Group #10602017 — subgroups

Subgroups — #10602017

Order 640.

186 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 57 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×#487	C₂	1	1	yes	maximal, characteristic, complemented
2	#65382	C₂	1	1	yes	maximal, characteristic, complemented
3	#65377	C₂	1	1	yes	maximal, characteristic
4	#65374	C₂	1	1	yes	maximal, characteristic
5	#65367	C₂	1	1	yes	maximal, characteristic, complemented
6	#64983	C₂	1	1	yes	maximal, characteristic
7	#64675	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×#688	—	2	2	no	—
11	#4020	C₂²	1	1	yes	characteristic
12	#4015	C₂²	1	1	yes	characteristic
13	C₅×#103	—	2	2	no	—
14	C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	#3953	—	2	2	no	—
16	#3942	—	2	2	no	—
17	#3941	—	2	2	no	—
18	#3939	—	2	2	no	—
19	C₄×#168	C₂²	1	1	yes	characteristic
20	C₄×#168	C₂²	1	1	yes	characteristic
21	#1719	—	5	1	no	maximal, Sylow
22	C₂×C₅×Q₈	—	2	1	no	—
23	C₂×C₅×Q₈	—	2	1	no	—
24	C₂×D₂₀	—	2	1	no	—
25	C₂×D₂₀	—	2	1	no	—
26	C₂×D₂₀	—	2	1	no	—
27	C₂×Dic₁₀	—	2	1	no	—
28	C₂×C₅×C₈	D₄	1	2	yes	—
29	C₅×C₄⋊C₄	—	2	1	no	—
30	C₅×C₄⋊C₄	—	2	1	no	—
31	C₅×C₄⋊C₄	—	2	1	no	—
32	C₅×C₄⋊C₄	—	2	1	no	—
33	C₄²×C₅	C₂³	1	1	yes	characteristic
34	#688	—	4	4	no	—
35	C₄⋊Dic₅	—	2	1	no	—
36	C₄⋊Dic₅	—	2	1	no	—
37	C₂×#168	D₄	1	2	yes	—
38	C₂×#168	D₄	1	2	yes	—
39	#500	—	5	1	no	—
40	#492	—	5	1	no	—
41	#487	—	5	1	no	—
42	#487	D₅	1	1	yes	characteristic, complemented
43	#486	—	5	1	no	—
44	#486	—	5	1	no	—
45	#322	—	5	1	no	—
46	C₂²×D₅	—	4	1	no	—
47	C₅×Q₈	—	4	2	no	—
48	C₅×Q₈	—	4	2	no	—
49	C₂×C₄×C₅	—	4	1	no	—
50	C₂×C₄×C₅	—	4	1	no	—
51	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
54	C₂×Dic₅	—	4	1	no	—
55	D₂₀	—	4	2	no	—
56	D₂₀	—	4	2	no	—
57	D₂₀	—	4	2	no	—
58	Dic₁₀	—	4	2	no	—
59	C₅×C₈	—	2	2	no	—
60	#168	—	2	2	no	—
61	#168	—	2	2	no	—
62	C₂×SD₁₆	—	10	2	no	—
63	C₄⋊Q₈	—	5	1	no	—
64	C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic
65	C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic
66	#127	—	5	1	no	—
67	#107	—	10	2	no	—
68	#103	—	2	2	no	—
69	#103	—	10	2	no	—
70	#102	—	10	2	no	—
71	#102	—	10	2	no	—
72	C₄×C₈	—	5	1	no	—
73	C₄×C₈	—	5	1	no	—
74	C₄×C₈	C₂×D₅	1	1	yes	characteristic
75	C₂²×C₅	#127	1	1	yes	characteristic
76	C₂×D₅	—	8	1	no	—
77	C₂×D₅	—	8	1	no	—
78	C₂×D₅	—	8	1	no	—
79	C₄×C₅	—	8	1	no	—
80	C₄×C₅	—	8	1	no	—
81	C₄×C₅	#135	1	2	yes	—
82	C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
83	C₄×C₅	#137	1	2	yes	—
84	Dic₅	—	8	1	no	—
85	C₂×Q₈	—	2	1	no	—
86	C₂×Q₈	—	2	1	no	—
87	C₂×Q₈	—	10	1	no	—
88	C₂×D₄	—	10	1	no	—
89	C₂×D₄	—	10	1	no	—
90	C₂×D₄	—	10	1	no	—
91	SD₁₆	—	20	4	no	—
92	C₂×C₈	—	5	2	no	—
93	C₂×C₈	—	5	2	no	—
94	C₂×C₈	C₂²⋊D₅	1	2	yes	—
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	—	2	1	no	—
97	C₄⋊C₄	—	2	1	no	—
98	C₄⋊C₄	—	2	1	no	—
99	C₄⋊C₄	—	10	1	no	—
100	C₄⋊C₄	—	10	1	no	—
101	C₄²	C₂²×D₅	1	1	yes	characteristic
102	C₂×C₅	#495	1	1	yes	characteristic
103	C₂×C₅	#496	1	1	yes	characteristic
104	C₂×C₅	#497	1	1	yes	characteristic
105	D₅	—	16	1	no	—
106	C₂³	—	20	1	no	—
107	Q₈	—	4	2	no	—
108	Q₈	—	4	2	no	—
109	Q₈	—	20	2	no	—
110	D₄	—	20	2	no	—
111	D₄	—	20	2	no	—
112	D₄	—	20	2	no	—
113	C₂×C₄	—	4	1	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	20	1	no	—
116	C₂×C₄	D₄×D₅	1	1	yes	characteristic
117	C₂×C₄	D₄×D₅	1	1	yes	characteristic
118	C₂×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
119	C₈	—	2	2	no	—
120	C₈	—	10	2	no	—
121	C₈	—	10	2	no	—
122	C₅	#1719	1	1	yes	characteristic, complemented, Sylow
123	C₂²	—	40	1	no	—
124	C₂²	—	40	1	no	—
125	C₂²	—	40	1	no	—
126	C₂²	#4086	1	1	yes	characteristic
127	C₄	—	8	1	no	—
128	C₄	—	8	1	no	—
129	C₄	—	40	1	no	—
130	C₄	#4060	1	2	yes	—
131	C₄	#4064	1	2	yes	—
132	C₄	#4096	1	2	yes	—
133	C₂	—	80	1	no	—
134	C₂	#65348	1	1	yes	characteristic
135	C₂	#65465	1	1	yes	characteristic
136	C₂	#65479	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#487

