Group #10604584 — subgroups

Subgroups — #10604584

Order 640.

202 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 60 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	#66022	C₂	1	1	yes	maximal, characteristic
2	#65678	C₂	1	2	yes	maximal
3	#64873	C₂	1	4	yes	maximal, complemented
4	C₅×#125	C₄	1	1	yes	characteristic
5	#4042	—	2	2	no	—
6	#4038	C₄	1	2	yes	—
7	#4034	C₂²	1	2	yes	—
8	#4033	—	2	2	no	—
9	C₄²×D₅	C₂²	1	1	yes	characteristic
10	#4016	C₄	1	1	yes	characteristic
11	#4004	C₂²	1	4	yes	—
12	#4003	—	2	2	no	—
13	C₄×#170	—	2	2	no	—
14	#1544	—	5	1	no	maximal, Sylow
15	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×D₅	D₄	1	2	yes	—
17	C₂×Dic₁₀	—	2	2	no	—
18	C₂×#170	—	2	4	no	—
19	C₅×C₄⋊C₄	—	2	2	no	—
20	C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
21	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
22	#685	—	2	2	no	—
23	C₄⋊Dic₅	C₂×C₄	1	2	yes	—
24	#683	—	2	2	no	—
25	#683	—	2	2	no	—
26	C₄×Dic₅	—	2	2	no	—
27	C₄×Dic₅	D₄	1	2	yes	—
28	C₄×Dic₅	C₂³	1	1	yes	characteristic
29	#533	—	5	1	no	—
30	#431	—	5	2	no	—
31	#332	—	5	4	no	—
32	C₂²×D₅	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₅	—	2	2	no	—
34	C₂×C₄×C₅	C₂²⋊C₄	1	2	yes	—
35	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
36	C₂×Dic₅	—	2	2	no	—
37	C₂×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×Dic₅	C₂×D₄	1	2	yes	—
39	C₄×D₅	—	2	2	no	—
40	C₄×D₅	—	2	4	no	—
41	Dic₁₀	—	4	4	no	—
42	#170	—	4	4	no	—
43	C₄⋊Q₈	—	5	2	no	—
44	#125	—	5	1	no	—
45	#125	AGL(1,𝔽₅)	1	1	yes	characteristic
46	C₂²⋊Q₈	—	10	2	no	—
47	C₄×Q₈	—	10	2	no	—
48	#117	—	5	2	no	—
49	C₂×C₄²	—	5	1	no	—
50	C₄⋊C₈	—	5	4	no	—
51	C₂²⋊C₈	—	10	2	no	—
52	C₄×C₈	—	10	2	no	—
53	C₂²×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂×D₅	—	2	1	no	—
55	C₂×D₅	—	2	2	no	—
56	C₄×C₅	—	2	1	no	—
57	C₄×C₅	—	2	2	no	—
58	C₄×C₅	—	4	2	no	—
59	Dic₅	—	4	2	no	—
60	Dic₅	#101	1	2	yes	—
61	Dic₅	#135	1	4	yes	—
62	C₂×Q₈	—	10	2	no	—
63	C₂²×C₄	—	5	2	no	—
64	C₂²×C₄	—	5	1	no	—
65	C₂×C₈	—	10	4	no	—
66	C₄⋊C₄	—	2	2	no	—
67	C₄⋊C₄	—	5	2	no	—
68	C₄⋊C₄	—	10	2	no	—
69	C₄⋊C₄	—	10	2	no	—
70	C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
71	C₂²⋊C₄	—	10	2	no	—
72	C₄²	—	5	2	no	—
73	C₄²	—	5	1	no	—
74	C₄²	—	10	2	no	—
75	C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
76	C₂×C₅	C₂×#101	1	1	yes	characteristic
77	C₂×C₅	#416	1	2	yes	—
78	D₅	—	4	1	no	—
79	C₂³	—	5	1	no	—
80	Q₈	—	20	4	no	—
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	5	1	no	—
83	C₂×C₄	—	5	2	no	—
84	C₂×C₄	—	10	4	no	—
85	C₂×C₄	—	10	2	no	—
86	C₂×C₄	—	10	2	no	—
87	C₂×C₄	C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
88	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
89	C₈	—	20	4	no	—
90	C₅	#1544	1	1	yes	characteristic, complemented, Sylow
91	C₂²	—	10	2	no	—
92	C₂²	—	10	1	no	—
93	C₂²	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
94	C₄	—	2	2	no	—
95	C₄	—	2	1	no	—
96	C₄	—	4	2	no	—
97	C₄	—	5	4	no	—
98	C₄	—	5	2	no	—
99	C₄	—	20	2	no	—
100	C₂	—	20	1	no	—
101	C₂	D₅⋊#101	1	1	yes	characteristic
102	C₂	#65794	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#66022

