Group #13102761 — subgroups

Subgroups — #13102761

Order 1600.

432 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 109 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	#11717228	C₂	1	2	yes	maximal, complemented
2	C₂×#87801	C₂	1	1	yes	maximal, characteristic, complemented
3	#11717221	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅²×#108	C₂	1	1	yes	maximal, characteristic, complemented
5	#11716690	C₂	1	2	yes	maximal, complemented
6	#87900	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	2	yes	—
8	C₅²×M₁₆	C₂²	1	2	yes	—
9	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
10	#87804	C₂²	1	1	yes	characteristic
11	#87801	—	2	2	no	—
12	#87801	—	2	1	no	—
13	#87801	—	2	1	no	—
14	#87800	—	2	2	no	—
15	#87799	—	2	1	no	—
16	C₈×C₅⋊D₅	—	2	1	no	—
17	#65186	—	5	6	no	maximal
18	C₂²×C₅⋊D₅	—	2	2	no	—
19	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
20	C₂²⋊(C₅⋊D₅)	—	2	1	no	—
21	C₄⋊(C₅⋊D₅)	—	2	2	no	—
22	C₄⋊(C₅⋊D₅)	—	2	2	no	—
23	C₄⋊(C₅⋊D₅)	D₄	1	1	yes	characteristic
24	C₄×C₅⋊D₅	—	2	1	no	—
25	#6045	D₄	1	1	yes	characteristic
26	C₅²×C₈	—	2	2	no	—
27	C₅²×C₈	D₄	1	1	yes	characteristic
28	C₅²×C₈	D₄	1	1	yes	characteristic
29	#6029	—	2	1	no	—
30	#4054	—	5	12	no	—
31	C₂×D₄₀	—	5	6	no	—
32	#4047	—	5	6	no	—
33	C₅×#108	D₅	1	6	yes	complemented
34	#3955	—	5	12	no	—
35	C₂²×C₅²	#41	1	1	yes	characteristic
36	C₂×C₅⋊D₅	—	2	1	no	—
37	C₂×C₅⋊D₅	—	4	2	no	—
38	C₂×C₅⋊D₅	—	8	2	no	—
39	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
40	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
41	#1039	—	2	1	no	—
42	#709	—	5	6	no	—
43	C₂×D₂₀	—	5	12	no	—
44	C₅×M₁₆	C₂×D₅	1	12	yes	—
45	C₂×C₅×C₈	C₂×D₅	1	6	yes	—
46	#681	—	5	6	no	—
47	D₄₀	—	10	12	no	—
48	D₄₀	—	10	6	no	—
49	D₄₀	—	10	6	no	—
50	#677	—	10	12	no	—
51	#676	—	10	6	no	—
52	C₈×D₅	—	10	6	no	—
53	#472	—	25	1	no	Sylow
54	C₂×C₅²	—	2	1	no	—
55	C₂×C₅²	#121	1	1	yes	characteristic
56	C₅⋊D₅	—	4	1	no	—
57	C₅⋊D₅	—	8	2	no	—
58	C₂²×D₅	—	10	12	no	—
59	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
60	C₂²⋊D₅	—	10	6	no	—
61	D₂₀	—	5	6	no	—
62	D₂₀	—	10	12	no	—
63	D₂₀	—	10	12	no	—
64	C₄×D₅	—	10	6	no	—
65	Dic₁₀	—	5	6	no	—
66	C₅×C₈	—	2	12	no	—
67	C₅×C₈	D₂₀	1	6	yes	—
68	C₅×C₈	D₂₀	1	6	yes	—
69	#168	—	10	6	no	—
70	AGL(1,ℤ/8)	—	25	2	no	—
71	C₂×D₈	—	25	1	no	—
72	#131	—	25	1	no	—
73	#108	C₅⋊D₅	1	1	yes	characteristic, complemented
74	#100	—	25	2	no	—
75	C₅²	#472	1	1	yes	characteristic, complemented, Sylow
76	C₂²×C₅	#713	1	6	yes	—
77	C₂×D₅	—	10	6	no	—
78	C₂×D₅	—	20	12	no	—
79	C₂×D₅	—	40	12	no	—
80	C₄×C₅	C₂×D₂₀	1	6	yes	—
81	C₄×C₅	D₄×D₅	1	6	yes	—
82	Dic₅	—	10	6	no	—
83	#41	—	25	1	no	—
84	C₂×D₄	—	25	2	no	—
85	SD₁₆	—	50	2	no	—
86	D₈	—	50	2	no	—
87	D₈	—	50	1	no	—
88	D₈	—	50	1	no	—
89	M₁₆	—	25	1	no	—
90	M₁₆	—	50	1	no	—
91	M₁₆	C₂×C₅⋊D₅	1	2	yes	—
92	C₂×C₈	—	50	1	no	—
93	C₂×C₈	C₂×C₅⋊D₅	1	1	yes	characteristic
94	C₂×C₅	—	2	6	no	—
95	C₂×C₅	#4041	1	6	yes	—
96	D₅	—	20	6	no	—
97	D₅	—	40	12	no	—
98	C₂³	—	50	2	no	—
99	Q₈	—	25	1	no	—
100	D₄	—	25	1	no	—
101	D₄	—	50	1	no	—
102	D₄	—	50	2	no	—
103	D₄	—	50	2	no	—
104	C₂×C₄	—	50	1	no	—
105	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
106	C₈	—	2	2	no	—
107	C₈	—	50	1	no	—
108	C₈	C₄⋊(C₅⋊D₅)	1	1	yes	characteristic
109	C₈	C₄⋊(C₅⋊D₅)	1	1	yes	characteristic
110	C₅	#65186	1	6	yes	complemented
111	C₂²	—	50	1	no	—
112	C₂²	—	100	2	no	—
113	C₂²	—	200	2	no	—
114	C₂²	#87904	1	1	yes	characteristic
115	C₄	—	50	1	no	—
116	C₄	C₂×C₄⋊(C₅⋊D₅)	1	1	yes	characteristic
117	C₄	D₄×C₅⋊D₅	1	1	yes	characteristic
118	C₂	—	2	1	no	—
119	C₂	—	100	1	no	—
120	C₂	—	200	2	no	—
121	C₂	#11717215	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11717228

