Group #10603883 — subgroups

Subgroups — #10603883

Order 640.

256 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 94 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×#3994	C₂	1	2	yes	maximal
2	C₂×#3993	C₂	1	2	yes	maximal
3	#65689	C₂	1	1	yes	maximal, characteristic
4	#65688	C₂	1	1	yes	maximal, characteristic
5	D₅×#97	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
7	C₂×C₄⋊AGL(1,𝔽₅)	C₂²	1	2	yes	—
8	C₂×C₄⋊AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
10	C₂×C₈×D₅	C₂²	1	2	yes	—
11	C₄²×D₅	C₂²	1	1	yes	characteristic
12	(C₂×AGL(1,𝔽₅))⋊C₄	—	2	2	no	—
13	#3994	—	2	4	no	—
14	#3993	—	2	4	no	—
15	C₅×#97	C₄	1	1	yes	characteristic, complemented
16	#3947	C₄	1	2	yes	complemented
17	#3935	C₄	1	1	yes	characteristic, complemented
18	#1847	—	5	1	no	maximal, Sylow
19	C₂²×AGL(1,𝔽₅)	—	2	2	no	—
20	C₂²×AGL(1,𝔽₅)	—	2	1	no	—
21	C₂²×AGL(1,𝔽₅)	—	2	1	no	—
22	C₂×C₄×D₅	D₄	1	2	yes	—
23	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
24	C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
25	C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
26	C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
27	C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
28	C₄⋊AGL(1,𝔽₅)	—	4	4	no	—
29	C₂×C₅×C₈	C₂×C₄	1	2	yes	—
30	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
32	C₄×Dic₅	Q₈	1	2	yes	—
33	C₂×#168	C₂×C₄	1	2	yes	—
34	C₈×D₅	D₄	1	4	yes	—
35	C₈×D₅	Q₈	1	4	yes	—
36	C₂×#107	—	5	2	no	—
37	C₂×#106	—	5	2	no	—
38	C₂×#97	—	5	1	no	—
39	#384	—	5	1	no	—
40	#382	—	5	1	no	—
41	C₂²×D₅	C₂×D₄	1	1	yes	characteristic
42	C₂×AGL(1,𝔽₅)	—	4	2	no	—
43	C₂×AGL(1,𝔽₅)	—	4	1	no	—
44	C₂×AGL(1,𝔽₅)	—	4	1	no	—
45	C₂×AGL(1,𝔽₅)	—	4	4	no	—
46	C₂×AGL(1,𝔽₅)	—	4	2	no	—
47	C₂×AGL(1,𝔽₅)	—	4	2	no	—
48	C₂×C₄×C₅	C₄⋊C₄	1	2	yes	—
49	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
50	C₂×Dic₅	C₄⋊C₄	1	2	yes	—
51	C₂×Dic₅	C₂×Q₈	1	1	yes	characteristic
52	C₄×D₅	—	2	2	no	—
53	C₄×D₅	—	2	2	no	—
54	C₄×D₅	C₂×D₄	1	2	yes	—
55	C₄×D₅	C₂×Q₈	1	2	yes	—
56	C₅×C₈	C₄⋊C₄	1	4	yes	complemented
57	#168	C₄⋊C₄	1	4	yes	complemented
58	C₂²×C₈	—	5	2	no	—
59	C₂×C₄⋊C₄	—	5	1	no	—
60	C₂×C₄⋊C₄	—	5	1	no	—
61	C₂×C₄⋊C₄	—	5	2	no	—
62	C₂×C₄⋊C₄	—	10	2	no	—
