Group #13023075 — subgroups

Subgroups — #13023075

Order 1280.

228 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 92 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	#10603830	C₂	1	2	yes	maximal
2	#10598840	C₂	1	1	yes	maximal, characteristic
3	#10598515	C₂	1	4	yes	maximal, complemented
4	#64974	C₄	1	2	yes	—
5	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
6	#64888	C₄	1	4	yes	—
7	#64887	C₄	1	4	yes	—
8	#64859	C₂²	1	4	yes	—
9	C₅×C₈²	C₄	1	1	yes	characteristic
10	#64674	C₂²	1	2	yes	—
11	C₈×#168	C₄	1	1	yes	characteristic
12	#11416	—	5	1	no	maximal, Sylow
13	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
14	C₄²×D₅	C₂×C₄	1	1	yes	characteristic
15	C₂×#674	C₂×C₄	1	8	yes	—
16	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
17	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
18	#3952	—	2	2	no	—
19	#3946	—	2	2	no	—
20	C₈×Dic₅	C₂×C₄	1	2	yes	—
21	C₄×#168	C₂×C₄	1	2	yes	—
22	C₄×#168	C₂³	1	1	yes	characteristic
23	#2163	—	5	2	no	—
24	#1450	—	5	1	no	—
25	#1313	—	5	4	no	—
26	C₂×C₄×D₅	C₄²	1	1	yes	characteristic
27	C₂×C₄×D₅	M₁₆	1	2	yes	—
28	C₂×C₅×C₈	—	2	2	no	—
29	C₂×C₅×C₈	C₄²	1	2	yes	—
30	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
31	C₄×Dic₅	C₄²	1	1	yes	characteristic
32	C₄×Dic₅	M₁₆	1	2	yes	—
33	C₂×#168	—	2	2	no	—
34	C₂×#168	C₂²×C₄	1	2	yes	—
35	C₈×D₅	—	2	4	no	—
36	#674	—	2	8	no	—
37	#431	—	5	2	no	—
38	C₂×C₄×C₈	—	5	1	no	—
39	C₄⋊C₁₆	—	5	4	no	—
40	C₂²⋊C₁₆	—	5	4	no	—
41	#346	—	5	4	no	—
42	#322	—	5	2	no	—
43	C₈²	—	5	1	no	—
44	C₈²	AGL(1,𝔽₅)	1	1	yes	characteristic
45	C₂²×D₅	#97	1	1	yes	characteristic
46	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
47	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
48	C₂×Dic₅	#97	1	2	yes	—
49	C₂×Dic₅	#97	1	1	yes	characteristic
50	C₄×D₅	—	2	2	no	—
51	C₄×D₅	—	4	2	no	—
52	C₅×C₈	—	2	2	no	—
53	C₅×C₈	—	4	2	no	—
54	#168	—	4	2	no	—
55	#168	#131	1	4	yes	—
56	C₂²×C₈	—	5	2	no	—
57	C₂×C₄²	—	5	1	no	—
58	C₂×C₁₆	—	5	8	no	—
59	C₄⋊C₈	—	10	2	no	—
60	C₂²⋊C₈	—	10	2	no	—
61	C₄×C₈	—	5	2	no	—
62	C₄×C₈	—	5	1	no	—
63	C₄×C₈	—	5	2	no	—
64	C₄×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
65	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
66	C₂²×C₅	C₂×#97	1	1	yes	characteristic
67	C₂×D₅	—	2	2	no	—
68	C₂×D₅	—	2	1	no	—
69	C₄×C₅	—	2	2	no	—
70	C₄×C₅	#405	1	2	yes	—
71	Dic₅	—	2	2	no	—
72	Dic₅	—	2	1	no	—
73	C₂²×C₄	—	5	1	no	—
74	C₂²×C₄	—	5	2	no	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	5	2	no	—
77	C₂×C₈	—	10	4	no	—
78	C₂×C₈	—	10	2	no	—
79	C₂×C₈	C₄×AGL(1,𝔽₅)	1	2	yes	—
80	C₄²	—	5	2	no	—
81	C₄²	—	5	1	no	—
82	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
83	C₁₆	—	10	8	no	—
84	C₂×C₅	#1452	1	1	yes	characteristic
85	C₂×C₅	#2110	1	2	yes	—
86	D₅	—	4	1	no	—
87	C₂³	—	5	1	no	—
88	C₂×C₄	—	5	1	no	—
89	C₂×C₄	—	5	2	no	—
90	C₂×C₄	—	10	2	no	—
91	C₂×C₄	—	20	2	no	—
92	C₂×C₄	D₅⋊M₁₆	1	2	yes	—
93	C₂×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
94	C₈	—	2	2	no	—
95	C₈	—	4	2	no	—
96	C₈	—	5	4	no	—
97	C₈	—	20	2	no	—
98	C₅	#11416	1	1	yes	characteristic, complemented, Sylow
99	C₂²	—	10	1	no	—
100	C₂²	—	10	2	no	—
101	C₂²	#65676	1	1	yes	characteristic
102	C₄	—	2	2	no	—
103	C₄	—	10	1	no	—
104	C₄	—	10	2	no	—
105	C₄	#65718	1	2	yes	—
106	C₂	—	20	1	no	—
107	C₂	#10603811	1	1	yes	characteristic
108	C₂	#10603832	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10603830

