Group #10604191 — subgroups

Subgroups — #10604191

Order 640.

300 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 138 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×#4004	C₂	1	4	yes	maximal
2	C₂×C₄⋊#170	C₂	1	2	yes	maximal
3	C₄²×Dic₅	C₂	1	1	yes	maximal, characteristic
4	C₂²×#170	C₂²	1	4	yes	—
5	C₂×C₄²×C₅	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₅	C₄	1	2	yes	—
7	C₂×C₄×Dic₅	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₅	C₂²	1	1	yes	characteristic
9	C₂×C₄×Dic₅	C₂²	1	2	yes	—
10	#4004	—	2	8	no	—
11	C₄⋊#170	—	2	4	no	—
12	#1842	—	5	1	no	maximal, Sylow
13	C₂²×C₄×C₅	C₂×C₄	1	2	yes	—
14	C₂²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
15	C₂²×Dic₅	C₂×C₄	1	1	yes	characteristic
16	C₂²×Dic₅	C₂×C₄	1	2	yes	—
17	C₂²×Dic₅	C₂³	1	1	yes	characteristic
18	C₂×#170	—	2	4	no	—
19	C₂×#170	—	2	8	no	—
20	C₄²×C₅	—	2	2	no	—
21	C₄×Dic₅	—	2	4	no	—
22	C₄×Dic₅	C₈	1	4	yes	complemented
23	C₄×Dic₅	D₄	1	2	yes	—
24	C₄×Dic₅	D₄	1	4	yes	—
25	C₄×Dic₅	Q₈	1	2	yes	—
26	C₄×Dic₅	Q₈	1	4	yes	—
27	C₂×C₄⋊C₈	—	5	2	no	—
28	C₂×C₄⋊C₈	—	5	4	no	—
29	C₄³	—	5	1	no	—
30	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₅	—	2	2	no	—
32	C₂×C₄×C₅	—	2	2	no	—
33	C₂×C₄×C₅	C₄⋊C₄	1	4	yes	—
34	C₂×C₄×C₅	C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₅	C₂×C₈	1	2	yes	—
36	C₂×C₄×C₅	M₁₆	1	2	yes	—
37	C₂×Dic₅	—	2	2	no	—
38	C₂×Dic₅	C₄⋊C₄	1	2	yes	—
39	C₂×Dic₅	C₄⋊C₄	1	4	yes	—
40	C₂×Dic₅	C₂×C₈	1	4	yes	—
41	C₂×Dic₅	M₁₆	1	4	yes	—
42	C₂×Dic₅	C₂×D₄	1	1	yes	characteristic
43	C₂×Dic₅	C₂×D₄	1	2	yes	—
44	C₂×Dic₅	C₂×Q₈	1	2	yes	—
45	C₂×Dic₅	C₂×Q₈	1	1	yes	characteristic
46	#170	—	4	4	no	—
47	C₂²×C₈	—	5	4	no	—
48	C₂×C₄²	—	5	2	no	—
49	C₂×C₄²	—	5	1	no	—
50	C₂×C₄²	—	5	1	no	—
51	C₂×C₄²	—	5	2	no	—
52	C₂×C₄²	AGL(1,𝔽₅)	1	1	yes	characteristic
53	C₄⋊C₈	—	10	4	no	—
54	C₄⋊C₈	—	10	8	no	—
55	C₂²×C₅	C₂×C₄⋊C₄	1	2	yes	—
56	C₂²×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₅	C₂²×C₈	1	1	yes	characteristic
58	C₂²×C₅	C₂×M₁₆	1	2	yes	—
59	C₂²×C₅	C₂×M₁₆	1	1	yes	characteristic
60	C₄×C₅	—	2	4	no	—
61	C₄×C₅	C₄⋊C₈	1	4	yes	complemented
62	Dic₅	—	2	2	no	—
63	Dic₅	C₄⋊C₈	1	8	yes	complemented
64	Dic₅	#127	1	1	yes	characteristic
65	Dic₅	C₄⋊Q₈	1	1	yes	characteristic
66	Dic₅	C₄⋊Q₈	1	2	yes	—
67	C₂²×C₄	—	5	1	no	—
68	C₂²×C₄	—	5	2	no	—
69	C₂²×C₄	—	5	1	no	—
70	C₂²×C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
71	C₂²×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
72	C₂×C₈	—	10	4	no	—
73	C₂×C₈	—	10	8	no	—
74	C₄²	—	2	2	no	—
75	C₄²	—	5	4	no	—
76	C₄²	—	5	4	no	—
77	C₄²	—	5	2	no	—
78	C₄²	—	5	4	no	—
79	C₄²	—	5	2	no	—
80	C₄²	—	10	4	no	—
81	C₂×C₅	#384	1	1	yes	characteristic
82	C₂×C₅	C₂×C₄⋊C₈	1	2	yes	—
83	C₂×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
84	C₂×C₅	#423	1	1	yes	characteristic
85	C₂×C₅	#423	1	2	yes	—
86	C₂³	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	5	4	no	—
90	C₂×C₄	—	5	4	no	—
91	C₂×C₄	—	5	2	no	—
92	C₂×C₄	—	5	1	no	—
93	C₂×C₄	—	5	4	no	—
94	C₂×C₄	—	5	2	no	—
95	C₂×C₄	—	5	1	no	—
96	C₂×C₄	—	5	2	no	—
97	C₂×C₄	—	10	2	no	—
98	C₂×C₄	D₅⋊C₈	1	2	yes	—
99	C₂×C₄	C₄⋊AGL(1,𝔽₅)	1	4	yes	—
100	C₂×C₄	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
101	C₂×C₄	#704	1	2	yes	—
102	C₈	—	20	4	no	—
103	C₅	#1842	1	1	yes	characteristic, complemented, Sylow
104	C₂²	C₂×D₅⋊C₈	1	1	yes	characteristic
105	C₂²	D₅⋊M₁₆	1	2	yes	—
106	C₂²	C₂×C₄⋊AGL(1,𝔽₅)	1	2	yes	—
107	C₂²	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
108	C₂²	C₂×#704	1	1	yes	characteristic
109	C₄	—	2	4	no	—
110	C₄	—	5	8	no	—
111	C₄	—	5	1	no	—
112	C₄	—	5	2	no	—
113	C₄	—	5	1	no	—
114	C₄	—	10	2	no	—
115	C₄	#4004	1	4	yes	complemented
116	C₂	D₅⋊(C₄⋊C₈)	1	2	yes	—
117	C₂	#65682	1	1	yes	characteristic
118	C₂	#65688	1	1	yes	characteristic
119	C₂	C₂×#4004	1	1	yes	characteristic
120	C₂	#65758	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#4004

