Group #10617523 — subgroups

Subgroups — #10617523

Order 640.

364 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 168 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×Q₈×Dic₅	C₂	1	1	yes	maximal, characteristic
2	#65788	C₂	1	6	yes	maximal, complemented
3	Q₈×#170	C₂	1	2	yes	maximal, complemented
4	#65759	C₂	1	3	yes	maximal
5	#65754	C₂	1	3	yes	maximal
6	C₂²×C₅×Q₈	C₄	1	1	yes	characteristic
7	Q₈×Dic₅	C₄	1	3	yes	—
8	Q₈×Dic₅	C₄	1	1	yes	characteristic
9	Q₈×Dic₅	C₂²	1	1	yes	characteristic
10	Q₈×Dic₅	C₂²	1	3	yes	—
11	C₂×C₄⋊Dic₅	C₄	1	3	yes	—
12	C₂×C₄×Dic₅	C₂²	1	3	yes	—
13	C₂²⋊#170	C₂²	1	1	yes	characteristic
14	C₂²⋊#170	C₂²	1	3	yes	—
15	#4004	C₂²	1	6	yes	—
16	#4002	C₂²	1	6	yes	—
17	C₄⋊#170	C₂²	1	6	yes	—
18	C₄×#170	C₂²	1	6	yes	—
19	#2984	—	5	1	no	maximal, Sylow
20	C₂×C₅×Q₈	—	2	4	no	—
21	C₂×C₅×Q₈	C₂×C₄	1	3	yes	—
22	C₂×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₅	C₂×C₄	1	3	yes	—
24	C₂²×Dic₅	C₂×C₄	1	3	yes	—
25	C₂²×Dic₅	C₂³	1	1	yes	characteristic
26	C₂×#170	C₂³	1	2	yes	—
27	C₂×#170	C₂³	1	6	yes	—
28	C₄⋊Dic₅	C₂×C₄	1	6	yes	—
29	C₄⋊Dic₅	C₂×C₄	1	3	yes	—
30	C₄⋊Dic₅	C₂×C₄	1	3	yes	—
31	C₄×Dic₅	C₂×C₄	1	3	yes	—
32	C₄×Dic₅	C₂×C₄	1	3	yes	—
33	C₄×Dic₅	C₂³	1	3	yes	—
34	C₄×Dic₅	C₂³	1	3	yes	—
35	C₂×C₄×Q₈	—	5	1	no	—
36	#446	—	5	6	no	—
37	C₈×Q₈	—	5	2	no	—
38	#432	—	5	3	no	—
39	#431	—	5	3	no	—
40	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
41	C₅×Q₈	—	2	6	no	—
42	C₅×Q₈	M₁₆	1	4	yes	—
43	C₂×C₄×C₅	—	2	6	no	—
44	C₂×C₄×C₅	C₂²×C₄	1	3	yes	—
45	C₂×C₄×C₅	C₂²×C₄	1	3	yes	—
46	C₂×Dic₅	—	2	6	no	—
47	C₂×Dic₅	—	2	2	no	—
48	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
49	C₂×Dic₅	C₂²×C₄	1	3	yes	—
50	C₂×Dic₅	C₂²×C₄	1	3	yes	—
51	C₂×Dic₅	C₂⁴	1	1	yes	characteristic
52	#170	—	2	2	no	—
53	#170	—	2	6	no	—
54	C₂²×Q₈	AGL(1,𝔽₅)	1	1	yes	characteristic
55	C₄×Q₈	—	5	1	no	—
56	C₄×Q₈	—	5	3	no	—
57	C₄×Q₈	—	5	3	no	—
