Group #10598789 — subgroups

Subgroups — #10598789

Order 640.

190 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 48 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	D₅⋊(C₂²⋊C₈)	C₂	1	2	yes	maximal
2	#65415	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×D₅	C₂²	1	1	yes	characteristic
4	C₂×D₅⋊C₈	—	2	2	no	—
5	C₂×C₅×M₁₆	C₄	1	1	yes	characteristic
6	C₂×#681	C₄	1	1	yes	characteristic
7	C₂²⋊#170	—	2	2	no	—
8	#4003	C₄	1	4	yes	—
9	#3952	—	2	2	no	—
10	#1298	—	5	1	no	maximal, Sylow
11	C₂³×D₅	Q₈	1	1	yes	characteristic
12	C₂²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
13	C₂²×Dic₅	D₄	1	1	yes	characteristic
14	C₂×C₄×D₅	—	4	1	no	—
15	C₂×C₄×D₅	—	4	1	no	—
16	C₂×C₄×D₅	C₂×C₄	1	2	yes	—
17	C₂×C₄×D₅	D₄	1	2	yes	—
18	C₂×#170	—	2	4	no	—
19	D₅⋊C₈	—	4	2	no	—
20	D₅⋊C₈	—	4	2	no	—
21	C₅×M₁₆	—	4	1	no	—
22	C₂×C₅×C₈	—	2	1	no	—
23	#681	—	4	1	no	—
24	C₂×#168	—	2	1	no	—
25	#407	—	5	1	no	—
26	C₂×C₂²⋊C₈	—	5	2	no	—
27	C₂³×C₅	C₄⋊C₄	1	1	yes	characteristic
28	C₂²×D₅	—	2	4	no	—
29	C₂²×D₅	—	4	1	no	—
30	C₂²×D₅	C₄⋊C₄	1	2	yes	—
31	C₂×C₄×C₅	—	2	1	no	—
32	C₂×C₄×C₅	—	2	1	no	—
33	C₂×C₄×C₅	C₄²	1	1	yes	characteristic
34	C₂×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×Dic₅	—	4	1	no	—
36	C₂×Dic₅	C₂²⋊C₄	1	2	yes	—
37	C₄×D₅	—	2	2	no	—
38	C₄×D₅	—	2	2	no	—
39	C₄×D₅	—	4	2	no	—
40	#170	—	4	4	no	—
41	C₅×C₈	—	4	1	no	—
42	#168	—	4	1	no	—
43	C₂³×C₄	—	5	1	no	—
44	C₂×M₁₆	—	5	1	no	—
45	C₂×M₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic
46	C₂²×C₈	—	10	2	no	—
47	C₂²⋊C₈	—	5	4	no	—
48	C₂²⋊C₈	—	10	2	no	—
49	C₂²⋊C₈	—	10	2	no	—
50	C₂²×C₅	—	2	2	no	—
51	C₂²×C₅	—	2	1	no	—
52	C₂²×C₅	#95	1	1	yes	characteristic
53	C₂×D₅	—	2	2	no	—
54	C₂×D₅	—	2	2	no	—
55	C₂×D₅	—	4	4	no	—
56	C₂×D₅	#108	1	4	yes	—
57	C₄×C₅	—	2	1	no	—
58	C₄×C₅	#99	1	1	yes	characteristic
59	C₄×C₅	#99	1	1	yes	characteristic
60	Dic₅	—	2	2	no	—
61	C₂⁴	—	5	1	no	—
62	C₂²×C₄	—	5	1	no	—
63	C₂²×C₄	—	5	2	no	—
64	C₂²×C₄	—	5	2	no	—
65	C₂²×C₄	—	20	1	no	—
66	C₂²×C₄	—	20	1	no	—
67	C₂²×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
68	M₁₆	—	4	1	no	—
69	M₁₆	—	20	1	no	—
70	C₂×C₈	—	2	1	no	—
71	C₂×C₈	—	10	4	no	—
72	C₂×C₈	—	10	1	no	—
73	C₂×C₈	—	20	2	no	—
74	C₂×C₈	—	20	2	no	—
75	C₂×C₅	—	4	1	no	—
76	C₂×C₅	#341	1	2	yes	—
77	C₂×C₅	#342	1	1	yes	characteristic
78	D₅	—	2	4	no	—
79	C₂³	—	5	2	no	—
80	C₂³	—	10	4	no	—
81	C₂³	—	20	1	no	—
82	C₂³	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	5	2	no	—
86	C₂×C₄	—	10	2	no	—
87	C₂×C₄	—	10	2	no	—
88	C₂×C₄	—	20	2	no	—
89	C₂×C₄	—	20	1	no	—
90	C₂×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
91	C₂×C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
92	C₈	—	4	1	no	—
93	C₈	—	20	4	no	—
94	C₈	—	20	1	no	—
95	C₅	#1298	1	1	yes	characteristic, complemented, Sylow
96	C₂²	—	2	2	no	—
97	C₂²	—	2	1	no	—
98	C₂²	—	5	4	no	—
99	C₂²	—	10	2	no	—
100	C₂²	—	10	2	no	—
101	C₂²	—	20	4	no	—
102	C₂²	(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	10	2	no	—
105	C₄	#3999	1	1	yes	characteristic
106	C₄	#4011	1	1	yes	characteristic
107	C₂	—	4	1	no	—
108	C₂	—	10	4	no	—
109	C₂	#64901	1	2	yes	—
110	C₂	#64917	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₅⋊(C₂²⋊C₈)

