Group #13102542 — subgroups

Subgroups — #13102542

Order 1600.

376 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 163 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	C₄×#6045	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅²×C₈	C₂	1	1	yes	maximal, characteristic
3	#11716682	C₂	1	2	yes	maximal
4	#11716681	C₂	1	2	yes	maximal
5	C₄⋊#6029	C₂	1	1	yes	maximal, characteristic
6	C₂×#6045	C₄	1	1	yes	characteristic
7	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
9	C₄²×C₅²	C₂²	1	1	yes	characteristic
10	C₄⋊#1039	C₄	1	1	yes	characteristic
11	#87806	C₄	1	2	yes	—
12	C₄×#1039	C₂²	1	2	yes	—
13	C₂×#6029	C₂²	1	2	yes	—
14	#64968	—	5	6	no	maximal
15	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
18	C₂×#1039	C₂×C₄	1	2	yes	—
19	C₂×#1039	C₂×C₄	1	2	yes	—
20	#6045	—	2	2	no	—
21	C₅²×C₈	—	2	1	no	—
22	C₅²×C₈	Q₈	1	2	yes	complemented
23	#6029	—	2	2	no	—
24	C₄×Dic₁₀	—	5	6	no	—
25	C₄×C₅×C₈	D₅	1	6	yes	—
26	#3947	—	5	12	no	—
27	#3946	—	5	12	no	—
28	C₄⋊#168	—	5	6	no	—
29	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₅²	—	2	1	no	—
31	C₄×C₅²	M₁₆	1	2	yes	—
32	C₄×C₅²	C₂×Q₈	1	1	yes	characteristic
33	C₄×C₅²	#41	1	1	yes	characteristic
34	#1039	—	2	2	no	—
35	#1039	—	2	2	no	—
36	C₂×Dic₁₀	—	5	6	no	—
37	C₂×C₅×C₈	C₂×D₅	1	6	yes	—
38	C₂×C₅×C₈	C₂×D₅	1	6	yes	—
39	C₄²×C₅	C₂×D₅	1	6	yes	—
40	C₄⋊Dic₅	—	5	6	no	—
41	#683	—	5	12	no	—
42	C₄×Dic₅	—	5	12	no	—
43	C₂×#168	—	5	12	no	—
44	#446	—	25	1	no	Sylow
45	C₂×C₅²	C₄×Q₈	1	1	yes	characteristic
46	C₂×C₅²	C₂×M₁₆	1	1	yes	characteristic
47	C₂×C₅²	#131	1	1	yes	characteristic
48	C₂×C₄×C₅	C₄×D₅	1	6	yes	—
49	C₂×C₄×C₅	C₄×D₅	1	6	yes	—
50	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
51	C₂×Dic₅	—	5	12	no	—
52	C₂×Dic₅	—	5	12	no	—
53	Dic₁₀	—	10	12	no	—
54	C₅×C₈	—	2	6	no	—
55	C₅×C₈	Dic₁₀	1	12	yes	complemented
56	#168	—	10	12	no	—
57	C₄×Q₈	—	25	1	no	—
58	C₄⋊C₈	—	25	2	no	—
59	C₄⋊C₈	—	25	1	no	—
60	#97	—	25	2	no	—
61	C₄×C₈	C₅⋊D₅	1	1	yes	characteristic
62	C₅²	#446	1	1	yes	characteristic, complemented, Sylow
63	C₂²×C₅	C₂×C₄×D₅	1	6	yes	—
64	C₄×C₅	—	2	6	no	—
65	C₄×C₅	#676	1	12	yes	—
66	C₄×C₅	C₂×Dic₁₀	1	6	yes	—
67	C₄×C₅	#709	1	6	yes	—
68	Dic₅	—	10	12	no	—
69	Dic₅	—	10	12	no	—
70	C₂×Q₈	—	25	1	no	—
71	C₂×C₈	—	25	2	no	—
72	C₂×C₈	C₂×C₅⋊D₅	1	1	yes	characteristic
73	C₂×C₈	C₂×C₅⋊D₅	1	1	yes	characteristic
74	C₄⋊C₄	—	25	2	no	—
75	C₄⋊C₄	—	25	1	no	—
76	C₄²	—	25	2	no	—
77	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
78	C₂×C₅	C₄×Dic₁₀	1	6	yes	—
79	C₂×C₅	C₂×#676	1	6	yes	—
80	C₂×C₅	#4047	1	6	yes	—
81	Q₈	—	50	2	no	—
82	C₂×C₄	—	25	2	no	—
83	C₂×C₄	—	25	2	no	—
84	C₂×C₄	C₄×C₅⋊D₅	1	1	yes	characteristic
85	C₂×C₄	C₄×C₅⋊D₅	1	1	yes	characteristic
86	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
87	C₈	—	2	1	no	—
88	C₈	—	50	2	no	—
89	C₈	#6045	1	2	yes	complemented
90	C₅	#64968	1	6	yes	complemented
91	C₂²	C₂×C₄×C₅⋊D₅	1	1	yes	characteristic
92	C₄	—	2	1	no	—
93	C₄	—	50	2	no	—
94	C₄	—	50	2	no	—
95	C₄	#87799	1	2	yes	—
96	C₄	C₂×#6045	1	1	yes	characteristic
97	C₄	#87900	1	1	yes	characteristic
98	C₂	C₄×#6045	1	1	yes	characteristic
99	C₂	C₂×#87799	1	1	yes	characteristic
100	C₂	#11717221	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#6045

