Group #10598598 — subgroups

Subgroups — #10598598

Order 640.

176 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 54 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	C₄×C₄⋊AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic
2	#65678	C₂	1	1	yes	maximal, characteristic
3	#65126	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊AGL(1,𝔽₅)	—	2	1	no	—
5	C₂×C₄×AGL(1,𝔽₅)	—	2	1	no	—
6	C₂×#676	—	2	1	no	—
7	C₄²×D₅	C₂²	1	1	yes	characteristic
8	(C₂×AGL(1,𝔽₅))⋊C₄	C₄	1	2	yes	—
9	#4004	C₄	1	2	yes	complemented
10	#4003	—	2	1	no	—
11	C₄×#170	—	2	1	no	—
12	C₅×C₄⋊C₈	C₄	1	1	yes	characteristic, complemented
13	#3946	—	2	1	no	—
14	C₄⋊#168	C₄	1	1	yes	characteristic, complemented
15	#1303	—	5	1	no	maximal, Sylow
16	C₂²×AGL(1,𝔽₅)	—	2	2	no	—
17	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×D₅	D₄	1	1	yes	characteristic
19	C₂×C₄×D₅	D₄	1	1	yes	characteristic
20	C₂×#170	—	2	2	no	—
21	C₄⋊AGL(1,𝔽₅)	—	2	4	no	—
22	C₄×AGL(1,𝔽₅)	—	4	1	no	—
23	C₄×AGL(1,𝔽₅)	—	4	1	no	—
24	C₂×C₅×C₈	—	2	1	no	—
25	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
26	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₅	D₄	1	1	yes	characteristic
28	C₄×Dic₅	Q₈	1	1	yes	characteristic
29	C₂×#168	—	2	1	no	—
30	#676	—	4	1	no	—
31	#676	—	4	1	no	—
32	#431	—	5	1	no	—
33	#423	—	5	1	no	—
34	C₄×C₄⋊C₄	—	5	1	no	—
35	C₂²×D₅	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×AGL(1,𝔽₅)	—	2	4	no	—
37	C₂×AGL(1,𝔽₅)	—	4	2	no	—
38	C₂×AGL(1,𝔽₅)	—	4	2	no	—
39	C₂×C₄×C₅	C₄²	1	1	yes	characteristic
40	C₂×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
42	C₂×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×Dic₅	C₄⋊C₄	1	1	yes	characteristic
44	C₂×Dic₅	C₄⋊C₄	1	1	yes	characteristic
45	C₄×D₅	—	2	2	no	—
46	C₄×D₅	—	4	1	no	—
47	C₄×D₅	D₈	1	1	yes	characteristic
48	C₄×D₅	SD₁₆	1	1	yes	characteristic
49	C₄×D₅	SD₁₆	1	1	yes	characteristic
50	C₄×D₅	Q₁₆	1	1	yes	characteristic
51	#170	—	4	2	no	—
52	C₅×C₈	—	4	1	no	—
53	#168	—	4	1	no	—
54	C₂×M₁₆	—	10	1	no	—
55	C₂×C₄⋊C₄	—	10	1	no	—
56	C₂×C₄²	—	5	1	no	—
57	C₂×C₄²	—	10	1	no	—
58	C₄⋊C₈	—	5	2	no	—
59	C₄⋊C₈	—	5	1	no	—
60	C₄⋊C₈	—	10	1	no	—
61	C₄⋊C₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
