Group #12014004 — subgroups

Subgroups — #12014004

Order 1280.

290 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 83 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	#10599851	C₂	1	4	yes	maximal, complemented
2	#10598857	C₂	1	1	yes	maximal, characteristic, complemented
3	#10598855	C₂	1	1	yes	maximal, characteristic
4	C₅×#1311	C₂	1	1	yes	maximal, characteristic, complemented
5	#64984	—	2	2	no	—
6	#64984	C₂²	1	2	yes	—
7	#64971	C₂²	1	2	yes	—
8	#64970	—	2	2	no	—
9	C₄×C₅×C₁₆	C₂²	1	2	yes	—
10	C₅×C₈²	C₂²	1	1	yes	characteristic
11	#64729	—	2	8	no	—
12	#12513	—	5	1	no	maximal, Sylow
13	C₂×D₄₀	—	2	4	no	—
14	C₂×D₄₀	—	4	4	no	—
15	#4020	—	2	2	no	—
16	#4015	—	2	2	no	—
17	C₂×C₅×C₁₆	D₄	1	4	yes	—
18	C₄×C₅×C₈	D₄	1	2	yes	—
19	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
20	#3950	—	2	4	no	—
21	#3948	—	4	4	no	—
22	#2240	—	5	4	no	—
23	#1712	—	5	1	no	—
24	#1710	—	5	1	no	—
25	#1311	D₅	1	1	yes	characteristic, complemented
26	C₂×D₂₀	—	4	2	no	—
27	C₂×D₂₀	—	4	4	no	—
28	C₂×Dic₁₀	—	4	2	no	—
29	C₂×C₅×C₈	D₈	1	4	yes	—
30	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
31	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
32	C₄⋊Dic₅	—	4	2	no	—
33	C₄⋊Dic₅	—	4	2	no	—
34	D₄₀	—	4	8	no	—
35	D₄₀	—	8	8	no	—
36	C₅×C₁₆	—	2	4	no	—
37	#500	—	5	2	no	—
38	#493	—	5	2	no	—
39	#493	—	10	2	no	—
40	#487	—	10	2	no	—
41	#357	—	10	8	no	—
42	C₄×C₁₆	C₂×D₅	1	2	yes	—
43	C₈²	C₂×D₅	1	1	yes	characteristic
44	C₂²×D₅	—	8	2	no	—
45	C₂×C₄×C₅	#127	1	1	yes	characteristic
46	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
47	C₂×Dic₅	—	8	2	no	—
48	D₂₀	—	8	4	no	—
49	D₂₀	—	8	8	no	—
50	Dic₁₀	—	8	4	no	—
51	C₅×C₈	—	2	4	no	—
52	C₅×C₈	#135	1	4	yes	—
53	C₂×D₈	—	10	4	no	—
54	C₂×D₈	—	20	4	no	—
55	C₄⋊Q₈	—	10	2	no	—
56	#127	—	10	2	no	—
57	C₂×C₁₆	D₂₀	1	4	yes	—
58	#107	—	10	4	no	—
59	#103	—	20	4	no	—
60	C₄×C₈	D₂₀	1	2	yes	—
61	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
62	C₂²×C₅	#493	1	1	yes	characteristic
63	C₂×D₅	—	16	2	no	—
64	C₂×D₅	—	16	4	no	—
65	C₄×C₅	#495	1	2	yes	—
66	C₄×C₅	#509	1	4	yes	—
67	Dic₅	—	16	2	no	—
68	C₂×Q₈	—	20	2	no	—
69	C₂×D₄	—	20	2	no	—
70	C₂×D₄	—	20	4	no	—
71	D₈	—	20	8	no	—
72	D₈	—	40	8	no	—
73	C₂×C₈	D₄₀	1	4	yes	—
74	C₂×C₈	C₂×D₂₀	1	2	yes	—
75	C₄⋊C₄	—	20	2	no	—
76	C₄⋊C₄	—	20	2	no	—
77	C₄²	C₂×D₂₀	1	1	yes	characteristic
78	C₁₆	—	2	4	no	—
79	C₂×C₅	#1709	1	1	yes	characteristic
80	C₂×C₅	#2250	1	2	yes	—
81	D₅	—	32	2	no	—
82	C₂³	—	40	2	no	—
83	Q₈	—	40	4	no	—
84	D₄	—	40	8	no	—
85	D₄	—	40	4	no	—
86	C₂×C₄	—	40	2	no	—
87	C₂×C₄	#4020	1	1	yes	characteristic
88	C₂×C₄	C₂×D₄₀	1	2	yes	—
89	C₈	—	2	4	no	—
90	C₈	#4050	1	4	yes	—
91	C₅	#12513	1	1	yes	characteristic, complemented, Sylow
92	C₂²	—	80	4	no	—
93	C₂²	—	80	2	no	—
94	C₂²	#64984	1	1	yes	characteristic
95	C₄	—	80	2	no	—
96	C₄	#64985	1	2	yes	—
97	C₄	#65197	1	4	yes	—
98	C₂	—	160	2	no	—
99	C₂	#10598853	1	1	yes	characteristic
100	C₂	#10599874	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10599851

