃₂	—	2	2	no	—
64	C₂×C₁₆	—	5	2	no	—
65	C₂×C₁₆	—	5	2	no	—
66	C₂×C₁₆	—	5	1	no	—
67	C₂×C₁₆	—	5	1	no	—
68	C₂×C₁₆	C₂×AGL(1,𝔽₅)	1	2	yes	—
69	#97	—	5	2	no	—
70	C₄×C₈	—	5	2	no	—
71	C₃₂	—	10	4	no	—
72	C₂²×C₅	—	4	1	no	—
73	C₂²×C₅	C₄⋊C₁₆	1	2	yes	—
74	C₂²×C₅	C₂×M₃₂	1	1	yes	characteristic
75	C₄×C₅	C₄⋊C₁₆	1	2	yes	—
76	C₄×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
77	C₄×C₅	C₂²×C₁₆	1	1	yes	characteristic
78	Dic₅	—	4	2	no	—
79	C₂²×C₄	D₅⋊C₈	1	1	yes	characteristic
80	M₁₆	—	5	4	no	—
81	C₂×C₈	—	5	2	no	—
82	C₂×C₈	D₅⋊C₈	1	1	yes	characteristic
83	C₂×C₈	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
84	C₂×C₈	#704	1	2	yes	—
85	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
86	C₄⋊C₄	—	10	1	no	—
87	C₂²⋊C₄	—	10	1	no	—
88	C₄²	—	5	2	no	—
89	C₁₆	—	2	2	no	—
90	C₁₆	—	5	1	no	—
91	C₁₆	—	5	2	no	—
92	C₁₆	—	5	1	no	—
93	C₂×C₅	—	2	1	no	—
94	C₂×C₅	—	2	2	no	—
95	C₂×C₅	C₂×C₄⋊C₁₆	1	1	yes	characteristic
96	C₂³	#3990	1	1	yes	characteristic
97	C₂×C₄	—	10	2	no	—
98	C₂×C₄	D₅⋊C₁₆	1	2	yes	—
99	C₂×C₄	#3990	1	1	yes	characteristic
100	C₂×C₄	#4004	1	2	yes	—
101	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
102	C₈	—	5	4	no	—
103	C₈	#4004	1	2	yes	—
104	C₈	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
105	C₈	C₂×#704	1	1	yes	characteristic
106	C₅	#13630	1	1	yes	characteristic, complemented, Sylow
107	C₂²	—	4	1	no	—
108	C₂²	#64888	1	2	yes	—
109	C₂²	C₂×#3990	1	1	yes	characteristic
110	C₄	—	20	2	no	—
111	C₄	#64888	1	2	yes	—
112	C₄	C₂×D₅⋊C₁₆	1	1	yes	characteristic
113	C₄	C₂×#4004	1	1	yes	characteristic
114	C₂	—	2	1	no	—
115	C₂	—	2	2	no	—
116	C₂	C₂×#64888	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#64877

C₂

yes

maximal

#10602212

C₂

yes

maximal, characteristic

#10598658

C₂

yes

maximal, complemented

C₂×C₅×M₃₂

C₄

yes

characteristic

#65390

C₄

yes

characteristic

C₂²×#672

C₂²

yes

characteristic

#64877

—

#64877

C₂²

yes

—

C₂×#3928

—

#64722

C₄

yes

—

C₁₆×Dic₅

C₂²

yes

—

#13630

—

maximal, Sylow

C₂²×C₅×C₈

C₂×C₄

yes

characteristic

#4072

C₈

yes

characteristic

C₂×#681

C₈

yes

characteristic

C₅×M₃₂

—

C₂×C₅×C₁₆

C₂×C₄

yes

—

#3947

C₈

yes

—

C₈×Dic₅

C₂×C₄

yes

—

C₂×#672

C₂×C₄

yes

characteristic

C₂×#672

D₄

yes

—

C₂×#672

Q₈

yes

—

C₂×#672

C₂³

yes

characteristic

#3928

—

C₂×M₆₄

—

#2108

—

#1422

—

C₂²×C₄×C₅

C₂×C₈

yes

characteristic

C₂×C₅×C₈

C₄⋊C₄

yes

—

C₂×C₅×C₈

C₂×C₈

yes

characteristic

C₂×C₅×C₈

M₁₆

yes

—

C₂×C₅×C₈

C₂²×C₄

yes

characteristic

C₂²⋊Dic₅

—

C₄⋊Dic₅

—

C₄×Dic₅

C₂×C₈

yes

—

#681

C₁₆

yes

—

C₂×#168

C₂×C₈

yes

—

C₅×C₁₆

—

#672

C₄⋊C₄

yes

—

#672

C₂×D₄

yes

characteristic

#672

C₂×Q₈

yes

characteristic

C₂×M₃₂

AGL(1,𝔽₅)

