Group #10604587 — subgroups

Subgroups — #10604587

Order 640.

284 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 74 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	#66022	C₂	1	1	yes	maximal, characteristic
2	C₄²×AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic
3	#65682	C₂	1	1	yes	maximal, characteristic
4	#64875	C₂	1	4	yes	maximal
5	C₂×C₄×AGL(1,𝔽₅)	—	2	2	no	—
6	C₂×C₄×AGL(1,𝔽₅)	C₂²	1	2	yes	—
7	C₂×#700	C₂²	1	2	yes	—
8	C₅×#125	C₄	1	1	yes	characteristic, complemented
9	#4042	—	2	2	no	—
10	#4038	C₂²	1	2	yes	—
11	#4034	C₄	1	2	yes	complemented
12	#4033	—	2	2	no	—
13	C₄²×D₅	C₂²	1	1	yes	characteristic
14	#4016	C₄	1	1	yes	characteristic, complemented
15	#4004	—	2	2	no	—
16	#1994	—	5	1	no	maximal, Sylow
17	C₂²×AGL(1,𝔽₅)	—	2	2	no	—
18	C₂×C₄×D₅	D₄	1	2	yes	—
19	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
20	C₂×Dic₁₀	—	2	2	no	—
21	C₂×#170	—	2	2	no	—
22	C₄×AGL(1,𝔽₅)	—	2	8	no	—
23	C₄×AGL(1,𝔽₅)	—	2	2	no	—
24	C₄×AGL(1,𝔽₅)	Q₈	1	4	yes	complemented
25	#700	—	2	4	no	—
26	C₅×C₄⋊C₄	—	2	2	no	—
27	C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
28	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
29	#685	—	2	2	no	—
30	C₄⋊Dic₅	C₂×C₄	1	2	yes	—
31	#683	—	2	2	no	—
32	#683	—	2	2	no	—
33	C₄×Dic₅	—	2	2	no	—
34	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
35	C₄×Dic₅	D₄	1	2	yes	—
36	#533	—	5	1	no	—
37	#423	—	5	1	no	—
38	C₄³	—	5	1	no	—
39	#339	—	5	4	no	—
40	C₂²×D₅	C₂×D₄	1	1	yes	characteristic
41	C₂×AGL(1,𝔽₅)	—	2	8	no	—
42	C₂×AGL(1,𝔽₅)	—	2	4	no	—
43	C₂×C₄×C₅	—	2	2	no	—
44	C₂×C₄×C₅	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
46	C₂×Dic₅	—	2	2	no	—
47	C₂×Dic₅	C₂²⋊C₄	1	2	yes	—
48	C₂×Dic₅	C₂×D₄	1	1	yes	characteristic
49	C₄×D₅	—	2	4	no	—
50	C₄×D₅	C₂×Q₈	1	2	yes	—
51	C₄×D₅	#41	1	2	yes	—
52	Dic₁₀	—	4	4	no	—
53	#170	—	4	2	no	—
54	C₂×M₁₆	—	5	2	no	—
55	C₄⋊Q₈	—	5	2	no	—
56	#125	—	5	1	no	—
57	#125	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
58	C₂²⋊Q₈	—	10	2	no	—
59	C₄×Q₈	—	10	2	no	—
60	#117	—	5	2	no	—
61	C₂×C₄²	—	5	1	no	—
62	C₂×C₄²	—	5	2	no	—
63	C₂×C₄²	—	10	2	no	—
64	C₄⋊C₈	—	10	2	no	—
65	C₂²×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×D₅	—	2	2	no	—
67	C₂×D₅	C₂²⋊Q₈	1	2	yes	—
68	AGL(1,𝔽₅)	—	2	8	no	—
69	C₄×C₅	—	2	2	no	—
70	C₄×C₅	—	4	2	no	—
71	C₄×C₅	C₄×Q₈	1	2	yes	—
72	Dic₅	—	4	2	no	—
73	Dic₅	#104	1	4	yes	—
74	Dic₅	#124	1	1	yes	characteristic
75	Dic₅	C₄⋊Q₈	1	1	yes	characteristic
76	C₂×Q₈	—	10	2	no	—
77	C₂²×C₄	—	5	1	no	—
78	C₂²×C₄	—	5	2	no	—
79	C₂²×C₄	—	10	2	no	—
80	M₁₆	—	10	4	no	—
81	C₂×C₈	—	10	2	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	5	2	no	—
84	C₄⋊C₄	—	10	2	no	—
85	C₄⋊C₄	—	10	2	no	—
86	C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
87	C₂²⋊C₄	—	10	2	no	—
88	C₄²	—	5	2	no	—
89	C₄²	—	5	1	no	—
90	C₄²	—	5	4	no	—
91	C₄²	—	10	2	no	—
92	C₄²	—	10	2	no	—
93	C₄²	—	10	8	no	—
94	C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
95	C₂×C₅	#391	1	1	yes	characteristic
96	C₂×C₅	C₂×#104	1	2	yes	—
97	D₅	—	2	2	no	—
98	C₂³	—	5	1	no	—
99	Q₈	—	20	4	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	5	1	no	—
102	C₂×C₄	—	5	2	no	—
103	C₂×C₄	—	5	2	no	—
104	C₂×C₄	—	5	2	no	—
105	C₂×C₄	—	10	4	no	—
106	C₂×C₄	—	10	4	no	—
107	C₂×C₄	—	10	8	no	—
108	C₂×C₄	—	10	2	no	—
109	C₂×C₄	C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
110	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
111	C₈	—	20	2	no	—
112	C₅	#1994	1	1	yes	characteristic, complemented, Sylow
113	C₂²	—	5	2	no	—
114	C₂²	—	10	2	no	—
115	C₂²	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	4	2	no	—
118	C₄	—	5	4	no	—
119	C₄	—	5	1	no	—
120	C₄	—	5	1	no	—
121	C₄	—	10	8	no	—
122	C₄	—	20	2	no	—
123	C₄	Q₈×AGL(1,𝔽₅)	1	2	yes	—
124	C₂	—	10	2	no	—
125	C₂	(C₂×AGL(1,𝔽₅))⋊Q₈	1	1	yes	characteristic
126	C₂	D₅⋊#104	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#66022