C₂

yes

maximal, characteristic, complemented

#65382

C₂

yes

maximal, characteristic, complemented

#65377

C₂

yes

maximal, characteristic

#65374

C₂

yes

maximal, characteristic

#65367

C₂

yes

maximal, characteristic, complemented

#64983

C₂

yes

maximal, characteristic

#64675

C₂

yes

maximal, characteristic, complemented

C₅×C₄⋊Q₈

C₂²

yes

characteristic

C₅×C₄⋊Q₈

C₂²

yes

characteristic

C₂×#688

—

#4020

C₂²

yes

characteristic

#4015

C₂²

yes

characteristic

C₅×#103

—

C₄×C₅×C₈

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₄×#168

C₂²

yes

characteristic

#1719

—

maximal, Sylow

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×Dic₁₀

—

C₂×C₅×C₈

D₄

yes

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₂³

yes

characteristic

#688

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₂×#168

D₄

yes

—

C₂×#168

D₄

yes

—

—

—

—

D₅

yes

characteristic, complemented

#486

—

#486

—

#322

—

C₂²×D₅

—

C₅×Q₈

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×D₄

yes

characteristic

C₂×C₄×C₅

C₂×D₄

yes

characteristic

C₂×C₄×C₅

C₂×D₄

yes

characteristic

C₂×Dic₅

—

D₂₀

—

D₂₀

—

D₂₀

—

Dic₁₀

—

C₅×C₈

—

#168

—

#168

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅

yes

characteristic

C₄⋊Q₈

C₂×D₅

yes

—

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₅

yes

characteristic

C₂²×C₅

#127

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#135

yes

—

C₄×C₅

AGL(1,ℤ/8)

yes

—

C₄×C₅

#137

yes

—

Dic₅

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₅

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄²

C₂²×D₅

yes

characteristic

102

C₂×C₅

#495

yes

characteristic

103

C₂×C₅

#496

yes

characteristic

104

C₂×C₅

#497

yes

characteristic

105

D₅

—

106

C₂³

—

107

Q₈

—

108

Q₈

—

109

Q₈

—

110

D₄

—

111

D₄

—

112

D₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

D₄×D₅

yes

characteristic

117

C₂×C₄

D₄×D₅

yes

characteristic

118

C₂×C₄

C₂×C₂²⋊D₅

yes

characteristic

119

C₈

—

120

C₈

—

121

C₈

—

122

C₅

#1719

yes

characteristic, complemented, Sylow

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

#4086

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

#4060

yes

—

131

C₄

#4064

yes

—

132

C₄

#4096

yes

—

133

C₂

—

134

C₂

#65348

yes

characteristic

135

C₂

#65465

yes

characteristic

136

C₂

#65479

yes

characteristic

	subgroup	G/H
center	C₂²	#4086
commutator, cosocle	C₄²×C₅	C₂³
Frattini	C₄²	C₂²×D₅
Fitting	C₅×#487	C₂
socle	C₂²×C₅	#127
prime radical	#4020	C₂²
hypercenter	#487	D₅
hypocenter	C₅	#1719
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #487

subgroup

G/H

center

C₂²

#4086

commutator, cosocle

C₄²×C₅

C₂³

Frattini

C₄²

C₂²×D₅

C₂

C₂²×C₅

C₂²

D₅

C₅

G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #487

N	H ≅ G/N	type
1. C₅×#487	C₂	semidirect
2. #65382	C₂	semidirect
5. #65367	C₂	semidirect
7. #64675	C₂	semidirect
42. #487	D₅	semidirect
122. C₅	#1719	semidirect

H ≅ G/N

type

1. C₅×#487

C₂

semidirect

2. #65382

C₂

semidirect

5. #65367

C₂

semidirect

7. #64675

C₂

semidirect

42. #487

D₅

semidirect

122. C₅

#1719

semidirect

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

subgroup	index
C₅×#487	2
#65382	2
#65377	2
#65374	2
#65367	2
#64983	2
#64675	2
#1719	5

subgroup

index

Subgroups — #10602017

Conjugacy classes of subgroups

Complements

Maximal subgroups