C₂

yes

maximal

C₂×C₄⋊#170

C₂

yes

maximal

C₄²×Dic₅

C₂

yes

maximal, characteristic

C₂²×#170

C₂²

yes

—

C₂×C₄²×C₅

C₄

yes

characteristic

C₂×C₄×Dic₅

C₄

yes

—

C₂×C₄×Dic₅

C₄

yes

characteristic

C₂×C₄×Dic₅

C₂²

yes

characteristic

C₂×C₄×Dic₅

C₂²

yes

—

—

—

—

C₂²×C₄×C₅

C₂×C₄

yes

—

C₂²×C₄×C₅

C₂×C₄

yes

characteristic

C₂²×Dic₅

C₂×C₄

yes

characteristic

C₂²×Dic₅

C₂×C₄

yes

—

C₂²×Dic₅

C₂³

yes

characteristic

C₂×#170

—

C₂×#170

—

C₄²×C₅

—

C₄×Dic₅

—

C₄×Dic₅

C₈

yes

complemented

C₄×Dic₅

D₄

yes

—

C₄×Dic₅

D₄

yes

—

C₄×Dic₅

Q₈

yes

—

C₄×Dic₅

Q₈

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₄³

—

C₂³×C₅

C₂²×C₄

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₄⋊C₄

yes

—

C₂×C₄×C₅

C₄⋊C₄

yes

—

C₂×C₄×C₅

C₂×C₈

yes

—

C₂×C₄×C₅

M₁₆

yes

—

C₂×Dic₅

—

C₂×Dic₅

C₄⋊C₄

yes

—

C₂×Dic₅

C₄⋊C₄

yes

—

C₂×Dic₅

C₂×C₈

yes

—

C₂×Dic₅

M₁₆

yes

—

C₂×Dic₅

C₂×D₄

yes

characteristic

C₂×Dic₅

C₂×D₄

yes

—

C₂×Dic₅

C₂×Q₈

yes

—

C₂×Dic₅

C₂×Q₈

yes

characteristic

#170

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

AGL(1,𝔽₅)