C₂

yes

maximal

#10598840

C₂

yes

maximal, characteristic

#10598515

C₂

yes

maximal, complemented

#64974

C₄

yes

—

C₄×C₈×D₅

C₂²

yes

characteristic

#64888

C₄

yes

—

#64887

C₄

yes

—

#64859

C₂²

yes

—

C₅×C₈²

C₄

yes

characteristic

#64674

C₂²

yes

—

C₈×#168

C₄

yes

characteristic

#11416

—

maximal, Sylow

C₂×C₈×D₅

C₂×C₄

yes

—

C₄²×D₅

C₂×C₄

yes

characteristic

C₂×#674

C₂×C₄

yes

—

C₄×C₅×C₈

C₂×C₄

yes

—

C₄×C₅×C₈

C₂×C₄

yes

characteristic

#3952

—

#3946

—

C₈×Dic₅

C₂×C₄

yes

—

C₄×#168

C₂×C₄

yes

—

C₄×#168

C₂³

yes

—

—

—

C₂×C₄×D₅

C₄²

yes

characteristic

C₂×C₄×D₅

M₁₆

yes

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₄²

yes

—

C₄²×C₅

C₂²×C₄

yes

characteristic

C₄×Dic₅

C₄²

yes

characteristic

C₄×Dic₅

M₁₆

yes

—

C₂×#168

—

C₂×#168

C₂²×C₄

yes

—

C₈×D₅

—

#674

—

#431

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

#322

—

C₈²

—

C₈²

yes

C₂²×D₅

yes

C₂×C₄×C₅

C₂×C₄²

yes

characteristic

C₂×C₄×C₅

C₂×M₁₆

yes

—

C₂×Dic₅

#97

yes

—

C₂×Dic₅

#97

yes

characteristic

C₄×D₅

—

C₄×D₅

—

C₅×C₈

—

C₅×C₈

—

#168

—

#168

#131

yes

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×AGL(1,𝔽₅)

yes

—

C₄×C₈

C₂×AGL(1,𝔽₅)

yes

characteristic

C₂²×C₅

C₂×#97

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

#405

yes

—

Dic₅

—

Dic₅

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×AGL(1,𝔽₅)

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×AGL(1,𝔽₅)

yes

characteristic

C₁₆

—

C₂×C₅

#1452

yes

characteristic

C₂×C₅

#2110

yes

—

D₅

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₅⋊M₁₆

yes

—

C₂×C₄

C₂×C₄×AGL(1,𝔽₅)

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

C₅

#11416

yes

characteristic, complemented, Sylow

C₂²

—

100

C₂²

—

101

C₂²

#65676

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

#65718

yes

—

106

C₂

—

107

C₂

#10603811

yes

characteristic

108

C₂

#10603832

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×AGL(1,𝔽₅)
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₄²	C₂²×AGL(1,𝔽₅)
Fitting	C₅×C₈²	C₄
socle	C₂²×C₅	C₂×#97
prime radical	C₂²×D₅	#97
cosocle	C₄×#168	C₂³
hypercenter	C₈²	AGL(1,𝔽₅)
hypocenter	C₅	#11416
LCS	G ▷ C₂×C₄×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₈²

subgroup

G/H

center

C₂×C₄

C₂×C₄×AGL(1,𝔽₅)

commutator

C₂×C₄×C₅

C₂×C₄²

Frattini

C₄²

C₂²×AGL(1,𝔽₅)

Fitting

C₅×C₈²

C₄

socle

C₂²×C₅

C₂×#97

C₂²×D₅

C₂³

C₈²

C₅

G ▷ C₂×C₄×C₅ ▷ C₅

C₁ ◁ C₂×C₄ ◁ C₈²

N	H ≅ G/N	type
3. #10598515	C₂	semidirect
98. C₅	#11416	semidirect

H ≅ G/N

type

3. #10598515

C₂

semidirect

98. C₅

#11416

semidirect

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

subgroup	index
#10603830	2
#10598840	2
#10598515	2
#11416	5

subgroup

index

Subgroups — #13023075

Conjugacy classes of subgroups

Complements

Maximal subgroups