Group #10600282 — subgroups

Subgroups — #10600282

Order 640.

292 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 125 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	C₂×#4015	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊#168	C₂	1	1	yes	maximal, characteristic
4	#64749	C₂	1	4	yes	maximal
5	C₂²×Dic₁₀	—	2	1	no	—
6	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
7	C₂×C₄²×C₅	—	2	1	no	—
8	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₅	—	2	1	no	—
10	C₂×C₄⋊Dic₅	C₂²	1	2	yes	—
11	C₂²×#168	C₂²	1	2	yes	—
12	#4015	—	2	2	no	—
13	#4015	C₄	1	4	yes	complemented
14	C₅×#95	—	2	1	no	—
15	C₄⋊#168	—	2	2	no	—
16	#1987	—	5	1	no	maximal, Sylow
17	C₂²×C₄×C₅	—	2	2	no	—
18	C₂²×C₄×C₅	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₅	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
21	C₂²×Dic₅	—	2	2	no	—
22	C₂×Dic₁₀	—	2	4	no	—
23	C₂×Dic₁₀	—	4	4	no	—
24	C₅×C₄⋊C₄	—	2	2	no	—
25	C₅×C₄⋊C₄	Q₈	1	4	yes	complemented
26	C₄²×C₅	—	4	2	no	—
27	C₄²×C₅	C₂×C₄	1	2	yes	—
28	C₄²×C₅	D₄	1	2	yes	—
29	C₄⋊Dic₅	—	2	4	no	—
30	C₄⋊Dic₅	—	2	2	no	—
31	C₄⋊Dic₅	C₂×C₄	1	4	yes	—
32	C₂×#168	—	2	2	no	—
33	C₂×#168	—	2	4	no	—
34	C₂×C₄⋊Q₈	—	5	1	no	—
35	C₂×C₄⋊C₈	—	5	1	no	—
36	C₄×C₄⋊C₄	D₅	1	1	yes	characteristic
37	#340	—	5	4	no	—
38	C₂³×C₅	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₅	—	2	2	no	—
40	C₂×C₄×C₅	—	2	4	no	—
41	C₂×C₄×C₅	—	4	2	no	—
42	C₂×C₄×C₅	—	4	2	no	—
43	C₂×C₄×C₅	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₅	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₅	SD₁₆	1	2	yes	—
46	C₂×C₄×C₅	Q₁₆	1	2	yes	—
47	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₅	C₂×Q₈	1	2	yes	—
50	C₂×C₄×C₅	#41	1	2	yes	—
51	C₂×Dic₅	—	2	4	no	—
52	C₂×Dic₅	—	4	4	no	—
53	Dic₁₀	—	4	8	no	—
54	#168	—	4	2	no	—
55	C₂²×Q₈	—	10	1	no	—
56	C₂²×C₈	—	5	2	no	—
57	C₄⋊Q₈	—	5	4	no	—
58	C₄⋊Q₈	—	10	2	no	—
59	C₂×C₄⋊C₄	—	5	2	no	—
60	C₂×C₄⋊C₄	—	10	1	no	—
61	C₂×C₄⋊C₄	C₂×D₅	1	2	yes	—
62	C₂×C₄²	—	2	1	no	—
63	C₂×C₄²	C₂×D₅	1	1	yes	characteristic
64	C₄⋊C₈	—	10	2	no	—
65	#95	—	2	1	no	—