58	C₄×Q₈	—	5	1	no	—
59	C₂×C₄⋊C₄	—	5	3	no	—
60	C₂×C₄²	—	5	3	no	—
61	C₄⋊C₈	—	5	6	no	—
62	C₄⋊C₈	—	5	6	no	—
63	C₂²⋊C₈	—	5	1	no	—
64	C₂²⋊C₈	—	5	3	no	—
65	#97	—	5	6	no	—
66	C₄×C₈	—	5	6	no	—
67	C₂²×C₅	—	2	1	no	—
68	C₂²×C₅	—	2	1	no	—
69	C₂²×C₅	#131	1	2	yes	—
70	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
71	C₄×C₅	—	2	3	no	—
72	C₄×C₅	C₂×M₁₆	1	6	yes	—
73	Dic₅	—	2	1	no	—
74	Dic₅	—	2	3	no	—
75	Dic₅	—	2	3	no	—
76	Dic₅	2₋¹⁺⁴	1	2	yes	—
77	C₂×Q₈	—	2	4	no	—
78	C₂×Q₈	C₂×AGL(1,𝔽₅)	1	3	yes	—
79	C₂×Q₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
80	C₂²×C₄	—	5	1	no	—
81	C₂²×C₄	—	5	3	no	—
82	C₂²×C₄	C₂×AGL(1,𝔽₅)	1	3	yes	—
83	C₂×C₈	—	5	2	no	—
84	C₂×C₈	—	5	6	no	—
85	C₄⋊C₄	—	5	6	no	—
86	C₄⋊C₄	—	5	3	no	—
87	C₄⋊C₄	—	5	3	no	—
88	C₄²	—	5	3	no	—
89	C₄²	—	5	3	no	—
90	C₄²	—	5	3	no	—
91	C₄²	—	5	3	no	—
92	C₂×C₅	—	2	2	no	—
93	C₂×C₅	#519	1	1	yes	characteristic
94	C₂×C₅	C₂²×M₁₆	1	1	yes	characteristic
95	C₂×C₅	C₂×#131	1	1	yes	characteristic
96	C₂³	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
97	Q₈	—	2	6	no	—
98	Q₈	#704	1	4	yes	—
99	C₂×C₄	—	2	6	no	—
100	C₂×C₄	—	5	1	no	—
101	C₂×C₄	—	5	1	no	—
102	C₂×C₄	—	5	3	no	—
103	C₂×C₄	—	5	3	no	—
104	C₂×C₄	—	10	2	no	—
105	C₂×C₄	—	10	6	no	—
106	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	3	yes	—
107	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	3	yes	—
108	C₈	—	10	2	no	—
109	C₈	—	10	6	no	—
110	C₅	#2984	1	1	yes	characteristic, complemented, Sylow
111	C₂²	—	2	1	no	—
112	C₂²	—	2	1	no	—
113	C₂²	#4133	1	2	yes	—
114	C₂²	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
115	C₄	—	2	3	no	—
116	C₄	—	5	2	no	—
117	C₄	—	10	3	no	—
118	C₄	—	10	3	no	—
119	C₄	—	10	1	no	—
120	C₄	C₂×#704	1	6	yes	—
121	C₂	—	2	2	no	—
122	C₂	C₂×#4133	1	1	yes	characteristic
123	C₂	#66262	1	1	yes	characteristic
124	C₂	C₂²×#704	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×Dic₅

