Group #10604019 — subgroups

Subgroups — #10604019

Order 640.

158 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 64 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	#65681	C₂	1	1	yes	maximal, characteristic
2	#65680	C₂	1	1	yes	maximal, characteristic
3	#65122	C₂	1	1	yes	maximal, characteristic
4	#64884	C₂	1	2	yes	maximal, complemented
5	#64881	C₂	1	2	yes	maximal, complemented
6	C₂×#700	—	2	1	no	—
7	C₄²×D₅	C₂²	1	1	yes	characteristic
8	#4003	—	2	1	no	—
9	C₄⋊#170	—	2	1	no	—
10	C₄⋊#170	C₂²	1	1	yes	characteristic
11	C₄⋊#170	C₂²	1	2	yes	—
12	C₄⋊#170	C₂²	1	1	yes	characteristic
13	C₄×#170	—	2	1	no	—
14	C₅×C₄⋊C₈	C₄	1	1	yes	characteristic
15	#3952	C₄	1	2	yes	—
16	C₈×Dic₅	C₂²	1	2	yes	—
17	C₄⋊#168	C₄	1	1	yes	characteristic
18	#1574	—	5	1	no	maximal, Sylow
19	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×D₅	D₄	1	1	yes	characteristic
21	C₂×C₄×D₅	Q₈	1	1	yes	characteristic
22	C₂×#170	—	2	2	no	—
23	C₂×#170	—	2	1	no	—
24	C₂×#170	—	2	1	no	—
25	#700	—	4	2	no	—
26	C₂×C₅×C₈	C₂×C₄	1	2	yes	—
27	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
28	C₄×Dic₅	D₄	1	1	yes	characteristic
29	C₄×Dic₅	Q₈	1	1	yes	characteristic
30	C₄×Dic₅	C₂³	1	1	yes	characteristic
31	C₂×#168	C₂×C₄	1	2	yes	—
32	#431	—	5	1	no	—
33	#431	—	5	1	no	—
34	#423	—	5	1	no	—
35	#335	—	5	2	no	—
36	#334	—	5	2	no	—
37	C₂²×D₅	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
41	C₂×Dic₅	C₄⋊C₄	1	1	yes	characteristic
42	C₂×Dic₅	C₂×D₄	1	1	yes	characteristic
43	C₂×Dic₅	C₂×Q₈	1	1	yes	characteristic
44	C₄×D₅	—	2	2	no	—
45	C₄×D₅	—	2	1	no	—
46	C₄×D₅	—	2	1	no	—
47	C₄×D₅	—	2	1	no	—
48	C₄×D₅	—	2	1	no	—
49	#170	—	4	2	no	—
50	#170	—	4	1	no	—
51	#170	—	4	1	no	—
52	C₅×C₈	—	2	2	no	—
53	#168	—	2	2	no	—
54	C₂×M₁₆	—	10	1	no	—
55	C₂×C₄²	—	5	1	no	—
56	C₄⋊C₈	—	5	1	no	—
57	C₄⋊C₈	—	5	2	no	—
58	C₄⋊C₈	—	5	1	no	—
59	C₄⋊C₈	—	5	1	no	—
60	C₄⋊C₈	—	10	1	no	—
61	C₄⋊C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
62	C₂²⋊C₈	—	5	2	no	—
63	C₂²⋊C₈	—	10	1	no	—
64	C₄×C₈	—	5	2	no	—
65	C₄×C₈	—	10	1	no	—
66	C₂²×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×D₅	—	2	1	no	—
68	C₂×D₅	—	2	1	no	—
69	C₂×D₅	—	2	1	no	—
70	C₄×C₅	—	2	1	no	—
71	C₄×C₅	#108	1	2	yes	—
72	C₄×C₅	#131	1	1	yes	characteristic
73	C₄×C₅	#131	1	1	yes	characteristic
74	Dic₅	—	2	1	no	—
75	Dic₅	—	2	1	no	—
76	Dic₅	AGL(1,ℤ/8)	1	1	yes	characteristic
77	Dic₅	#137	1	1	yes	characteristic
78	C₂²×C₄	—	5	1	no	—
79	C₂²×C₄	—	5	1	no	—
80	C₂²×C₄	—	5	1	no	—
81	M₁₆	—	20	2	no	—
82	C₂×C₈	—	5	2	no	—
83	C₂×C₈	—	10	1	no	—
84	C₂×C₈	—	10	1	no	—
85	C₂×C₈	—	10	2	no	—
86	C₂×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
87	C₄²	—	5	1	no	—
88	C₄²	—	5	1	no	—
89	C₄²	—	5	1	no	—
90	C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
91	C₂×C₅	#424	1	1	yes	characteristic
92	C₂×C₅	#428	1	1	yes	characteristic
93	C₂×C₅	C₂×#108	1	1	yes	characteristic
94	D₅	—	4	1	no	—
95	C₂³	—	5	1	no	—
96	C₂×C₄	—	5	1	no	—
97	C₂×C₄	—	5	1	no	—
98	C₂×C₄	—	5	1	no	—
99	C₂×C₄	—	10	1	no	—
100	C₂×C₄	—	10	1	no	—
101	C₂×C₄	—	10	1	no	—
102	C₂×C₄	—	10	1	no	—
103	C₂×C₄	—	10	2	no	—
104	C₂×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
105	C₂×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
106	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
107	C₈	—	2	2	no	—
108	C₈	—	10	2	no	—
109	C₈	—	20	2	no	—
110	C₈	—	20	1	no	—
111	C₈	—	20	1	no	—
112	C₅	#1574	1	1	yes	characteristic, complemented, Sylow
113	C₂²	—	10	1	no	—
114	C₂²	—	10	1	no	—
115	C₂²	—	10	1	no	—
116	C₂²	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
117	C₄	—	2	1	no	—
118	C₄	—	5	1	no	—
119	C₄	—	5	1	no	—
120	C₄	—	10	1	no	—
121	C₄	—	10	1	no	—
122	C₄	#3995	1	2	yes	—
123	C₄	#4131	1	1	yes	characteristic
124	C₄	#4133	1	1	yes	characteristic
125	C₂	—	20	1	no	—
126	C₂	#65708	1	1	yes	characteristic
127	C₂	C₂×#3995	1	1	yes	characteristic
128	C₂	#65727	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#65681

