Group #13102920 — subgroups

Subgroups — #13102920

Order 1600.

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

normal subgroups only

This group has 496 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×#124	C₂	1	1	yes	maximal, characteristic
2	C₂×#87812	C₂	1	2	yes	maximal, complemented
3	C₂×#87810	C₂	1	2	yes	maximal
4	#11717189	C₂	1	1	yes	maximal, characteristic, complemented
5	#11716670	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₅²×D₄	C₂²	1	1	yes	characteristic
8	C₂×#6045	C₂²	1	2	yes	—
9	C₅²×C₂²⋊C₄	—	2	2	no	—
10	C₄²×C₅²	C₂²	1	1	yes	characteristic
11	#87812	—	2	4	no	—
12	#87810	—	2	4	no	—
13	C₄⋊#1039	—	2	2	no	—
14	C₂×#6029	C₂²	1	2	yes	—
15	#65349	—	5	6	no	maximal
16	C₂³×C₅²	—	2	1	no	—
17	C₅²×Q₈	—	2	2	no	—
18	C₅²×D₄	—	2	2	no	—
19	C₂×C₄×C₅²	—	2	1	no	—
20	C₂×C₄×C₅²	D₄	1	2	yes	—
21	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
22	C₂×#1039	—	2	2	no	—
23	#6045	—	2	2	no	—
24	#6045	—	2	2	no	—
25	#6029	D₄	1	4	yes	—
26	C₅×#124	D₅	1	6	yes	—
27	C₂×#689	—	5	12	no	—
28	C₂×#687	—	5	12	no	—
29	#4015	—	5	6	no	—
30	#3935	—	5	6	no	—
31	C₂²×C₅²	—	4	1	no	—
32	C₂²×C₅²	—	4	2	no	—
33	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
34	C₄×C₅²	—	2	2	no	—
35	C₄×C₅²	—	4	1	no	—
36	C₄×C₅²	C₂×D₄	1	2	yes	—
37	#1039	—	4	2	no	—
38	C₂×C₅×Q₈	C₂×D₅	1	6	yes	—
39	C₂×C₅×D₄	C₂×D₅	1	6	yes	—
40	C₂×Dic₁₀	—	5	12	no	—
41	C₅×C₂²⋊C₄	—	2	12	no	—
42	C₄²×C₅	C₂×D₅	1	6	yes	—
43	#689	—	10	24	no	—
44	#687	—	10	24	no	—
45	C₄⋊Dic₅	—	10	12	no	—
46	C₂×#168	—	5	12	no	—
47	#497	—	25	1	no	Sylow
48	C₂×C₅²	—	8	1	no	—
49	C₂×C₅²	#127	1	1	yes	characteristic
50	C₂×C₅²	#137	1	2	yes	—
51	C₂³×C₅	—	2	6	no	—
52	C₅×Q₈	—	2	12	no	—
53	C₅×D₄	—	2	12	no	—
54	C₂×C₄×C₅	—	2	6	no	—
55	C₂×C₄×C₅	C₂²⋊D₅	1	12	yes	—
56	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
57	C₂×Dic₅	—	10	12	no	—
58	Dic₁₀	—	10	12	no	—
59	Dic₁₀	—	10	12	no	—
60	#168	—	5	24	no	—
61	C₂×Q₁₆	—	25	2	no	—
62	C₂×SD₁₆	—	25	2	no	—
63	C₄⋊Q₈	—	25	1	no	—
64	#124	C₅⋊D₅	1	1	yes	characteristic
65	#97	—	25	1	no	—
66	C₅²	#497	1	1	yes	characteristic, complemented, Sylow
67	C₂²×C₅	—	4	12	no	—
68	C₂²×C₅	—	4	6	no	—
69	C₂²×C₅	C₂×C₂²⋊D₅	1	6	yes	—
70	C₄×C₅	—	2	12	no	—
71	C₄×C₅	—	4	6	no	—
72	C₄×C₅	D₄×D₅	1	12	yes	—
73	Dic₅	—	20	12	no	—
74	C₂×Q₈	—	25	2	no	—
75	C₂×Q₈	C₂×C₅⋊D₅	1	1	yes	characteristic
76	C₂×D₄	C₂×C₅⋊D₅	1	1	yes	characteristic
77	Q₁₆	—	50	4	no	—
78	SD₁₆	—	50	4	no	—
79	C₂×C₈	—	25	2	no	—
80	C₄⋊C₄	—	50	2	no	—
81	C₂²⋊C₄	—	2	2	no	—
82	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
83	C₂×C₅	—	8	6	no	—
84	C₂×C₅	#4086	1	6	yes	—
85	C₂×C₅	#4097	1	12	yes	—
86	C₂³	—	2	1	no	—
87	Q₈	—	2	2	no	—
88	Q₈	—	50	2	no	—
89	Q₈	—	50	2	no	—
90	D₄	—	2	2	no	—
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	50	2	no	—
93	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	2	yes	—
94	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
95	C₈	—	25	4	no	—
96	C₅	#65349	1	6	yes	complemented
97	C₂²	—	4	1	no	—
98	C₂²	—	4	2	no	—
99	C₂²	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
100	C₄	—	2	2	no	—
101	C₄	—	4	1	no	—
102	C₄	—	100	2	no	—
103	C₄	D₄×C₅⋊D₅	1	2	yes	—
104	C₂	—	8	1	no	—
105	C₂	#11717260	1	1	yes	characteristic
106	C₂	#11717271	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅²×#124