yes

characteristic

C₂²×C₁₆

—

#405

—

M₆₄

—

M₆₄

—

C₂×C₃₂

—

#346

—

C₄×C₁₆

—

C₂³×C₅

M₃₂

yes

characteristic

C₂×C₄×C₅

C₄⋊C₈

yes

—

C₂×C₄×C₅

C₂×C₁₆

yes

—

C₂×C₄×C₅

M₃₂

yes

characteristic

C₂×C₄×C₅

C₂²×C₈

yes

characteristic

C₂×Dic₅

—

C₅×C₈

C₄⋊C₈

yes

—

C₅×C₈

C₂×C₄⋊C₄

yes

characteristic

C₅×C₈

C₂×M₁₆

yes

characteristic

#168

C₂×C₁₆

yes

—

C₂×M₁₆

—

C₂²×C₈

C₂×AGL(1,𝔽₅)

yes

characteristic

#117

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×AGL(1,𝔽₅)

yes

—

#97

—

C₄×C₈

—

C₃₂

—

C₂²×C₅

—

C₂²×C₅

C₄⋊C₁₆

yes

—

C₂²×C₅

C₂×M₃₂

yes

characteristic

C₄×C₅

C₄⋊C₁₆

yes

—

C₄×C₅

C₂×C₄⋊C₈

yes

characteristic

C₄×C₅

C₂²×C₁₆

yes

characteristic

Dic₅

—

C₂²×C₄

D₅⋊C₈

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

D₅⋊C₈

yes

characteristic

C₂×C₈

C₄⋊AGL(1,𝔽₅)

yes

—

C₂×C₈

#704

yes

—

C₂×C₈

C₂²×AGL(1,𝔽₅)

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

C₂×C₄⋊C₁₆

yes

characteristic

C₂³

#3990

yes

characteristic

C₂×C₄

—

C₂×C₄

D₅⋊C₁₆

yes

—

C₂×C₄

#3990

yes

characteristic

100

C₂×C₄

#4004

yes

—

101

C₂×C₄

C₂×D₅⋊C₈

yes

characteristic

102

C₈

—

103

C₈

#4004

yes

—

104

C₈

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

105

C₈

C₂×#704

yes

characteristic

106

C₅

#13630

yes

characteristic, complemented, Sylow

107

C₂²

—

108

C₂²

#64888

yes

—

109

C₂²

C₂×#3990

yes

characteristic

110

C₄

—

111

C₄

#64888

yes

—

112

C₄

C₂×D₅⋊C₁₆

yes

characteristic

113

C₄

C₂×#4004

yes

characteristic

114

C₂

—

115

C₂

—

116

C₂

C₂×#64888

yes

characteristic

	subgroup	G/H
center	C₈	C₂×C₄⋊AGL(1,𝔽₅)
commutator	C₄×C₅	C₂²×C₁₆
Frattini	C₂×C₈	C₂²×AGL(1,𝔽₅)
Fitting	C₂×C₅×M₃₂	C₄
socle	C₂×C₅	C₂×C₄⋊C₁₆
prime radical	C₂³×C₅	M₃₂
cosocle	C₂×#672	C₂³
hypercenter	C₂×M₃₂	AGL(1,𝔽₅)
hypocenter	C₅	#13630
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×M₃₂

subgroup

G/H

center

C₈

C₂×C₄⋊AGL(1,𝔽₅)

commutator

C₄×C₅

C₂²×C₁₆

Frattini

C₂×C₈

C₂²×AGL(1,𝔽₅)

Fitting

C₂×C₅×M₃₂

C₄

socle

C₂×C₅

C₂×C₄⋊C₁₆

prime radical

C₂³×C₅

M₃₂

C₂³

C₂×M₃₂

C₅

G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
3. #10598658	C₂	semidirect
106. C₅	#13630	semidirect

H ≅ G/N

type

3. #10598658

C₂

semidirect

106. C₅

#13630

semidirect

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

subgroup	index
C₂×#64877	2
#10602212	2
#10598658	2
#13630	5

subgroup

index

Subgroups — #13027801

Conjugacy classes of subgroups

Complements

Maximal subgroups