Group #13102926 — subgroups

Subgroups — #13102926

Order 1600.

360 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 127 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	C₅²×#125	C₂	1	1	yes	maximal, characteristic
2	#11717189	C₂	1	1	yes	maximal, characteristic
3	#11716675	C₂	1	2	yes	maximal
4	#11716674	C₂	1	2	yes	maximal
5	#11716670	C₂	1	1	yes	maximal, characteristic
6	C₂×#6045	—	2	1	no	—
7	C₅²×C₄⋊C₄	—	2	1	no	—
8	C₅²×C₄⋊C₄	—	2	1	no	—
9	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic
10	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic
11	C₄²×C₅²	C₂²	1	1	yes	characteristic
12	C₄⋊#1039	—	2	1	no	—
13	C₄⋊#1039	C₂²	1	2	yes	—
14	C₂×#6029	C₂²	1	2	yes	—
15	#65354	—	5	6	no	maximal
16	C₂×C₄×C₅²	—	2	1	no	—
17	C₂×C₄×C₅²	—	2	1	no	—
18	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
19	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
20	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
21	C₂×#1039	—	2	2	no	—
22	#6045	—	4	2	no	—
23	#6029	Q₈	1	4	yes	complemented
24	C₅×#125	D₅	1	6	yes	—
25	#4015	—	5	6	no	—
26	#3940	—	5	12	no	—
27	#3939	—	5	12	no	—
28	#3935	—	5	6	no	—
29	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
30	C₄×C₅²	—	2	1	no	—
31	C₄×C₅²	—	2	1	no	—
32	C₄×C₅²	—	4	1	no	—
33	C₄×C₅²	—	4	1	no	—
34	C₄×C₅²	C₂×Q₈	1	2	yes	—
35	#1039	—	4	2	no	—
36	C₂×Dic₁₀	—	10	6	no	—
37	C₅×C₄⋊C₄	—	2	6	no	—
38	C₅×C₄⋊C₄	—	2	6	no	—
39	C₅×C₄⋊C₄	C₂×D₅	1	6	yes	—
40	C₅×C₄⋊C₄	C₂×D₅	1	6	yes	—
41	C₄²×C₅	C₂×D₅	1	6	yes	—
42	C₄⋊Dic₅	—	5	12	no	—
43	C₄⋊Dic₅	—	10	6	no	—
44	C₂×#168	—	5	12	no	—
45	#501	—	25	1	no	Sylow
46	C₂×C₅²	C₄⋊Q₈	1	1	yes	characteristic
47	C₂×C₅²	AGL(1,ℤ/8)	1	1	yes	characteristic
48	C₂×C₅²	#137	1	1	yes	characteristic
49	C₂×C₄×C₅	—	2	6	no	—
50	C₂×C₄×C₅	—	2	6	no	—
51	C₂×C₄×C₅	C₂²⋊D₅	1	6	yes	—
52	C₂×C₄×C₅	C₂²⋊D₅	1	6	yes	—
53	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
54	C₂×Dic₅	—	10	12	no	—
55	Dic₁₀	—	20	12	no	—
56	#168	—	5	24	no	—
57	C₄⋊Q₈	—	25	1	no	—
58	#125	C₅⋊D₅	1	1	yes	characteristic
59	#107	—	25	2	no	—
60	#106	—	25	2	no	—
61	#97	—	25	1	no	—
62	C₅²	#501	1	1	yes	characteristic, complemented, Sylow
63	C₂²×C₅	C₂×C₂²⋊D₅	1	6	yes	—
64	C₄×C₅	—	2	6	no	—
65	C₄×C₅	—	2	6	no	—
66	C₄×C₅	—	4	6	no	—
67	C₄×C₅	—	4	6	no	—
68	C₄×C₅	Q₈×D₅	1	12	yes	—
69	Dic₅	—	20	12	no	—
70	C₂×Q₈	—	50	1	no	—
71	C₂×C₈	—	25	2	no	—
72	C₄⋊C₄	—	2	1	no	—
73	C₄⋊C₄	—	2	1	no	—
74	C₄⋊C₄	—	25	2	no	—
75	C₄⋊C₄	—	50	1	no	—
76	C₄⋊C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
77	C₄⋊C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
78	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
79	C₂×C₅	#4090	1	6	yes	—
80	C₂×C₅	#4095	1	6	yes	—
81	C₂×C₅	#4097	1	6	yes	—
82	Q₈	—	100	2	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	50	2	no	—
86	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
87	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
88	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
89	C₈	—	25	4	no	—
90	C₅	#65354	1	6	yes	complemented
91	C₂²	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
92	C₄	—	2	1	no	—
93	C₄	—	2	1	no	—
94	C₄	—	4	1	no	—
95	C₄	—	4	1	no	—
96	C₄	—	100	2	no	—
97	C₄	Q₈×C₅⋊D₅	1	2	yes	—
98	C₂	#11717264	1	1	yes	characteristic
99	C₂	#11717269	1	1	yes	characteristic
100	C₂	#11717271	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅²×#125

