Group #13102943 — subgroups

Subgroups — #13102943

Order 1600.

384 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 141 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×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₄×#6045	C₂	1	1	yes	maximal, characteristic
3	#11716677	C₂	1	2	yes	maximal
4	#11716674	C₂	1	2	yes	maximal
5	C₄⋊#6029	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅²×Q₈	—	2	1	no	—
7	C₂×#6045	C₂²	1	1	yes	characteristic
8	C₅²×C₄⋊C₄	—	2	1	no	—
9	C₅²×C₄⋊C₄	C₂²	1	2	yes	—
10	C₄²×C₅²	C₂²	1	1	yes	characteristic
11	C₄⋊#1039	C₂²	1	1	yes	characteristic
12	#87806	—	2	1	no	—
13	C₄×#1039	—	2	1	no	—
14	C₂×#6029	C₂²	1	2	yes	—
15	#65380	—	5	6	no	maximal
16	C₅²×Q₈	—	4	2	no	—
17	C₂×C₄×C₅²	—	2	2	no	—
18	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
19	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
20	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
21	C₂×#1039	—	2	1	no	—
22	C₂×#1039	—	2	1	no	—
23	#6045	—	2	1	no	—
24	#6045	Q₈	1	2	yes	complemented
25	#6029	—	2	2	no	—
26	C₅×C₄⋊Q₈	D₅	1	6	yes	—
27	C₄×Dic₁₀	—	5	6	no	—
28	#3942	—	5	12	no	—
29	#3939	—	5	12	no	—
30	C₄⋊#168	—	5	6	no	—
31	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
32	C₄×C₅²	—	2	1	no	—
33	C₄×C₅²	—	4	2	no	—
34	C₄×C₅²	Q₁₆	1	2	yes	—
35	C₄×C₅²	C₂×Q₈	1	1	yes	characteristic
36	C₄×C₅²	#41	1	1	yes	characteristic
37	#1039	—	2	2	no	—
38	#1039	—	4	1	no	—
39	C₂×C₅×Q₈	—	2	6	no	—
40	C₂×Dic₁₀	—	5	6	no	—
41	C₅×C₄⋊C₄	—	2	6	no	—
42	C₅×C₄⋊C₄	C₂×D₅	1	12	yes	—
43	C₄²×C₅	C₂×D₅	1	6	yes	—
44	C₄⋊Dic₅	—	5	6	no	—
45	#683	—	10	6	no	—
46	C₄×Dic₅	—	10	6	no	—
47	C₂×#168	—	5	12	no	—
48	#477	—	25	1	no	Sylow
49	C₂×C₅²	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂×C₅²	C₂×Q₁₆	1	1	yes	characteristic
51	C₂×C₅²	AGL(1,ℤ/8)	1	1	yes	characteristic
52	C₅×Q₈	—	4	12	no	—
53	C₂×C₄×C₅	—	2	12	no	—
54	C₂×C₄×C₅	C₂²⋊D₅	1	6	yes	—
55	C₂×C₄×C₅	C₂²⋊D₅	1	6	yes	—
56	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
57	C₂×Dic₅	—	10	6	no	—
58	C₂×Dic₅	—	10	6	no	—
59	Dic₁₀	—	5	12	no	—
60	Dic₁₀	—	10	6	no	—
61	#168	—	10	12	no	—
62	C₄⋊Q₈	C₅⋊D₅	1	1	yes	characteristic
63	C₄×Q₈	—	25	1	no	—
64	#107	—	25	2	no	—
65	C₄⋊C₈	—	25	1	no	—
66	#103	—	25	2	no	—
67	C₅²	#477	1	1	yes	characteristic, complemented, Sylow
68	C₂²×C₅	C₂×C₂²⋊D₅	1	6	yes	—
69	C₄×C₅	—	2	6	no	—
70	C₄×C₅	—	4	12	no	—
71	C₄×C₅	#689	1	12	yes	—
72	C₄×C₅	Q₈×D₅	1	6	yes	—
73	C₄×C₅	#713	1	6	yes	—
74	Dic₅	—	10	12	no	—
75	Dic₅	—	20	6	no	—
76	C₂×Q₈	—	2	1	no	—
77	C₂×Q₈	—	25	1	no	—
78	C₂×C₈	—	25	2	no	—
79	C₄⋊C₄	—	2	1	no	—
80	C₄⋊C₄	—	25	1	no	—
81	C₄⋊C₄	—	50	1	no	—
82	C₄⋊C₄	C₂×C₅⋊D₅	1	2	yes	—
83	C₄²	—	50	1	no	—
84	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
85	C₂×C₅	#4078	1	6	yes	—
86	C₂×C₅	C₂×#689	1	6	yes	—
87	C₂×C₅	#4092	1	6	yes	—
88	Q₈	—	4	2	no	—
89	Q₈	—	25	2	no	—
90	Q₈	—	50	1	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	50	1	no	—
93	C₂×C₄	—	50	1	no	—
94	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
95	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
96	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
97	C₈	—	50	2	no	—
98	C₅	#65380	1	6	yes	complemented
99	C₂²	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	4	2	no	—
102	C₄	—	50	2	no	—
103	C₄	—	100	1	no	—
104	C₄	#87812	1	2	yes	—
105	C₄	Q₈×C₅⋊D₅	1	1	yes	characteristic
106	C₄	#87904	1	1	yes	characteristic
107	C₂	#11717252	1	1	yes	characteristic
108	C₂	C₂×#87812	1	1	yes	characteristic
109	C₂	#11717266	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅²×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₄×#6045