C₂

yes

maximal, characteristic

#65788

C₂

yes

maximal, complemented

Q₈×#170

C₂

yes

maximal, complemented

#65759

C₂

yes

maximal

#65754

C₂

yes

maximal

C₂²×C₅×Q₈

C₄

yes

characteristic

Q₈×Dic₅

C₄

yes

—

Q₈×Dic₅

C₄

yes

characteristic

Q₈×Dic₅

C₂²

yes

characteristic

Q₈×Dic₅

C₂²

yes

—

C₂×C₄⋊Dic₅

C₄

yes

—

C₂×C₄×Dic₅

C₂²

yes

—

C₂²⋊#170

C₂²

yes

characteristic

C₂²⋊#170

C₂²

yes

—

#4004

C₂²

yes

—

#4002

C₂²

yes

—

C₄⋊#170

C₂²

yes

—

C₄×#170

C₂²

yes

—

#2984

—

maximal, Sylow

C₂×C₅×Q₈

—

C₂×C₅×Q₈

C₂×C₄

yes

—

C₂×C₅×Q₈

C₂×C₄

yes

characteristic

C₂²×C₄×C₅

C₂×C₄

yes

—

C₂²×Dic₅

C₂×C₄

yes

—

C₂²×Dic₅

C₂³

yes

characteristic

C₂×#170

C₂³

yes

—

C₂×#170

C₂³

yes

—

C₄⋊Dic₅

C₂×C₄

yes

—

C₄⋊Dic₅

C₂×C₄

yes

—

C₄⋊Dic₅

C₂×C₄

yes

—

C₄×Dic₅

C₂×C₄

yes

—

C₄×Dic₅

C₂×C₄

yes

—

C₄×Dic₅

C₂³

yes

—

C₄×Dic₅

C₂³

yes

—

C₂×C₄×Q₈

—

#446

—

C₈×Q₈

—

#432

—

#431

—

C₂³×C₅

C₂²×C₄

yes

characteristic

C₅×Q₈

—

C₅×Q₈

M₁₆

yes

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²×C₄

yes

—

C₂×C₄×C₅

C₂²×C₄

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

C₂²×C₄

yes

characteristic

C₂×Dic₅

C₂²×C₄

yes

—

C₂×Dic₅

C₂²×C₄

yes

—

C₂×Dic₅

C₂⁴

yes

characteristic

#170

—

#170

—

C₂²×Q₈

AGL(1,𝔽₅)

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

#131

yes

—

C₂²×C₅

C₂³×C₄

yes

characteristic

C₄×C₅

—

C₄×C₅

C₂×M₁₆

yes

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

2₋¹⁺⁴

yes

—

C₂×Q₈

—

C₂×Q₈

C₂×AGL(1,𝔽₅)

yes

—

C₂×Q₈

C₂×AGL(1,𝔽₅)

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×AGL(1,𝔽₅)

yes

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₂×C₅

—

C₂×C₅

#519

yes

characteristic

C₂×C₅

C₂²×M₁₆

yes

characteristic

C₂×C₅

C₂×#131

yes

characteristic

C₂³

C₂²×AGL(1,𝔽₅)

yes

characteristic

Q₈

—

Q₈

#704

yes

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₂²×AGL(1,𝔽₅)

yes

—

107

C₂×C₄

C₂²×AGL(1,𝔽₅)

yes

—

108

C₈

—

109

C₈

—

110

C₅

#2984

yes

characteristic, complemented, Sylow

111

C₂²

—

112

C₂²

—

113

C₂²

#4133

yes

—

114

C₂²

C₂³×AGL(1,𝔽₅)

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

C₂×#704

yes

—

121

C₂

—

122

C₂

C₂×#4133

yes

characteristic

123

C₂

#66262

yes

characteristic

124

C₂

C₂²×#704

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅	C₂³×C₄
Fitting	C₂²×C₅×Q₈	C₄
prime radical	C₂³×C₅	C₂²×C₄
cosocle	C₂×Dic₅	C₂⁴
hypercenter	C₂²×Q₈	AGL(1,𝔽₅)
hypocenter	C₅	#2984
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×AGL(1,𝔽₅)

commutator, socle

C₂²×C₅

C₂³×C₄

Fitting

C₂²×C₅×Q₈

C₄

prime radical

C₂³×C₅

C₂²×C₄

cosocle

C₂×Dic₅

C₂⁴

C₂²×Q₈

C₅

G ▷ C₂²×C₅ ▷ C₅

C₁ ◁ C₂² ◁ C₂²×Q₈

N	H ≅ G/N	type
2. #65788	C₂	semidirect
3. Q₈×#170	C₂	semidirect
110. C₅	#2984	semidirect

H ≅ G/N

type

2. #65788

C₂

semidirect

3. Q₈×#170

C₂

semidirect

110. C₅

#2984

semidirect

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

subgroup	index
C₂×Q₈×Dic₅	2
#65788	2
Q₈×#170	2
#65759	2
#65754	2
#2984	5

subgroup

index

C₂×Q₈×Dic₅

Subgroups — #10617523

Conjugacy classes of subgroups

Complements

Maximal subgroups