Group #14278615 — subgroups

Subgroups — #14278615

Order 1920.

120 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 70 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	#11746284	C₂	1	1	yes	maximal, characteristic
2	C₅×#4461	C₂	1	1	yes	maximal, characteristic
3	C₃×#64683	C₂	1	1	yes	maximal, characteristic
4	#10598005	—	3	1	no	maximal, Hall
5	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
6	#91096	—	2	1	no	—
7	C₂×C₅×C₃⋊C₁₆	C₄	1	1	yes	characteristic
8	C₂×C₅×C₃⋊C₁₆	C₄	1	1	yes	characteristic
9	C₃×#3944	—	2	1	no	—
10	#67567	—	5	1	no	maximal, Hall
11	C₅×#346	—	3	1	no	—
12	#64683	—	3	1	no	—
13	#64683	D₃	1	1	yes	characteristic
14	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
15	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
16	C₃×C₄²×C₅	Q₈	1	1	yes	characteristic
17	C₃⋊#672	—	4	1	no	—
18	C₃×#672	—	4	1	no	—
19	C₅×C₃⋊C₁₆	—	2	1	no	—
20	C₅×C₃⋊C₁₆	C₈	1	2	yes	—
21	C₃×#364	—	5	1	no	—
22	#4463	—	5	1	no	—
23	#4461	D₅	1	1	yes	characteristic
24	C₂×C₅×C₁₆	—	3	1	no	—
25	C₂×C₅×C₁₆	—	3	1	no	—
26	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
27	#3944	—	2	1	no	—
28	#3944	—	6	1	no	—
29	#1369	—	15	1	no	Sylow
30	C₂×C₃×C₄×C₅	—	2	1	no	—
31	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
32	C₃×C₅×C₈	C₂×C₈	1	1	yes	characteristic
33	C₃×C₅×C₈	M₁₆	1	1	yes	characteristic
34	C₃×C₅×C₈	SD₁₆	1	1	yes	characteristic
35	C₃×C₅×C₈	SD₁₆	1	1	yes	characteristic
36	C₃×M₃₂	—	10	1	no	—
37	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
38	#812	—	10	1	no	—
39	C₂×C₃⋊C₁₆	Dic₅	1	1	yes	characteristic
40	C₂×C₃⋊C₁₆	Dic₅	1	1	yes	characteristic
41	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
42	C₂×C₅×C₈	#67	1	1	yes	characteristic
43	C₄²×C₅	Dic₆	1	1	yes	characteristic
44	C₅×C₁₆	—	3	2	no	—
45	C₅×C₁₆	—	6	1	no	—
46	#672	—	4	1	no	—
47	#672	—	12	1	no	—
48	#364	—	5	1	no	—
49	#364	—	15	1	no	—
50	#346	—	3	1	no	—
51	C₂²×C₃×C₅	#108	1	1	yes	characteristic
52	C₃×C₄×C₅	—	4	1	no	—
53	C₃×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
54	C₃×C₄×C₅	#106	1	1	yes	characteristic
55	C₂×C₃×C₈	D₂₀	1	1	yes	characteristic
