Group #12013990 — subgroups

Subgroups — #12013990

Order 1280.

262 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 99 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	#10599858	C₂	1	4	yes	maximal
2	#10598858	C₂	1	1	yes	maximal, characteristic
3	#10598858	C₂	1	1	yes	maximal, characteristic
4	C₅×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic
5	#64988	—	2	1	no	—
6	#64988	C₂²	1	2	yes	—
7	#64971	—	2	1	no	—
8	#64971	—	2	1	no	—
9	#64971	—	2	1	no	—
10	#64971	C₂²	1	2	yes	—
11	C₄×C₅×C₁₆	C₂²	1	2	yes	—
12	C₅×C₈²	C₂²	1	1	yes	characteristic
13	#64723	—	2	8	no	—
14	#12510	—	5	1	no	maximal, Sylow
15	C₂×Dic₂₀	—	2	4	no	—
16	C₂×Dic₂₀	—	4	2	no	—
17	#4015	—	2	2	no	—
18	#4015	—	2	2	no	—
19	C₂×C₅×C₁₆	D₄	1	4	yes	—
20	C₄×C₅×C₈	D₄	1	1	yes	characteristic
21	C₄×C₅×C₈	D₄	1	1	yes	characteristic
22	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
23	#3950	—	2	4	no	—
24	#3950	—	4	2	no	—
25	#3950	—	4	2	no	—
26	#3950	—	4	2	no	—
27	#2241	—	5	4	no	—
28	#1713	—	5	1	no	—
29	#1713	—	5	1	no	—
30	C₈×C₁₆	D₅	1	1	yes	characteristic
31	C₂×Dic₁₀	—	4	2	no	—
32	C₂×Dic₁₀	—	4	2	no	—
33	C₂×C₅×C₈	D₈	1	2	yes	—
34	C₂×C₅×C₈	D₈	1	2	yes	—
35	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
36	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
37	C₄⋊Dic₅	—	4	2	no	—
38	C₄⋊Dic₅	—	4	2	no	—
39	C₄⋊Dic₅	—	4	2	no	—
40	C₄⋊Dic₅	—	4	2	no	—
41	Dic₂₀	—	4	8	no	—
42	Dic₂₀	—	8	4	no	—
43	C₅×C₁₆	—	2	4	no	—
44	#500	—	5	2	no	—
45	#500	—	10	1	no	—
46	#500	—	10	1	no	—
47	#500	—	10	1	no	—
48	#494	—	5	2	no	—
49	#494	—	10	1	no	—
50	#358	—	10	8	no	—
51	C₄×C₁₆	C₂×D₅	1	2	yes	—
52	C₈²	C₂×D₅	1	1	yes	characteristic
53	C₂×C₄×C₅	#127	1	1	yes	characteristic
54	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
56	C₂×Dic₅	—	8	2	no	—
57	C₂×Dic₅	—	8	2	no	—
58	Dic₁₀	—	8	4	no	—
59	Dic₁₀	—	8	4	no	—
60	C₅×C₈	SD₃₂	1	4	yes	—
61	C₅×C₈	Q₃₂	1	4	yes	—
62	C₅×C₈	#135	1	4	yes	—
63	C₂×Q₁₆	—	10	4	no	—
64	C₂×Q₁₆	—	20	2	no	—
65	C₄⋊Q₈	—	10	2	no	—
66	C₄⋊Q₈	—	10	2	no	—
67	C₂×C₁₆	D₂₀	1	4	yes	—
68	#107	—	10	4	no	—
69	#107	—	20	2	no	—
70	#107	—	20	2	no	—
71	#107	—	20	2	no	—
72	C₄×C₈	D₂₀	1	1	yes	characteristic
73	C₄×C₈	D₂₀	1	1	yes	characteristic
74	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
75	C₂²×C₅	#493	1	1	yes	characteristic
76	C₄×C₅	#495	1	2	yes	—
77	C₄×C₅	C₂×SD₃₂	1	2	yes	—
78	C₄×C₅	C₂×Q₃₂	1	2	yes	—
79	Dic₅	—	16	2	no	—
80	Dic₅	—	16	2	no	—
81	C₂×Q₈	—	20	2	no	—
82	C₂×Q₈	—	20	2	no	—
83	Q₁₆	—	20	8	no	—
84	Q₁₆	—	40	4	no	—
85	C₂×C₈	D₄₀	1	2	yes	—
86	C₂×C₈	D₄₀	1	2	yes	—
87	C₂×C₈	C₂×D₂₀	1	2	yes	—
88	C₄⋊C₄	—	20	2	no	—
89	C₄⋊C₄	—	20	2	no	—
90	C₄⋊C₄	—	20	2	no	—
91	C₄⋊C₄	—	20	2	no	—
92	C₄²	C₂×D₂₀	1	1	yes	characteristic
93	C₁₆	—	2	4	no	—
94	C₂×C₅	#1709	1	1	yes	characteristic
95	C₂×C₅	#2247	1	1	yes	characteristic
96	C₂×C₅	#2248	1	1	yes	characteristic
97	Q₈	—	40	4	no	—
98	Q₈	—	40	4	no	—
99	C₂×C₄	—	40	2	no	—
100	C₂×C₄	—	40	2	no	—
101	C₂×C₄	#4020	1	1	yes	characteristic
102	C₂×C₄	C₂×D₄₀	1	1	yes	characteristic
103	C₂×C₄	C₂×D₄₀	1	1	yes	characteristic
104	C₈	#3932	1	4	yes	—
105	C₈	Dic₄₀	1	4	yes	—
106	C₈	#4050	1	4	yes	—
107	C₅	#12510	1	1	yes	characteristic, complemented, Sylow
108	C₂²	#64984	1	1	yes	characteristic
109	C₄	—	80	2	no	—
110	C₄	—	80	2	no	—
111	C₄	#64985	1	2	yes	—
112	C₄	C₂×#3932	1	2	yes	—
113	C₄	C₂×Dic₄₀	1	2	yes	—
114	C₂	#10598853	1	1	yes	characteristic
115	C₂	#10599854	1	1	yes	characteristic
116	C₂	#10599859	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10599858

