Group #10604657 — subgroups

Subgroups — D₅⋊#333 #10604657

Order 640.

256 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 84 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×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	D₅⋊(C₄⋊C₈)	C₂	1	2	yes	maximal
3	#64931	C₂	1	4	yes	maximal, complemented
4	C₂×Q₈×D₅	C₄	1	2	yes	—
5	C₂×D₅⋊C₈	—	2	2	no	—
6	C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic
7	#4090	C₂²	1	2	yes	—
8	D₅×C₄⋊C₄	—	2	2	no	—
9	#4034	—	2	2	no	—
10	C₄²×D₅	C₂²	1	1	yes	characteristic
11	#4015	C₄	1	1	yes	characteristic
12	#4004	C₂²	1	4	yes	—
13	C₄⋊#170	—	2	2	no	—
14	C₂×#333	—	5	1	no	maximal, Sylow
15	C₂×C₅×Q₈	C₂×C₄	1	2	yes	—
16	Q₈×D₅	—	4	4	no	—
17	C₂×C₄×D₅	—	2	2	no	—
18	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×D₅	D₄	1	2	yes	—
20	C₂×Dic₁₀	—	2	2	no	—
21	C₂×Dic₁₀	C₂×C₄	1	2	yes	—
22	C₂×#170	—	2	4	no	—
23	D₅⋊C₈	—	4	4	no	—
24	C₅×C₄⋊C₄	—	2	2	no	—
25	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
26	C₄⋊Dic₅	—	2	2	no	—
27	#683	—	2	4	no	—
28	C₄×Dic₅	D₄	1	2	yes	—
29	C₄×Dic₅	C₂³	1	1	yes	characteristic
30	C₂×C₄⋊Q₈	—	5	1	no	—
31	C₂×C₄⋊C₈	—	5	2	no	—
32	#333	—	5	4	no	—
33	C₂²×D₅	C₂²⋊C₄	1	1	yes	characteristic
34	C₅×Q₈	—	4	2	no	—
35	C₂×C₄×C₅	—	2	2	no	—
36	C₂×C₄×C₅	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
38	C₂×Dic₅	—	2	2	no	—
39	C₂×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×Dic₅	C₂×D₄	1	2	yes	—
41	C₄×D₅	—	2	2	no	—
42	C₄×D₅	—	4	4	no	—
43	C₄×D₅	SD₁₆	1	4	yes	—
44	C₄×D₅	Q₁₆	1	4	yes	—
45	Dic₁₀	—	4	4	no	—
46	Dic₁₀	—	4	2	no	—
47	#170	—	4	4	no	—
48	C₂²×Q₈	—	5	2	no	—
49	C₂²×C₈	—	10	2	no	—
50	C₄⋊Q₈	—	5	2	no	—
51	C₄⋊Q₈	—	5	1	no	—
52	C₄⋊Q₈	—	10	2	no	—
53	C₄⋊Q₈	AGL(1,𝔽₅)	1	1	yes	characteristic
54	C₂×C₄⋊C₄	—	10	2	no	—
55	C₂×C₄²	—	5	1	no	—
56	C₄⋊C₈	—	5	4	no	—
57	C₄⋊C₈	—	10	2	no	—
58	C₂²×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂×D₅	#100	1	2	yes	—
60	C₂×D₅	#103	1	4	yes	—
61	C₄×C₅	—	2	1	no	—
62	C₄×C₅	—	4	2	no	—
63	C₄×C₅	#103	1	4	yes	—
64	Dic₅	—	2	1	no	—
65	Dic₅	—	4	2	no	—
66	Dic₅	C₂×SD₁₆	1	2	yes	—
67	Dic₅	C₂×Q₁₆	1	2	yes	—
68	C₂×Q₈	—	5	2	no	—
69	C₂×Q₈	—	10	2	no	—
70	C₂×Q₈	—	20	4	no	—
71	C₂×Q₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
72	C₂²×C₄	—	5	2	no	—
73	C₂²×C₄	—	5	1	no	—
74	C₂²×C₄	—	10	2	no	—
75	C₂×C₈	—	10	4	no	—
76	C₂×C₈	—	20	4	no	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄⋊C₄	—	10	4	no	—
79	C₄⋊C₄	—	10	2	no	—
80	C₄²	—	5	2	no	—
81	C₄²	—	5	1	no	—
82	C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
83	C₂×C₅	C₂×#100	1	1	yes	characteristic
84	C₂×C₅	C₂×#103	1	2	yes	—
85	D₅	#333	1	4	yes	complemented
86	C₂³	—	5	1	no	—
87	Q₈	—	4	2	no	—
88	Q₈	—	20	2	no	—
89	Q₈	—	20	4	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	5	2	no	—
92	C₂×C₄	—	5	4	no	—
93	C₂×C₄	—	5	4	no	—
94	C₂×C₄	—	5	1	no	—
95	C₂×C₄	—	10	2	no	—
96	C₂×C₄	—	10	2	no	—
97	C₂×C₄	—	20	4	no	—
98	C₂×C₄	C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
99	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
100	C₈	—	20	4	no	—
101	C₅	C₂×#333	1	1	yes	characteristic, complemented, Sylow
102	C₂²	—	5	2	no	—
103	C₂²	—	5	4	no	—
104	C₂²	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
105	C₄	—	2	1	no	—
106	C₄	—	4	2	no	—
107	C₄	—	5	2	no	—
108	C₄	—	5	2	no	—
109	C₄	—	10	1	no	—
110	C₄	—	20	2	no	—
111	C₄	#4009	1	4	yes	—
112	C₂	—	5	4	no	—
113	C₂	D₅⋊#100	1	1	yes	characteristic
114	C₂	C₂×#4009	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