C₂

yes

maximal, complemented

C₂×#87801

C₂

yes

maximal, characteristic, complemented

#11717221

C₂

yes

maximal, characteristic, complemented

C₅²×#108

C₂

yes

maximal, characteristic, complemented

#11716690

C₂

yes

maximal, complemented

#87900

C₂²

yes

characteristic

C₂×C₄⋊(C₅⋊D₅)

C₂²

yes

—

C₅²×M₁₆

C₂²

yes

—

C₂×C₅²×C₈

C₂²

yes

characteristic

#87804

C₂²

yes

—

—

—

—

—

C₈×C₅⋊D₅

—

#65186

—

maximal

C₂²×C₅⋊D₅

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂²⋊(C₅⋊D₅)

—

C₄⋊(C₅⋊D₅)

—

C₄⋊(C₅⋊D₅)

—

C₄⋊(C₅⋊D₅)

D₄

yes

characteristic

C₄×C₅⋊D₅

—

#6045

D₄

yes

characteristic

C₅²×C₈

—

C₅²×C₈

D₄

yes

characteristic

C₅²×C₈

D₄

yes

characteristic

#6029

—

#4054

—

C₂×D₄₀

—

#4047

—

C₅×#108

D₅

yes

complemented

#3955

—

C₂²×C₅²

#41

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₄×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