62	C₂²⋊C₈	—	10	1	no	—
63	C₄×C₈	—	10	1	no	—
64	#95	—	5	2	no	—
65	C₂²×C₅	#95	1	1	yes	characteristic
66	C₂×D₅	—	2	1	no	—
67	C₂×D₅	—	2	1	no	—
68	C₂×D₅	#102	1	1	yes	characteristic
69	C₂×D₅	#103	1	1	yes	characteristic
70	AGL(1,𝔽₅)	—	4	4	no	—
71	C₄×C₅	—	2	1	no	—
72	C₄×C₅	—	2	1	no	—
73	C₄×C₅	#102	1	1	yes	characteristic
74	C₄×C₅	#103	1	1	yes	characteristic
75	Dic₅	—	2	1	no	—
76	Dic₅	#104	1	2	yes	—
77	Dic₅	#106	1	1	yes	characteristic
78	Dic₅	#107	1	1	yes	characteristic
79	C₂²×C₄	—	5	1	no	—
80	C₂²×C₄	—	5	1	no	—
81	C₂²×C₄	—	5	1	no	—
82	C₂²×C₄	—	10	2	no	—
83	M₁₆	—	20	1	no	—
84	M₁₆	—	20	1	no	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	10	2	no	—
87	C₂×C₈	—	10	1	no	—
88	C₄⋊C₄	—	10	4	no	—
89	C₄²	—	5	1	no	—
90	C₄²	—	5	1	no	—
91	C₄²	—	5	1	no	—
92	C₄²	—	20	1	no	—
93	C₄²	—	20	1	no	—
94	C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
95	C₂×C₅	#339	1	1	yes	characteristic
96	C₂×C₅	#340	1	1	yes	characteristic
97	C₂×C₅	#344	1	1	yes	characteristic
98	D₅	—	2	1	no	—
99	D₅	—	2	1	no	—
100	C₂³	—	5	1	no	—
101	C₂×C₄	—	5	1	no	—
102	C₂×C₄	—	5	1	no	—
103	C₂×C₄	—	5	1	no	—
104	C₂×C₄	—	5	1	no	—
105	C₂×C₄	—	5	1	no	—
106	C₂×C₄	—	5	1	no	—
107	C₂×C₄	—	5	1	no	—
108	C₂×C₄	—	10	2	no	—
109	C₂×C₄	—	10	4	no	—
110	C₂×C₄	—	20	1	no	—
111	C₂×C₄	—	20	2	no	—
112	C₂×C₄	—	20	2	no	—
113	C₂×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
114	C₂×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
115	C₂×C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
116	C₈	—	4	1	no	—
117	C₈	—	20	2	no	—
118	C₈	—	20	1	no	—
119	C₅	#1303	1	1	yes	characteristic, complemented, Sylow
120	C₂²	—	5	1	no	—
121	C₂²	—	5	1	no	—
122	C₂²	—	10	1	no	—
123	C₂²	—	10	1	no	—
124	C₂²	(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic
125	C₄	—	2	1	no	—
126	C₄	—	2	1	no	—
127	C₄	—	5	1	no	—
128	C₄	—	5	1	no	—
129	C₄	—	5	2	no	—
130	C₄	—	10	1	no	—
131	C₄	—	20	4	no	—
132	C₄	#4007	1	1	yes	characteristic
133	C₄	#4009	1	1	yes	characteristic
134	C₂	—	10	1	no	—
135	C₂	—	10	1	no	—
136	C₂	#64874	1	1	yes	characteristic
137	C₂	#64894	1	1	yes	characteristic
138	C₂	#64900	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₄⋊AGL(1,𝔽₅)