C₂

yes

maximal, characteristic

#11717189

C₂

yes

maximal, characteristic

#11716675

C₂

yes

maximal

#11716674

C₂

yes

maximal

#11716670

C₂

yes

maximal, characteristic

C₂×#6045

—

C₅²×C₄⋊C₄

—

C₅²×C₄⋊C₄

—

C₅²×C₄⋊C₄

C₂²

yes

characteristic

C₅²×C₄⋊C₄

C₂²

yes

characteristic

C₄²×C₅²

C₂²

yes

characteristic

C₄⋊#1039

—

C₄⋊#1039

C₂²

yes

—

C₂×#6029

C₂²

yes

—

#65354

—

maximal

C₂×C₄×C₅²

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

D₄

yes

characteristic

C₂×C₄×C₅²

D₄

yes

characteristic

C₂×C₄×C₅²

C₂³

yes

—

—

Q₈

yes

complemented

C₅×#125

D₅

yes

—

—

—

—

—

C₂²×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

—

C₄×C₅²

—

C₄×C₅²

—

C₄×C₅²

—

C₄×C₅²

C₂×Q₈

yes

—

#1039

—

C₂×Dic₁₀

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

C₂×D₅

yes

—

C₅×C₄⋊C₄

C₂×D₅

yes

—

C₄²×C₅

C₂×D₅

yes

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₂×#168

—

#501

—

Sylow

C₂×C₅²

C₄⋊Q₈

yes

C₂×C₅²

yes

C₂×C₅²

yes

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

Dic₁₀

—

#168

—

C₄⋊Q₈

—

#125

C₅⋊D₅

yes

—

—

—

C₅²

#501

yes

characteristic, complemented, Sylow

C₂²×C₅

C₂×C₂²⋊D₅

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

Q₈×D₅

yes

—

Dic₅

—

C₂×Q₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×C₅⋊D₅

yes

characteristic

C₄⋊C₄

C₂×C₅⋊D₅

yes

characteristic

C₄²

C₂×C₅⋊D₅

yes

characteristic

C₂×C₅

#4090

yes

—

C₂×C₅

#4095

yes

—

C₂×C₅

#4097

yes

—

Q₈

—

100

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊(C₅⋊D₅)

yes

characteristic

C₂×C₄

C₂²⋊(C₅⋊D₅)

yes

characteristic

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

C₈

—

C₅

#65354

yes

complemented

C₂²

C₂×C₂²⋊(C₅⋊D₅)

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

100

—

C₄

Q₈×C₅⋊D₅

yes

—

C₂

yes

C₂

yes

100

C₂

#11717271

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊(C₅⋊D₅)
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂²×C₅⋊D₅
Fitting	C₅²×#125	C₂
socle, prime radical	C₂²×C₅²	C₂×D₄
hypercenter	#125	C₅⋊D₅
hypocenter	C₅²	#501
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

subgroup

G/H

center

C₂²

C₂×C₂²⋊(C₅⋊D₅)

commutator, cosocle

C₂×C₄×C₅²

C₂³

Frattini

C₂×C₄

C₂²×C₅⋊D₅

Fitting

C₅²×#125

C₂

socle, prime radical

C₂²×C₅²

C₂×D₄

C₅⋊D₅

C₅²

G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #125

N	H ≅ G/N	type
23. #6029	Q₈	semidirect
62. C₅²	#501	semidirect
90. C₅	#65354	semidirect

H ≅ G/N

type

23. #6029

Q₈

semidirect

62. C₅²

#501

semidirect

90. C₅

#65354

semidirect

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

subgroup	index
C₅²×#125	2
#11717189	2
#11716675	2
#11716674	2
#11716670	2
#65354	5

subgroup

index

C₅²×#125

Subgroups — #13102926

Conjugacy classes of subgroups

Complements

Maximal subgroups