C₂

yes

maximal, characteristic

C₂×#87812

C₂

yes

maximal, complemented

C₂×#87810

C₂

yes

maximal

#11717189

C₂

yes

maximal, characteristic, complemented

#11716670

C₂

yes

maximal, characteristic, complemented

C₂×C₅²×Q₈

C₂²

yes

characteristic

C₂×C₅²×D₄

C₂²

yes

characteristic

C₂×#6045

C₂²

yes

—

C₅²×C₂²⋊C₄

—

C₄²×C₅²

C₂²

yes

—

—

—

C₂²

yes

—

#65349

—

maximal

C₂³×C₅²

—

C₅²×Q₈

—

C₅²×D₄

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

D₄

yes

—

C₂×C₄×C₅²

C₂³

yes

—

—

—

D₄

yes

—

C₅×#124

D₅

yes

—

—

—

—

—

C₂²×C₅²

—

C₂²×C₅²

—

C₂²×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

—

C₄×C₅²

—

C₄×C₅²

C₂×D₄

yes

—

#1039

—

C₂×C₅×Q₈

C₂×D₅

yes

—

C₂×C₅×D₄

C₂×D₅

yes

—

C₂×Dic₁₀

—

C₅×C₂²⋊C₄

—

C₄²×C₅

C₂×D₅

yes

—

#689

—

#687

—

C₄⋊Dic₅

—

C₂×#168

—

#497

—

Sylow

C₂×C₅²

—

C₂×C₅²

#127

yes

characteristic

C₂×C₅²

#137

yes

—

C₂³×C₅

—

C₅×Q₈

—

C₅×D₄

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

Dic₁₀

—

Dic₁₀

—

#168

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₄⋊Q₈

—

#124

C₅⋊D₅

yes

characteristic

#97

—

C₅²

#497

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₂×C₂²⋊D₅

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

D₄×D₅

yes

—

Dic₅

—

C₂×Q₈

—

C₂×Q₈

C₂×C₅⋊D₅

yes

characteristic

C₂×D₄

C₂×C₅⋊D₅

yes

characteristic

Q₁₆

—

SD₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×C₅⋊D₅

yes

characteristic

C₂×C₅

—

C₂×C₅

#4086

yes

—

C₂×C₅

#4097

yes

—

C₂³

—

Q₈

—

Q₈

—

Q₈

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊(C₅⋊D₅)

yes

—

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

C₈

—

C₅

#65349

yes

complemented

C₂²

—

C₂²

—

C₂²

C₂×C₂²⋊(C₅⋊D₅)

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

100

—

103

C₄

D₄×C₅⋊D₅

yes

—

104

C₂

—

105

C₂

#11717260

yes

characteristic

106

C₂

#11717271

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊(C₅⋊D₅)
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂²×C₅⋊D₅
Fitting	C₅²×#124	C₂
socle	C₂²×C₅²	C₂×D₄
prime radical	C₂×C₅²×D₄	C₂²
hypercenter	#124	C₅⋊D₅
hypocenter	C₅²	#497
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #124

subgroup

G/H

center

C₂²

C₂×C₂²⋊(C₅⋊D₅)

commutator, cosocle

C₂×C₄×C₅²

C₂³

Frattini

C₂×C₄

C₂²×C₅⋊D₅

Fitting

C₅²×#124

C₂

socle

C₂²×C₅²

C₂×D₄

prime radical

C₂×C₅²×D₄

C₂²

C₅⋊D₅

C₅²

G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #124

N	H ≅ G/N	type
2. C₂×#87812	C₂	semidirect
4. #11717189	C₂	semidirect
5. #11716670	C₂	semidirect
66. C₅²	#497	semidirect
96. C₅	#65349	semidirect

H ≅ G/N

type

2. C₂×#87812

C₂

semidirect

4. #11717189

C₂

semidirect

5. #11716670

C₂

semidirect

66. C₅²

#497

semidirect

96. C₅

#65349

semidirect

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

subgroup	index
C₅²×#124	2
C₂×#87812	2
C₂×#87810	2
#11717189	2
#11716670	2
#65349	5

subgroup

index

C₅²×#124

Subgroups — #13102920

Conjugacy classes of subgroups

Complements

Maximal subgroups