C₂

yes

maximal, characteristic

#65678

C₂

yes

maximal, characteristic

#65126

C₂

yes

maximal, characteristic

C₂×C₄⋊AGL(1,𝔽₅)

—

C₂×C₄×AGL(1,𝔽₅)

—

C₂×#676

—

C₄²×D₅

C₂²

yes

characteristic

(C₂×AGL(1,𝔽₅))⋊C₄

C₄

yes

—

#4004

C₄

yes

complemented

#4003

—

C₄×#170

—

C₅×C₄⋊C₈

C₄

yes

characteristic, complemented

#3946

—

C₄⋊#168

C₄

yes

characteristic, complemented

#1303

—

maximal, Sylow

C₂²×AGL(1,𝔽₅)

—

C₂×C₄×D₅

C₂×C₄

yes

characteristic

C₂×C₄×D₅

D₄

yes

characteristic

C₂×C₄×D₅

D₄

yes

—

—

—

—

C₂×C₅×C₈

—

C₄²×C₅

C₂×C₄

yes

characteristic

C₄×Dic₅

C₂×C₄

yes

characteristic

C₄×Dic₅

D₄

yes

characteristic

C₄×Dic₅

Q₈

yes

—

—

—

—

—

C₄×C₄⋊C₄

—

C₂²×D₅

C₂²⋊C₄

yes

—

—

—

C₂×C₄×C₅

C₄²

yes

characteristic

C₂×C₄×C₅

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅

C₄⋊C₄

yes

characteristic

C₂×Dic₅

C₂²⋊C₄

yes

characteristic

C₂×Dic₅

C₄⋊C₄

yes

characteristic

C₂×Dic₅

C₄⋊C₄

yes

characteristic

C₄×D₅

—

C₄×D₅

—

C₄×D₅

D₈

yes

characteristic

C₄×D₅

SD₁₆

yes

characteristic

C₄×D₅

SD₁₆

yes

characteristic

C₄×D₅

Q₁₆

yes

characteristic

#170

—

C₅×C₈

—

#168

—

C₂×M₁₆

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

AGL(1,𝔽₅)

yes

characteristic, complemented

C₂²⋊C₈

—

C₄×C₈

—

#95

—

C₂²×C₅

#95

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₂×D₅

yes

C₂×D₅

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

yes

C₄×C₅

yes

Dic₅

—

Dic₅

#104

yes

—

Dic₅

yes

Dic₅

yes

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

yes

C₂×C₅

yes

C₂×C₅

yes

C₂×C₅

yes

D₅

—

D₅

—

100

C₂³

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₄×AGL(1,𝔽₅)

yes

characteristic

114

C₂×C₄

C₄⋊AGL(1,𝔽₅)

yes

characteristic

115

C₂×C₄

C₂²⋊AGL(1,𝔽₅)

yes

characteristic

116

C₈

—

117

C₈

—

118

C₈

—

119

C₅

#1303

yes

characteristic, complemented, Sylow

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

(C₂×AGL(1,𝔽₅))⋊C₄

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

#4007

yes

characteristic

133

C₄

#4009

yes

characteristic

134

C₂

—

135

C₂

—

136

C₂

#64874

yes

characteristic

137

C₂

#64894

yes

characteristic

138

C₂

#64900

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊AGL(1,𝔽₅)
commutator	C₂×C₄×C₅	C₄²
Frattini	C₄²	C₂×AGL(1,𝔽₅)
Fitting	C₅×C₄⋊C₈	C₄
socle	C₂²×C₅	#95
prime radical	C₂²×D₅	C₂²⋊C₄
cosocle	C₄²×D₅	C₂²
hypercenter	C₄⋊C₈	AGL(1,𝔽₅)
hypocenter	C₅	#1303
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊C₈

subgroup

G/H

center

C₂×C₄

C₂²⋊AGL(1,𝔽₅)

commutator

C₂×C₄×C₅

C₄²

Frattini

C₄²

C₂×AGL(1,𝔽₅)

Fitting

C₅×C₄⋊C₈

C₄

socle

C₂²×C₅

#95

prime radical

C₂²×D₅

C₂²⋊C₄

cosocle

C₄²×D₅

C₂²

C₄⋊C₈

C₅

G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊C₈

N	H ≅ G/N	type
9. #4004	C₄	semidirect
12. C₅×C₄⋊C₈	C₄	semidirect
14. C₄⋊#168	C₄	semidirect
61. C₄⋊C₈	AGL(1,𝔽₅)	semidirect
119. C₅	#1303	semidirect

H ≅ G/N

type

9. #4004

C₄

semidirect

12. C₅×C₄⋊C₈

C₄

semidirect

14. C₄⋊#168

C₄

semidirect

61. C₄⋊C₈

AGL(1,𝔽₅)

semidirect

119. C₅

#1303

semidirect

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

subgroup	index
C₄×C₄⋊AGL(1,𝔽₅)	2
#65678	2
#65126	2
#1303	5

subgroup

index

C₄×C₄⋊AGL(1,𝔽₅)

#65678

#65126

#1303

Subgroups — #10598598

Conjugacy classes of subgroups

Complements

Maximal subgroups