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²×C₅

C₂×C₄⋊C₄

yes

—

C₂²×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅

C₂²×C₈

yes

characteristic

C₂²×C₅

C₂×M₁₆

yes

—

C₂²×C₅

C₂×M₁₆

yes

characteristic

C₄×C₅

—

C₄×C₅

C₄⋊C₈

yes

complemented

Dic₅

—

Dic₅

C₄⋊C₈

yes

complemented

Dic₅

#127

yes

characteristic

Dic₅

C₄⋊Q₈

yes

characteristic

Dic₅

C₄⋊Q₈

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×AGL(1,𝔽₅)

yes

—

C₂²×C₄

C₂×AGL(1,𝔽₅)

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₂×C₅

#384

yes

characteristic

C₂×C₅

C₂×C₄⋊C₈

yes

—

C₂×C₅

C₂×C₄⋊C₈

yes

C₂×C₅

yes

C₂×C₅

yes

—

C₂³

C₂²×AGL(1,𝔽₅)

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₅⋊C₈

yes

—

C₂×C₄

C₄⋊AGL(1,𝔽₅)

yes

—

100

C₂×C₄

C₄⋊AGL(1,𝔽₅)

yes

—

101

C₂×C₄

#704

yes

—

102

C₈

—

103

C₅

#1842

yes

characteristic, complemented, Sylow

104

C₂²

C₂×D₅⋊C₈

yes

characteristic

105

C₂²

D₅⋊M₁₆

yes

—

106

C₂²

C₂×C₄⋊AGL(1,𝔽₅)

yes

—

107

C₂²

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

108

C₂²

C₂×#704

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

#4004

yes

complemented

116

C₂

D₅⋊(C₄⋊C₈)

yes

—

117

C₂

#65682

yes

characteristic

118

C₂

#65688

yes

characteristic

119

C₂

C₂×#4004

yes

characteristic

120

C₂

#65758

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂²×AGL(1,𝔽₅)
commutator	C₂²×C₅	C₂²×C₈
Fitting	C₂×C₄²×C₅	C₄
socle, prime radical	C₂³×C₅	C₂²×C₄
cosocle	C₂²×Dic₅	C₂³
hypercenter	C₂×C₄²	AGL(1,𝔽₅)
hypocenter	C₅	#1842
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

subgroup

G/H

center, Frattini

C₂³

C₂²×AGL(1,𝔽₅)

commutator

C₂²×C₅

C₂²×C₈

Fitting

C₂×C₄²×C₅

C₄

socle, prime radical

C₂³×C₅

C₂²×C₄

cosocle

C₂²×Dic₅

C₂³

C₂×C₄²

C₅

G ▷ C₂²×C₅ ▷ C₅

C₁ ◁ C₂³ ◁ C₂×C₄²

N	H ≅ G/N	type
22. C₄×Dic₅	C₈	semidirect
61. C₄×C₅	C₄⋊C₈	semidirect
63. Dic₅	C₄⋊C₈	semidirect
103. C₅	#1842	semidirect
115. C₄	#4004	semidirect

H ≅ G/N

type

22. C₄×Dic₅

C₈

semidirect

61. C₄×C₅

C₄⋊C₈

semidirect

63. Dic₅

C₄⋊C₈

semidirect

103. C₅

#1842

semidirect

115. C₄

#4004

semidirect

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

subgroup	index
C₂×#4004	2
C₂×C₄⋊#170	2
C₄²×Dic₅	2
#1842	5

subgroup

index

C₂×#4004

C₂×C₄⋊#170

C₄²×Dic₅

#1842

Subgroups — #10604191

Conjugacy classes of subgroups

Complements

Maximal subgroups