C₂

yes

maximal

#10598858

C₂

yes

maximal, characteristic

#10598858

C₂

yes

maximal, characteristic

C₅×C₈×C₁₆

C₂

yes

maximal, characteristic

#64988

—

#64988

C₂²

yes

—

—

—

—

C₂²

yes

—

C₄×C₅×C₁₆

C₂²

yes

—

C₅×C₈²

C₂²

yes

—

—

C₂×Dic₂₀

—

C₂×Dic₂₀

—

#4015

—

#4015

—

C₂×C₅×C₁₆

D₄

yes

—

C₄×C₅×C₈

D₄

yes

characteristic

C₄×C₅×C₈

D₄

yes

characteristic

C₄×C₅×C₈

C₂³

yes

—

—

—

—

—

—

—

C₈×C₁₆

D₅

yes

characteristic

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×C₅×C₈

D₈

yes

—

C₂×C₅×C₈

D₈

yes

—

C₂×C₅×C₈

C₂×D₄

yes

—

C₄²×C₅

C₂×D₄

yes

characteristic

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

Dic₂₀

—

Dic₂₀

—

C₅×C₁₆

—

—

—

—

—

—

—

—

C₄×C₁₆

C₂×D₅

yes

—

C₈²

C₂×D₅

yes

characteristic

C₂×C₄×C₅

#127

yes

characteristic

C₂×C₄×C₅

C₂×D₈

yes

characteristic

C₂×C₄×C₅

C₂×D₈

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

SD₃₂

yes

—

C₅×C₈

Q₃₂

yes

—

C₅×C₈

#135

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₁₆

D₂₀

yes

—

—

—

—

—

C₄×C₈

D₂₀

yes

characteristic

C₄×C₈

D₂₀

yes

characteristic

C₄×C₈

C₂²×D₅

yes

C₂²×C₅

yes

C₄×C₅

yes

—

C₄×C₅

C₂×SD₃₂

yes

—

C₄×C₅

C₂×Q₃₂

yes

—

Dic₅

—

Dic₅

—

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

D₄₀

yes

—

C₂×C₈

D₄₀

yes

—

C₂×C₈

C₂×D₂₀

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×D₂₀

yes

characteristic

C₁₆

—

C₂×C₅

yes

C₂×C₅

yes

C₂×C₅

yes

Q₈

—

Q₈

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

#4020

yes

characteristic

102

C₂×C₄

C₂×D₄₀

yes

characteristic

103

C₂×C₄

C₂×D₄₀

yes

characteristic

104

C₈

#3932

yes

—

105

C₈

Dic₄₀

yes

—

106

C₈

#4050

yes

—

107

C₅

#12510

yes

characteristic, complemented, Sylow

108

C₂²

#64984

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

#64985

yes

—

112

C₄

C₂×#3932

yes

—

113

C₄

C₂×Dic₄₀

yes

—

114

C₂

#10598853

yes

characteristic

115

C₂

#10599854

yes

characteristic

116

C₂

#10599859

yes

characteristic

	subgroup	G/H
center	C₂²	#64984
commutator, cosocle	C₄×C₅×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×C₈×C₁₆	C₂
socle, prime radical	C₂²×C₅	#493
hypercenter	C₈×C₁₆	D₅
hypocenter	C₅	#12510
LCS	G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

subgroup

G/H

center

C₂²

#64984

commutator, cosocle

C₄×C₅×C₈

C₂³

Frattini

C₄×C₈

C₂²×D₅

Fitting

C₅×C₈×C₁₆

C₂

socle, prime radical

C₂²×C₅

#493

hypercenter

C₈×C₁₆

D₅

hypocenter

C₅

#12510

LCS

G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
107. C₅	#12510	semidirect

H ≅ G/N

type

107. C₅

#12510

semidirect

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

subgroup	index
#10599858	2
#10598858	2
C₅×C₈×C₁₆	2
#12510	5

subgroup

index

#10599858

#10598858

C₅×C₈×C₁₆

#12510

Subgroups — #12013990

Conjugacy classes of subgroups

Complements

Maximal subgroups