C₂

yes

maximal, characteristic

#65680

C₂

yes

maximal, characteristic

#65122

C₂

yes

maximal, characteristic

#64884

C₂

yes

maximal, complemented

#64881

C₂

yes

maximal, complemented

C₂×#700

—

C₄²×D₅

C₂²

yes

—

—

C₂²

yes

characteristic

C₄⋊#170

C₂²

yes

—

C₄⋊#170

C₂²

yes

characteristic

C₄×#170

—

C₅×C₄⋊C₈

C₄

yes

characteristic

#3952

C₄

yes

—

C₈×Dic₅

C₂²

yes

—

C₄⋊#168

C₄

yes

characteristic

#1574

—

maximal, Sylow

C₂×C₄×D₅

C₂×C₄

yes

characteristic

C₂×C₄×D₅

D₄

yes

characteristic

C₂×C₄×D₅

Q₈

yes

—

—

—

—

C₂×C₅×C₈

C₂×C₄

yes

—

C₄²×C₅

C₂×C₄

yes

characteristic

C₄×Dic₅

D₄

yes

characteristic

C₄×Dic₅

Q₈

yes

characteristic

C₄×Dic₅

C₂³

yes

characteristic

C₂×#168

C₂×C₄

yes

—

—

—

—

—

—

C₂²×D₅

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×Dic₅

C₄⋊C₄

yes

characteristic

C₂×Dic₅

C₂×D₄

yes

characteristic

C₂×Dic₅

C₂×Q₈

yes

characteristic

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

#170

—

#170

—

#170

—

C₅×C₈

—

#168

—

C₂×M₁₆

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

AGL(1,𝔽₅)

yes

characteristic

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

#108

yes

—

C₄×C₅

yes

C₄×C₅

yes

Dic₅

—

Dic₅

—

Dic₅

yes

Dic₅

yes

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×AGL(1,𝔽₅)

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

yes

C₂×C₅

yes

C₂×C₅

yes

C₂×C₅

yes

D₅

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₄⋊AGL(1,𝔽₅)

yes

characteristic

105

C₂×C₄

C₄⋊AGL(1,𝔽₅)

yes

characteristic

106

C₂×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

107

C₈

—

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

—

112

C₅

#1574

yes

characteristic, complemented, Sylow

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

#3995

yes

—

123

C₄

#4131

yes

characteristic

124

C₄

#4133

yes

characteristic

125

C₂

—

126

C₂

#65708

yes

characteristic

127

C₂

C₂×#3995

yes

characteristic

128

C₂

#65727

yes

characteristic

	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×C₄⋊C₈	C₄
socle	C₂²×C₅	C₂×C₄⋊C₄
prime radical	C₂²×D₅	C₄⋊C₄
cosocle	C₄×Dic₅	C₂³
hypercenter	C₄⋊C₈	AGL(1,𝔽₅)
hypocenter	C₅	#1574
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

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×C₄⋊C₈

C₄

socle

C₂²×C₅

C₂×C₄⋊C₄

prime radical

C₂²×D₅

C₄⋊C₄

cosocle

C₄×Dic₅

C₂³

C₄⋊C₈

C₅

G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

N	H ≅ G/N	type
4. #64884	C₂	semidirect
5. #64881	C₂	semidirect
112. C₅	#1574	semidirect

H ≅ G/N

type

4. #64884

C₂

semidirect

5. #64881

C₂

semidirect

112. C₅

#1574

semidirect

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

subgroup	index
#65681	2
#65680	2
#65122	2
#64884	2
#64881	2
#1574	5

subgroup

index

Subgroups — #10604019

Conjugacy classes of subgroups

Complements

Maximal subgroups