Group #13102921 — subgroups

Subgroups — #13102921

Order 1600.

576 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 127 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	C₅²×#124	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#87811	C₂	1	2	yes	maximal, complemented
3	C₂×#87809	C₂	1	2	yes	maximal, complemented
4	#11717194	C₂	1	1	yes	maximal, characteristic, complemented
5	#11716670	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₅²×D₄	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊(C₅⋊D₅)	—	2	2	no	—
9	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	2	yes	—
10	C₅²×C₂²⋊C₄	—	2	2	no	—
11	C₄²×C₅²	C₂²	1	1	yes	characteristic
12	#87811	—	2	4	no	—
13	#87809	—	2	4	no	—
14	C₂×#6029	C₂²	1	2	yes	—
15	#65347	—	5	6	no	maximal
16	C₂³×C₅²	—	2	1	no	—
17	C₂²×C₅⋊D₅	—	2	2	no	—
18	C₅²×Q₈	—	2	2	no	—
19	C₅²×D₄	—	2	2	no	—
20	C₂×C₄×C₅²	—	2	1	no	—
21	C₂×C₄×C₅²	D₄	1	2	yes	—
22	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
23	C₄⋊(C₅⋊D₅)	—	2	2	no	—
24	C₄⋊(C₅⋊D₅)	—	2	2	no	—
25	C₄⋊(C₅⋊D₅)	—	4	4	no	—
26	#6029	D₄	1	4	yes	complemented
27	C₅×#124	D₅	1	6	yes	complemented
28	C₂×#688	—	5	12	no	—
29	C₂×#686	—	5	12	no	—
30	#4020	—	5	6	no	—
31	#3935	—	5	6	no	—
32	C₂²×C₅²	—	4	1	no	—
33	C₂²×C₅²	—	4	2	no	—
34	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
35	C₂×C₅⋊D₅	—	4	2	no	—
36	C₂×C₅⋊D₅	—	4	4	no	—
37	C₄×C₅²	—	2	2	no	—
38	C₄×C₅²	—	4	1	no	—
39	C₄×C₅²	C₂×D₄	1	2	yes	—
40	C₂×C₅×Q₈	C₂×D₅	1	6	yes	—
41	C₂×C₅×D₄	C₂×D₅	1	6	yes	—
42	C₂×D₂₀	—	5	12	no	—
43	C₂×D₂₀	—	10	12	no	—
44	C₅×C₂²⋊C₄	—	2	12	no	—
45	C₄²×C₅	C₂×D₅	1	6	yes	—
46	#688	—	10	24	no	—
47	#686	—	10	24	no	—
48	C₂×#168	—	5	12	no	—
49	#496	—	25	1	no	Sylow
50	C₂×C₅²	—	8	1	no	—
51	C₂×C₅²	#127	1	1	yes	characteristic
52	C₂×C₅²	AGL(1,ℤ/8)	1	2	yes	—
53	C₅⋊D₅	—	8	2	no	—
54	C₂³×C₅	—	2	6	no	—
55	C₂²×D₅	—	10	12	no	—
56	C₅×Q₈	—	2	12	no	—
57	C₅×D₄	—	2	12	no	—
58	C₂×C₄×C₅	—	2	6	no	—
59	C₂×C₄×C₅	C₂²⋊D₅	1	12	yes	—
60	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
61	D₂₀	—	10	12	no	—
62	D₂₀	—	10	12	no	—
63	D₂₀	—	20	24	no	—
64	#168	—	5	24	no	—
65	C₂×SD₁₆	—	25	2	no	—
66	C₂×D₈	—	25	2	no	—
67	#127	—	25	1	no	—
68	#124	C₅⋊D₅	1	1	yes	characteristic, complemented
69	#97	—	25	1	no	—
70	C₅²	#496	1	1	yes	characteristic, complemented, Sylow
71	C₂²×C₅	—	4	6	no	—
72	C₂²×C₅	—	4	12	no	—
73	C₂²×C₅	C₂×C₂²⋊D₅	1	6	yes	—
74	C₂×D₅	—	20	12	no	—
75	C₂×D₅	—	20	24	no	—
76	C₄×C₅	—	2	12	no	—
77	C₄×C₅	—	4	6	no	—
78	C₄×C₅	D₄×D₅	1	12	yes	—
79	C₂×Q₈	C₂×C₅⋊D₅	1	1	yes	characteristic
80	C₂×D₄	—	25	2	no	—
81	C₂×D₄	—	50	2	no	—
82	C₂×D₄	C₂×C₅⋊D₅	1	1	yes	characteristic
83	SD₁₆	—	50	4	no	—
84	D₈	—	50	4	no	—
85	C₂×C₈	—	25	2	no	—
86	C₂²⋊C₄	—	2	2	no	—
87	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
88	C₂×C₅	—	8	6	no	—
89	C₂×C₅	#4086	1	6	yes	—
90	C₂×C₅	#4095	1	12	yes	—
91	D₅	—	40	12	no	—
92	C₂³	—	2	1	no	—
93	C₂³	—	50	2	no	—
94	Q₈	—	2	2	no	—
95	D₄	—	2	2	no	—
96	D₄	—	50	2	no	—
97	D₄	—	50	2	no	—
98	D₄	—	100	4	no	—
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	2	yes	—
101	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
102	C₈	—	25	4	no	—
103	C₅	#65347	1	6	yes	complemented
104	C₂²	—	4	1	no	—
105	C₂²	—	4	2	no	—
106	C₂²	—	100	2	no	—
107	C₂²	—	100	4	no	—
108	C₂²	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
109	C₄	—	2	2	no	—
110	C₄	—	4	1	no	—
111	C₄	D₄×C₅⋊D₅	1	2	yes	—
112	C₂	—	8	1	no	—
113	C₂	—	200	2	no	—
114	C₂	#11717260	1	1	yes	characteristic
115	C₂	#11717269	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅²×#124

