Group #11930571 — subgroups

Subgroups — C₅×#7851 #11930571

Order 1280.

610 conjugacy classes of subgroups; only the 286 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 286 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×C₅×#431	C₂	1	3	yes	maximal
2	C₅×#1273	C₂	1	4	yes	maximal, complemented
3	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
4	C₅×#431	—	2	12	no	—
5	C₂×C₅×C₄⋊C₈	C₄	1	6	yes	—
6	C₂×C₅×C₂²⋊C₈	C₄	1	6	yes	—
7	C₂×C₄×C₅×C₈	C₂²	1	6	yes	—
8	#7851	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂³×C₄×C₅	C₂×C₄	1	3	yes	—
10	C₂²×C₅×C₈	C₂×C₄	1	12	yes	—
11	C₂×C₄²×C₅	C₂×C₄	1	3	yes	—
12	C₂×C₄²×C₅	D₄	1	6	yes	—
13	C₂×C₄²×C₅	Q₈	1	2	yes	—
14	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
15	C₅×C₄⋊C₈	—	2	12	no	—
16	C₅×C₂²⋊C₈	—	2	12	no	—
17	C₄×C₅×C₈	—	2	12	no	—
18	C₂×#431	C₂×C₅	1	3	yes	—
19	#1273	C₂×C₅	1	4	yes	complemented
20	C₂⁴×C₅	C₄²	1	1	yes	characteristic
21	C₂²×C₄×C₅	—	2	12	no	—
22	C₂²×C₄×C₅	C₄²	1	3	yes	—
23	C₂²×C₄×C₅	C₂²⋊C₄	1	6	yes	—
24	C₂²×C₄×C₅	C₄⋊C₄	1	6	yes	—
25	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
26	C₂×C₅×C₈	—	2	12	no	—
27	C₂×C₅×C₈	—	2	24	no	—
28	C₄²×C₅	C₂²⋊C₄	1	6	yes	—
29	C₄²×C₅	C₄⋊C₄	1	6	yes	—
30	C₄²×C₅	C₂×D₄	1	3	yes	—
31	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
32	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
33	#431	—	2	12	no	—
34	C₂×C₄⋊C₈	C₄×C₅	1	6	yes	—
35	C₂×C₂²⋊C₈	C₄×C₅	1	6	yes	—
36	C₂×C₄×C₈	C₂²×C₅	1	6	yes	—
37	C₂³×C₅	—	2	6	no	—
38	C₂³×C₅	#95	1	2	yes	—
39	C₂³×C₅	C₂×C₄²	1	1	yes	characteristic
40	C₂×C₄×C₅	—	2	6	no	—
41	C₂×C₄×C₅	—	2	12	no	—
42	C₂×C₄×C₅	—	2	12	no	—
43	C₂×C₄×C₅	#95	1	6	yes	—
44	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	3	yes	—
45	C₂×C₄×C₅	C₂×C₄⋊C₄	1	3	yes	—
46	C₂×C₄×C₅	#131	1	12	yes	—
47	C₅×C₈	—	4	12	no	—
48	C₂³×C₄	C₂×C₄×C₅	1	3	yes	—
49	C₂²×C₈	C₂×C₄×C₅	1	12	yes	—
50	C₂×C₄²	C₂×C₄×C₅	1	3	yes	—
51	C₂×C₄²	C₅×D₄	1	6	yes	—
52	C₂×C₄²	C₅×Q₈	1	2	yes	—
53	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
54	C₄⋊C₈	—	2	12	no	—
55	C₂²⋊C₈	—	2	12	no	—
56	C₄×C₈	—	2	12	no	—
57	C₂²×C₅	—	2	6	no	—
58	C₂²×C₅	—	4	4	no	—
59	C₂²×C₅	C₂×#95	1	1	yes	characteristic
60	C₂²×C₅	#405	1	6	yes	—
61	C₄×C₅	—	2	6	no	—
62	C₄×C₅	#408	1	6	yes	—
63	C₄×C₅	#424	1	6	yes	—
64	C₂⁴	C₄²×C₅	1	1	yes	characteristic
65	C₂²×C₄	—	2	12	no	—
66	C₂²×C₄	C₄²×C₅	1	3	yes	—
67	C₂²×C₄	C₅×C₂²⋊C₄	1	6	yes	—
68	C₂²×C₄	C₅×C₄⋊C₄	1	6	yes	—
69	C₂²×C₄	C₂²×C₄×C₅	1	3	yes	—
70	C₂×C₈	—	2	12	no	—
71	C₂×C₈	—	2	24	no	—
72	C₄²	C₅×C₂²⋊C₄	1	6	yes	—
73	C₄²	C₅×C₄⋊C₄	1	6	yes	—
74	C₄²	C₂×C₅×D₄	1	3	yes	—
75	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
76	C₂×C₅	—	4	2	no	—
77	C₂×C₅	#1454	1	4	yes	—
78	C₂×C₅	#1729	1	3	yes	—
79	C₂³	—	2	6	no	—
80	C₂³	C₅×#95	1	2	yes	—
81	C₂³	C₂×C₄²×C₅	1	1	yes	characteristic
82	C₂×C₄	—	2	6	no	—
83	C₂×C₄	—	2	12	no	—
84	C₂×C₄	—	2	12	no	—
85	C₂×C₄	C₅×#95	1	6	yes	—
86	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	3	yes	—
87	C₂×C₄	C₂×C₅×C₄⋊C₄	1	3	yes	—
88	C₂×C₄	C₅×#131	1	12	yes	—
89	C₈	—	4	12	no	—
90	C₅	#7851	1	1	yes	characteristic, direct product factor, Sylow
91	C₂²	—	2	6	no	—
92	C₂²	—	4	4	no	—
93	C₂²	C₂×C₅×#95	1	1	yes	characteristic
94	C₂²	C₅×#405	1	6	yes	—
95	C₄	—	2	6	no	—
96	C₄	C₅×#408	1	6	yes	—
97	C₄	C₅×#424	1	6	yes	—
98	C₂	—	4	2	no	—
99	C₂	C₅×#1454	1	4	yes	—
100	C₂	C₅×#1729	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅×#431

