Group #13099261 — subgroups

Subgroups — #13099261

Order 1600.

276 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 60 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	#11717194	C₂	1	1	yes	maximal, characteristic
2	C₅×C₄⋊#168	C₂	1	2	yes	maximal, complemented
3	C₂×C₄⋊(C₅⋊D₅)	—	2	2	no	—
4	C₂×C₄⋊(C₅⋊D₅)	C₄	1	2	yes	—
5	C₄²×C₅²	C₂²	1	1	yes	characteristic
6	C₂×C₅×#168	—	2	2	no	—
7	#64705	—	5	2	no	maximal
8	C₂²×C₅⋊D₅	—	2	2	no	—
9	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
10	C₂×C₄×C₅²	D₄	1	2	yes	—
11	C₄⋊(C₅⋊D₅)	—	4	2	no	—
12	C₄⋊(C₅⋊D₅)	—	4	2	no	—
13	C₄⋊(C₅⋊D₅)	—	4	2	no	—
14	C₅×#168	—	4	2	no	—
15	#4020	—	5	2	no	—
16	#4020	—	10	2	no	—
17	C₅×C₄⋊C₈	—	5	2	no	—
18	C₄⋊#168	D₅	1	2	yes	complemented
19	C₂²×C₅²	C₂²⋊C₄	1	1	yes	characteristic
20	C₂×C₅⋊D₅	—	4	2	no	—
21	C₂×C₅⋊D₅	—	4	4	no	—
22	C₄×C₅²	—	2	1	no	—
23	C₄×C₅²	D₈	1	2	yes	—
24	C₄×C₅²	SD₁₆	1	2	yes	—
25	C₂×D₂₀	—	5	4	no	—
26	C₂×D₂₀	—	10	4	no	—
27	C₂×D₂₀	—	10	8	no	—
28	C₂×D₂₀	—	10	2	no	—
29	C₂×D₂₀	—	10	2	no	—
30	C₂×C₅×C₈	—	10	2	no	—
31	C₄²×C₅	—	2	2	no	—
32	C₄²×C₅	C₂×D₅	1	2	yes	—
33	C₂×#168	—	2	2	no	—
34	#331	—	25	1	no	Sylow
35	C₂×C₅²	#100	1	1	yes	characteristic
36	C₂×C₅²	#102	1	2	yes	—
37	C₅⋊D₅	—	8	2	no	—
38	C₂²×D₅	—	10	8	no	—
39	C₂²×D₅	—	10	4	no	—
40	C₂×C₄×C₅	—	2	2	no	—
41	C₂×C₄×C₅	—	2	4	no	—
42	C₂×C₄×C₅	C₄×D₅	1	2	yes	—
43	C₂×C₄×C₅	D₂₀	1	2	yes	—
44	C₂×C₄×C₅	C₂²⋊D₅	1	2	yes	—
45	D₂₀	—	20	4	no	—
46	D₂₀	—	20	8	no	—
47	D₂₀	—	20	8	no	—
48	D₂₀	—	20	8	no	—
49	D₂₀	—	20	2	no	—
50	D₂₀	—	20	2	no	—
51	D₂₀	—	20	2	no	—
52	D₂₀	—	20	2	no	—
53	C₅×C₈	—	20	2	no	—
54	#168	—	4	2	no	—
55	#127	—	25	1	no	—
56	C₄⋊C₈	—	5	2	no	—
57	C₅²	#331	1	1	yes	characteristic, complemented, Sylow
58	C₂²×C₅	—	2	2	no	—
59	C₂²×C₅	#685	1	2	yes	—
60	C₂×D₅	—	20	8	no	—
61	C₂×D₅	—	20	16	no	—
62	C₂×D₅	—	20	4	no	—
63	C₂×D₅	—	20	4	no	—
64	C₂×D₅	—	20	4	no	—
65	C₄×C₅	—	2	4	no	—
66	C₄×C₅	—	2	2	no	—
67	C₄×C₅	—	2	4	no	—
68	C₄×C₅	—	2	4	no	—
69	C₄×C₅	#677	1	2	yes	—
70	C₄×C₅	D₄₀	1	2	yes	—
71	C₄×C₅	#686	1	2	yes	—
72	C₄×C₅	#688	1	2	yes	—
73	C₂×D₄	—	25	2	no	—
74	C₂×D₄	—	50	2	no	—
75	C₂×C₈	—	10	2	no	—
76	C₄²	D₅²	1	1	yes	characteristic
77	C₂×C₅	—	2	2	no	—
78	C₂×C₅	—	2	4	no	—
79	C₂×C₅	#3941	1	2	yes	—
80	C₂×C₅	#3953	1	2	yes	—
81	C₂×C₅	#3955	1	2	yes	—
82	D₅	—	40	8	no	—
83	D₅	—	40	4	no	—
84	C₂³	—	50	2	no	—
85	D₄	—	100	2	no	—
86	D₄	—	100	2	no	—
87	D₄	—	100	2	no	—
88	C₂×C₄	#6036	1	1	yes	characteristic
89	C₂×C₄	#6038	1	2	yes	—
90	C₈	—	20	2	no	—
91	C₅	—	2	2	no	—
92	C₅	#64705	1	2	yes	complemented
93	C₂²	—	100	2	no	—
94	C₂²	—	100	4	no	—
95	C₂²	#87779	1	1	yes	characteristic
96	C₄	—	2	1	no	—
97	C₄	#87771	1	2	yes	—
98	C₄	#87774	1	2	yes	—
99	C₂	—	200	2	no	—
100	C₂	#11716600	1	1	yes	characteristic
101	C₂	#11716605	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11717194