66	C₂²×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₅	C₂²⋊Q₈	1	2	yes	—
68	C₂²×C₅	C₂×SD₁₆	1	1	yes	characteristic
69	C₂²×C₅	C₂×Q₁₆	1	1	yes	characteristic
70	C₂²×C₅	AGL(1,ℤ/8)	1	2	yes	—
71	C₄×C₅	—	2	2	no	—
72	C₄×C₅	—	4	4	no	—
73	C₄×C₅	#103	1	4	yes	—
74	C₄×C₅	C₄×Q₈	1	2	yes	—
75	C₄×C₅	#124	1	1	yes	characteristic
76	C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
77	Dic₅	—	4	4	no	—
78	C₂×Q₈	—	10	4	no	—
79	C₂×Q₈	—	20	4	no	—
80	C₂²×C₄	—	2	2	no	—
81	C₂²×C₄	—	10	2	no	—
82	C₂²×C₄	C₂²⋊D₅	1	1	yes	characteristic
83	C₂²×C₄	C₂²⋊D₅	1	1	yes	characteristic
84	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
85	C₂×C₈	—	10	2	no	—
86	C₂×C₈	—	10	4	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	5	4	no	—
89	C₄⋊C₄	—	10	4	no	—
90	C₄⋊C₄	—	10	2	no	—
91	C₄⋊C₄	Dic₁₀	1	4	yes	complemented
92	C₄²	—	4	2	no	—
93	C₄²	C₄×D₅	1	2	yes	—
94	C₄²	D₂₀	1	2	yes	—
95	C₂×C₅	#391	1	1	yes	characteristic
96	C₂×C₅	C₂×#103	1	1	yes	characteristic
97	C₂×C₅	#418	1	1	yes	characteristic
98	C₂×C₅	#476	1	2	yes	—
99	C₂×C₅	#477	1	2	yes	—
100	C₂³	C₂×C₂²⋊D₅	1	1	yes	characteristic
101	Q₈	—	20	8	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	10	4	no	—
107	C₂×C₄	—	20	4	no	—
108	C₂×C₄	#685	1	2	yes	—
109	C₂×C₄	#685	1	2	yes	—
110	C₂×C₄	#687	1	2	yes	—
111	C₂×C₄	#689	1	2	yes	—
112	C₂×C₄	C₂×Dic₁₀	1	2	yes	—
113	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
114	C₂×C₄	C₂×D₂₀	1	1	yes	characteristic
115	C₂×C₄	#709	1	2	yes	—
116	C₈	—	20	2	no	—
117	C₅	#1987	1	1	yes	characteristic, complemented, Sylow
118	C₂²	#4070	1	2	yes	—
119	C₂²	C₂×#685	1	1	yes	characteristic
120	C₂²	C₂×#687	1	1	yes	characteristic
121	C₂²	C₂×#689	1	1	yes	characteristic
122	C₂²	#4095	1	2	yes	—
123	C₄	—	2	2	no	—
124	C₄	—	4	4	no	—
125	C₄	—	20	4	no	—
126	C₄	#3942	1	4	yes	—
127	C₄	C₄×Dic₁₀	1	2	yes	—
128	C₄	#4015	1	1	yes	characteristic
129	C₄	#4021	1	1	yes	characteristic
130	C₂	#65218	1	1	yes	characteristic
131	C₂	C₂×#3942	1	1	yes	characteristic
132	C₂	#65290	1	1	yes	characteristic
133	C₂	#65297	1	2	yes	—
134	C₂	#65310	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#4015

