Group #10617337 — subgroups

Subgroups — #10617337

Order 640.

330 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 148 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	#65707	C₂	1	4	yes	maximal, complemented
3	#65706	C₂	1	4	yes	maximal
4	#65703	C₂	1	4	yes	maximal, complemented
5	#65678	C₂	1	2	yes	maximal
6	C₅×#125	C₄	1	1	yes	characteristic
7	#4042	C₄	1	4	yes	—
8	#4038	C₂²	1	2	yes	—
9	#4034	C₄	1	2	yes	—
10	#4033	C₂²	1	4	yes	—
11	C₄²×D₅	C₂²	1	1	yes	characteristic
12	#4016	C₄	1	1	yes	characteristic
13	#4004	C₂²	1	2	yes	—
14	#4004	C₂²	1	2	yes	—
15	#4004	C₂²	1	4	yes	—
16	#4003	C₂²	1	4	yes	—
17	#4002	C₂²	1	4	yes	—
18	C₄⋊#170	C₂²	1	4	yes	—
19	C₄×#170	C₂²	1	4	yes	—
20	C₄×#170	C₂²	1	4	yes	—
21	#2986	—	5	1	no	maximal, Sylow
22	C₂×C₄×D₅	C₂×C₄	1	2	yes	—
23	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
24	C₂×Dic₁₀	C₂×C₄	1	4	yes	—
25	C₂×#170	C₂³	1	4	yes	—
26	C₂×#170	C₂³	1	4	yes	—
27	C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
28	C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
29	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
30	#685	C₂×C₄	1	4	yes	—
31	C₄⋊Dic₅	C₂×C₄	1	2	yes	—
32	#683	C₂×C₄	1	4	yes	—
33	#683	C₂×C₄	1	4	yes	—
34	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
35	C₄×Dic₅	C₂³	1	2	yes	—
36	C₄×Dic₅	C₂³	1	4	yes	—
37	#533	—	5	1	no	—
38	#446	—	5	4	no	—
39	C₈×Q₈	—	5	4	no	—
40	#433	—	5	4	no	—
41	#431	—	5	2	no	—
42	C₂²×D₅	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
45	C₂×C₄×C₅	C₂²×C₄	1	4	yes	—
46	C₂×Dic₅	C₂²×C₄	1	4	yes	—
47	C₂×Dic₅	C₂²×C₄	1	2	yes	—
48	C₂×Dic₅	C₂⁴	1	1	yes	characteristic
49	C₄×D₅	—	2	4	no	—
50	C₄×D₅	—	2	2	no	—
51	Dic₁₀	—	2	8	no	—
52	#170	—	2	4	no	—
53	#170	—	2	4	no	—
54	C₄⋊Q₈	—	5	2	no	—
55	#125	—	5	1	no	—
56	#125	AGL(1,𝔽₅)	1	1	yes	characteristic
57	C₂²⋊Q₈	—	5	4	no	—
58	C₄×Q₈	—	5	4	no	—
59	#117	—	5	2	no	—
60	C₂×C₄²	—	5	1	no	—
61	C₄⋊C₈	—	5	4	no	—
62	C₄⋊C₈	—	5	2	no	—
63	C₄⋊C₈	—	5	2	no	—
64	C₄⋊C₈	—	5	4	no	—
65	C₂²⋊C₈	—	5	4	no	—
66	#97	—	5	4	no	—
67	C₄×C₈	—	5	4	no	—
68	C₄×C₈	—	5	4	no	—
69	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
70	C₂×D₅	—	2	1	no	—
71	C₂×D₅	—	2	2	no	—
72	C₄×C₅	—	2	1	no	—
73	C₄×C₅	—	2	4	no	—
74	C₄×C₅	—	2	2	no	—
75	Dic₅	—	2	4	no	—
76	Dic₅	#131	1	4	yes	—
77	Dic₅	2₋¹⁺⁴	1	2	yes	—
78	C₂×Q₈	—	5	4	no	—
79	C₂²×C₄	—	5	1	no	—
80	C₂²×C₄	—	5	2	no	—
81	C₂×C₈	—	5	4	no	—
82	C₂×C₈	—	5	4	no	—
83	C₄⋊C₄	—	5	2	no	—
84	C₄⋊C₄	—	5	4	no	—
85	C₄⋊C₄	—	5	4	no	—
86	C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
87	C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	4	yes	—
88	C₂²⋊C₄	—	5	4	no	—
89	C₄²	—	5	2	no	—
90	C₄²	—	5	4	no	—
91	C₄²	—	5	1	no	—
92	C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
93	C₂×C₅	#519	1	1	yes	characteristic
94	C₂×C₅	C₂×#131	1	2	yes	—
95	D₅	—	4	1	no	—
96	C₂³	—	5	1	no	—
97	Q₈	—	10	8	no	—
98	C₂×C₄	—	5	1	no	—
99	C₂×C₄	—	5	2	no	—
100	C₂×C₄	—	5	4	no	—
101	C₂×C₄	—	10	2	no	—
102	C₂×C₄	—	10	4	no	—
103	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
104	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	2	yes	—
105	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	4	yes	—
106	C₈	—	10	4	no	—
107	C₈	—	10	4	no	—
108	C₅	#2986	1	1	yes	characteristic, complemented, Sylow
109	C₂²	—	10	1	no	—
110	C₂²	—	10	2	no	—
111	C₂²	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	2	2	no	—
114	C₄	—	2	4	no	—
115	C₄	—	5	2	no	—
116	C₄	—	5	4	no	—
117	C₄	—	10	4	no	—
118	C₂	—	20	1	no	—
119	C₂	#66262	1	1	yes	characteristic
120	C₂	D₅⋊#131	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#66022

