Group #10604058 — subgroups

Subgroups — #10604058

Order 640.

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

normal subgroups only

This group has 200 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×#3992	C₂	1	1	yes	maximal, characteristic
3	#65708	C₂	1	2	yes	maximal
4	#65149	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	#3992	—	2	2	no	—
15	C₅×#106	C₄	1	1	yes	characteristic, complemented
16	#3949	C₄	1	1	yes	characteristic, complemented
17	#3939	C₄	1	2	yes	complemented
18	#1946	—	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×#97	—	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)	—	4	2	no	—
46	C₂×C₄×C₅	—	2	2	no	—
47	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
48	C₂×Dic₅	—	2	2	no	—
49	C₂×Dic₅	C₂×D₄	1	1	yes	characteristic
50	C₄×D₅	C₂×D₄	1	1	yes	characteristic
51	C₄×D₅	C₂×Q₈	1	1	yes	characteristic
52	C₄×D₅	C₂×Q₈	1	1	yes	characteristic
53	C₄×D₅	#41	1	1	yes	characteristic
54	#170	—	2	1	no	—
55	#170	—	2	1	no	—
56	#170	—	4	2	no	—
57	C₅×C₈	C₄⋊C₄	1	2	yes	—
58	#168	C₄⋊C₄	1	2	yes	—
59	C₂×M₁₆	—	5	2	no	—
60	C₂²×C₈	—	5	1	no	—
61	C₂²×C₈	—	5	1	no	—
62	#117	—	5	2	no	—
63	C₂×C₄²	—	5	1	no	—
64	#108	—	10	2	no	—
65	#107	—	5	2	no	—
66	#106	—	5	1	no	—
67	#106	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
68	C₄⋊C₈	—	10	2	no	—
69	C₄⋊C₈	—	10	2	no	—
70	#97	—	10	2	no	—
71	C₂²×C₅	C₄×D₄	1	1	yes	characteristic
72	C₂×D₅	—	2	2	no	—
73	C₂×D₅	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂×D₅	#125	1	1	yes	characteristic
75	AGL(1,𝔽₅)	—	4	2	no	—
76	C₄×C₅	—	4	2	no	—
77	C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₄×C₅	C₄×Q₈	1	1	yes	characteristic
79	Dic₅	—	4	2	no	—
80	Dic₅	#121	1	1	yes	characteristic
81	Dic₅	C₄⋊Q₈	1	1	yes	characteristic
82	C₂²×C₄	—	5	1	no	—
83	C₂²×C₄	—	10	1	no	—
84	M₁₆	—	10	2	no	—
85	M₁₆	—	10	2	no	—
86	C₂×C₈	—	5	1	no	—
87	C₂×C₈	—	5	2	no	—
88	C₂×C₈	—	5	2	no	—
89	C₂×C₈	—	5	1	no	—
90	C₂×C₈	—	5	1	no	—
91	C₂×C₈	—	10	1	no	—
92	C₂×C₈	—	10	1	no	—
93	C₂×C₈	—	10	2	no	—
94	C₂×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
95	C₄⋊C₄	—	5	2	no	—
96	C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
97	C₂²⋊C₄	—	10	2	no	—
98	C₄²	—	5	2	no	—
99	C₄²	—	10	2	no	—
100	C₄²	—	10	1	no	—
101	C₂×C₅	#389	1	1	yes	characteristic
102	C₂×C₅	#444	1	2	yes	—
103	D₅	—	2	1	no	—
104	D₅	—	2	1	no	—
105	C₂³	—	5	1	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	5	1	no	—
108	C₂×C₄	—	5	1	no	—
109	C₂×C₄	—	5	1	no	—
110	C₂×C₄	—	5	1	no	—
111	C₂×C₄	—	5	1	no	—
112	C₂×C₄	—	10	2	no	—
113	C₂×C₄	—	10	2	no	—
114	C₂×C₄	—	20	2	no	—
115	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
116	C₈	—	5	2	no	—
117	C₈	—	10	1	no	—
118	C₈	—	10	1	no	—
119	C₈	—	20	2	no	—
120	C₈	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
121	C₅	#1946	1	1	yes	characteristic, complemented, Sylow
122	C₂²	—	5	1	no	—
123	C₂²	—	5	1	no	—
124	C₂²	—	10	2	no	—
125	C₂²	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
126	C₄	—	4	2	no	—
127	C₄	—	5	1	no	—
128	C₄	—	5	1	no	—
129	C₄	—	20	2	no	—
130	C₄	—	20	2	no	—
131	C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
132	C₄	Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
133	C₂	—	10	1	no	—
134	C₂	—	10	1	no	—
135	C₂	#65712	1	1	yes	characteristic
136	C₂	#65737	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#3996

C₂

yes

maximal, characteristic

C₂×#3992

C₂

yes

maximal, characteristic

#65708

C₂

yes

maximal

#65149

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

#3949

C₄

yes

characteristic, complemented

#3939

C₄

yes

complemented

#1946

—

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×#97

—

#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

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₂²×C₅

C₄×D₄

yes

characteristic

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

—

C₄²

—

C₄²

—

100

C₄²

—

101

C₂×C₅

#389

yes

characteristic

102

C₂×C₅

#444

yes

—

103

D₅

—

104

D₅

—

105

C₂³

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

116

C₈

—

117

C₈

—

118

C₈

—

119

C₈

—

120

C₈

C₄⋊AGL(1,𝔽₅)

yes

—

121

C₅

#1946

yes

characteristic, complemented, Sylow

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

D₄×AGL(1,𝔽₅)

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

132

C₄

Q₈×AGL(1,𝔽₅)

yes

characteristic

133

C₂

—

134

C₂

—

135

C₂

#65712

yes

characteristic

136

C₂

#65737

yes

—

	subgroup	G/H
center	C₂²	D₄×AGL(1,𝔽₅)
commutator	C₂×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#106	C₄
socle	C₂²×C₅	C₄×D₄
prime radical	C₂²×D₅	#41
cosocle	C₂×C₄×D₅	C₂³
hypercenter	#106	AGL(1,𝔽₅)
hypocenter	C₅	#1946
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

subgroup

G/H

center

C₂²

D₄×AGL(1,𝔽₅)

commutator

C₂×C₄×C₅

C₂²×C₄

Frattini

C₂×C₄

C₂²×AGL(1,𝔽₅)

Fitting

C₅×#106

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 ◁ #106

N	H ≅ G/N	type
15. C₅×#106	C₄	semidirect
16. #3949	C₄	semidirect
17. #3939	C₄	semidirect
67. #106	AGL(1,𝔽₅)	semidirect
121. C₅	#1946	semidirect

H ≅ G/N

type

15. C₅×#106

C₄

semidirect

16. #3949

C₄

semidirect

17. #3939

C₄

semidirect

67. #106

AGL(1,𝔽₅)

semidirect

121. C₅

#1946

semidirect

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

subgroup	index
C₂×#3996	2
C₂×#3992	2
#65708	2
#65149	2
#64875	2
#1946	5

subgroup

index

Subgroups — #10604058

Conjugacy classes of subgroups

Complements

Maximal subgroups