C₂

yes

maximal, characteristic, complemented

C₂×#87811

C₂

yes

maximal, complemented

C₂×#87809

C₂

yes

maximal, complemented

#11717194

C₂

yes

maximal, characteristic, complemented

#11716670

C₂

yes

maximal, characteristic, complemented

C₂×C₅²×Q₈

C₂²

yes

characteristic

C₂×C₅²×D₄

C₂²

yes

characteristic

C₂×C₄⋊(C₅⋊D₅)

—

C₂×C₄⋊(C₅⋊D₅)

C₂²

yes

—

C₅²×C₂²⋊C₄

—

C₄²×C₅²

C₂²

yes

—

—

C₂²

yes

—

#65347

—

maximal

C₂³×C₅²

—

C₂²×C₅⋊D₅

—

C₅²×Q₈

—

C₅²×D₄

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

D₄

yes

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₄⋊(C₅⋊D₅)

—

C₄⋊(C₅⋊D₅)

—

C₄⋊(C₅⋊D₅)

—

#6029

D₄

yes

complemented

C₅×#124

D₅

yes

—

—

—

—

C₂²×C₅²

—

C₂²×C₅²

—

C₂²×C₅²

C₂×D₄

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₄×C₅²

—

C₄×C₅²

—

C₄×C₅²

C₂×D₄

yes

—

C₂×C₅×Q₈

C₂×D₅

yes

—

C₂×C₅×D₄

C₂×D₅

yes

—

C₂×D₂₀

—

C₂×D₂₀

—

C₅×C₂²⋊C₄

—

C₄²×C₅

C₂×D₅

yes

—

—

—

—

—

C₂×C₅²

—

C₂×C₅²

#127

yes

characteristic

C₂×C₅²

AGL(1,ℤ/8)

yes

—

C₅⋊D₅

—

C₂³×C₅

—

C₂²×D₅

—

C₅×Q₈

—

C₅×D₄

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

—

D₂₀

—

D₂₀

—

D₂₀

—

#168

—

C₂×SD₁₆

—

C₂×D₈

—

#127

—

#124

C₅⋊D₅

yes

characteristic, complemented

#97

—

C₅²

#496

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₂×C₂²⋊D₅

yes

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

D₄×D₅

yes

—

C₂×Q₈

C₂×C₅⋊D₅

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×C₅⋊D₅

yes

characteristic

SD₁₆

—

D₈

—

C₂×C₈

—

C₂²⋊C₄

—

C₄²

C₂×C₅⋊D₅

yes

characteristic

C₂×C₅

—

C₂×C₅

#4086

yes

—

C₂×C₅

#4095

yes

—

D₅

—

C₂³

—

C₂³

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

100

—

C₂×C₄

—

100

C₂×C₄

C₂²⋊(C₅⋊D₅)

yes

—

101

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

102

C₈

—

103

C₅

#65347

yes

complemented

104

C₂²

—

105

C₂²

—

106

C₂²

—

100

—

107

C₂²

—

100

—

108

C₂²

C₂×C₂²⋊(C₅⋊D₅)

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

D₄×C₅⋊D₅

yes

—

112

C₂

—

113

C₂

—

200

—

114

C₂

#11717260

yes

characteristic

115

C₂

#11717269

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊(C₅⋊D₅)
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂²×C₅⋊D₅
Fitting	C₅²×#124	C₂
socle	C₂²×C₅²	C₂×D₄
hypercenter	#124	C₅⋊D₅
hypocenter	C₅²	#496
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #124

subgroup

G/H

center

C₂²

C₂×C₂²⋊(C₅⋊D₅)

commutator, cosocle

C₂×C₄×C₅²

C₂³

Frattini

C₂×C₄

C₂²×C₅⋊D₅

Fitting

C₅²×#124

C₂

socle

C₂²×C₅²

C₂×D₄

C₅⋊D₅

C₅²

G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #124

N	H ≅ G/N	type
1. C₅²×#124	C₂	semidirect
2. C₂×#87811	C₂	semidirect
3. C₂×#87809	C₂	semidirect
4. #11717194	C₂	semidirect
5. #11716670	C₂	semidirect
26. #6029	D₄	semidirect
27. C₅×#124	D₅	semidirect
68. #124	C₅⋊D₅	semidirect
70. C₅²	#496	semidirect
103. C₅	#65347	semidirect

H ≅ G/N

type

1. C₅²×#124

C₂

semidirect

2. C₂×#87811

C₂

semidirect

3. C₂×#87809

C₂

semidirect

4. #11717194

C₂

semidirect

5. #11716670

C₂

semidirect

26. #6029

D₄

semidirect

27. C₅×#124

D₅

semidirect

68. #124

C₅⋊D₅

semidirect

70. C₅²

#496

semidirect

103. C₅

#65347

semidirect

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

subgroup	index
C₅²×#124	2
C₂×#87811	2
C₂×#87809	2
#11717194	2
#11716670	2
#65347	5

subgroup

index

C₅²×#124

Subgroups — #13102921

Conjugacy classes of subgroups

Complements

Maximal subgroups