C₂

yes

maximal, characteristic

C₄×C₅×C₄⋊C₄

C₂

yes

maximal, characteristic

C₂×C₄⋊#168

C₂

yes

maximal, characteristic

#64749

C₂

yes

maximal

C₂²×Dic₁₀

—

C₂×C₅×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₅

—

C₂×C₄²×C₅

C₂²

yes

characteristic

C₂×C₄⋊Dic₅

—

C₂×C₄⋊Dic₅

C₂²

yes

—

C₂²×#168

C₂²

yes

—

#4015

—

#4015

C₄

yes

complemented

C₅×#95

—

C₄⋊#168

—

#1987

—

maximal, Sylow

C₂²×C₄×C₅

—

C₂²×C₄×C₅

D₄

yes

characteristic

C₂²×C₄×C₅

D₄

yes

characteristic

C₂²×C₄×C₅

C₂³

yes

characteristic

C₂²×Dic₅

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

Q₈

yes

complemented

C₄²×C₅

—

C₄²×C₅

C₂×C₄

yes

—

C₄²×C₅

D₄

yes

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

C₂×C₄

yes

—

C₂×#168

—

C₂×#168

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊C₈

—

C₄×C₄⋊C₄

D₅

yes

characteristic

#340

—

C₂³×C₅

C₂×D₄

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²⋊C₄

yes

—

C₂×C₄×C₅

C₂²⋊C₄

yes

—

C₂×C₄×C₅

SD₁₆

yes

—

C₂×C₄×C₅

Q₁₆

yes

—

C₂×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×C₄×C₅

C₂×D₄

yes

characteristic

C₂×C₄×C₅

C₂×Q₈

yes

—

C₂×C₄×C₅

#41

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

#168

—

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅

yes

characteristic

C₄⋊C₈

—

#95

—

C₂²×C₅

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅

C₂²⋊Q₈

yes

—

C₂²×C₅

C₂×SD₁₆

yes

characteristic

C₂²×C₅

C₂×Q₁₆

yes

characteristic

C₂²×C₅

AGL(1,ℤ/8)

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#103

yes

—

C₄×C₅

C₄×Q₈

yes

—

C₄×C₅

#124

yes

characteristic

C₄×C₅

C₄⋊Q₈

yes

characteristic

Dic₅

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₅

yes

characteristic

C₂²×C₄

C₂²⋊D₅

yes

characteristic

C₂²×C₄

C₂²×D₅

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

Dic₁₀

yes

complemented

C₄²

—

C₄²

C₄×D₅

yes

—

C₄²

D₂₀

yes

—

C₂×C₅

yes

C₂×C₅

yes

C₂×C₅

yes

C₂×C₅

yes

—

C₂×C₅

#477

yes

—

100

C₂³

C₂×C₂²⋊D₅

yes

characteristic

101

Q₈

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

#685

yes

—

109

C₂×C₄

#685

yes

—

110

C₂×C₄

#687

yes

—

111

C₂×C₄

#689

yes

—

112

C₂×C₄

C₂×Dic₁₀

yes

—

113

C₂×C₄

C₂×C₄×D₅

yes

characteristic

114

C₂×C₄

C₂×D₂₀

yes

characteristic

115

C₂×C₄

#709

yes

—

116

C₈

—

117

C₅

#1987

yes

characteristic, complemented, Sylow

118

C₂²

#4070

yes

—

119

C₂²

C₂×#685

yes

characteristic

120

C₂²

C₂×#687

yes

characteristic

121

C₂²

C₂×#689

yes

characteristic

122

C₂²

#4095

yes

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#3942

yes

—

127

C₄

C₄×Dic₁₀

yes

—

128

C₄

#4015

yes

characteristic

129

C₄

#4021

yes

characteristic

130

C₂

#65218

yes

characteristic

131

C₂

C₂×#3942

yes

characteristic

132

C₂

#65290

yes

characteristic

133

C₂

#65297

yes

—

134

C₂

#65310

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅
commutator	C₂×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅
Fitting	C₄×C₅×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₅	C₂×D₄
cosocle	C₂²×C₄×C₅	C₂³
hypercenter	C₄×C₄⋊C₄	D₅
hypocenter	C₅	#1987
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅

commutator

C₂×C₄×C₅

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅

Fitting

C₄×C₅×C₄⋊C₄

C₂

socle, prime radical

C₂³×C₅

C₂×D₄

cosocle

C₂²×C₄×C₅

C₂³

hypercenter

C₄×C₄⋊C₄

D₅

hypocenter

C₅

#1987

LCS

G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
13. #4015	C₄	semidirect
25. C₅×C₄⋊C₄	Q₈	semidirect
91. C₄⋊C₄	Dic₁₀	semidirect
117. C₅	#1987	semidirect

H ≅ G/N

type

13. #4015

C₄

semidirect

25. C₅×C₄⋊C₄

Q₈

semidirect

91. C₄⋊C₄

Dic₁₀

semidirect

117. C₅

#1987

semidirect

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

subgroup	index
C₂×#4015	2
C₄×C₅×C₄⋊C₄	2
C₂×C₄⋊#168	2
#64749	2
#1987	5

subgroup

index

C₂×#4015

C₄×C₅×C₄⋊C₄

C₂×C₄⋊#168

#64749

#1987

Subgroups — #10600282

Conjugacy classes of subgroups

Complements

Maximal subgroups