Group #12013986 — subgroups

Subgroups — #12013986

Order 1280.

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

normal subgroups only

This group has 306 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	#10599851	C₂	1	4	yes	maximal, complemented
2	#10598858	C₂	1	1	yes	maximal, characteristic, complemented
3	#10598855	C₂	1	1	yes	maximal, characteristic
4	C₅×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#64988	—	2	1	no	—
6	#64984	—	2	1	no	—
7	#64984	—	2	1	no	—
8	#64984	C₂²	1	2	yes	—
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	#64729	—	2	8	no	—
14	#12509	—	5	1	no	maximal, Sylow
15	C₂×Dic₂₀	—	4	2	no	—
16	C₂×D₄₀	—	2	4	no	—
17	C₂×D₄₀	—	4	2	no	—
18	C₂×D₄₀	—	4	2	no	—
19	#4020	—	2	2	no	—
20	#4015	—	2	2	no	—
21	C₂×C₅×C₁₆	D₄	1	4	yes	—
22	C₄×C₅×C₈	D₄	1	1	yes	characteristic
23	C₄×C₅×C₈	D₄	1	1	yes	characteristic
24	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
25	#3950	—	2	4	no	—
26	#3950	—	4	2	no	—
27	#2240	—	5	4	no	—
28	#1713	—	5	1	no	—
29	#1712	—	5	1	no	—
30	C₈×C₁₆	D₅	1	1	yes	characteristic, complemented
31	C₂×D₂₀	—	4	2	no	—
32	C₂×D₂₀	—	4	2	no	—
33	C₂×D₂₀	—	4	2	no	—
34	C₂×Dic₁₀	—	4	2	no	—
35	C₂×C₅×C₈	D₈	1	2	yes	—
36	C₂×C₅×C₈	D₈	1	2	yes	—
37	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
38	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
39	C₄⋊Dic₅	—	4	2	no	—
40	C₄⋊Dic₅	—	4	2	no	—
41	Dic₂₀	—	8	4	no	—
42	D₄₀	—	4	8	no	—
43	D₄₀	—	8	4	no	—
44	D₄₀	—	8	4	no	—
45	C₅×C₁₆	—	2	4	no	—
46	#500	—	5	2	no	—
47	#500	—	10	1	no	—
48	#494	—	10	1	no	—
49	#493	—	5	2	no	—
50	#493	—	10	1	no	—
51	#493	—	10	1	no	—
52	#357	—	10	8	no	—
53	C₄×C₁₆	C₂×D₅	1	2	yes	—
54	C₈²	C₂×D₅	1	1	yes	characteristic
55	C₂²×D₅	—	8	2	no	—
56	C₂×C₄×C₅	#127	1	1	yes	characteristic
57	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
59	C₂×Dic₅	—	8	2	no	—
60	D₂₀	—	8	4	no	—
61	D₂₀	—	8	4	no	—
62	D₂₀	—	8	4	no	—
63	Dic₁₀	—	8	4	no	—
64	C₅×C₈	D₁₆	1	4	yes	—
65	C₅×C₈	SD₃₂	1	4	yes	—
66	C₅×C₈	#135	1	4	yes	—
67	C₂×Q₁₆	—	20	2	no	—
68	C₂×D₈	—	10	4	no	—
69	C₂×D₈	—	20	2	no	—
70	C₂×D₈	—	20	2	no	—
71	C₄⋊Q₈	—	10	2	no	—
72	#127	—	10	2	no	—
73	C₂×C₁₆	D₂₀	1	4	yes	—
74	#107	—	10	4	no	—
75	#107	—	20	2	no	—
76	C₄×C₈	D₂₀	1	1	yes	characteristic
77	C₄×C₈	D₂₀	1	1	yes	characteristic
78	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
79	C₂²×C₅	#493	1	1	yes	characteristic
80	C₂×D₅	—	16	2	no	—
81	C₂×D₅	—	16	2	no	—
82	C₂×D₅	—	16	2	no	—
83	C₄×C₅	#495	1	2	yes	—
84	C₄×C₅	C₂×D₁₆	1	2	yes	—
85	C₄×C₅	C₂×SD₃₂	1	2	yes	—
86	Dic₅	—	16	2	no	—
87	C₂×Q₈	—	20	2	no	—
88	C₂×D₄	—	20	2	no	—
89	C₂×D₄	—	20	2	no	—
90	C₂×D₄	—	20	2	no	—
91	Q₁₆	—	40	4	no	—
92	D₈	—	20	8	no	—
93	D₈	—	40	4	no	—
94	D₈	—	40	4	no	—
95	C₂×C₈	D₄₀	1	2	yes	—
96	C₂×C₈	D₄₀	1	2	yes	—
97	C₂×C₈	C₂×D₂₀	1	2	yes	—
98	C₄⋊C₄	—	20	2	no	—
99	C₄⋊C₄	—	20	2	no	—
100	C₄²	C₂×D₂₀	1	1	yes	characteristic
101	C₁₆	—	2	4	no	—
102	C₂×C₅	#1709	1	1	yes	characteristic
103	C₂×C₅	#2246	1	1	yes	characteristic
104	C₂×C₅	#2248	1	1	yes	characteristic
105	D₅	—	32	2	no	—
106	C₂³	—	40	2	no	—
107	Q₈	—	40	4	no	—
108	D₄	—	40	4	no	—
109	D₄	—	40	4	no	—
110	D₄	—	40	4	no	—
111	C₂×C₄	—	40	2	no	—
112	C₂×C₄	#4020	1	1	yes	characteristic
113	C₂×C₄	C₂×D₄₀	1	1	yes	characteristic
114	C₂×C₄	C₂×D₄₀	1	1	yes	characteristic
115	C₈	D₈₀	1	4	yes	—
116	C₈	#3932	1	4	yes	—
117	C₈	#4050	1	4	yes	—
118	C₅	#12509	1	1	yes	characteristic, complemented, Sylow
119	C₂²	—	80	2	no	—
120	C₂²	—	80	2	no	—
121	C₂²	—	80	2	no	—
122	C₂²	#64984	1	1	yes	characteristic
123	C₄	—	80	2	no	—
124	C₄	#64985	1	2	yes	—
125	C₄	C₂×D₈₀	1	2	yes	—
126	C₄	C₂×#3932	1	2	yes	—
127	C₂	—	160	2	no	—
128	C₂	#10598853	1	1	yes	characteristic
129	C₂	#10599853	1	1	yes	characteristic
130	C₂	#10599854	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10599851