56	C₂×C₃×C₈	C₂×Dic₅	1	1	yes	characteristic
57	C₃×C₄²	Dic₁₀	1	1	yes	characteristic
58	C₃×C₁₆	—	20	1	no	—
59	C₃⋊C₁₆	—	2	1	no	—
60	C₃⋊C₁₆	—	20	1	no	—
61	C₃⋊C₁₆	#168	1	2	yes	—
62	C₂×C₄×C₅	—	2	1	no	—
63	C₂×C₄×C₅	Dic₃⋊C₄	1	1	yes	characteristic
64	C₅×C₈	D₃×C₈	1	1	yes	characteristic
65	C₅×C₈	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
66	C₅×C₈	#214	1	1	yes	characteristic
67	C₅×C₈	#215	1	1	yes	characteristic
68	M₃₂	—	10	1	no	—
69	M₃₂	—	30	1	no	—
70	C₂×C₁₆	—	3	1	no	—
71	C₂×C₁₆	—	3	1	no	—
72	C₄×C₈	D₃×D₅	1	1	yes	characteristic
73	C₂×C₃×C₅	—	2	1	no	—
74	C₂×C₃×C₅	#334	1	1	yes	characteristic
75	C₂×C₃×C₄	—	2	1	no	—
76	C₂×C₃×C₄	C₄⋊Dic₅	1	1	yes	characteristic
77	C₃×C₈	#677	1	1	yes	characteristic
78	C₃×C₈	#677	1	1	yes	characteristic
79	C₃×C₈	C₂×#168	1	1	yes	characteristic
80	C₃×C₈	#681	1	1	yes	characteristic
81	C₂²×C₅	#822	1	1	yes	characteristic
82	C₄×C₅	—	4	1	no	—
83	C₄×C₅	#808	1	1	yes	characteristic
84	C₄×C₅	Dic₃⋊C₈	1	1	yes	characteristic
85	C₂×C₈	D₃×Dic₅	1	1	yes	characteristic
86	C₂×C₈	#1202	1	1	yes	characteristic
87	C₄²	#1204	1	1	yes	characteristic
88	C₁₆	—	3	2	no	—
89	C₁₆	—	6	1	no	—
90	C₁₆	—	20	1	no	—
91	C₁₆	—	60	1	no	—
92	C₃×C₅	#1369	1	1	yes	characteristic, complemented, Hall
93	C₂²×C₃	#3951	1	1	yes	characteristic
94	C₃×C₄	—	4	1	no	—
95	C₃×C₄	C₄⋊#168	1	1	yes	characteristic
96	C₃×C₄	#3949	1	1	yes	characteristic
97	C₂×C₅	—	2	1	no	—
98	C₂×C₅	#4480	1	1	yes	characteristic
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	Dic₃⋊Dic₅	1	1	yes	characteristic
101	C₈	D₃×#168	1	1	yes	characteristic
102	C₈	#6650	1	1	yes	characteristic
103	C₈	#6657	1	1	yes	characteristic
104	C₈	#6658	1	1	yes	characteristic
105	C₂×C₃	—	2	1	no	—
106	C₂×C₃	#64677	1	1	yes	characteristic
107	C₅	#67567	1	1	yes	characteristic, complemented, Sylow
108	C₂²	#90993	1	1	yes	characteristic
109	C₄	—	4	1	no	—
110	C₄	Dic₃⋊#168	1	1	yes	characteristic
111	C₄	#90987	1	1	yes	characteristic
112	C₃	#10598005	1	1	yes	characteristic, complemented, Sylow
113	C₂	—	2	1	no	—
114	C₂	#11745866	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11746284

