Group #10604081 — subgroups

Subgroups — #10604081

Order 640.

208 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 70 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×#3996	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic
3	#65708	C₂	1	2	yes	maximal
4	#65169	C₂	1	1	yes	maximal, characteristic
5	#64875	C₂	1	2	yes	maximal
6	C₂×C₄×AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
7	C₂×#700	C₂²	1	2	yes	—
8	C₂×D₅⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₈×D₅	C₂²	1	1	yes	characteristic
10	#4038	C₂²	1	2	yes	—
11	#4004	—	2	2	no	—
12	C₄⋊#170	—	2	2	no	—
13	#3996	—	2	2	no	—
14	C₈×AGL(1,𝔽₅)	—	2	2	no	—
15	C₅×#107	C₄	1	1	yes	characteristic, complemented
16	#3950	C₄	1	1	yes	characteristic, complemented
17	#3940	C₄	1	2	yes	complemented
18	#1945	—	5	1	no	maximal, Sylow
19	C₂²×AGL(1,𝔽₅)	—	2	1	no	—
20	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
21	C₂×#170	—	2	2	no	—
22	C₂×#170	D₄	1	1	yes	characteristic
23	C₂×#170	D₄	1	1	yes	characteristic
24	C₄×AGL(1,𝔽₅)	—	2	1	no	—
25	C₄×AGL(1,𝔽₅)	Q₈	1	2	yes	—
26	#700	—	2	2	no	—
27	#700	—	2	2	no	—
28	D₅⋊C₈	—	2	1	no	—
29	D₅⋊C₈	—	2	1	no	—
30	C₂×C₅×C₈	C₂×C₄	1	1	yes	characteristic
31	C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
32	#685	—	2	2	no	—
33	C₄⋊Dic₅	C₂×C₄	1	2	yes	—
34	C₄×Dic₅	—	2	2	no	—
35	C₂×#168	C₂×C₄	1	1	yes	characteristic
36	C₈×D₅	D₄	1	2	yes	—
37	C₈×D₅	Q₈	1	2	yes	—
38	C₂×#108	—	5	1	no	—
39	#427	—	5	1	no	—
40	#424	—	5	2	no	—
41	C₂×C₄×C₈	—	5	1	no	—
42	#339	—	5	2	no	—
43	C₂²×D₅	#41	1	1	yes	characteristic
44	C₂×AGL(1,𝔽₅)	—	2	2	no	—
45	C₂×AGL(1,𝔽₅)	—	2	2	no	—
46	C₂×AGL(1,𝔽₅)	—	2	2	no	—
47	C₂×C₄×C₅	—	2	2	no	—
48	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
49	C₂×Dic₅	—	2	2	no	—
50	C₂×Dic₅	C₂×D₄	1	1	yes	characteristic
51	C₄×D₅	C₂×D₄	1	1	yes	characteristic
52	C₄×D₅	C₂×Q₈	1	1	yes	characteristic
53	C₄×D₅	C₂×Q₈	1	1	yes	characteristic
54	C₄×D₅	#41	1	1	yes	characteristic
55	#170	—	2	1	no	—
56	#170	—	2	1	no	—
57	#170	—	4	2	no	—
58	C₅×C₈	C₄⋊C₄	1	2	yes	—
59	#168	C₄⋊C₄	1	2	yes	—
60	C₂×M₁₆	—	5	2	no	—
61	C₂²×C₈	—	5	1	no	—
62	C₂²×C₈	—	5	1	no	—
63	#117	—	5	2	no	—
64	C₂×C₄²	—	5	1	no	—
65	#108	—	10	2	no	—
66	#107	—	5	1	no	—
67	#107	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
