Group #14293602 — subgroups

Subgroups — #14293602

Order 1920.

356 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 111 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	#11750795	C₂	1	3	yes	maximal
2	#11750782	C₂	1	3	yes	maximal, complemented
3	C₃×C₅×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
4	#10598854	—	3	1	no	maximal, Hall
5	#91763	C₂²	1	1	yes	characteristic
6	#91758	C₂²	1	3	yes	—
7	C₃×C₄×C₅×C₈	C₂²	1	3	yes	—
8	#91105	—	2	6	no	—
9	#91100	—	2	6	no	—
10	#68061	—	5	1	no	maximal, Hall
11	#64983	—	3	3	no	—
12	#64970	—	3	3	no	—
13	C₅×C₈²	D₃	1	1	yes	characteristic, complemented
14	C₂×D₆₀	—	2	3	no	—
15	C₂×Dic₃₀	—	2	3	no	—
16	C₂×C₃×C₅×C₈	D₄	1	6	yes	—
17	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
18	C₄⋊Dic₁₅	—	2	3	no	—
19	C₄⋊Dic₁₅	—	2	3	no	—
20	#4694	—	5	3	no	—
21	#4681	—	5	3	no	—
22	C₃×C₈²	D₅	1	1	yes	characteristic, complemented
23	#4020	—	3	1	no	—
24	#4015	—	3	3	no	—
25	C₄×C₅×C₈	C₂×D₃	1	3	yes	—
26	#3953	—	6	6	no	—
27	#3948	—	6	6	no	—
28	#1711	—	15	1	no	Sylow
29	C₂²×D₁₅	—	4	1	no	—
30	C₂×C₃×C₄×C₅	C₂×D₄	1	3	yes	—
31	C₂×Dic₁₅	—	4	3	no	—
32	D₆₀	—	4	6	no	—
33	Dic₃₀	—	4	6	no	—
34	C₃×C₅×C₈	—	2	6	no	—
35	#874	—	5	1	no	—
36	#869	—	5	3	no	—
37	C₃×C₄×C₈	C₂×D₅	1	3	yes	—
38	#821	—	10	6	no	—
39	#816	—	10	6	no	—
40	C₂×D₂₀	—	6	3	no	—
41	C₂×Dic₁₀	—	6	3	no	—
42	C₂×C₅×C₈	D₁₂	1	6	yes	—
43	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
44	C₄⋊Dic₅	—	6	3	no	—
45	C₄⋊Dic₅	—	6	3	no	—
46	#487	—	15	3	no	—
47	#486	—	15	3	no	—
48	C₈²	D₁₅	1	1	yes	characteristic, complemented
49	C₂²×C₃×C₅	#127	1	1	yes	characteristic
50	C₂×D₁₅	—	8	3	no	—
51	C₃×C₄×C₅	#135	1	6	yes	—
52	Dic₁₅	—	8	3	no	—
53	C₂×D₁₂	—	10	3	no	—
54	C₂×Dic₆	—	10	3	no	—
55	C₂×C₃×C₈	D₂₀	1	6	yes	—
56	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
57	C₄⋊Dic₃	—	10	3	no	—
58	C₄⋊Dic₃	—	10	3	no	—
59	C₂²×D₅	—	12	1	no	—
60	C₂×C₄×C₅	C₂×D₁₂	1	3	yes	—
61	C₂×Dic₅	—	12	3	no	—
62	D₂₀	—	12	6	no	—
63	Dic₁₀	—	12	6	no	—
64	C₅×C₈	—	2	6	no	—
65	C₄⋊Q₈	—	15	3	no	—
66	#127	—	15	1	no	—
67	#103	—	30	6	no	—
68	#102	—	30	6	no	—
69	C₄×C₈	C₂×D₁₅	1	3	yes	—
70	C₂×C₃×C₅	#495	1	3	yes	—
71	D₁₅	—	16	1	no	—
72	C₂²×D₃	—	20	1	no	—
73	C₂×C₃×C₄	C₂×D₂₀	1	3	yes	—
74	C₂×Dic₃	—	20	3	no	—
75	D₁₂	—	20	6	no	—
76	Dic₆	—	20	6	no	—
77	C₃×C₈	—	2	6	no	—
78	C₂²×C₅	#874	1	1	yes	characteristic
79	C₂×D₅	—	24	3	no	—
80	C₄×C₅	#904	1	6	yes	—
81	Dic₅	—	24	3	no	—
82	C₂×Q₈	—	30	3	no	—
83	C₂×D₄	—	30	3	no	—
84	C₂×C₈	D₆₀	1	6	yes	—
85	C₄⋊C₄	—	30	3	no	—
86	C₄⋊C₄	—	30	3	no	—
87	C₄²	C₂²×D₁₅	1	1	yes	characteristic
88	C₃×C₅	#1711	1	1	yes	characteristic, complemented, Hall
89	C₂²×C₃	#4020	1	1	yes	characteristic
90	C₂×D₃	—	40	3	no	—
91	C₃×C₄	#4050	1	6	yes	—
92	Dic₃	—	40	3	no	—
93	C₂×C₅	#4696	1	3	yes	—
94	D₅	—	48	1	no	—
95	C₂³	—	60	1	no	—
96	Q₈	—	60	6	no	—
97	D₄	—	60	6	no	—
98	C₂×C₄	—	60	3	no	—
99	C₂×C₄	C₂×D₆₀	1	3	yes	—
100	C₈	—	2	6	no	—
101	C₂×C₃	#64985	1	3	yes	—
102	D₃	—	80	1	no	—
103	C₅	#68061	1	1	yes	characteristic, complemented, Sylow
104	C₂²	—	120	3	no	—
105	C₂²	#91763	1	1	yes	characteristic
106	C₄	—	120	3	no	—
107	C₄	#91793	1	6	yes	—
108	C₃	#10598854	1	1	yes	characteristic, complemented, Sylow
109	C₂	—	240	1	no	—
110	C₂	#11750797	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11750795

