Group #13098360 — subgroups

Subgroups — #13098360

Order 1600.

135 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 135 conjugacy classes of subgroups — too many to draw. Only its 57 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	C₂₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#11716432	C₂	1	1	yes	maximal, characteristic
3	#11716356	C₂	1	2	yes	maximal
4	#11716355	C₂	1	2	yes	maximal
5	#11716351	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₈×C₂₅	—	2	1	no	—
7	C₄⋊C₄×C₂₅	—	2	1	no	—
8	C₄⋊C₄×C₂₅	C₂²	1	2	yes	—
9	C₄²×C₂₅	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₂₅	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₂₅	C₂²	1	1	yes	characteristic
12	#87718	—	2	1	no	—
13	#87718	—	2	1	no	—
14	C₂×#6014	C₂²	1	2	yes	—
15	#65369	—	5	1	no	maximal
16	Q₈×C₂₅	—	4	2	no	—
17	C₂×C₄×C₂₅	—	2	2	no	—
18	C₂×C₄×C₂₅	D₄	1	1	yes	characteristic
19	C₂×C₄×C₂₅	D₄	1	1	yes	characteristic
20	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
21	C₂×Dic₂₅	—	2	1	no	—
22	C₂×Dic₂₅	—	2	1	no	—
23	#6014	Q₈	1	4	yes	complemented
24	C₅×C₄⋊Q₈	D₅	1	1	yes	characteristic
25	#4016	—	5	1	no	—
26	#3940	—	5	2	no	—
27	#3939	—	5	2	no	—
28	#3935	—	5	1	no	—
29	C₂²×C₂₅	C₂×D₄	1	1	yes	characteristic
30	C₄×C₂₅	—	2	1	no	—
31	C₄×C₂₅	—	2	1	no	—
32	C₄×C₂₅	—	4	2	no	—
33	C₄×C₂₅	C₂×Q₈	1	2	yes	—
34	Dic₂₅	—	4	1	no	—
35	Dic₂₅	—	4	1	no	—
36	C₂×C₅×Q₈	—	2	1	no	—
37	C₅×C₄⋊C₄	—	2	1	no	—
38	C₅×C₄⋊C₄	C₂×D₅	1	2	yes	—
39	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
40	C₄⋊Dic₅	—	5	1	no	—
41	C₄⋊Dic₅	—	5	1	no	—
42	#683	—	10	1	no	—
43	#683	—	10	1	no	—
44	C₂×#168	—	5	2	no	—
45	#501	—	25	1	no	Sylow
46	C₂×C₂₅	C₄⋊Q₈	1	1	yes	characteristic
47	C₂×C₂₅	AGL(1,ℤ/8)	1	1	yes	characteristic
48	C₂×C₂₅	#137	1	1	yes	characteristic
49	C₅×Q₈	—	4	2	no	—
50	C₂×C₄×C₅	—	2	2	no	—
51	C₂×C₄×C₅	C₂²⋊D₅	1	1	yes	characteristic
52	C₂×C₄×C₅	C₂²⋊D₅	1	1	yes	characteristic
53	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
54	C₂×Dic₅	—	10	1	no	—
55	C₂×Dic₅	—	10	1	no	—
56	#168	—	5	4	no	—
57	C₄⋊Q₈	D₂₅	1	1	yes	characteristic
58	#125	—	25	1	no	—
59	#107	—	25	2	no	—
60	#106	—	25	2	no	—
61	#97	—	25	1	no	—
62	C₂₅	#501	1	1	yes	characteristic, complemented, Sylow
63	C₂²×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
64	C₄×C₅	—	2	1	no	—
65	C₄×C₅	—	2	1	no	—
66	C₄×C₅	—	4	2	no	—
67	C₄×C₅	Q₈×D₅	1	2	yes	—
68	Dic₅	—	20	1	no	—
69	Dic₅	—	20	1	no	—
70	C₂×Q₈	—	2	1	no	—
71	C₂×C₈	—	25	2	no	—
72	C₄⋊C₄	—	2	1	no	—
73	C₄⋊C₄	—	25	1	no	—
74	C₄⋊C₄	—	25	1	no	—
75	C₄⋊C₄	—	50	1	no	—
76	C₄⋊C₄	—	50	1	no	—
77	C₄⋊C₄	C₂×D₂₅	1	2	yes	—
78	C₄²	C₂×D₂₅	1	1	yes	characteristic
79	C₂×C₅	#4078	1	1	yes	characteristic
80	C₂×C₅	#4088	1	1	yes	characteristic
81	C₂×C₅	#4090	1	1	yes	characteristic
82	Q₈	—	4	2	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	50	1	no	—
85	C₂×C₄	—	50	1	no	—
86	C₂×C₄	C₂²⋊D₂₅	1	1	yes	characteristic
87	C₂×C₄	C₂²⋊D₂₅	1	1	yes	characteristic
88	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
89	C₈	—	25	4	no	—
90	C₅	#65369	1	1	yes	characteristic
91	C₂²	C₂×C₂²⋊D₂₅	1	1	yes	characteristic
92	C₄	—	2	1	no	—
93	C₄	—	2	1	no	—
94	C₄	—	4	2	no	—
95	C₄	—	100	1	no	—
96	C₄	—	100	1	no	—
97	C₄	Q₈×D₂₅	1	2	yes	—
98	C₂	#11716495	1	1	yes	characteristic
99	C₂	#11716505	1	1	yes	characteristic
100	C₂	#11716506	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₅×C₄⋊Q₈

