Group #44560869 — subgroups

Subgroups — C₂×#30632089 #44560869

Order 32128.

360 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 181 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	C₂×#22548707	C₂	1	2	yes	maximal
2	#30632089	C₂	1	8	yes	maximal, direct product factor
3	C₂×#22548649	C₂	1	2	yes	maximal
4	C₂×#22548647	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₂₅₁	C₂	1	1	yes	maximal, characteristic
6	#22548707	—	2	4	no	—
7	#22548707	C₂²	1	8	yes	—
8	C₂×C₄⋊Dic₂₅₁	C₂²	1	4	yes	—
9	C₂×Dic₂₅₁⋊C₄	—	2	4	no	—
10	#22548649	C₂²	1	8	yes	—
11	#22548647	C₂²	1	8	yes	—
12	C₂²×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₂₅₁	C₂²	1	4	yes	—
16	C₂²×Dic₂₅₁	—	2	4	no	—
17	C₄⋊Dic₂₅₁	—	2	4	no	—
18	C₄⋊Dic₂₅₁	C₂³	1	8	yes	—
19	Dic₂₅₁⋊C₄	—	2	16	no	—
20	C₂²×C₄×C₂₅₁	D₄	1	2	yes	—
21	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
22	C₂×C₈×C₂₅₁	Q₈	1	4	yes	—
23	C₂×C₈×C₂₅₁	Q₈	1	4	yes	—
24	C₂×C₈×C₂₅₁	C₂³	1	2	yes	—
25	C₂×C₈×C₂₅₁	C₂³	1	2	yes	—
26	C₄²×C₂₅₁	D₄	1	2	yes	—
27	C₄²×C₂₅₁	C₂³	1	2	yes	—
28	C₂×Dic₂₅₁	—	2	8	no	—
29	C₂×Dic₂₅₁	—	4	8	no	—
30	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₂₅₁	—	2	4	no	—
32	C₂×C₄×C₂₅₁	C₂×D₄	1	4	yes	—
33	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₂₅₁	C₂×Q₈	1	2	yes	—
35	C₂×C₄×C₂₅₁	C₂×Q₈	1	2	yes	—
36	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
37	C₈×C₂₅₁	C₂×Q₈	1	4	yes	—
38	C₈×C₂₅₁	C₂×Q₈	1	4	yes	—
39	C₂²×C₂₅₁	C₄⋊Q₈	1	2	yes	—
40	C₂²×C₂₅₁	#135	1	4	yes	—
41	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
42	C₄×C₂₅₁	—	2	4	no	—
43	C₄×C₂₅₁	C₄⋊Q₈	1	2	yes	—
44	C₄×C₂₅₁	C₂²×Q₈	1	1	yes	characteristic
45	C₄×C₂₅₁	C₂²×Q₈	1	1	yes	characteristic
46	Dic₂₅₁	—	4	8	no	—
47	C₂×C₂₅₁	#499	1	4	yes	complemented
48	C₂×C₂₅₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
49	C₂×C₂₅₁	C₂×#135	1	2	yes	—
50	C₂₅₁	C₂×#499	1	1	yes	characteristic, complemented, Sylow
51	C₂×#499	—	251	1	no	maximal, Sylow
52	C₂×#125	—	251	2	no	—
53	#499	—	251	8	no	—
54	C₂×#107	—	251	2	no	—
55	C₂×#106	—	251	2	no	—
56	C₂×C₄×C₈	D₂₅₁	1	1	yes	characteristic
57	C₂²×C₈	C₂×D₂₅₁	1	1	yes	characteristic
58	C₂²×C₈	C₂×D₂₅₁	1	1	yes	characteristic
59	#125	—	502	4	no	—
60	#125	—	251	8	no	—
61	C₂×C₄⋊C₄	—	502	4	no	—
62	C₂×C₄⋊C₄	—	251	4	no	—
63	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
64	#107	—	251	8	no	—
65	#106	—	251	8	no	—
66	C₄×C₈	C₂×D₂₅₁	1	4	yes	—
67	C₂²×C₄	—	502	4	no	—
68	C₂²×C₄	D₁₀₀₄	1	2	yes	—
69	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
70	C₂×C₈	Dic₅₀₂	1	4	yes	—
71	C₂×C₈	Dic₅₀₂	1	4	yes	—
72	C₂×C₈	C₂²×D₂₅₁	1	2	yes	—
73	C₂×C₈	C₂²×D₂₅₁	1	2	yes	—
74	C₄⋊C₄	—	502	16	no	—
75	C₄⋊C₄	—	502	4	no	—
76	C₄⋊C₄	—	251	8	no	—
77	C₄²	D₁₀₀₄	1	2	yes	—
78	C₄²	C₂²×D₂₅₁	1	2	yes	—
79	C₂³	C₂×D₁₀₀₄	1	1	yes	characteristic
80	C₂×C₄	—	2	4	no	—
81	C₂×C₄	—	1004	8	no	—
82	C₂×C₄	—	502	8	no	—
83	C₂×C₄	C₂×D₁₀₀₄	1	4	yes	—
84	C₂×C₄	C₂×D₁₀₀₄	1	1	yes	characteristic
85	C₂×C₄	C₂×Dic₅₀₂	1	2	yes	—
86	C₂×C₄	C₂×Dic₅₀₂	1	2	yes	—
87	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
88	C₈	C₂×Dic₅₀₂	1	4	yes	—
89	C₈	C₂×Dic₅₀₂	1	4	yes	—
90	C₂²	#22548715	1	2	yes	—
91	C₂²	#22548737	1	4	yes	—
92	C₂²	C₂²×D₁₀₀₄	1	1	yes	characteristic
93	C₄	—	2	4	no	—
94	C₄	—	1004	8	no	—
95	C₄	#22548715	1	2	yes	—
96	C₄	C₂²×Dic₅₀₂	1	1	yes	characteristic
97	C₄	C₂²×Dic₅₀₂	1	1	yes	characteristic
98	C₂	#30632089	1	4	yes	direct product factor
99	C₂	C₂×#22548715	1	1	yes	characteristic
100	C₂	C₂×#22548737	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22548707

