Group #13013805 — subgroups

Subgroups — #13013805

Order 1280.

148 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 50 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	#10604715	C₂	1	2	yes	maximal
2	#10602194	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₅×C₁₆	C₄	1	1	yes	characteristic
4	#65390	C₂²	1	1	yes	characteristic
5	C₂×#3944	C₄	1	1	yes	characteristic
6	#64912	—	2	2	no	—
7	#64890	C₄	1	4	yes	—
8	#64889	—	2	2	no	—
9	#64721	—	2	2	no	—
10	#7240	—	5	1	no	maximal, Sylow
11	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
12	#4072	Q₈	1	1	yes	characteristic
13	C₂×#681	D₄	1	1	yes	characteristic
14	#3998	—	2	4	no	—
15	C₂×C₅×C₁₆	—	2	1	no	—
16	C₂×C₅×C₁₆	—	4	1	no	—
17	#3947	C₂×C₄	1	2	yes	—
18	C₈×Dic₅	D₄	1	2	yes	—
19	#3944	—	4	1	no	—
20	C₂×#672	—	2	1	no	—
21	#2166	—	5	2	no	—
22	#2151	—	5	1	no	—
23	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
24	C₂×C₅×C₈	—	2	1	no	—
25	C₂×C₅×C₈	—	2	1	no	—
26	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
27	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
28	C₂²⋊Dic₅	—	2	1	no	—
29	C₄⋊Dic₅	—	2	1	no	—
30	C₄×Dic₅	C₄⋊C₄	1	2	yes	—
31	#681	—	4	1	no	—
32	C₂×#168	C₂²⋊C₄	1	2	yes	—
33	#674	—	4	4	no	—
34	C₅×C₁₆	—	4	1	no	—
35	#672	—	4	1	no	—
36	C₂×M₃₂	—	5	1	no	—
37	C₂²×C₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic
38	#405	—	5	1	no	—
39	#364	—	10	2	no	—
40	C₄⋊C₁₆	—	10	2	no	—
41	#349	—	10	2	no	—
42	#347	—	5	4	no	—
43	C₂³×C₅	#108	1	1	yes	characteristic
44	C₂×C₄×C₅	—	2	1	no	—
45	C₂×C₄×C₅	—	2	1	no	—
46	C₂×C₄×C₅	#95	1	1	yes	characteristic
47	C₂×C₄×C₅	#108	1	1	yes	characteristic
48	C₂×Dic₅	—	2	2	no	—
49	C₅×C₈	—	2	1	no	—
50	C₅×C₈	#100	1	1	yes	characteristic
51	C₅×C₈	#101	1	1	yes	characteristic
52	#168	—	2	2	no	—
53	C₂×M₁₆	—	5	1	no	—
54	C₂²×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
55	#117	—	5	1	no	—
56	M₃₂	—	10	4	no	—
57	M₃₂	—	20	1	no	—
58	C₂×C₁₆	—	2	1	no	—
59	C₂×C₁₆	—	4	1	no	—
60	C₂×C₁₆	—	10	1	no	—
61	#97	—	5	2	no	—
62	C₄×C₈	—	5	2	no	—
63	C₂²×C₅	—	2	1	no	—
64	C₂²×C₅	—	4	1	no	—
65	C₂²×C₅	#341	1	1	yes	characteristic
66	C₄×C₅	—	2	1	no	—
67	C₄×C₅	H(3,ℤ/4)	1	1	yes	characteristic
68	C₄×C₅	#343	1	1	yes	characteristic
69	Dic₅	—	4	2	no	—
70	C₂²×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
71	M₁₆	—	20	1	no	—
72	C₂×C₈	—	2	1	no	—
73	C₂×C₈	—	2	1	no	—
74	C₂×C₈	—	5	2	no	—
75	C₂×C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
76	C₂×C₈	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
77	C₄⋊C₄	—	10	1	no	—
78	C₂²⋊C₄	—	10	1	no	—
79	C₄²	—	5	2	no	—
80	C₁₆	—	4	1	no	—
81	C₁₆	—	20	4	no	—
82	C₁₆	—	20	1	no	—
83	C₂×C₅	—	2	1	no	—
84	C₂×C₅	—	4	1	no	—
85	C₂×C₅	#1290	1	1	yes	characteristic
86	C₂³	#3995	1	1	yes	characteristic
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	10	2	no	—
90	C₂×C₄	#3996	1	1	yes	characteristic
91	C₂×C₄	(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic
92	C₈	—	2	1	no	—
93	C₈	—	10	2	no	—
94	C₈	#4005	1	1	yes	characteristic
95	C₈	#4013	1	1	yes	characteristic
96	C₅	#7240	1	1	yes	characteristic, complemented, Sylow
97	C₂²	—	2	1	no	—
98	C₂²	—	4	1	no	—
99	C₂²	#64896	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	20	2	no	—
102	C₄	#64895	1	1	yes	characteristic
103	C₄	#64919	1	1	yes	characteristic
104	C₂	—	2	1	no	—
105	C₂	—	4	1	no	—
106	C₂	#10598776	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10604715