C₂

yes

maximal

#11750782

C₂

yes

maximal, complemented

C₃×C₅×C₈²

C₂

yes

maximal, characteristic, complemented

#10598854

—

maximal, Hall

#91763

C₂²

yes

characteristic

#91758

C₂²

yes

—

C₃×C₄×C₅×C₈

C₂²

yes

—

—

—

—

—

—

C₅×C₈²

D₃

yes

characteristic, complemented

C₂×D₆₀

—

C₂×Dic₃₀

—

C₂×C₃×C₅×C₈

D₄

yes

—

C₃×C₄²×C₅

C₂³

yes

characteristic

C₄⋊Dic₁₅

—

C₄⋊Dic₁₅

—

#4694

—

#4681

—

C₃×C₈²

D₅

yes

characteristic, complemented

#4020

—

#4015

—

C₄×C₅×C₈

C₂×D₃

yes

—

—

—

—

C₂²×D₁₅

—

C₂×C₃×C₄×C₅

C₂×D₄

yes

—

C₂×Dic₁₅

—

D₆₀

—

Dic₃₀

—

C₃×C₅×C₈

—

#874

—

#869

—

C₃×C₄×C₈

C₂×D₅

yes

—

#821

—

#816

—

C₂×D₂₀

—

C₂×Dic₁₀

—

C₂×C₅×C₈

D₁₂

yes

—

C₄²×C₅

C₂²×D₃

yes

characteristic

C₄⋊Dic₅

—

C₄⋊Dic₅

—

#487

—

#486

—

C₈²

D₁₅

yes

characteristic, complemented

C₂²×C₃×C₅

#127

yes

characteristic

C₂×D₁₅

—

C₃×C₄×C₅

#135

yes

—

Dic₁₅

—

C₂×D₁₂

—

C₂×Dic₆

—

C₂×C₃×C₈

D₂₀

yes

—

C₃×C₄²

C₂²×D₅

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₂²×D₅

—

C₂×C₄×C₅

C₂×D₁₂

yes

—

C₂×Dic₅

—

D₂₀

—

Dic₁₀

—

C₅×C₈

—

C₄⋊Q₈

—

#127

—

#103

—

#102

—

C₄×C₈

C₂×D₁₅

yes

—

C₂×C₃×C₅

#495

yes

—

D₁₅

—

C₂²×D₃

—

C₂×C₃×C₄

C₂×D₂₀

yes

—

C₂×Dic₃

—

D₁₂

—

Dic₆

—

C₃×C₈

—

C₂²×C₅

#874

yes

characteristic

C₂×D₅

—

C₄×C₅

#904

yes

—

Dic₅

—

C₂×Q₈

—

C₂×D₄

—

C₂×C₈

D₆₀

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×D₁₅

yes

characteristic

C₃×C₅

#1711

yes

characteristic, complemented, Hall

C₂²×C₃

#4020

yes

characteristic

C₂×D₃

—

C₃×C₄

#4050

yes

—

Dic₃

—

C₂×C₅

#4696

yes

—

D₅

—

C₂³

—

Q₈

—

D₄

—

C₂×C₄

—

C₂×C₄

C₂×D₆₀

yes

—

100

C₈

—

101

C₂×C₃

#64985

yes

—

102

D₃

—

103

C₅

#68061

yes

characteristic, complemented, Sylow

104

C₂²

—

120

—

105

C₂²

#91763

yes

characteristic

106

C₄

—

120

—

107

C₄

#91793

yes

—

108

C₃

#10598854

yes

characteristic, complemented, Sylow

109

C₂

—

240

—

110

C₂

#11750797

yes

—

	subgroup	G/H
center	C₂²	#91763
commutator, cosocle	C₃×C₄²×C₅	C₂³
Frattini	C₄²	C₂²×D₁₅
Fitting	C₃×C₅×C₈²	C₂
socle	C₂²×C₃×C₅	#127
prime radical	#91763	C₂²
hypercenter	C₈²	D₁₅
hypocenter	C₃×C₅	#1711
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

C₂²×C₃×C₅

C₂²

C₈²

D₁₅

hypocenter

C₃×C₅

#1711

LCS

G ▷ C₃×C₄²×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈²

N	H ≅ G/N	type
2. #11750782	C₂	semidirect
3. C₃×C₅×C₈²	C₂	semidirect
13. C₅×C₈²	D₃	semidirect
22. C₃×C₈²	D₅	semidirect
48. C₈²	D₁₅	semidirect
88. C₃×C₅	#1711	semidirect
103. C₅	#68061	semidirect
108. C₃	#10598854	semidirect

H ≅ G/N

type

2. #11750782

C₂

semidirect

3. C₃×C₅×C₈²

C₂

semidirect

13. C₅×C₈²

D₃

semidirect

22. C₃×C₈²

D₅

semidirect

48. C₈²

D₁₅

semidirect

88. C₃×C₅

#1711

semidirect

103. C₅

#68061

semidirect

108. C₃

#10598854

semidirect

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

subgroup	index
#11750795	2
#11750782	2
C₃×C₅×C₈²	2
#10598854	3
#68061	5

subgroup

index

C₃×C₅×C₈²

Subgroups — #14293602

Conjugacy classes of subgroups

Complements

Maximal subgroups