C₂

yes

maximal, characteristic

C₅×#4461

C₂

yes

maximal, characteristic

C₃×#64683

C₂

yes

maximal, characteristic

#10598005

—

maximal, Hall

C₃×C₄×C₅×C₈

C₂²

yes

characteristic

#91096

—

C₂×C₅×C₃⋊C₁₆

C₄

yes

characteristic

C₂×C₅×C₃⋊C₁₆

C₄

yes

—

—

—

—

D₃

yes

characteristic

C₂×C₃×C₅×C₈

C₂×C₄

yes

characteristic

C₂×C₃×C₅×C₈

D₄

yes

characteristic

C₃×C₄²×C₅

Q₈

yes

characteristic

C₃⋊#672

—

C₃×#672

—

C₅×C₃⋊C₁₆

—

C₅×C₃⋊C₁₆

C₈

yes

—

C₃×#364

—

#4463

—

#4461

D₅

yes

characteristic

C₂×C₅×C₁₆

—

C₂×C₅×C₁₆

—

C₄×C₅×C₈

C₂×D₃

yes

—

—

—

C₂×C₃×C₄×C₅

—

C₂×C₃×C₄×C₅

C₄⋊C₄

yes

characteristic

C₃×C₅×C₈

C₂×C₈

yes

characteristic

C₃×C₅×C₈

M₁₆

yes

characteristic

C₃×C₅×C₈

SD₁₆

yes

characteristic

C₃×C₅×C₈

SD₁₆

yes

characteristic

C₃×M₃₂

—

C₃×C₄×C₈

C₂×D₅

yes

characteristic

#812

—

C₂×C₃⋊C₁₆

Dic₅

yes

characteristic

C₂×C₃⋊C₁₆

Dic₅

yes

characteristic

C₂×C₅×C₈

C₄×D₃

yes

characteristic

C₂×C₅×C₈

#67

yes

characteristic

C₄²×C₅

Dic₆

yes

characteristic

C₅×C₁₆

—

C₅×C₁₆

—

—

—

—

—

—

C₂²×C₃×C₅

#108

yes

characteristic

C₃×C₄×C₅

—

C₃×C₄×C₅

C₄⋊C₈

yes

characteristic

C₃×C₄×C₅

#106

yes

characteristic

C₂×C₃×C₈

D₂₀

yes

characteristic

C₂×C₃×C₈

C₂×Dic₅

yes

characteristic

C₃×C₄²

Dic₁₀

yes

characteristic

C₃×C₁₆

—

C₃⋊C₁₆

—

C₃⋊C₁₆

—

C₃⋊C₁₆

#168

yes

—

C₂×C₄×C₅

—

C₂×C₄×C₅

Dic₃⋊C₄

yes

characteristic

C₅×C₈

D₃×C₈

yes

characteristic

C₅×C₈

ΓL(1,𝔽₂₅)

yes

C₅×C₈

yes

C₅×C₈

yes

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄×C₈

D₃×D₅

yes

characteristic

C₂×C₃×C₅

—

C₂×C₃×C₅

#334

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

C₄⋊Dic₅

yes

C₃×C₈

yes

C₃×C₈

yes

C₃×C₈

yes

C₃×C₈

yes

C₂²×C₅

yes

C₄×C₅

—

C₄×C₅

#808

yes

characteristic

C₄×C₅

Dic₃⋊C₈

yes

characteristic

C₂×C₈

D₃×Dic₅

yes

C₂×C₈

yes

C₄²

yes

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

C₃×C₅

#1369

yes

characteristic, complemented, Hall

C₂²×C₃

#3951

yes

characteristic

C₃×C₄

—

C₃×C₄

yes

C₃×C₄

yes

C₂×C₅

—

C₂×C₅

#4480

yes

characteristic

C₂×C₄

—

100

C₂×C₄

Dic₃⋊Dic₅

yes

characteristic

101

C₈

D₃×#168

yes

characteristic

102

C₈

#6650

yes

characteristic

103

C₈

#6657

yes

characteristic

104

C₈

#6658

yes

characteristic

105

C₂×C₃

—

106

C₂×C₃

#64677

yes

characteristic

107

C₅

#67567

yes

characteristic, complemented, Sylow

108

C₂²

#90993

yes

characteristic

109

C₄

—

110

C₄

Dic₃⋊#168

yes

characteristic

111

C₄

#90987

yes

characteristic

112

C₃

#10598005

yes

characteristic, complemented, Sylow

113

C₂

—

114

C₂

#11745866

yes

characteristic

	subgroup	G/H
center	C₄	#90987
commutator	C₃×C₅×C₈	C₂×C₈
Frattini, hypercenter	C₄×C₈	D₃×D₅
Fitting, cosocle	C₃×C₄×C₅×C₈	C₂²
socle	C₂×C₃×C₅	#334
prime radical	C₂²×C₃×C₅	#108
hypocenter	C₃×C₅	#1369
LCS	G ▷ C₃×C₅×C₈ ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈

subgroup

G/H

center

C₄

#90987

commutator

C₃×C₅×C₈

C₂×C₈

Frattini, hypercenter

C₄×C₈

D₃×D₅

Fitting, cosocle

C₃×C₄×C₅×C₈

C₂²

C₂×C₃×C₅

C₂²×C₃×C₅

C₃×C₅

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

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈

N	H ≅ G/N	type
92. C₃×C₅	#1369	semidirect
107. C₅	#67567	semidirect
112. C₃	#10598005	semidirect

H ≅ G/N

type

92. C₃×C₅

#1369

semidirect

107. C₅

#67567

semidirect

112. C₃

#10598005

semidirect

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

subgroup	index
#11746284	2
C₅×#4461	2
C₃×#64683	2
#10598005	3
#67567	5

subgroup

index

Subgroups — #14278615

Conjugacy classes of subgroups

Complements

Maximal subgroups