C₂

yes

maximal, characteristic

#65707

C₂

yes

maximal, complemented

#65706

C₂

yes

maximal

#65703

C₂

yes

maximal, complemented

#65678

C₂

yes

maximal

C₅×#125

C₄

yes

characteristic

#4042

C₄

yes

—

#4038

C₂²

yes

—

#4034

C₄

yes

—

#4033

C₂²

yes

—

C₄²×D₅

C₂²

yes

characteristic

#4016

C₄

yes

characteristic

#4004

C₂²

yes

—

#4004

C₂²

yes

—

#4004

C₂²

yes

—

#4003

C₂²

yes

—

#4002

C₂²

yes

—

C₄⋊#170

C₂²

yes

—

C₄×#170

C₂²

yes

—

C₄×#170

C₂²

yes

—

#2986

—

maximal, Sylow

C₂×C₄×D₅

C₂×C₄

yes

—

C₂×C₄×D₅

C₂³

yes

characteristic

C₂×Dic₁₀

C₂×C₄

yes

—

C₂×#170

C₂³

yes

—

C₂×#170

C₂³

yes

—

C₅×C₄⋊C₄

C₂×C₄

yes

—

C₅×C₄⋊C₄

C₂×C₄

yes

—

C₄²×C₅

C₂×C₄

yes

characteristic

#685

C₂×C₄

yes

—

C₄⋊Dic₅

C₂×C₄

yes

—

#683

C₂×C₄

yes

—

#683

C₂×C₄

yes

—

C₄×Dic₅

C₂×C₄

yes

characteristic

C₄×Dic₅

C₂³

yes

—

C₄×Dic₅

C₂³

yes

—

#533

—

#446

—

C₈×Q₈

—

#433

—

#431

—

C₂²×D₅

C₂²×C₄

yes

characteristic

C₂×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×C₄×C₅

C₂²×C₄

yes

—

C₂×C₄×C₅

C₂²×C₄

yes

—

C₂×Dic₅

C₂²×C₄

yes

—

C₂×Dic₅

C₂²×C₄

yes

—

C₂×Dic₅

C₂⁴

yes

characteristic

C₄×D₅

—

C₄×D₅

—

Dic₁₀

—

#170

—

#170

—

C₄⋊Q₈

—

—

yes

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₅

C₂³×C₄

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

Dic₅

—

Dic₅

#131

yes

—

Dic₅

2₋¹⁺⁴

yes

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×AGL(1,𝔽₅)

yes

—

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

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

104

C₂×C₄

C₂²×AGL(1,𝔽₅)

yes

—

105

C₂×C₄

C₂²×AGL(1,𝔽₅)

yes

—

106

C₈

—

107

C₈

—

108

C₅

#2986

yes

characteristic, complemented, Sylow

109

C₂²

—

110

C₂²

—

111

C₂²

C₂³×AGL(1,𝔽₅)

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₂

—

119

C₂

#66262

yes

characteristic

120

C₂

D₅⋊#131

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅	C₂³×C₄
Fitting	C₅×#125	C₄
prime radical	C₂²×D₅	C₂²×C₄
cosocle	C₂×Dic₅	C₂⁴
hypercenter	#125	AGL(1,𝔽₅)
hypocenter	C₅	#2986
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×AGL(1,𝔽₅)

commutator, socle

C₂²×C₅

C₂³×C₄

Fitting

C₅×#125

C₄

prime radical

C₂²×D₅

C₂²×C₄

cosocle

C₂×Dic₅

C₂⁴

C₅

G ▷ C₂²×C₅ ▷ C₅

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
2. #65707	C₂	semidirect
4. #65703	C₂	semidirect
108. C₅	#2986	semidirect

H ≅ G/N

type

2. #65707

C₂

semidirect

4. #65703

C₂

semidirect

108. C₅

#2986

semidirect

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

subgroup	index
#66022	2
#65707	2
#65706	2
#65703	2
#65678	2
#2986	5

subgroup

index

Subgroups — #10617337

Conjugacy classes of subgroups

Complements

Maximal subgroups