Group #12014005 — subgroups

Subgroups — #12014005

Order 1280.

290 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 79 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	C₅×#1710	C₂	1	1	yes	maximal, characteristic, complemented
2	#10602174	C₂	1	4	yes	maximal, complemented
3	#10598855	C₂	1	1	yes	maximal, characteristic
4	#10597928	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#500	C₂²	1	2	yes	—
6	C₅×#487	—	2	2	no	—
7	#64984	—	2	2	no	—
8	#64984	C₂²	1	2	yes	—
9	C₅×C₈²	C₂²	1	1	yes	characteristic
10	#64711	—	2	8	no	—
11	C₄×#672	C₂²	1	2	yes	—
12	#12513	—	5	1	no	maximal, Sylow
13	C₅×C₄⋊Q₈	—	2	2	no	—
14	C₂×D₄₀	—	2	4	no	—
15	C₂×D₄₀	—	4	4	no	—
16	#4020	—	2	2	no	—
17	C₅×#107	—	2	4	no	—
18	C₅×#103	—	4	4	no	—
19	C₄×C₅×C₈	D₄	1	2	yes	—
20	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
21	C₂×#672	D₄	1	4	yes	—
22	#2240	—	5	4	no	—
23	#1712	—	5	1	no	—
24	#1710	D₅	1	1	yes	characteristic, complemented
25	#1311	—	5	1	no	—
26	C₂×C₅×Q₈	—	4	2	no	—
27	C₂×D₂₀	—	4	2	no	—
28	C₂×D₂₀	—	4	4	no	—
29	C₂×C₅×C₈	D₈	1	4	yes	—
30	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
31	C₅×C₄⋊C₄	—	4	2	no	—
32	C₅×C₄⋊C₄	—	4	2	no	—
33	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
34	D₄₀	—	4	8	no	—
35	D₄₀	—	8	8	no	—
36	#672	—	2	4	no	—
37	#500	C₂×D₅	1	2	yes	—
38	#493	—	5	2	no	—
39	#493	—	10	2	no	—
40	#487	—	2	2	no	—
41	#357	—	10	8	no	—
42	C₄×C₁₆	—	5	2	no	—
43	C₈²	C₂×D₅	1	1	yes	characteristic
44	C₂²×D₅	—	8	2	no	—
45	C₅×Q₈	—	8	4	no	—
46	C₂×C₄×C₅	—	8	2	no	—
47	C₂×C₄×C₅	#127	1	1	yes	characteristic
48	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
49	D₂₀	—	8	4	no	—
50	D₂₀	—	8	8	no	—
51	C₅×C₈	—	2	4	no	—
52	C₅×C₈	#135	1	4	yes	—
53	C₂×D₈	—	10	4	no	—
54	C₂×D₈	—	20	4	no	—
55	C₄⋊Q₈	—	2	2	no	—
56	#127	—	10	2	no	—
57	C₂×C₁₆	—	5	4	no	—
58	#107	—	2	4	no	—
59	#103	—	4	4	no	—
60	C₄×C₈	C₂²⋊D₅	1	2	yes	—
61	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
62	C₂²×C₅	#493	1	1	yes	characteristic
63	C₂×D₅	—	16	2	no	—
64	C₂×D₅	—	16	4	no	—
65	C₄×C₅	—	16	2	no	—
66	C₄×C₅	#495	1	2	yes	—
67	C₄×C₅	#509	1	4	yes	—
68	C₂×Q₈	—	4	2	no	—
69	C₂×D₄	—	20	4	no	—
70	C₂×D₄	—	20	2	no	—
71	D₈	—	20	8	no	—
72	D₈	—	40	8	no	—
73	C₂×C₈	#686	1	4	yes	—
74	C₂×C₈	D₄×D₅	1	2	yes	—
75	C₄⋊C₄	—	4	2	no	—
76	C₄⋊C₄	—	4	2	no	—
77	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
78	C₁₆	—	10	4	no	—
79	C₂×C₅	#1709	1	1	yes	characteristic
80	C₂×C₅	#2250	1	2	yes	—
81	D₅	—	32	2	no	—
82	C₂³	—	40	2	no	—
83	Q₈	—	8	4	no	—
84	D₄	—	40	4	no	—
85	D₄	—	40	8	no	—
86	C₂×C₄	—	8	2	no	—
87	C₂×C₄	C₂×#686	1	2	yes	—
88	C₂×C₄	#4086	1	1	yes	characteristic
89	C₈	—	2	4	no	—
90	C₈	#4065	1	4	yes	—
91	C₅	#12513	1	1	yes	characteristic, complemented, Sylow
92	C₂²	—	80	2	no	—
93	C₂²	—	80	4	no	—
94	C₂²	#65362	1	1	yes	characteristic
95	C₄	—	16	2	no	—
96	C₄	#65480	1	2	yes	—
97	C₄	#65482	1	4	yes	—
98	C₂	—	160	2	no	—
99	C₂	#10602090	1	1	yes	characteristic
100	C₂	#10602114	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#1710

