Group #13101783 — subgroups

Subgroups — #13101783

Order 1600.

248 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 58 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	#11717264	C₂	1	1	yes	maximal, characteristic
2	#11716991	C₂	1	1	yes	maximal, characteristic
3	C₅×#3967	C₂	1	2	yes	maximal
4	#11716608	C₂	1	2	yes	maximal
5	#11716596	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
7	C₂×#6045	C₂²	1	1	yes	characteristic
8	#87806	—	2	2	no	—
9	C₄×#1039	C₂²	1	1	yes	characteristic
10	C₅×C₄⋊Dic₅	C₂²	1	2	yes	—
11	C₂×C₅×#168	C₂²	1	2	yes	—
12	#87780	—	2	2	no	—
13	#65086	—	5	2	no	maximal
14	C₅²×Q₈	—	2	2	no	—
15	C₂×C₄×C₅²	—	2	1	no	—
16	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
17	C₂×#1039	—	2	1	no	—
18	C₂×#1039	D₄	1	2	yes	—
19	#6045	—	2	2	no	—
20	C₂×C₅×Dic₅	—	2	2	no	—
21	C₅×#168	—	2	2	no	—
22	#4090	—	5	2	no	—
23	#4090	—	10	2	no	—
24	#4036	—	5	2	no	—
25	C₅×#103	—	5	2	no	—
26	#3967	D₅	1	2	yes	—
27	#3948	—	5	2	no	—
28	#3947	—	5	2	no	—
29	#3942	—	5	2	no	—
30	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
31	C₄×C₅²	—	4	1	no	—
32	C₄×C₅²	#41	1	2	yes	—
33	#1039	—	2	2	no	—
34	#1039	—	4	1	no	—
35	C₅×Dic₅	—	4	2	no	—
36	C₂×C₅×Q₈	—	2	2	no	—
37	C₂×C₅×Q₈	C₂×D₅	1	2	yes	—
38	C₂×Dic₁₀	—	5	2	no	—
39	C₂×Dic₁₀	—	10	2	no	—
40	C₂×C₅×C₈	—	5	2	no	—
41	C₅×C₄⋊C₄	—	5	2	no	—
42	C₄⋊Dic₅	—	10	2	no	—
43	C₄⋊Dic₅	C₂×D₅	1	2	yes	—
44	#683	—	10	2	no	—
45	#683	—	10	2	no	—
46	#683	—	10	2	no	—
47	#683	—	10	8	no	—
48	C₄×Dic₅	—	5	2	no	—
49	C₄×Dic₅	—	10	2	no	—
50	C₂×#168	C₂×D₅	1	2	yes	—
51	#491	—	25	1	no	Sylow
52	C₂×C₅²	#124	1	1	yes	characteristic
53	C₂×C₅²	#137	1	2	yes	—
54	C₅×Q₈	—	2	2	no	—
55	C₅×Q₈	—	2	2	no	—
56	C₅×Q₈	—	4	4	no	—
57	C₂×C₄×C₅	—	2	4	no	—
58	C₂×C₄×C₅	—	2	2	no	—
59	C₂×C₄×C₅	—	2	2	no	—
60	C₂×C₄×C₅	—	10	2	no	—
61	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
62	C₂×Dic₅	—	2	2	no	—
63	C₂×Dic₅	—	5	2	no	—
64	C₂×Dic₅	—	5	2	no	—
65	C₂×Dic₅	—	10	4	no	—
66	C₂×Dic₅	—	10	4	no	—
67	C₂×Dic₅	—	10	2	no	—
68	Dic₁₀	—	10	2	no	—
69	Dic₁₀	—	10	2	no	—
70	Dic₁₀	—	20	4	no	—
71	C₅×C₈	—	10	2	no	—
72	#168	—	2	2	no	—
73	C₄⋊Q₈	—	25	1	no	—
74	#125	—	25	1	no	—
75	#103	—	5	2	no	—
76	#103	—	25	2	no	—
77	#97	—	25	1	no	—
78	C₅²	#491	1	1	yes	characteristic, complemented, Sylow
79	C₂²×C₅	—	2	2	no	—
80	C₂²×C₅	D₄×D₅	1	2	yes	—
81	C₄×C₅	—	2	4	no	—
82	C₄×C₅	—	4	4	no	—
83	C₄×C₅	—	4	2	no	—
84	C₄×C₅	—	20	2	no	—
85	C₄×C₅	#709	1	2	yes	—
86	C₄×C₅	#711	1	2	yes	—
87	Dic₅	—	4	2	no	—
88	Dic₅	—	10	8	no	—
89	Dic₅	—	10	2	no	—
90	Dic₅	—	10	2	no	—
91	Dic₅	—	20	4	no	—
92	Dic₅	—	20	2	no	—
93	C₂×Q₈	—	25	1	no	—
94	C₂×Q₈	D₅²	1	1	yes	characteristic
95	C₂×C₈	—	5	2	no	—
96	C₄⋊C₄	—	5	2	no	—
97	C₄⋊C₄	—	50	2	no	—
98	C₄⋊C₄	—	50	2	no	—
99	C₄²	—	25	1	no	—
100	C₂×C₅	—	2	4	no	—
101	C₂×C₅	—	2	2	no	—
102	C₂×C₅	#4031	1	2	yes	—
103	C₂×C₅	#4061	1	2	yes	—
104	C₂×C₅	#4064	1	2	yes	—
105	Q₈	—	2	2	no	—
106	Q₈	—	50	2	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	10	2	no	—
109	C₂×C₄	—	25	2	no	—
110	C₂×C₄	—	50	1	no	—
111	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
112	C₈	—	10	2	no	—
113	C₅	—	2	2	no	—
114	C₅	#65086	1	2	yes	complemented
115	C₂²	#87886	1	1	yes	characteristic
116	C₄	—	4	1	no	—
117	C₄	—	20	2	no	—
118	C₄	—	50	2	no	—
119	C₄	—	100	1	no	—
120	C₄	#87879	1	2	yes	—
121	C₂	#11717069	1	2	yes	—
122	C₂	#11717071	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11717264