C₂

yes

maximal, characteristic

#11716432

C₂

yes

maximal, characteristic

#11716356

C₂

yes

maximal

#11716355

C₂

yes

maximal

#11716351

C₂

yes

maximal, characteristic

C₂×Q₈×C₂₅

—

C₄⋊C₄×C₂₅

—

C₄⋊C₄×C₂₅

C₂²

yes

—

C₄²×C₂₅

C₂²

yes

characteristic

C₄⋊Dic₂₅

C₂²

yes

characteristic

C₄⋊Dic₂₅

C₂²

yes

—

—

C₂²

yes

—

#65369

—

maximal

Q₈×C₂₅

—

C₂×C₄×C₂₅

—

C₂×C₄×C₂₅

D₄

yes

characteristic

C₂×C₄×C₂₅

D₄

yes

characteristic

C₂×C₄×C₂₅

C₂³

yes

characteristic

C₂×Dic₂₅

—

C₂×Dic₂₅

—

#6014

Q₈

yes

complemented

C₅×C₄⋊Q₈

D₅

yes

—

—

—

—

C₂²×C₂₅

C₂×D₄

yes

characteristic

C₄×C₂₅

—

C₄×C₂₅

—

C₄×C₂₅

—

C₄×C₂₅

C₂×Q₈

yes

—

Dic₂₅

—

Dic₂₅

—

C₂×C₅×Q₈

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

C₂×D₅

yes

—

C₄²×C₅

C₂×D₅

yes

characteristic

C₄⋊Dic₅

—

C₄⋊Dic₅

—

—

—

—

—

C₂×C₂₅

C₄⋊Q₈

yes

C₂×C₂₅

yes

C₂×C₂₅

yes

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²⋊D₅

yes

characteristic

C₂×C₄×C₅

C₂²⋊D₅

yes

characteristic

C₂×C₄×C₅

C₂²×D₅

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

—

#168

—

C₄⋊Q₈

D₂₅

yes

—

—

—

—

C₂₅

#501

yes

characteristic, complemented, Sylow

C₂²×C₅

C₂×C₂²⋊D₅

yes

characteristic

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

Q₈×D₅

yes

—

Dic₅

—

Dic₅

—

C₂×Q₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₂₅

yes

—

C₄²

C₂×D₂₅

yes

C₂×C₅

yes

C₂×C₅

yes

C₂×C₅

yes

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊D₂₅

yes

characteristic

C₂×C₄

C₂²⋊D₂₅

yes

characteristic

C₂×C₄

C₂²×D₂₅

yes

characteristic

C₈

—

C₅

#65369

yes

characteristic

C₂²

C₂×C₂²⋊D₂₅

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

100

—

C₄

—

100

—

C₄

Q₈×D₂₅

yes

—

C₂

yes

C₂

yes

100

C₂

#11716506

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₅
commutator, cosocle	C₂×C₄×C₂₅	C₂³
Frattini	C₂×C₄×C₅	C₂²×D₅
Fitting	C₂₅×C₄⋊Q₈	C₂
socle, prime radical	C₂²×C₅	C₂×C₂²⋊D₅
hypercenter	C₄⋊Q₈	D₂₅
hypocenter	C₂₅	#501
LCS	G ▷ C₂×C₄×C₂₅ ▷ C₂×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊D₂₅

commutator, cosocle

C₂×C₄×C₂₅

C₂³

Frattini

C₂×C₄×C₅

C₂²×D₅

Fitting

C₂₅×C₄⋊Q₈

C₂

socle, prime radical

C₂²×C₅

C₂×C₂²⋊D₅

hypercenter

C₄⋊Q₈

D₂₅

hypocenter

C₂₅

#501

LCS

G ▷ C₂×C₄×C₂₅ ▷ C₂×C₂₅ ▷ C₂₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
23. #6014	Q₈	semidirect
62. C₂₅	#501	semidirect

H ≅ G/N

type

23. #6014

Q₈

semidirect

62. C₂₅

#501

semidirect

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

subgroup	index
C₂₅×C₄⋊Q₈	2
#11716432	2
#11716356	2
#11716355	2
#11716351	2
#65369	5

subgroup

index

C₂₅×C₄⋊Q₈

Subgroups — #13098360

Conjugacy classes of subgroups

Complements

Maximal subgroups