C₂

yes

maximal, characteristic

C₄²×AGL(1,𝔽₅)

C₂

yes

maximal, characteristic

#65682

C₂

yes

maximal, characteristic

#64875

C₂

yes

maximal

C₂×C₄×AGL(1,𝔽₅)

—

C₂×C₄×AGL(1,𝔽₅)

C₂²

yes

—

C₂×#700

C₂²

yes

—

C₅×#125

C₄

yes

characteristic, complemented

#4042

—

#4038

C₂²

yes

—

#4034

C₄

yes

complemented

#4033

—

C₄²×D₅

C₂²

yes

characteristic

#4016

C₄

yes

characteristic, complemented

#4004

—

#1994

—

maximal, Sylow

C₂²×AGL(1,𝔽₅)

—

C₂×C₄×D₅

D₄

yes

—

C₂×C₄×D₅

C₂³

yes

characteristic

C₂×Dic₁₀

—

—

—

—

Q₈

yes

complemented

#700

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

C₂×C₄

yes

—

C₄²×C₅

C₂×C₄

yes

characteristic

#685

—

C₄⋊Dic₅

C₂×C₄

yes

—

#683

—

#683

—

C₄×Dic₅

—

C₄×Dic₅

C₂×C₄

yes

characteristic

C₄×Dic₅

D₄

yes

—

#533

—

#423

—

C₄³

—

#339

—

C₂²×D₅

C₂×D₄

yes

characteristic

C₂×AGL(1,𝔽₅)

—

C₂×AGL(1,𝔽₅)

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²⋊C₄

yes

—

C₂×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

C₂²⋊C₄

yes

—

C₂×Dic₅

C₂×D₄

yes

characteristic

C₄×D₅

—

C₄×D₅

C₂×Q₈

yes

—

C₄×D₅

#41

yes

—

Dic₁₀

—

#170

—

C₂×M₁₆

—

C₄⋊Q₈

—

#125

—

#125

AGL(1,𝔽₅)

yes

characteristic, complemented

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

C₂²×C₅

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₅

—

C₂×D₅

C₂²⋊Q₈

yes

—

AGL(1,𝔽₅)

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

C₄×Q₈

yes

—

Dic₅

—

Dic₅

#104

yes

—

Dic₅

#124

yes

characteristic

Dic₅

C₄⋊Q₈

yes

characteristic

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×AGL(1,𝔽₅)

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

yes

C₂×C₅

yes

C₂×C₅

yes

—

D₅

—

C₂³

—

Q₈

—

100

C₂×C₄

—

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

C₂²⋊AGL(1,𝔽₅)

yes

—

110

C₂×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

111

C₈

—

112

C₅

#1994

yes

characteristic, complemented, Sylow

113

C₂²

—

114

C₂²

—

115

C₂²

C₂×C₂²⋊AGL(1,𝔽₅)

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

Q₈×AGL(1,𝔽₅)

yes

—

124

C₂

—

125

C₂

(C₂×AGL(1,𝔽₅))⋊Q₈

yes

characteristic

126

C₂

D₅⋊#104

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊AGL(1,𝔽₅)
commutator	C₂×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#125	C₄
socle	C₂²×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×D₅	C₂×D₄
cosocle	C₂×C₄×D₅	C₂³
hypercenter	#125	AGL(1,𝔽₅)
hypocenter	C₅	#1994
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

subgroup

G/H

center

C₂²

C₂×C₂²⋊AGL(1,𝔽₅)

commutator

C₂×C₄×C₅

C₂²×C₄

Frattini

C₂×C₄

C₂²×AGL(1,𝔽₅)

Fitting

C₅×#125

C₄

socle

C₂²×C₅

C₂×C₂²⋊C₄

prime radical

C₂²×D₅

C₂×D₄

cosocle

C₂×C₄×D₅

C₂³

C₅

G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #125

N	H ≅ G/N	type
8. C₅×#125	C₄	semidirect
11. #4034	C₄	semidirect
14. #4016	C₄	semidirect
24. C₄×AGL(1,𝔽₅)	Q₈	semidirect
57. #125	AGL(1,𝔽₅)	semidirect
112. C₅	#1994	semidirect

H ≅ G/N

type

8. C₅×#125

C₄

semidirect

11. #4034

C₄

semidirect

14. #4016

C₄

semidirect

24. C₄×AGL(1,𝔽₅)

Q₈

semidirect

57. #125

AGL(1,𝔽₅)

semidirect

112. C₅

#1994

semidirect

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

subgroup	index
#66022	2
C₄²×AGL(1,𝔽₅)	2
#65682	2
#64875	2
#1994	5

subgroup

index

C₄²×AGL(1,𝔽₅)

Subgroups — #10604587

Conjugacy classes of subgroups

Complements

Maximal subgroups