Group #14293584 — subgroups

Subgroups — #14293584

Order 1920.

336 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 127 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	#11750783	C₂	1	2	yes	maximal
2	#11750782	C₂	1	4	yes	maximal
3	C₃×C₅×C₈²	C₂	1	1	yes	maximal, characteristic
4	#10598857	—	3	1	no	maximal, Hall
5	#91758	C₂²	1	4	yes	—
6	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
7	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
8	#91102	—	2	4	no	—
9	#91100	—	2	8	no	—
10	#68064	—	5	1	no	maximal, Hall
11	#64971	—	3	2	no	—
12	#64970	—	3	4	no	—
13	C₅×C₈²	D₃	1	1	yes	characteristic
14	C₂×Dic₃₀	—	2	4	no	—
15	C₂×C₃×C₅×C₈	D₄	1	4	yes	—
16	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
17	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
18	C₄⋊Dic₁₅	—	2	4	no	—
19	C₄⋊Dic₁₅	—	2	4	no	—
20	#4682	—	5	2	no	—
21	#4681	—	5	4	no	—
22	C₃×C₈²	D₅	1	1	yes	characteristic
23	#4015	—	3	4	no	—
24	C₄×C₅×C₈	C₂×D₃	1	2	yes	—
25	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
26	#3950	—	6	4	no	—
27	#3948	—	6	8	no	—
28	#1710	—	15	1	no	Sylow
29	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
30	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
31	C₂×Dic₁₅	—	4	4	no	—
32	Dic₃₀	—	4	8	no	—
33	C₃×C₅×C₈	—	2	4	no	—
34	C₃×C₅×C₈	Q₁₆	1	4	yes	—
35	#869	—	5	4	no	—
36	C₃×C₄×C₈	C₂×D₅	1	2	yes	—
37	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
38	#818	—	10	4	no	—
39	#816	—	10	8	no	—
40	C₂×Dic₁₀	—	6	4	no	—
41	C₂×C₅×C₈	D₁₂	1	4	yes	—
42	C₂×C₅×C₈	D₁₂	1	2	yes	—
43	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
44	C₄⋊Dic₅	—	6	4	no	—
45	C₄⋊Dic₅	—	6	4	no	—
46	#500	—	15	2	no	—
47	#487	—	15	4	no	—
48	C₈²	D₁₅	1	1	yes	characteristic
49	C₂²×C₃×C₅	#127	1	1	yes	characteristic
50	C₃×C₄×C₅	C₂×Q₁₆	1	2	yes	—
51	C₃×C₄×C₅	#135	1	4	yes	—
52	Dic₁₅	—	8	4	no	—
53	C₂×Dic₆	—	10	4	no	—
