Group #13103043 — subgroups

Subgroups — #13103043

Order 1600.

488 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 131 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	C₂×C₅²×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	#11717264	C₂	1	2	yes	maximal
3	C₂×#87812	C₂	1	2	yes	maximal
4	C₂×#87802	C₂	1	1	yes	maximal, characteristic
5	C₈×#1039	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅²×Q₈	C₂²	1	2	yes	—
7	C₂×#6045	C₂²	1	2	yes	—
8	C₅²×Q₁₆	—	2	2	no	—
9	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
10	#87812	—	2	4	no	—
11	#87806	—	2	4	no	—
12	C₄×#1039	C₂²	1	1	yes	characteristic
13	C₂×#6029	C₂²	1	1	yes	characteristic
14	#87802	—	2	2	no	—
15	#65470	—	5	6	no	maximal
16	C₅²×Q₈	—	2	2	no	—
17	C₅²×Q₈	—	2	2	no	—
18	C₂×C₄×C₅²	—	2	2	no	—
19	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
20	C₂×#1039	—	2	2	no	—
21	C₂×#1039	D₄	1	2	yes	—
22	#6045	—	2	2	no	—
23	#6045	—	2	2	no	—
24	C₅²×C₈	D₄	1	2	yes	—
25	#6029	D₄	1	2	yes	—
26	C₂×C₅×Q₁₆	D₅	1	6	yes	—
27	#4090	—	5	12	no	—
28	C₂×#689	—	5	12	no	—
29	C₂×Dic₂₀	—	5	6	no	—
30	C₈×Dic₅	—	5	6	no	—
31	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
32	C₄×C₅²	—	4	2	no	—
33	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
34	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
35	#1039	—	4	2	no	—
36	#1039	Q₁₆	1	4	yes	complemented
37	C₂×C₅×Q₈	C₂×D₅	1	12	yes	—
38	C₂×Dic₁₀	—	5	12	no	—
39	C₅×Q₁₆	—	2	12	no	—
40	C₂×C₅×C₈	C₂×D₅	1	6	yes	—
41	#689	—	10	24	no	—
42	#683	—	10	24	no	—
43	C₄×Dic₅	—	5	6	no	—
44	C₂×#168	—	5	6	no	—
45	Dic₂₀	—	10	12	no	—
46	#494	—	25	1	no	Sylow
47	C₂×C₅²	#127	1	1	yes	characteristic
48	C₂×C₅²	C₂×Q₁₆	1	2	yes	—
49	C₅×Q₈	—	2	12	no	—
50	C₅×Q₈	—	2	12	no	—
51	C₂×C₄×C₅	—	2	12	no	—
52	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
53	C₂×Dic₅	—	5	12	no	—
54	C₂×Dic₅	—	10	12	no	—
55	Dic₁₀	—	10	12	no	—
56	Dic₁₀	—	10	12	no	—
57	C₅×C₈	C₂²⋊D₅	1	12	yes	—
58	#168	—	5	12	no	—
59	C₂×Q₁₆	—	25	2	no	—
60	C₂×Q₁₆	—	25	1	no	—
61	C₂×Q₁₆	C₅⋊D₅	1	1	yes	characteristic
62	C₄⋊Q₈	—	25	2	no	—
63	C₄×C₈	—	25	1	no	—
64	C₅²	#494	1	1	yes	characteristic, complemented, Sylow
65	C₂²×C₅	D₄×D₅	1	6	yes	—
66	C₄×C₅	—	4	12	no	—
67	C₄×C₅	D₄×D₅	1	6	yes	—
68	C₄×C₅	C₂×C₂²⋊D₅	1	6	yes	—
69	Dic₅	—	5	24	no	—
70	Dic₅	—	20	12	no	—
71	C₂×Q₈	—	25	2	no	—
72	C₂×Q₈	C₂×C₅⋊D₅	1	2	yes	—
73	Q₁₆	—	2	2	no	—
74	Q₁₆	—	50	4	no	—
75	Q₁₆	—	50	2	no	—
76	C₂×C₈	—	25	1	no	—
77	C₂×C₈	C₂×C₅⋊D₅	1	1	yes	characteristic
78	C₄⋊C₄	—	50	4	no	—
79	C₄²	—	25	1	no	—
80	C₂×C₅	D₅×Q₁₆	1	12	yes	—
81	C₂×C₅	#4086	1	6	yes	—
82	Q₈	—	2	2	no	—
83	Q₈	—	2	2	no	—
84	Q₈	—	50	2	no	—
85	Q₈	—	50	2	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	25	2	no	—
88	C₂×C₄	—	50	2	no	—
89	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
90	C₈	—	25	2	no	—
91	C₈	C₂²⋊(C₅⋊D₅)	1	2	yes	—
92	C₅	#65470	1	6	yes	complemented
93	C₂²	D₄×C₅⋊D₅	1	1	yes	characteristic
94	C₄	—	4	2	no	—
95	C₄	—	25	4	no	—
96	C₄	—	100	2	no	—
97	C₄	D₄×C₅⋊D₅	1	1	yes	characteristic
98	C₄	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
99	C₂	Q₁₆×C₅⋊D₅	1	2	yes	—
100	C₂	#11717260	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅²×Q₁₆