C₂

yes

maximal

#10602194

C₂

yes

maximal, characteristic

C₂²×C₅×C₁₆

C₄

yes

characteristic

#65390

C₂²

yes

characteristic

C₂×#3944

C₄

yes

characteristic

#64912

—

#64890

C₄

yes

—

—

—

—

C₂²×C₅×C₈

C₂×C₄

yes

characteristic

#4072

Q₈

yes

characteristic

C₂×#681

D₄

yes

characteristic

#3998

—

C₂×C₅×C₁₆

—

C₂×C₅×C₁₆

—

#3947

C₂×C₄

yes

—

C₈×Dic₅

D₄

yes

—

—

—

—

—

C₂²×C₄×C₅

C₄⋊C₄

yes

characteristic

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₄²

yes

characteristic

C₂×C₅×C₈

C₂²⋊C₄

yes

characteristic

C₂²⋊Dic₅

—

C₄⋊Dic₅

—

C₄×Dic₅

C₄⋊C₄

yes

—

#681

—

C₂×#168

C₂²⋊C₄

yes

—

#674

—

C₅×C₁₆

—

#672

—

C₂×M₃₂

—

C₂²×C₁₆

yes

—

—

C₄⋊C₁₆

—

#349

—

#347

—

C₂³×C₅

#108

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

yes

C₂×C₄×C₅

yes

C₂×Dic₅

—

C₅×C₈

—

C₅×C₈

yes

C₅×C₈

yes

—

C₂×M₁₆

—

C₂²×C₈

C₂×AGL(1,𝔽₅)

yes

characteristic

#117

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#97

—

C₄×C₈

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

#341

yes

characteristic

C₄×C₅

—

C₄×C₅

yes

C₄×C₅

yes

Dic₅

—

C₂²×C₄

C₄⋊AGL(1,𝔽₅)

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×AGL(1,𝔽₅)

yes

characteristic

C₂×C₈

C₂²⋊AGL(1,𝔽₅)

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

C₁₆

—

C₁₆

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

yes

C₂³

yes

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#3996

yes

characteristic

C₂×C₄

(C₂×AGL(1,𝔽₅))⋊C₄

yes

characteristic

C₈

—

C₈

—

C₈

yes

C₈

yes

C₅

yes

characteristic, complemented, Sylow

C₂²

—

C₂²

—

C₂²

#64896

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

#64895

yes

characteristic

103

C₄

#64919

yes

characteristic

104

C₂

—

105

C₂

—

106

C₂

#10598776

yes

characteristic

	subgroup	G/H
center	C₄	#64895
commutator	C₂×C₅×C₈	C₄²
Frattini	C₂²×C₈	C₂×AGL(1,𝔽₅)
Fitting	C₂²×C₅×C₁₆	C₄
socle	C₂×C₅	#1290
prime radical	C₂³×C₅	#108
cosocle	#65390	C₂²
hypercenter	C₂²×C₁₆	AGL(1,𝔽₅)
hypocenter	C₅	#7240
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center

C₄

#64895

commutator

C₂×C₅×C₈

C₄²

Frattini

C₂²×C₈

C₂×AGL(1,𝔽₅)

Fitting

C₂²×C₅×C₁₆

C₄

C₂×C₅

C₂³×C₅

C₂²

C₂²×C₁₆

C₅

G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₅

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
96. C₅	#7240	semidirect

H ≅ G/N

type

96. C₅

#7240

semidirect

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

subgroup	index
#10604715	2
#10602194	2
#7240	5

subgroup

index

#10604715

#10602194

#7240

Subgroups — #13013805

Conjugacy classes of subgroups

Complements

Maximal subgroups