54	C₂×C₃×C₈	D₂₀	1	4	yes	—
55	C₂×C₃×C₈	D₂₀	1	2	yes	—
56	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
57	C₄⋊Dic₃	—	10	4	no	—
58	C₄⋊Dic₃	—	10	4	no	—
59	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
60	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
61	C₂×Dic₅	—	12	4	no	—
62	Dic₁₀	—	12	8	no	—
63	C₅×C₈	—	2	4	no	—
64	C₅×C₈	Dic₁₂	1	4	yes	—
65	C₄⋊Q₈	—	15	4	no	—
66	#107	—	30	4	no	—
67	#103	—	30	8	no	—
68	C₄×C₈	C₂×D₁₅	1	2	yes	—
69	C₄×C₈	C₂×D₁₅	1	1	yes	characteristic
70	C₂×C₃×C₅	#494	1	1	yes	characteristic
71	C₂×C₃×C₅	#495	1	2	yes	—
72	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
73	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
74	C₂×Dic₃	—	20	4	no	—
75	Dic₆	—	20	8	no	—
76	C₃×C₈	—	2	4	no	—
77	C₃×C₈	Dic₂₀	1	4	yes	—
78	C₂²×C₅	#874	1	1	yes	characteristic
79	C₄×C₅	#904	1	4	yes	—
80	C₄×C₅	C₂×Dic₁₂	1	2	yes	—
81	Dic₅	—	24	4	no	—
82	C₂×Q₈	—	30	4	no	—
83	C₂×C₈	D₆₀	1	4	yes	—
84	C₂×C₈	D₆₀	1	2	yes	—
85	C₄⋊C₄	—	30	4	no	—
86	C₄⋊C₄	—	30	4	no	—
87	C₄²	C₂²×D₁₅	1	1	yes	characteristic
88	C₃×C₅	#1710	1	1	yes	characteristic, complemented, Hall
89	C₂²×C₃	#4020	1	1	yes	characteristic
90	C₃×C₄	#4050	1	4	yes	—
91	C₃×C₄	C₂×Dic₂₀	1	2	yes	—
92	Dic₃	—	40	4	no	—
93	C₂×C₅	#4696	1	2	yes	—
94	C₂×C₅	#4699	1	1	yes	characteristic
95	Q₈	—	60	8	no	—
96	C₂×C₄	—	60	4	no	—
97	C₂×C₄	C₂×D₆₀	1	2	yes	—
98	C₂×C₄	C₂×D₆₀	1	1	yes	characteristic
99	C₈	—	2	4	no	—
100	C₈	Dic₆₀	1	4	yes	—
101	C₂×C₃	#64985	1	2	yes	—
102	C₂×C₃	#64988	1	1	yes	characteristic
103	C₅	#68064	1	1	yes	characteristic, complemented, Sylow
104	C₂²	#91763	1	1	yes	characteristic
105	C₄	—	120	4	no	—
106	C₄	#91793	1	4	yes	—
107	C₄	C₂×Dic₆₀	1	2	yes	—
108	C₃	#10598857	1	1	yes	characteristic, complemented, Sylow
109	C₂	#11750797	1	2	yes	—
110	C₂	#11750800	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11750783