C₂

yes

maximal, complemented

#10598858

C₂

yes

maximal, characteristic, complemented

#10598855

C₂

yes

maximal, characteristic

C₅×C₈×C₁₆

C₂

yes

maximal, characteristic, complemented

—

—

—

C₂²

yes

—

#64971

—

#64971

C₂²

yes

—

C₄×C₅×C₁₆

C₂²

yes

—

C₅×C₈²

C₂²

yes

—

—

C₂×Dic₂₀

—

C₂×D₄₀

—

C₂×D₄₀

—

C₂×D₄₀

—

#4020

—

#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, complemented

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

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

—

Dic₂₀

—

D₄₀

—

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

characteristic

C₂×C₄×C₅

C₂×D₈

yes

characteristic

C₂×Dic₅

—

D₂₀

—

D₂₀

—

D₂₀

—

Dic₁₀

—

C₅×C₈

D₁₆

yes

—

C₅×C₈

SD₃₂

yes

—

C₅×C₈

#135

yes

—

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

D₂₀

yes

—

#107

—

#107

—

C₄×C₈

D₂₀

yes

characteristic

C₄×C₈

D₂₀

yes

characteristic

C₄×C₈

C₂²×D₅

yes

characteristic

C₂²×C₅

#493

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

#495

yes

—

C₄×C₅

C₂×D₁₆

yes

—

C₄×C₅

C₂×SD₃₂

yes

—

Dic₅

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

D₈

—

D₈

—

D₈

—

C₂×C₈

D₄₀

yes

—

C₂×C₈

D₄₀

yes

—

C₂×C₈

C₂×D₂₀

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄²

C₂×D₂₀

yes

characteristic

101

C₁₆

—

102

C₂×C₅

#1709

yes

characteristic

103

C₂×C₅

#2246

yes

characteristic

104

C₂×C₅

#2248

yes

characteristic

105

D₅

—

106

C₂³

—

107

Q₈

—

108

D₄

—

109

D₄

—

110

D₄

—

111

C₂×C₄

—

112

C₂×C₄

#4020

yes

characteristic

113

C₂×C₄

C₂×D₄₀

yes

characteristic

114

C₂×C₄

C₂×D₄₀

yes

characteristic

115

C₈

D₈₀

yes

—

116

C₈

#3932

yes

—

117

C₈

#4050

yes

—

118

C₅

#12509

yes

characteristic, complemented, Sylow

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

#64984

yes

characteristic

123

C₄

—

124

C₄

#64985

yes

—

125

C₄

C₂×D₈₀

yes

—

126

C₄

C₂×#3932

yes

—

127

C₂

—

160

—

128

C₂

#10598853

yes

characteristic

129

C₂

#10599853

yes

characteristic

130

C₂

#10599854

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	C₂²×C₅	#493
prime radical	#10598855	C₂
hypercenter	C₈×C₁₆	D₅
hypocenter	C₅	#12509
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

C₂²×C₅

C₂

C₈×C₁₆

D₅

hypocenter

C₅

#12509

LCS

G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
1. #10599851	C₂	semidirect
2. #10598858	C₂	semidirect
4. C₅×C₈×C₁₆	C₂	semidirect
30. C₈×C₁₆	D₅	semidirect
118. C₅	#12509	semidirect

H ≅ G/N

type

1. #10599851

C₂

semidirect

2. #10598858

C₂

semidirect

4. C₅×C₈×C₁₆

C₂

semidirect

30. C₈×C₁₆

D₅

semidirect

118. C₅

#12509

semidirect

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

subgroup	index
#10599851	2
#10598858	2
#10598855	2
C₅×C₈×C₁₆	2
#12509	5

subgroup

index

C₅×C₈×C₁₆

Subgroups — #12013986

Conjugacy classes of subgroups

Complements

Maximal subgroups