Group #13100366 — subgroups

Subgroups — #13100366

Order 1600.

256 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 56 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×#87863	C₂	1	2	yes	maximal
2	C₂×C₈×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
4	C₂×#6034	—	2	2	no	—
5	#87863	—	2	2	no	—
6	#87863	C₄	1	4	yes	—
7	C₂×C₅²×C₈	C₄	1	1	yes	characteristic
8	C₂×#6029	C₄	1	1	yes	characteristic
9	C₈×C₅⋊D₅	—	2	2	no	—
10	#64896	—	5	2	no	maximal
11	C₂²×C₅⋊D₅	Q₈	1	1	yes	characteristic
12	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
13	C₂×#1039	D₄	1	1	yes	characteristic
14	C₄×C₅⋊D₅	C₂×C₄	1	2	yes	—
15	C₄×C₅⋊D₅	D₄	1	2	yes	—
16	#6034	—	2	4	no	—
17	C₅²×C₈	—	2	1	no	—
18	#6029	—	2	1	no	—
19	C₂×#700	—	5	4	no	—
20	C₂×C₈×D₅	—	5	2	no	—
21	C₂×C₈×D₅	—	10	2	no	—
22	C₂²×C₅²	C₄⋊C₄	1	1	yes	characteristic
23	C₂×C₅⋊D₅	—	2	1	no	—
24	C₂×C₅⋊D₅	—	2	1	no	—
25	C₂×C₅⋊D₅	C₄⋊C₄	1	2	yes	—
26	C₄×C₅²	C₄²	1	1	yes	characteristic
27	C₄×C₅²	C₂²⋊C₄	1	1	yes	characteristic
28	#1039	C₂²⋊C₄	1	2	yes	—
29	C₂×C₄×D₅	—	5	2	no	—
30	C₂×C₄×D₅	—	10	2	no	—
31	C₂×#170	—	10	4	no	—
32	#700	—	5	8	no	—
33	#700	—	10	4	no	—
34	C₂×C₅×C₈	—	2	2	no	—
35	C₂×C₅×C₈	AGL(1,𝔽₅)	1	2	yes	—
36	C₂×#168	—	5	2	no	—
37	C₂×#168	—	10	2	no	—
38	C₈×D₅	—	10	4	no	—
39	C₈×D₅	—	10	8	no	—
40	#341	—	25	1	no	Sylow
41	C₂×C₅²	#95	1	1	yes	characteristic
42	C₂×C₅²	#108	1	1	yes	characteristic
43	C₂×C₅²	#108	1	1	yes	characteristic
44	C₅⋊D₅	—	2	2	no	—
45	C₂²×D₅	—	5	2	no	—
46	C₂²×D₅	—	10	2	no	—
47	C₂×C₄×C₅	—	2	2	no	—
48	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	2	yes	—
49	C₂×Dic₅	—	5	2	no	—
50	C₂×Dic₅	—	10	2	no	—
51	C₄×D₅	—	5	4	no	—
52	C₄×D₅	—	5	4	no	—
53	C₄×D₅	—	10	4	no	—
54	C₄×D₅	—	10	4	no	—
55	#170	—	10	8	no	—
56	C₅×C₈	—	2	4	no	—
57	C₅×C₈	—	2	2	no	—
58	#168	—	10	4	no	—
59	#168	—	10	2	no	—
60	C₂×M₁₆	—	25	2	no	—
61	C₂²×C₈	—	25	1	no	—
62	C₅²	#341	1	1	yes	characteristic, complemented, Sylow
63	C₂²×C₅	—	2	2	no	—
64	C₂²×C₅	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
65	C₂×D₅	—	5	4	no	—
66	C₂×D₅	—	10	4	no	—
67	C₂×D₅	—	10	4	no	—
68	C₂×D₅	—	10	4	no	—
69	C₂×D₅	—	10	2	no	—
70	C₂×D₅	—	10	2	no	—
71	C₄×C₅	—	2	2	no	—
72	C₄×C₅	—	2	2	no	—
73	C₄×C₅	C₄×AGL(1,𝔽₅)	1	2	yes	—
74	C₄×C₅	C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
75	Dic₅	—	5	4	no	—
76	Dic₅	—	10	4	no	—
77	C₂²×C₄	—	25	1	no	—
78	M₁₆	—	25	4	no	—
79	M₁₆	—	50	2	no	—
80	C₂×C₈	—	25	1	no	—
81	C₂×C₈	—	50	2	no	—
82	C₂×C₈	—	50	2	no	—
83	C₂×C₈	#1044	1	1	yes	characteristic
84	C₂×C₅	—	2	2	no	—
85	C₂×C₅	—	2	2	no	—
86	C₂×C₅	—	2	2	no	—
87	C₂×C₅	#3995	1	2	yes	—
88	C₂×C₅	#3996	1	2	yes	—
89	C₂×C₅	(C₂×AGL(1,𝔽₅))⋊C₄	1	2	yes	—
90	D₅	—	10	8	no	—
91	D₅	—	10	4	no	—
92	C₂³	—	25	1	no	—
93	C₂×C₄	—	25	1	no	—
94	C₂×C₄	—	25	2	no	—
95	C₂×C₄	—	25	2	no	—
96	C₂×C₄	C₂×#1044	1	1	yes	characteristic
97	C₈	—	2	1	no	—
98	C₈	—	50	4	no	—
99	C₈	—	50	1	no	—
100	C₅	—	2	2	no	—
101	C₅	#64896	1	2	yes	complemented
102	C₂²	—	25	2	no	—
103	C₂²	—	50	1	no	—
104	C₂²	—	50	1	no	—
105	C₂²	C₄⋊#1044	1	1	yes	characteristic
106	C₄	—	25	2	no	—
107	C₄	C₄×#1044	1	1	yes	characteristic
108	C₄	C₂²⋊#1044	1	1	yes	characteristic
109	C₂	—	50	2	no	—
110	C₂	#11716888	1	1	yes	characteristic
111	C₂	#11716889	1	1	yes	characteristic
112	C₂	#11716899	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#87863