C₂

yes

maximal, characteristic

#11716991

C₂

yes

maximal, characteristic

C₅×#3967

C₂

yes

maximal

#11716608

C₂

yes

maximal

#11716596

C₂

yes

maximal, characteristic

C₂×C₅²×Q₈

C₂²

yes

characteristic

C₂×#6045

C₂²

yes

characteristic

#87806

—

C₄×#1039

C₂²

yes

characteristic

C₅×C₄⋊Dic₅

C₂²

yes

—

C₂×C₅×#168

C₂²

yes

—

#87780

—

#65086

—

maximal

C₅²×Q₈

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×#1039

—

C₂×#1039

D₄

yes

—

#6045

—

C₂×C₅×Dic₅

—

—

—

—

—

—

D₅

yes

—

#3948

—

#3947

—

#3942

—

C₂²×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

—

C₄×C₅²

#41

yes

—

#1039

—

#1039

—

C₅×Dic₅

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

C₂×D₅

yes

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×C₅×C₈

—

C₅×C₄⋊C₄

—

C₄⋊Dic₅

—

C₄⋊Dic₅

C₂×D₅

yes

—

—

—

—

—

C₄×Dic₅

—

C₄×Dic₅

—

C₂×#168

C₂×D₅

yes

—

—

C₂×C₅²

yes

C₂×C₅²

yes

—

C₅×Q₈

—

C₅×Q₈

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

—

#168

—

C₄⋊Q₈

—

—

—

—

—

C₅²

#491

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

D₄×D₅

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#709

yes

—

C₄×C₅

#711

yes

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

C₂×Q₈

—

C₂×Q₈

D₅²

yes

characteristic

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

100

C₂×C₅

—

101

C₂×C₅

—

102

C₂×C₅

#4031

yes

—

103

C₂×C₅

#4061

yes

—

104

C₂×C₅

#4064

yes

—

105

Q₈

—

106

Q₈

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂×D₅²

yes

characteristic

112

C₈

—

113

C₅

—

114

C₅

#65086

yes

complemented

115

C₂²

#87886

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

100

—

120

C₄

#87879

yes

—

121

C₂

#11717069

yes

—

122

C₂

#11717071

yes

characteristic

	subgroup	G/H
center	C₂²	#87886
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂×D₅²
Fitting	C₂×C₅²×Q₈	C₂²
socle, prime radical	C₂²×C₅²	C₂×D₄
hypercenter	C₂×Q₈	D₅²
hypocenter	C₅²	#491
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₈

subgroup

G/H

center

C₂²

#87886

commutator, cosocle

C₂×C₄×C₅²

C₂³

Frattini

C₂×C₄

C₂×D₅²

Fitting

C₂×C₅²×Q₈

C₂²

socle, prime radical

C₂²×C₅²

C₂×D₄

hypercenter

C₂×Q₈

D₅²

hypocenter

C₅²

#491

LCS

G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₈

N	H ≅ G/N	type
78. C₅²	#491	semidirect
114. C₅	#65086	semidirect

H ≅ G/N

type

78. C₅²

#491

semidirect

114. C₅

#65086

semidirect

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

subgroup	index
#11717264	2
#11716991	2
C₅×#3967	2
#11716608	2
#11716596	2
#65086	5

subgroup

index

Subgroups — #13101783

Conjugacy classes of subgroups

Complements

Maximal subgroups