C₂

yes

maximal

#30632089

C₂

yes

maximal, direct product factor

C₂×#22548649

C₂

yes

maximal

C₂×#22548647

C₂

yes

maximal

C₂×C₄×C₈×C₂₅₁

C₂

yes

maximal, characteristic

#22548707

—

#22548707

C₂²

yes

—

C₂×C₄⋊Dic₂₅₁

C₂²

yes

—

C₂×Dic₂₅₁⋊C₄

—

#22548649

C₂²

yes

—

#22548647

C₂²

yes

—

C₂²×C₈×C₂₅₁

C₂²

yes

characteristic

C₂²×C₈×C₂₅₁

C₂²

yes

characteristic

C₂×C₄²×C₂₅₁

C₂²

yes

characteristic

C₄×C₈×C₂₅₁

C₂²

yes

—

C₂²×Dic₂₅₁

—

C₄⋊Dic₂₅₁

—

C₄⋊Dic₂₅₁

C₂³

yes

—

Dic₂₅₁⋊C₄

—

C₂²×C₄×C₂₅₁

D₄

yes

—

C₂²×C₄×C₂₅₁

C₂³

yes

characteristic

C₂×C₈×C₂₅₁

Q₈

yes

—

C₂×C₈×C₂₅₁

Q₈

yes

—

C₂×C₈×C₂₅₁

C₂³

yes

—

C₂×C₈×C₂₅₁

C₂³

yes

—

C₄²×C₂₅₁

D₄

yes

—

C₄²×C₂₅₁

C₂³

yes

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂³×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂×Q₈