C₂×D₄

yes

characteristic

#1039

—

#709

—

C₂×D₂₀

—

C₅×M₁₆

C₂×D₅

yes

—

C₂×C₅×C₈

C₂×D₅

yes

—

#681

—

D₄₀

—

D₄₀

—

D₄₀

—

#677

—

#676

—

C₈×D₅

—

#472

—

Sylow

C₂×C₅²

—

C₂×C₅²

#121

yes

characteristic

C₅⋊D₅

—

C₅⋊D₅

—

C₂²×D₅

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂²⋊D₅

—

D₂₀

—

D₂₀

—

D₂₀

—

C₄×D₅

—

Dic₁₀

—

C₅×C₈

—

C₅×C₈

D₂₀

yes

—

C₅×C₈

D₂₀

yes

—

#168

—

AGL(1,ℤ/8)

—

C₂×D₈

—

#131

—

#108

C₅⋊D₅

yes

characteristic, complemented

#100

—

C₅²

#472

yes

characteristic, complemented, Sylow

C₂²×C₅

#713

yes

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

C₂×D₂₀

yes

—

C₄×C₅

D₄×D₅

yes

—

Dic₅

—

#41

—

C₂×D₄

—

SD₁₆

—

D₈

—

D₈

—

D₈

—

M₁₆

—

M₁₆

—

M₁₆

C₂×C₅⋊D₅

yes

—

C₂×C₈

—

C₂×C₈

C₂×C₅⋊D₅

yes

characteristic

C₂×C₅

—

C₂×C₅

#4041

yes

—

D₅

—

D₅

—

C₂³

—

Q₈

—

100

D₄

—

101

D₄

—

102

D₄

—

103

D₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

106

C₈

—

107

C₈

—

108

C₈

C₄⋊(C₅⋊D₅)

yes

characteristic

109

C₈

C₄⋊(C₅⋊D₅)

yes

characteristic

110

C₅

#65186

yes

complemented

111

C₂²

—

112

C₂²

—

100

—

113

C₂²

—

200

—

114

C₂²

#87904

yes

characteristic

115

C₄

—

116

C₄

C₂×C₄⋊(C₅⋊D₅)

yes

characteristic

117

C₄

D₄×C₅⋊D₅

yes

characteristic

118

C₂

—

119

C₂

—

100

—

120

C₂

—

200

—

121

C₂

#11717215

yes

characteristic

	subgroup	G/H
center	C₂	#11717215
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂²×C₅⋊D₅
Fitting	C₅²×#108	C₂
socle	C₂×C₅²	#121
hypercenter	#108	C₅⋊D₅
hypocenter	C₅²	#472
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #108

subgroup

G/H

center

C₂

#11717215

commutator, cosocle

C₂×C₄×C₅²

C₂³

Frattini

C₂×C₄

C₂²×C₅⋊D₅

Fitting

C₅²×#108

C₂

C₂×C₅²

C₅⋊D₅

C₅²

G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #108

N	H ≅ G/N	type
1. #11717228	C₂	semidirect
2. C₂×#87801	C₂	semidirect
3. #11717221	C₂	semidirect
4. C₅²×#108	C₂	semidirect
5. #11716690	C₂	semidirect
33. C₅×#108	D₅	semidirect
73. #108	C₅⋊D₅	semidirect
75. C₅²	#472	semidirect
110. C₅	#65186	semidirect

H ≅ G/N

type

1. #11717228

C₂

semidirect

2. C₂×#87801

C₂

semidirect

3. #11717221

C₂

semidirect

4. C₅²×#108

C₂

semidirect

5. #11716690

C₂

semidirect

33. C₅×#108

D₅

semidirect

73. #108

C₅⋊D₅

semidirect

75. C₅²

#472

semidirect

110. C₅

#65186

semidirect

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

subgroup	index
#11717228	2
C₂×#87801	2
#11717221	2
C₅²×#108	2
#11716690	2
#65186	5

subgroup

index

C₅²×#108

Subgroups — #13102761

Conjugacy classes of subgroups

Complements

Maximal subgroups