C₂

yes

maximal, characteristic

C₄×C₅²×C₈

C₂

yes

maximal, characteristic

#11716682

C₂

yes

maximal

#11716681

C₂

yes

maximal

C₄⋊#6029

C₂

yes

maximal, characteristic

C₂×#6045

C₄

yes

characteristic

C₂×C₅²×C₈

C₂²

yes

characteristic

C₂×C₅²×C₈

C₂²

yes

characteristic

C₄²×C₅²

C₂²

yes

characteristic

C₄⋊#1039

C₄

yes

characteristic

#87806

C₄

yes

—

C₄×#1039

C₂²

yes

—

C₂×#6029

C₂²

yes

—

#64968

—

maximal

C₂×C₄×C₅²

C₂×C₄

yes

characteristic

C₂×C₄×C₅²

C₂×C₄

yes

characteristic

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×#1039

C₂×C₄

yes

—

C₂×#1039

C₂×C₄

yes

—

#6045

—

C₅²×C₈

—

C₅²×C₈

Q₈

yes

complemented

#6029

—

C₄×Dic₁₀

—

C₄×C₅×C₈

D₅

yes

—

#3947

—

#3946

—

C₄⋊#168

—

C₂²×C₅²

C₂²×C₄

yes

characteristic

C₄×C₅²

—

C₄×C₅²

M₁₆

yes

—

C₄×C₅²

C₂×Q₈

yes

C₄×C₅²

yes

—

—

C₂×Dic₁₀

—

C₂×C₅×C₈

C₂×D₅

yes

—

C₂×C₅×C₈

C₂×D₅

yes

—

C₄²×C₅

C₂×D₅

yes

—

C₄⋊Dic₅

—

#683

—

C₄×Dic₅

—

C₂×#168

—

#446

—

Sylow

C₂×C₅²

C₄×Q₈

yes

characteristic

C₂×C₅²

C₂×M₁₆

yes

characteristic

C₂×C₅²

#131

yes

characteristic

C₂×C₄×C₅

C₄×D₅

yes

—

C₂×C₄×C₅

C₄×D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

C₅×C₈

—

C₅×C₈

Dic₁₀

yes

complemented

#168

—

C₄×Q₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

C₅⋊D₅

yes

characteristic

C₅²

#446

yes

characteristic, complemented, Sylow

C₂²×C₅

C₂×C₄×D₅

yes

—

C₄×C₅

—

C₄×C₅

#676

yes

—

C₄×C₅

C₂×Dic₁₀

yes

—

C₄×C₅

#709

yes

—

Dic₅

—

Dic₅

—

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

C₂×C₅⋊D₅

yes

characteristic

C₂×C₈

C₂×C₅⋊D₅

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₂×C₅⋊D₅

yes

characteristic

C₂×C₅

C₄×Dic₁₀

yes

—

C₂×C₅

C₂×#676

yes

—

C₂×C₅

#4047

yes

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×C₅⋊D₅

yes

characteristic

C₂×C₄

C₄×C₅⋊D₅

yes

characteristic

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

C₈

—

C₈

—

C₈

yes

C₅

yes

C₂²

C₂×C₄×C₅⋊D₅

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#87799

yes

—

C₄

yes

C₄

yes

C₂

yes

C₂

yes

100

C₂

#11717221

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×C₅⋊D₅
commutator, socle, prime radical	C₂²×C₅²	C₂²×C₄
Fitting	C₄×C₅²×C₈	C₂
cosocle	C₂×C₄×C₅²	C₂³
hypercenter	C₄×C₈	C₅⋊D₅
hypocenter	C₅²	#446
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×C₅⋊D₅

commutator, socle, prime radical

C₂²×C₅²

C₂²×C₄

Fitting

C₄×C₅²×C₈

C₂

cosocle

C₂×C₄×C₅²

C₂³

hypercenter

C₄×C₈

C₅⋊D₅

C₅²

G ▷ C₂²×C₅² ▷ C₅²

C₁ ◁ C₂×C₄ ◁ C₄×C₈

N	H ≅ G/N	type
22. C₅²×C₈	Q₈	semidirect
55. C₅×C₈	Dic₁₀	semidirect
62. C₅²	#446	semidirect
89. C₈	#6045	semidirect
90. C₅	#64968	semidirect

H ≅ G/N

type

22. C₅²×C₈

Q₈

semidirect

55. C₅×C₈

Dic₁₀

semidirect

62. C₅²

#446

semidirect

89. C₈

#6045

semidirect

90. C₅

#64968

semidirect

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

subgroup	index
C₄×#6045	2
C₄×C₅²×C₈	2
#11716682	2
#11716681	2
C₄⋊#6029	2
#64968	5

subgroup

index

C₄×C₅²×C₈

Subgroups — #13102542

Conjugacy classes of subgroups

Complements

Maximal subgroups