68	#106	—	5	2	no	—
69	C₄⋊C₈	—	10	2	no	—
70	C₄⋊C₈	—	10	2	no	—
71	C₄×C₈	—	10	2	no	—
72	C₂²×C₅	C₄×D₄	1	1	yes	characteristic
73	C₂×D₅	—	2	1	no	—
74	C₂×D₅	—	2	1	no	—
75	C₂×D₅	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂×D₅	#125	1	1	yes	characteristic
77	AGL(1,𝔽₅)	—	2	4	no	—
78	C₄×C₅	—	4	2	no	—
79	C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₄×C₅	C₄×Q₈	1	1	yes	characteristic
81	Dic₅	—	4	2	no	—
82	Dic₅	#121	1	1	yes	characteristic
83	Dic₅	C₄⋊Q₈	1	1	yes	characteristic
84	C₂²×C₄	—	5	1	no	—
85	C₂²×C₄	—	10	1	no	—
86	M₁₆	—	10	2	no	—
87	M₁₆	—	10	2	no	—
88	C₂×C₈	—	5	1	no	—
89	C₂×C₈	—	5	2	no	—
90	C₂×C₈	—	5	2	no	—
91	C₂×C₈	—	5	1	no	—
92	C₂×C₈	—	5	1	no	—
93	C₂×C₈	—	10	1	no	—
94	C₂×C₈	—	10	1	no	—
95	C₂×C₈	—	10	2	no	—
96	C₂×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
97	C₄⋊C₄	—	5	2	no	—
98	C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
99	C₂²⋊C₄	—	10	2	no	—
100	C₄²	—	5	2	no	—
101	C₄²	—	10	2	no	—
102	C₄²	—	10	1	no	—
103	C₂×C₅	#389	1	1	yes	characteristic
104	C₂×C₅	#443	1	1	yes	characteristic
105	C₂×C₅	#443	1	1	yes	characteristic
106	D₅	—	2	1	no	—
107	D₅	—	2	1	no	—
108	C₂³	—	5	1	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	5	1	no	—
111	C₂×C₄	—	5	1	no	—
112	C₂×C₄	—	5	1	no	—
113	C₂×C₄	—	5	1	no	—
114	C₂×C₄	—	5	1	no	—
115	C₂×C₄	—	10	2	no	—
116	C₂×C₄	—	10	2	no	—
117	C₂×C₄	—	10	2	no	—
118	C₂×C₄	—	10	2	no	—
119	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
120	C₈	—	5	2	no	—
121	C₈	—	10	1	no	—
122	C₈	—	10	1	no	—
123	C₈	—	20	2	no	—
124	C₈	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
125	C₅	#1945	1	1	yes	characteristic, complemented, Sylow
126	C₂²	—	5	1	no	—
127	C₂²	—	5	1	no	—
128	C₂²	—	10	1	no	—
129	C₂²	—	10	1	no	—
130	C₂²	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
131	C₄	—	4	2	no	—
132	C₄	—	5	1	no	—
133	C₄	—	5	1	no	—
134	C₄	—	10	4	no	—
135	C₄	—	20	2	no	—
136	C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
137	C₄	Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
138	C₂	—	10	1	no	—
139	C₂	—	10	1	no	—
140	C₂	#65712	1	1	yes	characteristic
141	C₂	#65732	1	1	yes	characteristic
142	C₂	#65740	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#3996

