Group #14278610 — subgroups

Subgroups — #14278610

Order 1920.

120 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 65 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	#11746760	C₂	1	2	yes	maximal
2	C₃×#64722	C₂	1	1	yes	maximal, characteristic
3	#10598648	—	3	1	no	maximal, Hall
4	#91227	—	2	2	no	—
5	C₂×C₃×C₅×C₁₆	C₄	1	1	yes	characteristic
6	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
7	C₂×C₃×#672	C₄	1	1	yes	characteristic
8	#67514	—	5	1	no	maximal, Hall
9	#64889	—	3	2	no	—
10	#64722	D₃	1	1	yes	characteristic
11	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
12	C₃×C₄×Dic₅	Q₈	1	1	yes	characteristic
13	C₂×C₃×#168	D₄	1	1	yes	characteristic
14	C₃⋊#674	—	4	2	no	—
15	C₃×C₅×C₁₆	C₈	1	2	yes	—
16	C₃×#672	—	2	1	no	—
17	C₃×#346	—	5	1	no	—
18	#4463	—	5	2	no	—
19	#3998	—	6	2	no	—
20	C₂×C₅×C₁₆	Dic₃	1	1	yes	characteristic
21	C₈×Dic₅	C₂×D₃	1	1	yes	characteristic
22	C₂×#672	Dic₃	1	1	yes	characteristic
23	#1369	—	15	1	no	Sylow
24	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
25	C₂×C₃×Dic₅	—	2	1	no	—
26	C₃×C₅×C₈	C₂×C₈	1	1	yes	characteristic
27	C₃×C₅×C₈	M₁₆	1	1	yes	characteristic
28	C₃×#168	SD₁₆	1	2	yes	—
29	C₂×C₃×C₁₆	—	5	1	no	—
30	C₂×C₃×C₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic
31	C₃×C₄×C₈	—	5	1	no	—
32	#812	—	10	2	no	—
33	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
34	C₄×Dic₅	Dic₆	1	1	yes	characteristic
35	C₂×#168	D₁₂	1	1	yes	characteristic
36	#674	—	12	2	no	—
37	C₅×C₁₆	C₃⋊C₈	1	2	yes	—
38	#672	—	2	1	no	—
39	#364	—	15	2	no	—
40	#346	—	5	1	no	—
41	C₂²×C₃×C₅	#108	1	1	yes	characteristic
42	C₃×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
43	C₃×C₄×C₅	#106	1	1	yes	characteristic
44	C₃×Dic₅	—	4	1	no	—
45	C₂×C₃×C₈	—	5	1	no	—
46	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
47	C₃×C₄²	—	5	1	no	—
48	C₃×C₁₆	—	10	1	no	—
49	C₃×C₁₆	#170	1	2	yes	—
50	C₃⋊C₁₆	—	20	2	no	—
51	C₂×C₄×C₅	C₄⋊Dic₃	1	1	yes	characteristic
52	C₂×Dic₅	—	2	1	no	—
53	C₅×C₈	C₂×C₃⋊C₈	1	1	yes	characteristic
54	C₅×C₈	#208	1	1	yes	characteristic
55	#168	#204	1	2	yes	—
56	M₃₂	—	30	2	no	—
57	C₂×C₁₆	—	5	1	no	—
58	C₂×C₁₆	ΓL(1,𝔽₁₆)	1	1	yes	characteristic
59	C₄×C₈	—	5	1	no	—
60	C₂×C₃×C₅	—	2	1	no	—
61	C₂×C₃×C₅	#334	1	1	yes	characteristic
62	C₂×C₃×C₄	—	10	1	no	—
63	C₂×C₃×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
64	C₃×C₈	—	5	2	no	—
65	C₃×C₈	#700	1	1	yes	characteristic
66	C₃×C₈	C₂×#170	1	1	yes	characteristic
67	C₂²×C₅	#819	1	1	yes	characteristic
68	C₄×C₅	C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
69	C₄×C₅	#817	1	1	yes	characteristic
70	Dic₅	—	4	1	no	—
71	C₂×C₈	—	5	1	no	—
72	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
73	C₄²	—	5	1	no	—
74	C₁₆	—	10	1	no	—
75	C₁₆	—	60	2	no	—
76	C₁₆	C₃⋊#170	1	2	yes	—
77	C₃×C₅	#1369	1	1	yes	characteristic, complemented, Hall
78	C₂²×C₃	#3995	1	1	yes	characteristic
79	C₃×C₄	—	20	1	no	—
80	C₃×C₄	#3993	1	1	yes	characteristic
81	C₃×C₄	C₄⋊#170	1	1	yes	characteristic
82	C₂×C₅	—	2	1	no	—
83	C₂×C₅	#4457	1	1	yes	characteristic
84	C₂×C₄	—	10	1	no	—
85	C₂×C₄	C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
86	C₈	—	5	2	no	—
87	C₈	#6758	1	1	yes	characteristic
88	C₈	C₂×C₃⋊#170	1	1	yes	characteristic
89	C₂×C₃	—	2	1	no	—
90	C₂×C₃	#64881	1	1	yes	characteristic
91	C₅	#67514	1	1	yes	characteristic, complemented, Sylow
92	C₂²	#91224	1	1	yes	characteristic
93	C₄	—	20	1	no	—
94	C₄	#91222	1	1	yes	characteristic
95	C₄	C₄⋊(C₃⋊#170)	1	1	yes	characteristic
96	C₃	#10598648	1	1	yes	characteristic, complemented, Sylow
97	C₂	—	2	1	no	—
98	C₂	#11746752	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11746760