C₂

yes

maximal, characteristic

#11716677

C₂

yes

maximal

#11716674

C₂

yes

maximal

C₄⋊#6029

C₂

yes

maximal, characteristic

C₂×C₅²×Q₈

—

C₂×#6045

C₂²

yes

characteristic

C₅²×C₄⋊C₄

—

C₅²×C₄⋊C₄

C₂²

yes

—

C₄²×C₅²

C₂²

yes

characteristic

C₄⋊#1039

C₂²

yes

—

—

C₂²

yes

—

#65380

—

maximal

C₅²×Q₈

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

D₄

yes

characteristic

C₂×C₄×C₅²

D₄

yes

characteristic

C₂×C₄×C₅²

C₂³

yes

—

—

—

Q₈

yes

complemented

#6029

—

C₅×C₄⋊Q₈

D₅

yes

—

C₄×Dic₁₀

—

#3942

—

#3939

—

C₄⋊#168

—

C₂²×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

—

C₄×C₅²

—

C₄×C₅²

Q₁₆

yes

—

C₄×C₅²

C₂×Q₈

yes

C₄×C₅²

yes

—

—

C₂×C₅×Q₈

—

C₂×Dic₁₀

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

C₂×D₅

yes

—

C₄²×C₅

C₂×D₅

yes

—

C₄⋊Dic₅

—

#683

—

C₄×Dic₅

—

C₂×#168

—

#477

—

Sylow

C₂×C₅²

C₂²⋊Q₈

yes

characteristic

C₂×C₅²

C₂×Q₁₆

yes

characteristic

C₂×C₅²

AGL(1,ℤ/8)

yes

characteristic

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

Dic₁₀

—

#168

—

C₄⋊Q₈

C₅⋊D₅

yes

characteristic

C₄×Q₈

—

#107

—

C₄⋊C₈

—

#103

—

C₅²

#477

yes

characteristic, complemented, Sylow

C₂²×C₅

C₂×C₂²⋊D₅

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#689

yes

—

C₄×C₅

Q₈×D₅

yes

—

C₄×C₅

#713

yes

—

Dic₅

—

Dic₅

—

C₂×Q₈

—

C₂×Q₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×C₅⋊D₅

yes

—

C₄²

—

C₄²

C₂×C₅⋊D₅

yes

characteristic

C₂×C₅

#4078

yes

—

C₂×C₅

C₂×#689

yes

—

C₂×C₅

#4092

yes

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊(C₅⋊D₅)

yes

characteristic

C₂×C₄

C₂²⋊(C₅⋊D₅)

yes

characteristic

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

C₈

—

C₅

#65380

yes

complemented

C₂²

C₂×C₂²⋊(C₅⋊D₅)

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

100

—

104

C₄

#87812

yes

—

105

C₄

Q₈×C₅⋊D₅

yes

characteristic

106

C₄

#87904

yes

characteristic

107

C₂

#11717252

yes

characteristic

108

C₂

C₂×#87812

yes

characteristic

109

C₂

#11717266

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊(C₅⋊D₅)
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂²×C₅⋊D₅
Fitting	C₅²×C₄⋊Q₈	C₂
socle, prime radical	C₂²×C₅²	C₂×D₄
hypercenter	C₄⋊Q₈	C₅⋊D₅
hypocenter	C₅²	#477
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊(C₅⋊D₅)

commutator, cosocle

C₂×C₄×C₅²

C₂³

Frattini

C₂×C₄

C₂²×C₅⋊D₅

Fitting

C₅²×C₄⋊Q₈

C₂

socle, prime radical

C₂²×C₅²

C₂×D₄

hypercenter

C₄⋊Q₈

C₅⋊D₅

hypocenter

C₅²

#477

LCS

G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
24. #6045	Q₈	semidirect
67. C₅²	#477	semidirect
98. C₅	#65380	semidirect

H ≅ G/N

type

24. #6045

Q₈

semidirect

67. C₅²

#477

semidirect

98. C₅

#65380

semidirect

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

subgroup	index
C₅²×C₄⋊Q₈	2
C₄×#6045	2
#11716677	2
#11716674	2
C₄⋊#6029	2
#65380	5

subgroup

index

C₅²×C₄⋊Q₈

Subgroups — #13102943

Conjugacy classes of subgroups

Complements

Maximal subgroups