C₂

yes

maximal, characteristic

C₂×C₈×AGL(1,𝔽₅)

C₂

yes

maximal, characteristic

#65708

C₂

yes

maximal

#65169

C₂

yes

maximal, characteristic

#64875

C₂

yes

maximal

C₂×C₄×AGL(1,𝔽₅)

C₂²

yes

characteristic

C₂×#700

C₂²

yes

—

C₂×D₅⋊C₈

C₂²

yes

characteristic

C₂×C₈×D₅

C₂²

yes

characteristic

#4038

C₂²

yes

—

—

—

—

—

C₄

yes

characteristic, complemented

#3950

C₄

yes

characteristic, complemented

#3940

C₄

yes

complemented

#1945

—

maximal, Sylow

C₂²×AGL(1,𝔽₅)

—

C₂×C₄×D₅

C₂³

yes

characteristic

C₂×#170

—

C₂×#170

D₄

yes

characteristic

C₂×#170

D₄

yes

characteristic

C₄×AGL(1,𝔽₅)

—

C₄×AGL(1,𝔽₅)

Q₈

yes

—

#700

—

#700

—

D₅⋊C₈

—

D₅⋊C₈

—

C₂×C₅×C₈

C₂×C₄

yes

characteristic

C₅×C₄⋊C₄

C₂×C₄

yes

—

#685

—

C₄⋊Dic₅

C₂×C₄

yes

—

C₄×Dic₅

—

C₂×#168

C₂×C₄

yes

characteristic

C₈×D₅

D₄

yes

—

C₈×D₅

Q₈

yes

—

C₂×#108

—

#427

—

#424

—

C₂×C₄×C₈

—

#339

—

C₂²×D₅

yes

—

—

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

C₂×D₄

yes

characteristic

C₄×D₅

C₂×D₄

yes

characteristic

C₄×D₅

C₂×Q₈

yes

characteristic

C₄×D₅

C₂×Q₈

yes

C₄×D₅

yes

—

—

—

C₅×C₈

C₄⋊C₄

yes

—

#168

C₄⋊C₄

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

#117

—

C₂×C₄²

—

—

—

yes

characteristic, complemented

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₂²×C₅

C₄×D₄

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₂×D₅

C₂²⋊Q₈

yes

C₂×D₅

yes

—

C₄×C₅

—

C₄×C₅

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅

C₄×Q₈

yes

characteristic

Dic₅

—

Dic₅

#121

yes

characteristic

Dic₅

C₄⋊Q₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×AGL(1,𝔽₅)

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

C₂×AGL(1,𝔽₅)

yes

—

C₂²⋊C₄

—

100

C₄²

—

101

C₄²

—

102

C₄²

—

103

C₂×C₅

#389

yes

characteristic

104

C₂×C₅

#443

yes

characteristic

105

C₂×C₅

#443

yes

characteristic

106

D₅

—

107

D₅

—

108

C₂³

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

C₄⋊AGL(1,𝔽₅)

yes

—

125

C₅

#1945

yes

characteristic, complemented, Sylow

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

D₄×AGL(1,𝔽₅)

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

137

C₄

Q₈×AGL(1,𝔽₅)

yes

characteristic

138

C₂

—

139

C₂

—

140

C₂

#65712

yes

characteristic

141

C₂

#65732

yes

characteristic

142

C₂

#65740

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×AGL(1,𝔽₅)
commutator	C₂×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#107	C₄
socle	C₂²×C₅	C₄×D₄
prime radical	C₂²×D₅	#41
cosocle	C₂×C₄×D₅	C₂³
hypercenter	#107	AGL(1,𝔽₅)
hypocenter	C₅	#1945
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

subgroup

G/H

center

C₂²

D₄×AGL(1,𝔽₅)

commutator

C₂×C₄×C₅

C₂²×C₄

Frattini

C₂×C₄

C₂²×AGL(1,𝔽₅)

Fitting

C₅×#107

C₄

socle

C₂²×C₅

C₄×D₄

prime radical

C₂²×D₅

#41

cosocle

C₂×C₄×D₅

C₂³

C₅

G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

N	H ≅ G/N	type
15. C₅×#107	C₄	semidirect
16. #3950	C₄	semidirect
17. #3940	C₄	semidirect
67. #107	AGL(1,𝔽₅)	semidirect
125. C₅	#1945	semidirect

H ≅ G/N

type

15. C₅×#107

C₄

semidirect

16. #3950

C₄

semidirect

17. #3940

C₄

semidirect

67. #107

AGL(1,𝔽₅)

semidirect

125. C₅

#1945

semidirect

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

subgroup	index
C₂×#3996	2
C₂×C₈×AGL(1,𝔽₅)	2
#65708	2
#65169	2
#64875	2
#1945	5

subgroup

index

C₂×C₈×AGL(1,𝔽₅)

Subgroups — #10604081

Conjugacy classes of subgroups

Complements

Maximal subgroups