C₂

yes

maximal, characteristic

#65678

C₂

yes

maximal

#64873

C₂

yes

maximal, complemented

C₅×#125

C₄

yes

characteristic

#4042

—

#4038

C₄

yes

—

#4034

C₂²

yes

—

#4033

—

C₄²×D₅

C₂²

yes

characteristic

#4016

C₄

yes

characteristic

#4004

C₂²

yes

—

—

—

—

C₂×C₄×D₅

C₂×C₄

yes

characteristic

C₂×C₄×D₅

D₄

yes

—

C₂×Dic₁₀

—

C₂×#170

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

C₂×C₄

yes

—

C₄²×C₅

C₂×C₄

yes

characteristic

#685

—

C₄⋊Dic₅

C₂×C₄

yes

—

#683

—

#683

—

C₄×Dic₅

—

C₄×Dic₅

D₄

yes

—

C₄×Dic₅

C₂³

yes

—

—

—

C₂²×D₅

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²⋊C₄

yes

—

C₂×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

C₂²⋊C₄

yes

characteristic

C₂×Dic₅

C₂×D₄

yes

—

C₄×D₅

—

C₄×D₅

—

Dic₁₀

—

#170

—

C₄⋊Q₈

—

—

yes

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₂²×C₅

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

Dic₅

—

Dic₅

#101

yes

—

Dic₅

#135

yes

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×AGL(1,𝔽₅)

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

yes

C₂×C₅

yes

C₂×C₅

yes

—

D₅

—

C₂³

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊AGL(1,𝔽₅)

yes

—

C₂×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

C₈

—

C₅

#1544

yes

characteristic, complemented, Sylow

C₂²

—

C₂²

—

C₂²

C₂×C₂²⋊AGL(1,𝔽₅)

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

100

C₂

—

101

C₂

D₅⋊#101

yes

characteristic

102

C₂

#65794

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊AGL(1,𝔽₅)
commutator	C₂×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#125	C₄
socle	C₂²×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×D₅	C₂²⋊C₄
cosocle	C₄×Dic₅	C₂³
hypercenter	#125	AGL(1,𝔽₅)
hypocenter	C₅	#1544
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

subgroup

G/H

center

C₂²

C₂×C₂²⋊AGL(1,𝔽₅)

commutator

C₂×C₄×C₅

C₂²×C₄

Frattini

C₂×C₄

C₂²×AGL(1,𝔽₅)

Fitting

C₅×#125

C₄

socle

C₂²×C₅

C₂×C₂²⋊C₄

prime radical

C₂²×D₅

C₂²⋊C₄

cosocle

C₄×Dic₅

C₂³

C₅

G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #125

N	H ≅ G/N	type
3. #64873	C₂	semidirect
90. C₅	#1544	semidirect

H ≅ G/N

type

3. #64873

C₂

semidirect

90. C₅

#1544

semidirect

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

subgroup	index
#66022	2
#65678	2
#64873	2
#1544	5

subgroup

index

Subgroups — #10604584

Conjugacy classes of subgroups

Complements

Maximal subgroups