C₂

yes

maximal

C₃×#64722

C₂

yes

maximal, characteristic

#10598648

—

maximal, Hall

#91227

—

C₂×C₃×C₅×C₁₆

C₄

yes

characteristic

C₃×C₈×Dic₅

C₂²

yes

characteristic

C₂×C₃×#672

C₄

yes

—

—

D₃

yes

characteristic

C₂×C₃×C₅×C₈

C₂×C₄

yes

characteristic

C₃×C₄×Dic₅

Q₈

yes

characteristic

C₂×C₃×#168

D₄

yes

characteristic

C₃⋊#674

—

C₃×C₅×C₁₆

C₈

yes

—

—

—

—

—

C₂×C₅×C₁₆

Dic₃

yes

characteristic

C₈×Dic₅

C₂×D₃

yes

characteristic

C₂×#672

Dic₃

yes

characteristic

#1369

—

Sylow

C₂×C₃×C₄×C₅

C₄⋊C₄

yes

characteristic

C₂×C₃×Dic₅

—

C₃×C₅×C₈

C₂×C₈

yes

characteristic

C₃×C₅×C₈

M₁₆

yes

characteristic

C₃×#168

SD₁₆

yes

—

C₂×C₃×C₁₆

—

C₂×C₃×C₁₆

AGL(1,𝔽₅)

yes

characteristic

C₃×C₄×C₈

—

#812

—

C₂×C₅×C₈

C₂×Dic₃

yes

characteristic

C₄×Dic₅

Dic₆

yes

characteristic

C₂×#168

D₁₂

yes

characteristic

#674

—

C₅×C₁₆

C₃⋊C₈

yes

—

#672

—

#364

—

#346

—

C₂²×C₃×C₅

#108

yes

characteristic

C₃×C₄×C₅

C₄⋊C₈

yes

characteristic

C₃×C₄×C₅

#106

yes

characteristic

C₃×Dic₅

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×AGL(1,𝔽₅)

yes

characteristic

C₃×C₄²

—

C₃×C₁₆

—

C₃×C₁₆

#170

yes

—

C₃⋊C₁₆

—

C₂×C₄×C₅

C₄⋊Dic₃

yes

characteristic

C₂×Dic₅

—

C₅×C₈

C₂×C₃⋊C₈

yes

C₅×C₈

yes

yes

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

ΓL(1,𝔽₁₆)

yes

characteristic

C₄×C₈

—

C₂×C₃×C₅

—

C₂×C₃×C₅

#334

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

C₄⋊AGL(1,𝔽₅)

yes

characteristic

C₃×C₈

—

C₃×C₈

yes

C₃×C₈

yes

C₂²×C₅

yes

C₄×C₅

C₄⋊(C₃⋊C₈)

yes

characteristic

C₄×C₅

#817

yes

characteristic

Dic₅

—

C₂×C₈

—

C₂×C₈

C₂×ΓL(1,𝔽₁₆)

yes

characteristic

C₄²

—

C₁₆

—

C₁₆

—

C₁₆

C₃⋊#170

yes

—

C₃×C₅

#1369

yes

characteristic, complemented, Hall

C₂²×C₃

#3995

yes

characteristic

C₃×C₄

—

C₃×C₄

yes

C₃×C₄

yes

C₂×C₅

—

C₂×C₅

#4457

yes

characteristic

C₂×C₄

—

C₂×C₄

C₄⋊ΓL(1,𝔽₁₆)

yes

characteristic

C₈

—

C₈

#6758

yes

characteristic

C₈

C₂×C₃⋊#170

yes

characteristic

C₂×C₃

—

C₂×C₃

#64881

yes

characteristic

C₅

#67514

yes

characteristic, complemented, Sylow

C₂²

#91224

yes

characteristic

C₄

—

C₄

yes

C₄

yes

C₃

yes

characteristic, complemented, Sylow

C₂

—

C₂

#11746752

yes

characteristic

	subgroup	G/H
center	C₄	#91222
commutator	C₃×C₅×C₈	C₂×C₈
Frattini	C₂×C₈	C₂×ΓL(1,𝔽₁₆)
Fitting	C₂×C₃×C₅×C₁₆	C₄
socle	C₂×C₃×C₅	#334
prime radical	C₂²×C₃×C₅	#108
cosocle	C₃×C₈×Dic₅	C₂²
hypercenter	C₂×C₁₆	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	#1369
LCS	G ▷ C₃×C₅×C₈ ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

subgroup

G/H

center

C₄

#91222

commutator

C₃×C₅×C₈

C₂×C₈

Frattini

C₂×C₈

C₂×ΓL(1,𝔽₁₆)

Fitting

C₂×C₃×C₅×C₁₆

C₄

C₂×C₃×C₅

C₂²×C₃×C₅

C₃×C₈×Dic₅

C₂²

hypercenter

C₂×C₁₆

ΓL(1,𝔽₁₆)

hypocenter

C₃×C₅

#1369

LCS

G ▷ C₃×C₅×C₈ ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

N	H ≅ G/N	type
77. C₃×C₅	#1369	semidirect
91. C₅	#67514	semidirect
96. C₃	#10598648	semidirect

H ≅ G/N

type

77. C₃×C₅

#1369

semidirect

91. C₅

#67514

semidirect

96. C₃

#10598648

semidirect

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

subgroup	index
#11746760	2
C₃×#64722	2
#10598648	3
#67514	5

subgroup

index

Subgroups — #14278610

Conjugacy classes of subgroups

Complements

Maximal subgroups