C₂

yes

maximal, characteristic

#11717264

C₂

yes

maximal

C₂×#87812

C₂

yes

maximal

C₂×#87802

C₂

yes

maximal, characteristic

C₈×#1039

C₂

yes

maximal, characteristic

C₂×C₅²×Q₈

C₂²

yes

—

C₂×#6045

C₂²

yes

—

C₅²×Q₁₆

—

C₂×C₅²×C₈

C₂²

yes

—

—

C₂²

yes

characteristic

C₂×#6029

C₂²

yes

—

—

C₅²×Q₈

—

C₅²×Q₈

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×#1039

—

C₂×#1039

D₄

yes

—

#6045

—

#6045

—

C₅²×C₈

D₄

yes

—

#6029

D₄

yes

—

C₂×C₅×Q₁₆

D₅

yes

—

#4090

—

C₂×#689

—

C₂×Dic₂₀

—

C₈×Dic₅

—

C₂²×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

—

C₄×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

C₂×D₄

yes

characteristic

#1039

—

#1039

Q₁₆

yes

complemented

C₂×C₅×Q₈

C₂×D₅

yes

—

C₂×Dic₁₀

—

C₅×Q₁₆

—

C₂×C₅×C₈

C₂×D₅

yes

—

#689

—

#683

—

C₄×Dic₅

—

C₂×#168

—

Dic₂₀

—

—

C₂×C₅²

yes

C₂×C₅²

C₂×Q₁₆

yes

—

C₅×Q₈

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

C₂²⋊D₅

yes

—

#168

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₅⋊D₅

yes

characteristic

C₄⋊Q₈

—

C₄×C₈

—

C₅²

#494

yes

characteristic, complemented, Sylow

C₂²×C₅

D₄×D₅

yes

—

C₄×C₅

—

C₄×C₅

D₄×D₅

yes

—

C₄×C₅

C₂×C₂²⋊D₅

yes

—

Dic₅

—

Dic₅

—

C₂×Q₈

—

C₂×Q₈

C₂×C₅⋊D₅

yes

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

C₂×C₅⋊D₅

yes

characteristic

C₄⋊C₄

—

C₄²

—

C₂×C₅

D₅×Q₁₆

yes

—

C₂×C₅

#4086

yes

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

C₈

—

C₈

C₂²⋊(C₅⋊D₅)

yes

—

C₅

#65470

yes

complemented

C₂²

D₄×C₅⋊D₅

yes

characteristic

C₄

—

C₄

—

C₄

—

100

—

C₄

D₄×C₅⋊D₅

yes

characteristic

C₄

C₂×C₂²⋊(C₅⋊D₅)

yes

characteristic

C₂

Q₁₆×C₅⋊D₅

yes

—

100

C₂

#11717260

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×C₅⋊D₅
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂²×C₅⋊D₅
Fitting	C₂×C₅²×Q₁₆	C₂
socle, prime radical	C₂²×C₅²	C₂×D₄
hypercenter	C₂×Q₁₆	C₅⋊D₅
hypocenter	C₅²	#494
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

subgroup

G/H

center

C₂²

D₄×C₅⋊D₅

commutator, cosocle

C₂×C₄×C₅²

C₂³

Frattini

C₂×C₄

C₂²×C₅⋊D₅

Fitting

C₂×C₅²×Q₁₆

C₂

socle, prime radical

C₂²×C₅²

C₂×D₄

hypercenter

C₂×Q₁₆

C₅⋊D₅

hypocenter

C₅²

#494

LCS

G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

N	H ≅ G/N	type
36. #1039	Q₁₆	semidirect
64. C₅²	#494	semidirect
92. C₅	#65470	semidirect

H ≅ G/N

type

36. #1039

Q₁₆

semidirect

64. C₅²

#494

semidirect

92. C₅

#65470

semidirect

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

subgroup	index
C₂×C₅²×Q₁₆	2
#11717264	2
C₂×#87812	2
C₂×#87802	2
C₈×#1039	2
#65470	5

subgroup

index

C₂×C₅²×Q₁₆

Subgroups — #13103043

Conjugacy classes of subgroups

Complements

Maximal subgroups