63	C₂×C₄²	—	5	1	no	—
64	#107	—	10	4	no	—
65	#106	—	10	4	no	—
66	#97	—	5	2	no	—
67	#97	—	5	1	no	—
68	#97	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
69	#95	—	10	2	no	—
70	C₂²×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×D₅	#127	1	1	yes	characteristic
72	C₂×D₅	C₄⋊Q₈	1	1	yes	characteristic
73	C₂×D₅	AGL(1,ℤ/8)	1	2	yes	—
74	C₂×D₅	#137	1	2	yes	—
75	AGL(1,𝔽₅)	—	8	2	no	—
76	AGL(1,𝔽₅)	—	8	1	no	—
77	AGL(1,𝔽₅)	—	8	1	no	—
78	C₄×C₅	—	2	2	no	—
79	C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
80	Dic₅	—	2	2	no	—
81	Dic₅	C₄⋊Q₈	1	2	yes	—
82	C₂²×C₄	—	5	1	no	—
83	C₂²×C₄	—	5	2	no	—
84	C₂²×C₄	—	10	1	no	—
85	C₂²×C₄	—	10	1	no	—
86	C₂²×C₄	—	10	2	no	—
87	C₂×C₈	—	5	2	no	—
88	C₂×C₈	—	5	4	no	—
89	C₂×C₈	—	5	4	no	—
90	C₂×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
91	C₄⋊C₄	—	10	2	no	—
92	C₄⋊C₄	—	10	2	no	—
93	C₄⋊C₄	—	10	2	no	—
94	C₄⋊C₄	—	10	2	no	—
95	C₄⋊C₄	—	20	4	no	—
96	C₄²	—	5	2	no	—
97	C₄²	—	5	1	no	—
98	C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
99	C₂×C₅	#384	1	1	yes	characteristic
100	C₂×C₅	#428	1	2	yes	—
101	D₅	#496	1	1	yes	characteristic
102	D₅	#497	1	1	yes	characteristic
103	D₅	#501	1	2	yes	—
104	C₂³	—	5	1	no	—
105	C₂×C₄	—	5	1	no	—
106	C₂×C₄	—	5	2	no	—
107	C₂×C₄	—	5	2	no	—
108	C₂×C₄	—	5	2	no	—
109	C₂×C₄	—	10	2	no	—
110	C₂×C₄	—	10	2	no	—
111	C₂×C₄	—	20	1	no	—
112	C₂×C₄	—	20	2	no	—
113	C₂×C₄	—	20	1	no	—
114	C₂×C₄	—	20	4	no	—
115	C₂×C₄	—	20	2	no	—
116	C₂×C₄	—	20	2	no	—
117	C₂×C₄	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
118	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
119	C₈	—	5	4	no	—
120	C₈	C₄⋊AGL(1,𝔽₅)	1	4	yes	complemented
121	C₅	#1847	1	1	yes	characteristic, complemented, Sylow
122	C₂²	—	5	1	no	—
123	C₂²	—	5	1	no	—
124	C₂²	—	5	2	no	—
125	C₂²	—	5	2	no	—
126	C₂²	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	5	2	no	—
129	C₄	—	10	2	no	—
130	C₄	—	40	2	no	—
131	C₄	—	40	1	no	—
132	C₄	—	40	1	no	—
133	C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	2	yes	—
134	C₂	—	5	1	no	—
135	C₂	—	5	1	no	—
136	C₂	—	5	2	no	—
137	C₂	#65688	1	1	yes	characteristic
138	C₂	#65727	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#3994

