Group #13014568 — subgroups

Subgroups — #13014568

Order 1280.

186 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 22 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	#10602069	C₂	1	1	yes	maximal, characteristic
2	#10598516	C₂	1	2	yes	maximal, complemented
3	C₅×#493	C₄	1	1	yes	characteristic, complemented
4	#65362	—	2	2	no	—
5	#64984	C₂²	1	1	yes	characteristic
6	#64853	—	2	2	no	—
7	C₈×#168	C₄	1	1	yes	characteristic, complemented
8	#7529	—	5	1	no	maximal, Sylow
9	C₂×C₅×D₈	—	4	1	no	—
10	C₅×#127	—	2	1	no	—
11	C₂×#686	—	4	4	no	—
12	C₂×D₄₀	—	4	1	no	—
13	#4020	D₄	1	2	yes	—
14	#3999	—	4	2	no	—
15	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
16	C₄×#168	—	2	1	no	—
17	#1712	—	5	1	no	—
18	#1408	—	5	2	no	—
19	C₂×C₅×D₄	—	4	2	no	—
20	C₂×C₅×D₄	—	4	1	no	—
21	C₂×D₂₀	—	2	2	no	—
22	C₂×D₂₀	—	4	2	no	—
23	C₂²⋊AGL(1,𝔽₅)	—	8	2	no	—
24	C₅×D₈	—	8	2	no	—
25	C₂×C₅×C₈	—	2	1	no	—
26	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
27	#686	—	8	8	no	—
28	C₂×#168	—	2	2	no	—
29	D₄₀	—	8	2	no	—
30	#493	—	5	1	no	—
31	#493	—	10	2	no	—
32	#493	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
33	#353	—	10	2	no	—
34	C₈²	—	5	1	no	—
35	C₂³×C₅	—	8	1	no	—
36	C₂²×D₅	—	4	2	no	—
37	C₂×AGL(1,𝔽₅)	—	16	2	no	—
38	C₅×D₄	—	8	4	no	—
39	C₅×D₄	—	8	2	no	—
40	C₂×C₄×C₅	—	2	1	no	—
41	C₂×C₄×C₅	#99	1	1	yes	characteristic
42	D₂₀	—	8	2	no	—
43	D₂₀	—	8	4	no	—
44	C₅×C₈	—	2	2	no	—
45	#168	—	2	4	no	—
46	C₂×D₈	—	4	1	no	—
47	C₂×D₈	—	20	1	no	—
48	C₂×D₈	—	20	4	no	—
49	#127	—	2	1	no	—
50	#127	—	5	2	no	—
51	#99	—	20	2	no	—
52	C₄×C₈	—	10	1	no	—
53	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
54	C₂²×C₅	—	16	2	no	—
55	C₂²×C₅	—	16	1	no	—
56	C₂²×C₅	#353	1	1	yes	characteristic
57	C₂×D₅	—	8	2	no	—
58	C₂×D₅	—	16	2	no	—
59	AGL(1,𝔽₅)	—	32	2	no	—
60	C₄×C₅	—	2	2	no	—
61	C₄×C₅	—	2	1	no	—
62	C₂×D₄	—	4	2	no	—
63	C₂×D₄	—	4	1	no	—
64	C₂×D₄	—	10	2	no	—
65	C₂×D₄	—	20	2	no	—
66	D₈	—	8	2	no	—
67	D₈	—	40	2	no	—
68	D₈	—	40	8	no	—
69	C₂×C₈	—	2	1	no	—
70	C₂×C₈	—	10	2	no	—
71	C₂²⋊C₄	—	40	2	no	—
72	C₄²	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
73	C₂×C₅	—	2	1	no	—
74	C₂×C₅	—	32	1	no	—
75	C₂×C₅	#1408	1	1	yes	characteristic
76	D₅	—	16	2	no	—
77	C₂³	—	8	1	no	—
78	C₂³	—	20	2	no	—
79	D₄	—	8	2	no	—
80	D₄	—	8	4	no	—
81	D₄	—	40	2	no	—
82	D₄	—	40	4	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	80	2	no	—
85	C₂×C₄	#4011	1	1	yes	characteristic
86	C₈	—	2	2	no	—
87	C₈	—	10	4	no	—
88	C₅	#7529	1	1	yes	characteristic, complemented, Sylow
89	C₂²	—	16	2	no	—
90	C₂²	—	16	1	no	—
91	C₂²	—	40	2	no	—
92	C₂²	—	80	2	no	—
93	C₂²	#64922	1	1	yes	characteristic
94	C₄	—	2	2	no	—
95	C₄	—	2	1	no	—
96	C₄	—	160	2	no	—
97	C₂	—	2	1	no	—
98	C₂	—	32	1	no	—
99	C₂	—	80	2	no	—
100	C₂	#10598747	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10602069