yes

—

C₂×C₄×C₂₅₁

C₂×Q₈

yes

—

C₂×C₄×C₂₅₁

C₂⁴

yes

characteristic

C₈×C₂₅₁

C₂×Q₈

yes

—

C₈×C₂₅₁

C₂×Q₈

yes

—

C₂²×C₂₅₁

C₄⋊Q₈

yes

—

C₂²×C₂₅₁

#135

yes

—

C₂²×C₂₅₁

C₂²×D₄

yes

characteristic

C₄×C₂₅₁

—

C₄×C₂₅₁

C₄⋊Q₈

yes

—

C₄×C₂₅₁

C₂²×Q₈

yes

characteristic

C₄×C₂₅₁

C₂²×Q₈

yes

characteristic

Dic₂₅₁

—

C₂×C₂₅₁

#499

yes

complemented

C₂×C₂₅₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₂₅₁

C₂×#135

yes

—

C₂₅₁

C₂×#499

yes

characteristic, complemented, Sylow

C₂×#499

—

251

maximal, Sylow

C₂×#125

—

251

—

#499

—

251

—

C₂×#107

—

251

—

C₂×#106

—

251

—

C₂×C₄×C₈

D₂₅₁

yes

characteristic

C₂²×C₈

C₂×D₂₅₁

yes

characteristic

C₂²×C₈

C₂×D₂₅₁

yes

characteristic

#125

—

502

—

#125

—

251

—

C₂×C₄⋊C₄

—

502

—

C₂×C₄⋊C₄

—

251

—

C₂×C₄²

C₂×D₂₅₁

yes

characteristic

#107

—

251

—

#106

—

251

—

C₄×C₈

C₂×D₂₅₁

yes

—

C₂²×C₄

—

502

—

C₂²×C₄

D₁₀₀₄

yes

—

C₂²×C₄

C₂²×D₂₅₁

yes

characteristic

C₂×C₈

Dic₅₀₂

yes

—

C₂×C₈

Dic₅₀₂

yes

—

C₂×C₈

C₂²×D₂₅₁

yes

—

C₂×C₈

C₂²×D₂₅₁

yes

—

C₄⋊C₄

—

502

—

C₄⋊C₄

—

502

—

C₄⋊C₄

—

251

—

C₄²

D₁₀₀₄

yes

—

C₄²

C₂²×D₂₅₁

yes

—

C₂³

C₂×D₁₀₀₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

1004

—

C₂×C₄

—

502

—

C₂×C₄

C₂×D₁₀₀₄

yes

—

C₂×C₄

C₂×D₁₀₀₄

yes

characteristic

C₂×C₄

C₂×Dic₅₀₂

yes

—

C₂×C₄

C₂×Dic₅₀₂

yes

—

C₂×C₄

C₂³×D₂₅₁

yes

characteristic

C₈

C₂×Dic₅₀₂

yes

—

C₈

C₂×Dic₅₀₂

yes

—

C₂²

#22548715

yes

—

C₂²

#22548737

yes

—

C₂²

C₂²×D₁₀₀₄

yes

characteristic

C₄

—

C₄

—

1004

—

C₄

#22548715

yes

—

C₄

C₂²×Dic₅₀₂

yes

characteristic

C₄

C₂²×Dic₅₀₂

yes

characteristic

C₂

#30632089

yes

direct product factor

C₂

C₂×#22548715

yes

characteristic

100

C₂

C₂×#22548737

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₀₀₄
commutator, cosocle	C₂×C₄×C₂₅₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	C₂×C₄×C₈×C₂₅₁	C₂
socle, prime radical	C₂³×C₂₅₁	C₂×D₄
hypercenter	C₂×C₄×C₈	D₂₅₁
hypocenter	C₂₅₁	C₂×#499
LCS	G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₁₀₀₄

commutator, cosocle

C₂×C₄×C₂₅₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₅₁

Fitting

C₂×C₄×C₈×C₂₅₁

C₂

socle, prime radical

C₂³×C₂₅₁

C₂×D₄

hypercenter

C₂×C₄×C₈

D₂₅₁

hypocenter

C₂₅₁

C₂×#499

LCS

G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #30632089	C₂	direct (N × H)
47. C₂×C₂₅₁	#499	semidirect
50. C₂₅₁	C₂×#499	semidirect
98. C₂	#30632089	direct (N × H)

H ≅ G/N

type

2. #30632089

C₂

direct (N × H)

47. C₂×C₂₅₁

#499

semidirect

50. C₂₅₁

C₂×#499

semidirect

98. C₂

#30632089

direct (N × H)

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

subgroup	index
C₂×#22548707	2
#30632089	2
C₂×#22548649	2
C₂×#22548647	2
C₂×C₄×C₈×C₂₅₁	2
C₂×#499	251

subgroup

index

C₂×C₄×C₈×C₂₅₁

251

Subgroups — C₂×#30632089 #44560869

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#30632089 #44560869

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#30632089 #44560869