C₂

yes

maximal

C₂×#3993

C₂

yes

maximal

#65689

C₂

yes

maximal, characteristic

#65688

C₂

yes

maximal, characteristic

D₅×#97

C₂

yes

maximal, characteristic

C₂×C₄⋊AGL(1,𝔽₅)

—

C₂×C₄⋊AGL(1,𝔽₅)

C₂²

yes

—

C₂×C₄⋊AGL(1,𝔽₅)

C₂²

yes

characteristic

C₂×C₄⋊AGL(1,𝔽₅)

C₂²

yes

characteristic

C₂×C₈×D₅

C₂²

yes

—

C₄²×D₅

C₂²

yes

characteristic

(C₂×AGL(1,𝔽₅))⋊C₄

—

#3994

—

#3993

—

C₅×#97

C₄

yes

characteristic, complemented

#3947

C₄

yes

complemented

#3935

C₄

yes

characteristic, complemented

#1847

—

maximal, Sylow

C₂²×AGL(1,𝔽₅)

—

C₂²×AGL(1,𝔽₅)

—

C₂²×AGL(1,𝔽₅)

—

C₂×C₄×D₅

D₄

yes

—

C₂×C₄×D₅

C₂³

yes

—

—

—

—

—

C₂×C₅×C₈

C₂×C₄

yes

—

C₄²×C₅

C₂×C₄

yes

characteristic

C₄×Dic₅

C₂×C₄

yes

characteristic

C₄×Dic₅

Q₈

yes

—

C₂×#168

C₂×C₄

yes

—

C₈×D₅

D₄

yes

—

C₈×D₅

Q₈

yes

—

C₂×#107

—

C₂×#106

—

C₂×#97

—

#384

—

#382

—

C₂²×D₅

C₂×D₄

yes

—

—

—

—

—

—

C₂×C₄×C₅

C₄⋊C₄

yes

—

C₂×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×Dic₅

C₄⋊C₄

yes

—

C₂×Dic₅

C₂×Q₈

yes

characteristic

C₄×D₅

—

C₄×D₅

—

C₄×D₅

C₂×D₄

yes

—

C₄×D₅

C₂×Q₈

yes

—

C₅×C₈

C₄⋊C₄

yes

complemented

#168

C₄⋊C₄

yes

complemented

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

—

—

—

—

yes

characteristic, complemented

#95

—

C₂²×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×D₅

#127

yes

characteristic

C₂×D₅

C₄⋊Q₈

yes

characteristic

C₂×D₅

AGL(1,ℤ/8)

yes

—

C₂×D₅

#137

yes

—

AGL(1,𝔽₅)

—

AGL(1,𝔽₅)

—

AGL(1,𝔽₅)

—

C₄×C₅

—

C₄×C₅

C₂×C₄⋊C₄

yes

—

Dic₅

—

Dic₅

C₄⋊Q₈

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×AGL(1,𝔽₅)

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

yes

C₂×C₅

yes

100

C₂×C₅

#428

yes

—

101

D₅

#496

yes

characteristic

102

D₅

#497

yes

characteristic

103

D₅

#501

yes

—

104

C₂³

—

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

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₄⋊AGL(1,𝔽₅)

yes

—

118

C₂×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

119

C₈

—

120

C₈

C₄⋊AGL(1,𝔽₅)

yes

complemented

121

C₅

#1847

yes

characteristic, complemented, Sylow

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

C₂×C₄⋊AGL(1,𝔽₅)

yes

—

134

C₂

—

135

C₂

—

136

C₂

—

137

C₂

#65688

yes

characteristic

138

C₂

#65727

yes

—

	subgroup	G/H
center	C₂²	C₂×C₄⋊AGL(1,𝔽₅)
commutator	C₂×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#97	C₄
socle	C₂²×C₅	C₂×C₄⋊C₄
prime radical	C₂²×D₅	C₂×D₄
cosocle	C₂×C₄×D₅	C₂³
hypercenter	#97	AGL(1,𝔽₅)
hypocenter	C₅	#1847
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

subgroup

G/H

center

C₂²

C₂×C₄⋊AGL(1,𝔽₅)

commutator

C₂×C₄×C₅

C₂²×C₄

Frattini

C₂×C₄

C₂²×AGL(1,𝔽₅)

Fitting

C₅×#97

C₄

socle

C₂²×C₅

C₂×C₄⋊C₄

prime radical

C₂²×D₅

C₂×D₄

cosocle

C₂×C₄×D₅

C₂³

C₅

G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #97

N	H ≅ G/N	type
15. C₅×#97	C₄	semidirect
16. #3947	C₄	semidirect
17. #3935	C₄	semidirect
56. C₅×C₈	C₄⋊C₄	semidirect
57. #168	C₄⋊C₄	semidirect
68. #97	AGL(1,𝔽₅)	semidirect
120. C₈	C₄⋊AGL(1,𝔽₅)	semidirect
121. C₅	#1847	semidirect

H ≅ G/N

type

15. C₅×#97

C₄

semidirect

16. #3947

C₄

semidirect

17. #3935

C₄

semidirect

56. C₅×C₈

C₄⋊C₄

semidirect

57. #168

C₄⋊C₄

semidirect

68. #97

AGL(1,𝔽₅)

semidirect

120. C₈

C₄⋊AGL(1,𝔽₅)

semidirect

121. C₅

#1847

semidirect

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

subgroup	index
C₂×#3994	2
C₂×#3993	2
#65689	2
#65688	2
D₅×#97	2
#1847	5

subgroup

index

D₅×#97

Subgroups — #10603883

Conjugacy classes of subgroups

Complements

Maximal subgroups