C₂

yes

maximal, characteristic

C₅×C₄⋊#168

C₂

yes

maximal, complemented

C₂×C₄⋊(C₅⋊D₅)

—

C₂×C₄⋊(C₅⋊D₅)

C₄

yes

—

C₄²×C₅²

C₂²

yes

characteristic

C₂×C₅×#168

—

#64705

—

maximal

C₂²×C₅⋊D₅

—

C₂×C₄×C₅²

C₂×C₄

yes

characteristic

C₂×C₄×C₅²

D₄

yes

—

C₄⋊(C₅⋊D₅)

—

C₄⋊(C₅⋊D₅)

—

C₄⋊(C₅⋊D₅)

—

C₅×#168

—

#4020

—

#4020

—

C₅×C₄⋊C₈

—

C₄⋊#168

D₅

yes

complemented

C₂²×C₅²

C₂²⋊C₄

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₄×C₅²

—

C₄×C₅²

D₈

yes

—

C₄×C₅²

SD₁₆

yes

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×C₅×C₈

—

C₄²×C₅

—

C₄²×C₅

C₂×D₅

yes

—

—

—

C₂×C₅²

yes

C₂×C₅²

yes

—

C₅⋊D₅

—

C₂²×D₅

—

C₂²×D₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₄×D₅

yes

—

C₂×C₄×C₅

D₂₀

yes

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

D₂₀

—

D₂₀

—

D₂₀

—

D₂₀

—

D₂₀

—

D₂₀

—

D₂₀

—

D₂₀

—

C₅×C₈

—

#168

—

#127

—

C₄⋊C₈

—

C₅²

#331

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

#685

yes

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#677

yes

—

C₄×C₅

D₄₀

yes

—

C₄×C₅

#686

yes

—

C₄×C₅

#688

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×C₈

—

C₄²

D₅²

yes

characteristic

C₂×C₅

—

C₂×C₅

—

C₂×C₅

#3941

yes

—

C₂×C₅

#3953

yes

—

C₂×C₅

#3955

yes

—

D₅

—

D₅

—

C₂³

—

D₄

—

100

—

D₄

—

100

—

D₄

—

100

—

C₂×C₄

#6036

yes

characteristic

C₂×C₄

#6038

yes

—

C₈

—

C₅

—

C₅

#64705

yes

complemented

C₂²

—

100

—

C₂²

—

100

—

C₂²

#87779

yes

characteristic

C₄

—

C₄

#87771

yes

—

C₄

#87774

yes

—

C₂

—

200

—

100

C₂

#11716600

yes

characteristic

101

C₂

#11716605

yes

—

	subgroup	G/H
center	C₂²	#87779
commutator	C₂×C₄×C₅²	C₂×C₄
Frattini, hypercenter	C₄²	D₅²
Fitting, cosocle	C₄²×C₅²	C₂²
socle	C₂²×C₅²	C₂²⋊C₄
prime radical	#11717194	C₂
hypocenter	C₅²	#331
LCS	G ▷ C₂×C₄×C₅² ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄²

subgroup

G/H

center

C₂²

#87779

commutator

C₂×C₄×C₅²

C₂×C₄

Frattini, hypercenter

C₄²

D₅²

Fitting, cosocle

C₄²×C₅²

C₂²

socle

C₂²×C₅²

C₂²⋊C₄

C₂

C₅²

G ▷ C₂×C₄×C₅² ▷ C₂²×C₅² ▷ C₅²

UCS

C₁ ◁ C₂² ◁ C₄²

N	H ≅ G/N	type
2. C₅×C₄⋊#168	C₂	semidirect
18. C₄⋊#168	D₅	semidirect
57. C₅²	#331	semidirect
92. C₅	#64705	semidirect

H ≅ G/N

type

2. C₅×C₄⋊#168

C₂

semidirect

18. C₄⋊#168

D₅

semidirect

57. C₅²

#331

semidirect

92. C₅

#64705

semidirect

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

subgroup	index
#11717194	2
C₅×C₄⋊#168	2
#64705	5

subgroup

index

#11717194

C₅×C₄⋊#168

#64705

Subgroups — #13099261

Conjugacy classes of subgroups

Complements

Maximal subgroups