C₂

yes

maximal

C₂×C₈×C₅⋊D₅

C₂

yes

maximal, characteristic

C₂×C₄×C₅⋊D₅

C₂²

yes

—

—

C₄

yes

—

C₂×C₅²×C₈

C₄

yes

characteristic

C₂×#6029

C₄

yes

characteristic

C₈×C₅⋊D₅

—

#64896

—

maximal

C₂²×C₅⋊D₅

Q₈

yes

characteristic

C₂×C₄×C₅²

C₂×C₄

yes

characteristic

C₂×#1039

D₄

yes

characteristic

C₄×C₅⋊D₅

C₂×C₄

yes

—

C₄×C₅⋊D₅

D₄

yes

—

#6034

—

C₅²×C₈

—

#6029

—

C₂×#700

—

C₂×C₈×D₅

—

C₂×C₈×D₅

—

C₂²×C₅²

C₄⋊C₄

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

C₄⋊C₄

yes

—

C₄×C₅²

C₄²

yes

characteristic

C₄×C₅²

C₂²⋊C₄

yes

characteristic

#1039

C₂²⋊C₄

yes

—

C₂×C₄×D₅

—

C₂×C₄×D₅

—

C₂×#170

—

#700

—

#700

—

C₂×C₅×C₈

—

C₂×C₅×C₈

AGL(1,𝔽₅)

yes

—

C₂×#168

—

C₂×#168

—

C₈×D₅

—

C₈×D₅

—

—

C₂×C₅²

yes

C₂×C₅²

yes

C₂×C₅²

yes

C₅⋊D₅

—

C₂²×D₅

—

C₂²×D₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×AGL(1,𝔽₅)

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

#170

—

C₅×C₈

—

C₅×C₈

—

#168

—

#168

—

C₂×M₁₆

—

C₂²×C₈

—

C₅²

#341

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

C₄⋊AGL(1,𝔽₅)

yes

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

C₄×AGL(1,𝔽₅)

yes

—

C₄×C₅

C₂²⋊AGL(1,𝔽₅)

yes

—

Dic₅

—

Dic₅

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#1044

yes

characteristic

C₂×C₅

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

#3995

yes

—

C₂×C₅

#3996

yes

—

C₂×C₅

(C₂×AGL(1,𝔽₅))⋊C₄

yes

—

D₅

—

D₅

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×#1044

yes

characteristic

C₈

—

C₈

—

C₈

—

100

C₅

—

101

C₅

#64896

yes

complemented

102

C₂²

—

103

C₂²

—

104

C₂²

—

105

C₂²

C₄⋊#1044

yes

characteristic

106

C₄

—

107

C₄

C₄×#1044

yes

characteristic

108

C₄

C₂²⋊#1044

yes

characteristic

109

C₂

—

110

C₂

#11716888

yes

characteristic

111

C₂

#11716889

yes

characteristic

112

C₂

#11716899

yes

characteristic

	subgroup	G/H
center	C₂²	C₄⋊#1044
commutator	C₄×C₅²	C₄²
Frattini	C₂×C₄	C₂×#1044
Fitting	C₂×C₅²×C₈	C₄
socle	C₂²×C₅²	C₄⋊C₄
prime radical	C₂²×C₅⋊D₅	Q₈
cosocle	C₂×C₄×C₅⋊D₅	C₂²
hypercenter	C₂×C₈	#1044
hypocenter	C₅²	#341
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

subgroup

G/H

center

C₂²

C₄⋊#1044

commutator

C₄×C₅²

C₄²

Frattini

C₂×C₄

C₂×#1044

Fitting

C₂×C₅²×C₈

C₄

socle

C₂²×C₅²

C₄⋊C₄

prime radical

C₂²×C₅⋊D₅

Q₈

cosocle

C₂×C₄×C₅⋊D₅

C₂²

C₂×C₈

C₅²

G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

N	H ≅ G/N	type
62. C₅²	#341	semidirect
101. C₅	#64896	semidirect

H ≅ G/N

type

62. C₅²

#341

semidirect

101. C₅

#64896

semidirect

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

subgroup	index
C₂×#87863	2
C₂×C₈×C₅⋊D₅	2
#64896	5

subgroup

index

C₂×#87863

C₂×C₈×C₅⋊D₅

#64896

Subgroups — #13100366

Conjugacy classes of subgroups

Complements

Maximal subgroups