Group #13018762 — subgroups

Subgroups — #13018762

Order 1280.

228 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 76 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	#10603821	C₂	1	2	yes	maximal
2	#10598840	C₂	1	1	yes	maximal, characteristic
3	#10598532	C₂	1	4	yes	maximal
4	#65683	C₂²	1	4	yes	—
5	#64974	C₂²	1	2	yes	—
6	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
7	#64884	—	2	2	no	—
8	#64883	—	2	2	no	—
9	C₅×C₈²	C₄	1	1	yes	characteristic
10	#64674	C₄	1	2	yes	—
11	C₈×#168	C₄	1	1	yes	characteristic
12	#8797	—	5	1	no	maximal, Sylow
13	C₂×C₈×D₅	Q₈	1	2	yes	—
14	C₄²×D₅	C₂³	1	1	yes	characteristic
15	#4003	—	2	4	no	—
16	#4002	—	2	4	no	—
17	C₄⋊#170	—	2	4	no	—
18	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
19	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
20	#3952	—	2	2	no	—
21	#3946	—	2	2	no	—
22	C₈×Dic₅	D₄	1	2	yes	—
23	C₄×#168	C₂×C₄	1	2	yes	—
24	C₄×#168	C₂×C₄	1	1	yes	characteristic
25	#1568	—	5	2	no	—
26	#1450	—	5	1	no	—
27	#1287	—	5	4	no	—
28	C₂×C₄×D₅	C₂×Q₈	1	1	yes	characteristic
29	C₂×C₄×D₅	#41	1	2	yes	—
30	C₂×#170	—	4	4	no	—
31	C₂×C₅×C₈	—	2	2	no	—
32	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
33	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
34	C₄×Dic₅	C₂×D₄	1	1	yes	characteristic
35	C₄×Dic₅	#41	1	2	yes	—
36	C₂×#168	—	2	2	no	—
37	C₂×#168	C₄⋊C₄	1	2	yes	—
38	C₈×D₅	—	2	4	no	—
39	#432	—	5	4	no	—
40	#431	—	5	2	no	—
41	C₂×C₄×C₈	—	5	1	no	—
42	#335	—	10	2	no	—
43	#334	—	10	2	no	—
44	#322	—	5	2	no	—
45	C₈²	—	5	1	no	—
46	C₈²	AGL(1,𝔽₅)	1	1	yes	characteristic
47	C₂²×D₅	#125	1	1	yes	characteristic
48	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₅	#117	1	2	yes	—
50	C₂×Dic₅	#124	1	2	yes	—
51	C₂×Dic₅	#125	1	1	yes	characteristic
52	C₄×D₅	—	2	2	no	—
53	C₄×D₅	—	4	2	no	—
54	#170	—	8	4	no	—
55	C₅×C₈	—	2	2	no	—
56	C₅×C₈	—	4	2	no	—
57	#168	—	4	2	no	—
58	#168	#108	1	4	yes	—
59	C₂²×C₈	—	5	2	no	—
60	C₂×C₄²	—	5	1	no	—
61	C₄⋊C₈	—	10	2	no	—
62	C₄⋊C₈	—	10	4	no	—
63	C₂²⋊C₈	—	10	2	no	—
64	C₂²⋊C₈	—	10	4	no	—
65	#97	—	10	4	no	—
66	C₄×C₈	—	5	2	no	—
67	C₄×C₈	—	5	1	no	—
68	C₄×C₈	—	5	2	no	—
69	C₄×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
70	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
71	C₂²×C₅	#382	1	1	yes	characteristic
72	C₂×D₅	—	2	2	no	—
73	C₂×D₅	—	2	1	no	—
74	C₄×C₅	—	2	2	no	—
75	C₄×C₅	C₂×#108	1	2	yes	—
76	Dic₅	—	2	2	no	—
77	Dic₅	—	2	1	no	—
78	C₂²×C₄	—	5	1	no	—
79	C₂²×C₄	—	5	2	no	—
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	5	2	no	—
82	C₂×C₈	—	10	4	no	—
83	C₂×C₈	—	10	2	no	—
84	C₂×C₈	—	20	4	no	—
85	C₂×C₈	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
86	C₄²	—	5	1	no	—
87	C₄²	—	5	2	no	—
88	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
89	C₂×C₅	#1833	1	2	yes	—
90	C₂×C₅	#1837	1	1	yes	characteristic
91	D₅	—	4	1	no	—
92	C₂³	—	5	1	no	—
93	C₂×C₄	—	5	2	no	—
94	C₂×C₄	—	5	1	no	—
95	C₂×C₄	—	10	2	no	—
96	C₂×C₄	—	20	2	no	—
97	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
98	C₂×C₄	#4130	1	2	yes	—
99	C₈	—	2	2	no	—
100	C₈	—	4	2	no	—
101	C₈	—	5	4	no	—
102	C₈	—	20	2	no	—
103	C₈	—	40	4	no	—
104	C₅	#8797	1	1	yes	characteristic, complemented, Sylow
105	C₂²	—	10	2	no	—
106	C₂²	—	10	1	no	—
107	C₂²	#65689	1	1	yes	characteristic
108	C₄	—	2	2	no	—
109	C₄	—	10	2	no	—
110	C₄	—	10	1	no	—
111	C₄	D₅⋊#108	1	2	yes	—
112	C₂	—	20	1	no	—
113	C₂	#10603852	1	2	yes	—
114	C₂	#10603857	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10603821

