Group #10602007 — subgroups

Subgroups — #10602007

Order 640.

168 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 61 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	C₅×#487	C₂	1	1	yes	maximal, characteristic
2	#65382	C₂	1	1	yes	maximal, characteristic
3	#65370	C₂	1	1	yes	maximal, characteristic
4	#65367	C₂	1	1	yes	maximal, characteristic
5	#65367	C₂	1	1	yes	maximal, characteristic
6	#64970	C₂	1	1	yes	maximal, characteristic
7	C₈×#168	C₂	1	1	yes	maximal, characteristic
8	C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	#4015	C₂²	1	1	yes	characteristic
11	#4015	C₂²	1	1	yes	characteristic
12	C₅×#103	—	2	2	no	—
13	C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	#3967	—	2	2	no	—
15	#3948	—	2	2	no	—
16	#3942	—	2	2	no	—
17	#3939	—	2	2	no	—
18	#3939	—	2	2	no	—
19	C₄×#168	C₂²	1	1	yes	characteristic
20	C₄×#168	C₂²	1	1	yes	characteristic
21	#1710	—	5	1	no	maximal, Sylow
22	C₂×C₅×Q₈	—	2	1	no	—
23	C₂×C₅×Q₈	—	2	1	no	—
24	C₂×Dic₁₀	—	2	1	no	—
25	C₂×Dic₁₀	—	2	1	no	—
26	C₂×C₅×C₈	D₄	1	2	yes	—
27	C₅×C₄⋊C₄	—	2	1	no	—
28	C₅×C₄⋊C₄	—	2	1	no	—
29	C₅×C₄⋊C₄	—	2	1	no	—
30	C₅×C₄⋊C₄	—	2	1	no	—
31	C₄²×C₅	C₂³	1	1	yes	characteristic
32	C₄⋊Dic₅	—	2	1	no	—
33	C₄⋊Dic₅	—	2	1	no	—
34	C₄⋊Dic₅	—	2	1	no	—
35	C₄⋊Dic₅	—	2	1	no	—
36	C₂×#168	D₄	1	2	yes	—
37	C₂×#168	D₄	1	2	yes	—
38	#500	—	5	1	no	—
39	#500	—	5	1	no	—
40	#487	—	5	1	no	—
41	#487	—	5	1	no	—
42	#487	—	5	1	no	—
43	#487	D₅	1	1	yes	characteristic
44	C₈²	—	5	1	no	—
45	C₅×Q₈	—	4	2	no	—
46	C₅×Q₈	—	4	2	no	—
47	C₂×C₄×C₅	—	4	1	no	—
48	C₂×C₄×C₅	—	4	1	no	—
49	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₅	—	4	1	no	—
53	C₂×Dic₅	—	4	1	no	—
54	Dic₁₀	—	4	2	no	—
55	Dic₁₀	—	4	2	no	—
56	C₅×C₈	—	2	2	no	—
57	#168	—	2	2	no	—
58	#168	Q₁₆	1	4	yes	—
59	C₄⋊Q₈	—	5	1	no	—
60	C₄⋊Q₈	—	5	1	no	—
61	C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic
62	C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic
63	#107	—	10	2	no	—
64	#107	—	10	2	no	—
65	#103	—	2	2	no	—
66	#103	—	10	2	no	—
67	#103	—	10	2	no	—
68	#103	—	10	2	no	—
69	C₄×C₈	—	5	1	no	—
70	C₄×C₈	—	5	1	no	—
71	C₄×C₈	C₂×D₅	1	1	yes	characteristic
72	C₂²×C₅	#127	1	1	yes	characteristic
73	C₄×C₅	—	8	1	no	—
74	C₄×C₅	—	8	1	no	—
75	C₄×C₅	C₂×Q₁₆	1	2	yes	—
76	C₄×C₅	#135	1	2	yes	—
77	C₄×C₅	#135	1	2	yes	—
78	Dic₅	—	8	1	no	—
79	Dic₅	—	8	1	no	—
80	C₂×Q₈	—	2	1	no	—
81	C₂×Q₈	—	2	1	no	—
82	C₂×Q₈	—	10	1	no	—
83	C₂×Q₈	—	10	1	no	—
84	C₂×C₈	—	5	2	no	—
85	C₂×C₈	—	5	2	no	—
86	C₂×C₈	C₂²⋊D₅	1	2	yes	—
87	C₄⋊C₄	—	2	1	no	—
88	C₄⋊C₄	—	2	1	no	—
89	C₄⋊C₄	—	2	1	no	—
90	C₄⋊C₄	—	2	1	no	—
91	C₄⋊C₄	—	10	1	no	—
92	C₄⋊C₄	—	10	1	no	—
93	C₄⋊C₄	—	10	1	no	—
94	C₄⋊C₄	—	10	1	no	—
95	C₄²	C₂²×D₅	1	1	yes	characteristic
96	C₂×C₅	#494	1	1	yes	characteristic
97	C₂×C₅	#495	1	1	yes	characteristic
98	C₂×C₅	#495	1	1	yes	characteristic
99	Q₈	—	4	2	no	—
100	Q₈	—	4	2	no	—
101	Q₈	—	20	2	no	—
102	Q₈	—	20	2	no	—
103	C₂×C₄	—	4	1	no	—
104	C₂×C₄	—	4	1	no	—
105	C₂×C₄	—	20	1	no	—
106	C₂×C₄	—	20	1	no	—
107	C₂×C₄	D₄×D₅	1	1	yes	characteristic
108	C₂×C₄	D₄×D₅	1	1	yes	characteristic
109	C₂×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
110	C₈	—	2	2	no	—
111	C₈	—	5	4	no	—
112	C₈	—	10	2	no	—
113	C₅	#1710	1	1	yes	characteristic, complemented, Sylow
114	C₂²	#4086	1	1	yes	characteristic
115	C₄	—	8	1	no	—
116	C₄	—	8	1	no	—
117	C₄	—	40	1	no	—
118	C₄	—	40	1	no	—
119	C₄	#4062	1	2	yes	—
120	C₄	D₅×Q₁₆	1	2	yes	—
121	C₄	#4096	1	2	yes	—
122	C₂	#65348	1	1	yes	characteristic
123	C₂	#65457	1	1	yes	characteristic
124	C₂	#65470	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#487