C₂

yes

maximal, characteristic, complemented

#10602174

C₂

yes

maximal, complemented

#10598855

C₂

yes

maximal, characteristic

#10597928

C₂

yes

maximal, characteristic, complemented

C₅×#500

C₂²

yes

—

C₅×#487

—

#64984

—

#64984

C₂²

yes

—

C₅×C₈²

C₂²

yes

characteristic

#64711

—

C₄×#672

C₂²

yes

—

#12513

—

maximal, Sylow

C₅×C₄⋊Q₈

—

C₂×D₄₀

—

C₂×D₄₀

—

#4020

—

C₅×#107

—

C₅×#103

—

C₄×C₅×C₈

D₄

yes

—

C₄×C₅×C₈

C₂³

yes

characteristic

C₂×#672

D₄

yes

—

#2240

—

#1712

—

#1710

D₅

yes

characteristic, complemented

#1311

—

C₂×C₅×Q₈

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×C₅×C₈

D₈

yes

—

C₂×C₅×C₈

C₂×D₄

yes

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₂×D₄

yes

characteristic

D₄₀

—

D₄₀

—

#672

—

#500

C₂×D₅

yes

—

—

—

—

—

C₄×C₁₆

—

C₈²

C₂×D₅

yes

characteristic

C₂²×D₅

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#127

yes

characteristic

C₂×C₄×C₅

C₂×D₈

yes

—

D₂₀

—

D₂₀

—

C₅×C₈

—

C₅×C₈

#135

yes

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

—

#107

—

#103

—

C₄×C₈

C₂²⋊D₅

yes

—

C₄×C₈

C₂²×D₅

yes

characteristic

C₂²×C₅

#493

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

#495

yes

—

C₄×C₅

#509

yes

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

D₈

—

D₈

—

C₂×C₈

#686

yes

—

C₂×C₈

D₄×D₅

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₂²⋊D₅

yes

characteristic

C₁₆

—

C₂×C₅

#1709

yes

characteristic

C₂×C₅

#2250

yes

—

D₅

—

C₂³

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

C₂×#686

yes

—

C₂×C₄

#4086

yes

characteristic

C₈

—

C₈

#4065

yes

—

C₅

#12513

yes

characteristic, complemented, Sylow

C₂²

—

C₂²

—

C₂²

#65362

yes

characteristic

C₄

—

C₄

#65480

yes

—

C₄

#65482

yes

—

C₂

—

160

—

C₂

#10602090

yes

characteristic

100

C₂

#10602114

yes

—

	subgroup	G/H
center	C₂²	#65362
commutator, cosocle	C₄×C₅×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×#1710	C₂
socle	C₂²×C₅	#493
prime radical	#10598855	C₂
hypercenter	#1710	D₅
hypocenter	C₅	#12513
LCS	G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

subgroup

G/H

center

C₂²

#65362

commutator, cosocle

C₄×C₅×C₈

C₂³

Frattini

C₄×C₈

C₂²×D₅

C₂

C₂²×C₅

C₂

D₅

C₅

G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

N	H ≅ G/N	type
1. C₅×#1710	C₂	semidirect
2. #10602174	C₂	semidirect
4. #10597928	C₂	semidirect
24. #1710	D₅	semidirect
91. C₅	#12513	semidirect

H ≅ G/N

type

1. C₅×#1710

C₂

semidirect

2. #10602174

C₂

semidirect

4. #10597928

C₂

semidirect

24. #1710

D₅

semidirect

91. C₅

#12513

semidirect

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

subgroup	index
C₅×#1710	2
#10602174	2
#10598855	2
#10597928	2
#12513	5

subgroup

index

Subgroups — #12014005

Conjugacy classes of subgroups

Complements

Maximal subgroups