C₂

yes

maximal, complemented

#10598857

C₂

yes

maximal, characteristic, complemented

#10598855

C₂

yes

maximal, characteristic

C₅×#1311

C₂

yes

maximal, characteristic, complemented

#64984

—

#64984

C₂²

yes

—

#64971

C₂²

yes

—

#64970

—

C₄×C₅×C₁₆

C₂²

yes

—

C₅×C₈²

C₂²

yes

—

—

C₂×D₄₀

—

C₂×D₄₀

—

#4020

—

#4015

—

C₂×C₅×C₁₆

D₄

yes

—

C₄×C₅×C₈

D₄

yes

—

C₄×C₅×C₈

C₂³

yes

—

—

—

—

—

D₅

yes

characteristic, complemented

C₂×D₂₀

—

C₂×D₂₀

—

C₂×Dic₁₀

—

C₂×C₅×C₈

D₈

yes

—

C₂×C₅×C₈

C₂×D₄

yes

—

C₄²×C₅

C₂×D₄

yes

characteristic

C₄⋊Dic₅

—

C₄⋊Dic₅

—

D₄₀

—

D₄₀

—

C₅×C₁₆

—

—

—

—

—

—

C₄×C₁₆

C₂×D₅

yes

—

C₈²

C₂×D₅

yes

characteristic

C₂²×D₅

—

C₂×C₄×C₅

#127

yes

characteristic

C₂×C₄×C₅

C₂×D₈

yes

—

C₂×Dic₅

—

D₂₀

—

D₂₀

—

Dic₁₀

—

C₅×C₈

—

C₅×C₈

#135

yes

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

D₂₀

yes

—

#107

—

#103

—

C₄×C₈

D₂₀

yes

—

C₄×C₈

C₂²×D₅

yes

characteristic

C₂²×C₅

#493

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₄×C₅

#495

yes

—

C₄×C₅

#509

yes

—

Dic₅

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

D₈

—

D₈

—

C₂×C₈

D₄₀

yes

—

C₂×C₈

C₂×D₂₀

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×D₂₀

yes

characteristic

C₁₆

—

C₂×C₅

#1709

yes

characteristic

C₂×C₅

#2250

yes

—

D₅

—

C₂³

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

#4020

yes

characteristic

C₂×C₄

C₂×D₄₀

yes

—

C₈

—

C₈

#4050

yes

—

C₅

#12513

yes

characteristic, complemented, Sylow

C₂²

—

C₂²

—

C₂²

#64984

yes

characteristic

C₄

—

C₄

#64985

yes

—

C₄

#65197

yes

—

C₂

—

160

—

C₂

#10598853

yes

characteristic

100

C₂

#10599874

yes

—

	subgroup	G/H
center	C₂²	#64984
commutator, cosocle	C₄×C₅×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×#1311	C₂
socle	C₂²×C₅	#493
prime radical	#10598855	C₂
hypercenter	#1311	D₅
hypocenter	C₅	#12513
LCS	G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂²

#64984

commutator, cosocle

C₄×C₅×C₈

C₂³

Frattini

C₄×C₈

C₂²×D₅

C₂

C₂²×C₅

C₂

D₅

C₅

G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

N	H ≅ G/N	type
1. #10599851	C₂	semidirect
2. #10598857	C₂	semidirect
4. C₅×#1311	C₂	semidirect
25. #1311	D₅	semidirect
91. C₅	#12513	semidirect

H ≅ G/N

type

1. #10599851

C₂

semidirect

2. #10598857

C₂

semidirect

4. C₅×#1311

C₂

semidirect

25. #1311

D₅

semidirect

91. C₅

#12513

semidirect

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

subgroup	index
#10599851	2
#10598857	2
#10598855	2
C₅×#1311	2
#12513	5

subgroup

index

Subgroups — #12014004

Conjugacy classes of subgroups

Complements

Maximal subgroups