C₂

yes

maximal, characteristic

#65382

C₂

yes

maximal, characteristic

#65370

C₂

yes

maximal, characteristic

#65367

C₂

yes

maximal, characteristic

#65367

C₂

yes

maximal, characteristic

#64970

C₂

yes

maximal, characteristic

C₈×#168

C₂

yes

maximal, characteristic

C₅×C₄⋊Q₈

C₂²

yes

characteristic

C₅×C₄⋊Q₈

C₂²

yes

characteristic

#4015

C₂²

yes

characteristic

#4015

C₂²

yes

characteristic

C₅×#103

—

C₄×C₅×C₈

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

C₄×#168

C₂²

yes

characteristic

#1710

—

maximal, Sylow

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×C₅×C₈

D₄

yes

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₂³

yes

characteristic

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₂×#168

D₄

yes

—

C₂×#168

D₄

yes

—

—

—

—

—

—

D₅

yes

characteristic

C₈²

—

C₅×Q₈

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×D₄

yes

characteristic

C₂×C₄×C₅

C₂×D₄

yes

characteristic

C₂×C₄×C₅

C₂×D₄

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

—

#168

—

#168

Q₁₆

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅

yes

characteristic

C₄⋊Q₈

C₂×D₅

yes

—

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₅

yes

characteristic

C₂²×C₅

#127

yes

characteristic

C₄×C₅

—

C₄×C₅

—

C₄×C₅

C₂×Q₁₆

yes

—

C₄×C₅

#135

yes

—

C₄×C₅

#135

yes

—

Dic₅

—

Dic₅

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₅

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×D₅

yes

C₂×C₅

yes

C₂×C₅

yes

C₂×C₅

yes

Q₈

—

100

Q₈

—

101

Q₈

—

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

D₄×D₅

yes

characteristic

108

C₂×C₄

D₄×D₅

yes

characteristic

109

C₂×C₄

C₂×C₂²⋊D₅

yes

characteristic

110

C₈

—

111

C₈

—

112

C₈

—

113

C₅

#1710

yes

characteristic, complemented, Sylow

114

C₂²

#4086

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

#4062

yes

—

120

C₄

D₅×Q₁₆

yes

—

121

C₄

#4096

yes

—

122

C₂

#65348

yes

characteristic

123

C₂

#65457

yes

characteristic

124

C₂

#65470

yes

characteristic

	subgroup	G/H
center	C₂²	#4086
commutator, cosocle	C₄²×C₅	C₂³
Frattini	C₄²	C₂²×D₅
Fitting	C₅×#487	C₂
socle, prime radical	C₂²×C₅	#127
hypercenter	#487	D₅
hypocenter	C₅	#1710
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #487

subgroup

G/H

center

C₂²

#4086

commutator, cosocle

C₄²×C₅

C₂³

Frattini

C₄²

C₂²×D₅

C₂

C₂²×C₅

D₅

C₅

G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #487

N	H ≅ G/N	type
113. C₅	#1710	semidirect

H ≅ G/N

type

113. C₅

#1710

semidirect

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

subgroup	index
C₅×#487	2
#65382	2
#65370	2
#65367	2
#64970	2
C₈×#168	2
#1710	5

subgroup

index

Subgroups — #10602007

Conjugacy classes of subgroups

Complements

Maximal subgroups