C₂

yes

maximal, characteristic

#10598516

C₂

yes

maximal, complemented

C₅×#493

C₄

yes

characteristic, complemented

#65362

—

#64984

C₂²

yes

characteristic

#64853

—

C₈×#168

C₄

yes

characteristic, complemented

#7529

—

maximal, Sylow

C₂×C₅×D₈

—

C₅×#127

—

C₂×#686

—

C₂×D₄₀

—

#4020

D₄

yes

—

#3999

—

C₄×C₅×C₈

C₂×C₄

yes

—

—

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂²⋊AGL(1,𝔽₅)

—

C₅×D₈

—

C₂×C₅×C₈

—

C₄²×C₅

C₂²⋊C₄

yes

characteristic

#686

—

C₂×#168

—

D₄₀

—

—

—

yes

characteristic, complemented

#353

—

C₈²

—

C₂³×C₅

—

C₂²×D₅

—

C₂×AGL(1,𝔽₅)

—

C₅×D₄

—

C₅×D₄

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#99

yes

characteristic

D₂₀

—

D₂₀

—

C₅×C₈

—

#168

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

#127

—

#127

—

#99

—

C₄×C₈

—

C₄×C₈

C₂×AGL(1,𝔽₅)

yes

characteristic

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

#353

yes

characteristic

C₂×D₅

—

C₂×D₅

—

AGL(1,𝔽₅)

—

C₄×C₅

—

C₄×C₅

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

D₈

—

D₈

—

D₈

—

C₂×C₈

—

C₂×C₈

—

C₂²⋊C₄

—

C₄²

C₂²⋊AGL(1,𝔽₅)

yes

characteristic

C₂×C₅

—

C₂×C₅

—

C₂×C₅

#1408

yes

characteristic

D₅

—

C₂³

—

C₂³

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#4011

yes

characteristic

C₈

—

C₈

—

C₅

#7529

yes

characteristic, complemented, Sylow

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#64922

yes

characteristic

C₄

—

C₄

—

C₄

—

160

—

C₂

—

C₂

—

C₂

—

100

C₂

#10598747

yes

characteristic

	subgroup	G/H
center	C₂	#10598747
commutator	C₄×C₅×C₈	C₂×C₄
Frattini	C₄×C₈	C₂×AGL(1,𝔽₅)
Fitting	C₅×#493	C₄
socle	C₂×C₅	#1408
prime radical	#10602069	C₂
cosocle	#64984	C₂²
hypercenter	#493	AGL(1,𝔽₅)
hypocenter	C₅	#7529
LCS	G ▷ C₄×C₅×C₈ ▷ C₄²×C₅ ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #493

subgroup

G/H

center

C₂

#10598747

commutator

C₄×C₅×C₈

C₂×C₄

C₄×C₈

C₄

C₂×C₅

C₂

C₂²

C₅

G ▷ C₄×C₅×C₈ ▷ C₄²×C₅ ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #493

N	H ≅ G/N	type
2. #10598516	C₂	semidirect
3. C₅×#493	C₄	semidirect
7. C₈×#168	C₄	semidirect
32. #493	AGL(1,𝔽₅)	semidirect
88. C₅	#7529	semidirect

H ≅ G/N

type

2. #10598516

C₂

semidirect

3. C₅×#493

C₄

semidirect

7. C₈×#168

C₄

semidirect

32. #493

AGL(1,𝔽₅)

semidirect

88. C₅

#7529

semidirect

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

subgroup	index
#10602069	2
#10598516	2
#7529	5

subgroup

index

#10602069

#10598516

#7529

Subgroups — #13014568

Conjugacy classes of subgroups

Complements

Maximal subgroups