C₂

yes

maximal

#11750782

C₂

yes

maximal

C₃×C₅×C₈²

C₂

yes

maximal, characteristic

#10598857

—

maximal, Hall

#91758

C₂²

yes

—

C₃×C₄×C₅×C₈

C₂²

yes

—

C₃×C₄×C₅×C₈

C₂²

yes

—

—

—

—

—

C₅×C₈²

D₃

yes

characteristic

C₂×Dic₃₀

—

C₂×C₃×C₅×C₈

D₄

yes

—

C₂×C₃×C₅×C₈

D₄

yes

—

C₃×C₄²×C₅

C₂³

yes

characteristic

C₄⋊Dic₁₅

—

C₄⋊Dic₁₅

—

#4682

—

#4681

—

C₃×C₈²

D₅

yes

characteristic

#4015

—

C₄×C₅×C₈

C₂×D₃

yes

—

C₄×C₅×C₈

C₂×D₃

yes

—

—

—

C₂×C₃×C₄×C₅

C₂×D₄

yes

characteristic

C₂×C₃×C₄×C₅

C₂×D₄

yes

—

C₂×Dic₁₅

—

Dic₃₀

—

C₃×C₅×C₈

—

C₃×C₅×C₈

Q₁₆

yes

—

#869

—

C₃×C₄×C₈

C₂×D₅

yes

—

C₃×C₄×C₈

C₂×D₅

yes

characteristic

#818

—

#816

—

C₂×Dic₁₀

—

C₂×C₅×C₈

D₁₂

yes

—

C₂×C₅×C₈

D₁₂

yes

—

C₄²×C₅

C₂²×D₃

yes

characteristic

C₄⋊Dic₅

—

C₄⋊Dic₅

—

#500

—

#487

—

C₈²

D₁₅

yes

characteristic

C₂²×C₃×C₅

#127

yes

characteristic

C₃×C₄×C₅

C₂×Q₁₆

yes

—

C₃×C₄×C₅

#135

yes

—

Dic₁₅

—

C₂×Dic₆

—

C₂×C₃×C₈

D₂₀

yes

—

C₂×C₃×C₈

D₂₀

yes

—

C₃×C₄²

C₂²×D₅

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₂×C₄×C₅

C₂×D₁₂

yes

—

C₂×C₄×C₅

C₂×D₁₂

yes

characteristic

C₂×Dic₅

—

Dic₁₀

—

C₅×C₈

—

C₅×C₈

Dic₁₂

yes

—

C₄⋊Q₈

—

#107

—

#103

—

C₄×C₈

C₂×D₁₅

yes

—

C₄×C₈

C₂×D₁₅

yes

C₂×C₃×C₅

yes

C₂×C₃×C₅

yes

—

C₂×C₃×C₄

C₂×D₂₀

yes

—

C₂×C₃×C₄

C₂×D₂₀

yes

characteristic

C₂×Dic₃

—

Dic₆

—

C₃×C₈

—

C₃×C₈

Dic₂₀

yes

—

C₂²×C₅

#874

yes

characteristic

C₄×C₅

#904

yes

—

C₄×C₅

C₂×Dic₁₂

yes

—

Dic₅

—

C₂×Q₈

—

C₂×C₈

D₆₀

yes

—

C₂×C₈

D₆₀

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×D₁₅

yes

characteristic

C₃×C₅

#1710

yes

characteristic, complemented, Hall

C₂²×C₃

#4020

yes

characteristic

C₃×C₄

#4050

yes

—

C₃×C₄

C₂×Dic₂₀

yes

—

Dic₃

—

C₂×C₅

#4696

yes

—

C₂×C₅

#4699

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

C₂×D₆₀

yes

—

C₂×C₄

C₂×D₆₀

yes

characteristic

C₈

—

100

C₈

Dic₆₀

yes

—

101

C₂×C₃

#64985

yes

—

102

C₂×C₃

#64988

yes

characteristic

103

C₅

#68064

yes

characteristic, complemented, Sylow

104

C₂²

#91763

yes

characteristic

105

C₄

—

120

—

106

C₄

#91793

yes

—

107

C₄

C₂×Dic₆₀

yes

—

108

C₃

#10598857

yes

characteristic, complemented, Sylow

109

C₂

#11750797

yes

—

110

C₂

#11750800

yes

characteristic

	subgroup	G/H
center	C₂²	#91763
commutator, cosocle	C₃×C₄²×C₅	C₂³
Frattini	C₄²	C₂²×D₁₅
Fitting	C₃×C₅×C₈²	C₂
socle, prime radical	C₂²×C₃×C₅	#127
hypercenter	C₈²	D₁₅
hypocenter	C₃×C₅	#1710
LCS	G ▷ C₃×C₄²×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

subgroup

G/H

center

C₂²

#91763

commutator, cosocle

C₃×C₄²×C₅

C₂³

Frattini

C₄²

C₂²×D₁₅

Fitting

C₃×C₅×C₈²

C₂

socle, prime radical

C₂²×C₃×C₅

#127

hypercenter

C₈²

D₁₅

hypocenter

C₃×C₅

#1710

LCS

G ▷ C₃×C₄²×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈²

N	H ≅ G/N	type
88. C₃×C₅	#1710	semidirect
103. C₅	#68064	semidirect
108. C₃	#10598857	semidirect

H ≅ G/N

type

88. C₃×C₅

#1710

semidirect

103. C₅

#68064

semidirect

108. C₃

#10598857

semidirect

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

subgroup	index
#11750783	2
#11750782	2
C₃×C₅×C₈²	2
#10598857	3
#68064	5

subgroup

index

C₃×C₅×C₈²

Subgroups — #14293584

Conjugacy classes of subgroups

Complements

Maximal subgroups