C₂

yes

maximal

#10598840

C₂

yes

maximal, characteristic

#10598532

C₂

yes

maximal

#65683

C₂²

yes

—

#64974

C₂²

yes

—

C₄×C₈×D₅

C₂²

yes

characteristic

#64884

—

#64883

—

C₅×C₈²

C₄

yes

characteristic

#64674

C₄

yes

—

C₈×#168

C₄

yes

characteristic

#8797

—

maximal, Sylow

C₂×C₈×D₅

Q₈

yes

—

C₄²×D₅

C₂³

yes

—

—

—

C₄×C₅×C₈

C₂×C₄

yes

—

C₄×C₅×C₈

C₂×C₄

yes

characteristic

#3952

—

#3946

—

C₈×Dic₅

D₄

yes

—

C₄×#168

C₂×C₄

yes

—

C₄×#168

C₂×C₄

yes

—

—

—

C₂×C₄×D₅

C₂×Q₈

yes

characteristic

C₂×C₄×D₅

#41

yes

—

C₂×#170

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₄⋊C₄

yes

—

C₄²×C₅

C₂²×C₄

yes

characteristic

C₄×Dic₅

C₂×D₄

yes

characteristic

C₄×Dic₅

#41

yes

—

C₂×#168

—

C₂×#168

C₄⋊C₄

yes

—

C₈×D₅

—

#432

—

#431

—

C₂×C₄×C₈

—

#335

—

#334

—

#322

—

C₈²

—

C₈²

yes

C₂²×D₅

yes

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅

#117

yes

—

C₂×Dic₅

#124

yes

—

C₂×Dic₅

#125

yes

characteristic

C₄×D₅

—

C₄×D₅

—

#170

—

C₅×C₈

—

C₅×C₈

—

#168

—

#168

#108

yes

—

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×AGL(1,𝔽₅)

yes

—

C₄×C₈

yes

C₂²×C₅

yes

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

C₂×#108

yes

—

Dic₅

—

Dic₅

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊AGL(1,𝔽₅)

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×AGL(1,𝔽₅)

yes

characteristic

C₂×C₅

#1833

yes

—

C₂×C₅

#1837

yes

characteristic

D₅

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

C₂×C₄

#4130

yes

—

C₈

—

100

C₈

—

101

C₈

—

102

C₈

—

103

C₈

—

104

C₅

#8797

yes

characteristic, complemented, Sylow

105

C₂²

—

106

C₂²

—

107

C₂²

#65689

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

D₅⋊#108

yes

—

112

C₂

—

113

C₂

#10603852

yes

—

114

C₂

#10603857

yes

characteristic

	subgroup	G/H
center	C₂²	#65689
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₄²	C₂²×AGL(1,𝔽₅)
Fitting	C₅×C₈²	C₄
socle	C₂²×C₅	#382
prime radical	C₂²×D₅	#125
cosocle	C₄²×D₅	C₂³
hypercenter	C₈²	AGL(1,𝔽₅)
hypocenter	C₅	#8797
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

subgroup

G/H

center

C₂²

#65689

commutator

C₄²×C₅

C₂²×C₄

Frattini

C₄²

C₂²×AGL(1,𝔽₅)

Fitting

C₅×C₈²

C₄

C₂²×C₅

C₂²×D₅

C₄²×D₅

C₂³

C₈²

C₅

G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈²

N	H ≅ G/N	type
104. C₅	#8797	semidirect

H ≅ G/N

type

104. C₅

#8797

semidirect

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

subgroup	index
#10603821	2
#10598840	2
#10598532	2
#8797	5

subgroup

index

Subgroups — #13018762

Conjugacy classes of subgroups

Complements

Maximal subgroups