C₂

yes

maximal

#65415

C₂

yes

maximal, characteristic

C₂²×C₄×D₅

C₂²

yes

characteristic

C₂×D₅⋊C₈

—

C₂×C₅×M₁₆

C₄

yes

characteristic

C₂×#681

C₄

yes

characteristic

C₂²⋊#170

—

#4003

C₄

yes

—

#3952

—

#1298

—

maximal, Sylow

C₂³×D₅

Q₈

yes

characteristic

C₂²×C₄×C₅

C₂×C₄

yes

characteristic

C₂²×Dic₅

D₄

yes

characteristic

C₂×C₄×D₅

—

C₂×C₄×D₅

—

C₂×C₄×D₅

C₂×C₄

yes

—

C₂×C₄×D₅

D₄

yes

—

C₂×#170

—

D₅⋊C₈

—

D₅⋊C₈

—

C₅×M₁₆

—

C₂×C₅×C₈

—

#681

—

C₂×#168

—

#407

—

C₂×C₂²⋊C₈

—

C₂³×C₅

C₄⋊C₄

yes

characteristic

C₂²×D₅

—

C₂²×D₅

—

C₂²×D₅

C₄⋊C₄

yes

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₄²

yes

characteristic

C₂×C₄×C₅

C₂²⋊C₄

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

C₂²⋊C₄

yes

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

#170

—

C₅×C₈

—

#168

—

C₂³×C₄

—

C₂×M₁₆

—

C₂×M₁₆

AGL(1,𝔽₅)

yes

characteristic

C₂²×C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

#95

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

#108

yes

—

C₄×C₅

—

C₄×C₅

yes

C₄×C₅

yes

Dic₅

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×AGL(1,𝔽₅)

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₅

—

C₂×C₅

#341

yes

—

C₂×C₅

#342

yes

characteristic

D₅

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₄⋊AGL(1,𝔽₅)

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×AGL(1,𝔽₅)

yes

characteristic

C₂×C₄

C₂²⋊AGL(1,𝔽₅)

yes

characteristic

C₈

—

C₈

—

C₈

—

C₅

#1298

yes

characteristic, complemented, Sylow

C₂²

—

C₂²

—

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

—

102

C₂²

(C₂×AGL(1,𝔽₅))⋊C₄

yes

characteristic

103

C₄

—

104

C₄

—

105

C₄

#3999

yes

characteristic

106

C₄

#4011

yes

characteristic

107

C₂

—

108

C₂

—

109

C₂

#64901

yes

—

110

C₂

#64917

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊AGL(1,𝔽₅)
commutator	C₂×C₄×C₅	C₄²
Frattini	C₂²×C₄	C₂×AGL(1,𝔽₅)
Fitting	C₂×C₅×M₁₆	C₄
socle	C₂²×C₅	#95
prime radical	C₂³×D₅	Q₈
cosocle	C₂²×C₄×D₅	C₂²
hypercenter	C₂×M₁₆	AGL(1,𝔽₅)
hypocenter	C₅	#1298
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

subgroup

G/H

center

C₂×C₄

C₂²⋊AGL(1,𝔽₅)

commutator

C₂×C₄×C₅

C₄²

Frattini

C₂²×C₄

C₂×AGL(1,𝔽₅)

Fitting

C₂×C₅×M₁₆

C₄

socle

C₂²×C₅

#95

prime radical

C₂³×D₅

Q₈

cosocle

C₂²×C₄×D₅

C₂²

C₂×M₁₆

C₅

G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
95. C₅	#1298	semidirect

H ≅ G/N

type

95. C₅

#1298

semidirect

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

subgroup	index
D₅⋊(C₂²⋊C₈)	2
#65415	2
#1298	5

subgroup

index

D₅⋊(C₂²⋊C₈)

#65415

#1298

Subgroups — #10598789

Conjugacy classes of subgroups

Complements

Maximal subgroups