C₂

yes

maximal

C₅×#1273

C₂

yes

maximal, complemented

C₂²×C₄²×C₅

C₂²

yes

characteristic

C₅×#431

—

C₂×C₅×C₄⋊C₈

C₄

yes

—

C₂×C₅×C₂²⋊C₈

C₄

yes

—

C₂×C₄×C₅×C₈

C₂²

yes

—

#7851

C₅

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₄×C₅

C₂×C₄

yes

—

C₂²×C₅×C₈

C₂×C₄

yes

—

C₂×C₄²×C₅

C₂×C₄

yes

—

C₂×C₄²×C₅

D₄

yes

—

C₂×C₄²×C₅

Q₈

yes

—

C₂×C₄²×C₅

C₂³

yes

characteristic

C₅×C₄⋊C₈

—

C₅×C₂²⋊C₈

—

C₄×C₅×C₈

—

C₂×#431

C₂×C₅

yes

—

#1273

C₂×C₅

yes

complemented

C₂⁴×C₅

C₄²

yes

characteristic

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₄²

yes

—

C₂²×C₄×C₅

C₂²⋊C₄

yes

—

C₂²×C₄×C₅

C₄⋊C₄

yes

—

C₂²×C₄×C₅

C₂²×C₄

yes

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₄²×C₅

C₂²⋊C₄

yes

—

C₄²×C₅

C₄⋊C₄

yes

—

C₄²×C₅

C₂×D₄

yes

—

C₄²×C₅

C₂×Q₈

yes

characteristic

C₂²×C₄²

C₂²×C₅

yes

characteristic

#431

—

C₂×C₄⋊C₈

C₄×C₅

yes

—

C₂×C₂²⋊C₈

C₄×C₅

yes

—

C₂×C₄×C₈

C₂²×C₅

yes

—

C₂³×C₅

—

C₂³×C₅

#95

yes

—

C₂³×C₅

C₂×C₄²

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#95

yes

—

C₂×C₄×C₅

C₂×C₂²⋊C₄

yes

—

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅

#131

yes

—

C₅×C₈

—

C₂³×C₄

C₂×C₄×C₅

yes

—

C₂²×C₈

C₂×C₄×C₅

yes

—

C₂×C₄²

C₂×C₄×C₅

yes

—

C₂×C₄²

C₅×D₄

yes

—

C₂×C₄²

C₅×Q₈

yes

—

C₂×C₄²

C₂³×C₅

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₂×#95

yes

characteristic

C₂²×C₅

#405

yes

—

C₄×C₅

—

C₄×C₅

#408

yes

—

C₄×C₅

#424

yes

—

C₂⁴

C₄²×C₅

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₄²×C₅

yes

—

C₂²×C₄

C₅×C₂²⋊C₄

yes

—

C₂²×C₄

C₅×C₄⋊C₄

yes

—

C₂²×C₄

C₂²×C₄×C₅

yes

—

C₂×C₈

—

C₂×C₈

—

C₄²

C₅×C₂²⋊C₄

yes

—

C₄²

C₅×C₄⋊C₄

yes

—

C₄²

C₂×C₅×D₄

yes

—

C₄²

C₂×C₅×Q₈

yes

characteristic

C₂×C₅

—

C₂×C₅

#1454

yes

—

C₂×C₅

#1729

yes

—

C₂³

—

C₂³

C₅×#95

yes

—

C₂³

C₂×C₄²×C₅

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₅×#95

yes

—

C₂×C₄

C₂×C₅×C₂²⋊C₄

yes

—

C₂×C₄

C₂×C₅×C₄⋊C₄

yes

—

C₂×C₄

C₅×#131

yes

—

C₈

—

C₅

#7851

yes

characteristic, direct product factor, Sylow

C₂²

—

C₂²

—

C₂²

C₂×C₅×#95

yes

characteristic

C₂²

C₅×#405

yes

—

C₄

—

C₄

C₅×#408

yes

—

C₄

C₅×#424

yes

—

C₂

—

C₂

C₅×#1454

yes

—

100

C₂

C₅×#1729

yes

—

	subgroup	G/H
center	C₂×C₄²×C₅	C₂³
commutator	C₂³	C₂×C₄²×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂³×C₅	C₂×C₄²
prime radical	C₂⁴×C₅	C₄²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₅ ◁ G

subgroup

G/H

center

C₂×C₄²×C₅

C₂³

commutator

C₂³

C₂×C₄²×C₅

Frattini, cosocle

C₂×C₄²

C₂³×C₅

socle

C₂³×C₅

C₂×C₄²

prime radical

C₂⁴×C₅

C₄²

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂×C₄²×C₅ ◁ G

N	H ≅ G/N	type
2. C₅×#1273	C₂	semidirect
8. #7851	C₅	direct (N × H)
19. #1273	C₂×C₅	semidirect
90. C₅	#7851	direct (N × H)

H ≅ G/N

type

2. C₅×#1273

C₂

semidirect

8. #7851

C₅

direct (N × H)

19. #1273

C₂×C₅

semidirect

90. C₅

#7851

direct (N × H)

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

subgroup	index
C₂×C₅×#431	2
C₅×#1273	2
#7851	5

subgroup

index

C₂×C₅×#431

C₅×#1273

#7851

Subgroups — C₅×#7851 #11930571

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C5×#7851 #11930571

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#7851 #11930571