D₅×C₄⋊Q₈

C₂

yes

maximal, characteristic

D₅⋊(C₄⋊C₈)

C₂

yes

maximal

#64931

C₂

yes

maximal, complemented

C₂×Q₈×D₅

C₄

yes

—

C₂×D₅⋊C₈

—

C₅×C₄⋊Q₈

C₄

yes

characteristic

#4090

C₂²

yes

—

D₅×C₄⋊C₄

—

#4034

—

C₄²×D₅

C₂²

yes

characteristic

#4015

C₄

yes

characteristic

#4004

C₂²

yes

—

C₄⋊#170

—

C₂×#333

—

maximal, Sylow

C₂×C₅×Q₈

C₂×C₄

yes

—

Q₈×D₅

—

C₂×C₄×D₅

—

C₂×C₄×D₅

C₂×C₄

yes

characteristic

C₂×C₄×D₅

D₄

yes

—

C₂×Dic₁₀

—

C₂×Dic₁₀

C₂×C₄

yes

—

C₂×#170

—

D₅⋊C₈

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₂×C₄

yes

characteristic

C₄⋊Dic₅

—

#683

—

C₄×Dic₅

D₄

yes

—

C₄×Dic₅

C₂³

yes

characteristic

C₂×C₄⋊Q₈

—

C₂×C₄⋊C₈

—

#333

—

C₂²×D₅

C₂²⋊C₄

yes

characteristic

C₅×Q₈

—

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

—

C₄×D₅

SD₁₆

yes

—

C₄×D₅

Q₁₆

yes

—

Dic₁₀

—

Dic₁₀

—

#170

—

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

AGL(1,𝔽₅)

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²×C₅

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₅

#100

yes

—

C₂×D₅

#103

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#103

yes

—

Dic₅

—

Dic₅

—

Dic₅

C₂×SD₁₆

yes

—

Dic₅

C₂×Q₁₆

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×AGL(1,𝔽₅)

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

yes

C₂×C₅

yes

C₂×C₅

yes

—

D₅

#333

yes

complemented

C₂³

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

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

100

C₈

—

101

C₅

C₂×#333

yes

characteristic, complemented, Sylow

102

C₂²

—

103

C₂²

—

104

C₂²

C₂×C₂²⋊AGL(1,𝔽₅)

yes

characteristic

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

#4009

yes

—

112

C₂

—

113

C₂

D₅⋊#100

yes

characteristic

114

C₂

C₂×#4009

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×C₄⋊Q₈	C₄
socle	C₂²×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×D₅	C₂²⋊C₄
cosocle	C₄×Dic₅	C₂³
hypercenter	C₄⋊Q₈	AGL(1,𝔽₅)
hypocenter	C₅	C₂×#333
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

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×C₄⋊Q₈

C₄

socle

C₂²×C₅

C₂×C₂²⋊C₄

prime radical

C₂²×D₅

C₂²⋊C₄

cosocle

C₄×Dic₅

C₂³

C₄⋊Q₈

C₅

G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
3. #64931	C₂	semidirect
85. D₅	#333	semidirect
101. C₅	C₂×#333	semidirect

H ≅ G/N

type

3. #64931

C₂

semidirect

85. D₅

#333

semidirect

101. C₅

C₂×#333

semidirect

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

subgroup	index
D₅×C₄⋊Q₈	2
D₅⋊(C₄⋊C₈)	2
#64931	2
C₂×#333	5

subgroup

index

D₅×C₄⋊Q₈

D₅⋊(C₄⋊C₈)

#64931

C₂×#333

Subgroups — D₅⋊#333 #10604657

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D5⋊#333 #10604657

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅⋊#333 #10604657