Group #44549694 — subgroups

Subgroups — #44549694

Order 32128.

172 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 45 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	C₂₅₁⋊#564	C₂	1	1	yes	maximal, characteristic
2	#30632096	C₂	1	2	yes	maximal
3	#30632057	C₂	1	1	yes	maximal, characteristic
4	#30631428	C₂	1	2	yes	maximal
5	#330×C₂₅₁	C₂	1	1	yes	maximal, characteristic
6	#22548715	C₂²	1	1	yes	characteristic
7	#22548707	C₂²	1	2	yes	—
8	#22548706	—	2	2	no	—
9	#22548705	—	2	2	no	—
10	#22548705	—	2	2	no	—
11	#22548649	—	2	2	no	—
12	#22548647	—	2	2	no	—
13	#22548638	—	2	2	no	—
14	#22548626	C₂²	1	1	yes	characteristic
15	#125×C₂₅₁	C₂²	1	1	yes	characteristic
16	#97×C₂₅₁	C₂²	1	2	yes	—
17	C₂×Dic₅₀₂	—	2	1	no	—
18	C₂×C₂₅₁⋊C₈	—	2	1	no	—
19	C₄⋊Dic₂₅₁	—	2	2	no	—
20	C₄⋊Dic₂₅₁	—	2	2	no	—
21	C₄⋊Dic₂₅₁	D₄	1	2	yes	—
22	C₄⋊Dic₂₅₁	D₄	1	2	yes	—
23	Dic₂₅₁⋊C₄	—	2	2	no	—
24	Dic₂₅₁⋊C₄	—	2	2	no	—
25	Dic₂₅₁⋊C₄	—	2	2	no	—
26	Dic₂₅₁⋊C₄	—	2	2	no	—
27	C₄×Dic₂₅₁	—	2	2	no	—
28	C₂×C₈×C₂₅₁	—	2	2	no	—
29	C₄⋊C₄×C₂₅₁	—	2	2	no	—
30	C₄⋊C₄×C₂₅₁	D₄	1	2	yes	—
31	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
32	Dic₅₀₂	—	4	2	no	—
33	C₂×Dic₂₅₁	—	2	2	no	—
34	C₂×Dic₂₅₁	—	2	2	no	—
35	C₂×Dic₂₅₁	—	4	1	no	—
36	C₂₅₁⋊C₈	—	4	1	no	—
37	C₂×C₄×C₂₅₁	—	2	2	no	—
38	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
40	C₈×C₂₅₁	—	2	4	no	—
41	C₂²×C₂₅₁	#120	1	1	yes	characteristic
42	C₄×C₂₅₁	—	2	1	no	—
43	C₄×C₂₅₁	—	2	2	no	—
44	C₄×C₂₅₁	—	4	2	no	—
45	Dic₂₅₁	—	4	2	no	—
46	Dic₂₅₁	—	4	2	no	—
47	Dic₂₅₁	—	8	1	no	—
48	C₂×C₂₅₁	#454	1	1	yes	characteristic
49	C₂×C₂₅₁	#456	1	2	yes	—
50	C₂₅₁	#1663	1	1	yes	characteristic, complemented, Sylow
51	#1663	—	251	1	no	maximal, Sylow
52	#564	—	251	1	no	—
53	#501	—	251	2	no	—
54	#491	—	251	1	no	—
55	#330	—	251	2	no	—
56	#330	D₂₅₁	1	1	yes	characteristic
57	C₄⋊Q₈	—	251	1	no	—
58	#125	—	502	2	no	—
59	#125	—	502	2	no	—
60	#125	—	502	2	no	—
61	#125	—	251	2	no	—
62	#125	C₂×D₂₅₁	1	1	yes	characteristic
63	#107	—	502	2	no	—
64	#106	—	502	2	no	—
65	#103	—	502	2	no	—
66	#97	—	251	1	no	—
67	#97	C₂×D₂₅₁	1	2	yes	—
68	C₂×Q₈	—	502	1	no	—
69	C₂×C₈	—	2	2	no	—
70	C₂×C₈	—	502	1	no	—
71	C₄⋊C₄	—	2	2	no	—
72	C₄⋊C₄	—	502	2	no	—
73	C₄⋊C₄	—	502	2	no	—
74	C₄⋊C₄	—	502	2	no	—
75	C₄⋊C₄	—	502	2	no	—
76	C₄⋊C₄	—	502	2	no	—
77	C₄⋊C₄	—	502	2	no	—
78	C₄⋊C₄	—	251	2	no	—
79	C₄⋊C₄	—	251	2	no	—
80	C₄⋊C₄	D₁₀₀₄	1	2	yes	—
81	C₄²	—	502	2	no	—
82	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
83	Q₈	—	1004	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	1004	1	no	—
86	C₂×C₄	—	502	2	no	—
87	C₂×C₄	—	502	2	no	—
88	C₂×C₄	D₄×D₂₅₁	1	2	yes	—
89	C₂×C₄	C₂×D₁₀₀₄	1	1	yes	characteristic
90	C₈	—	2	4	no	—
91	C₈	—	1004	1	no	—
92	C₂²	#22548685	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	2	2	no	—
95	C₄	—	4	2	no	—
96	C₄	—	2008	1	no	—
97	C₄	—	1004	2	no	—
98	C₄	—	1004	2	no	—
99	C₂	#30631862	1	1	yes	characteristic
100	C₂	#30631871	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₅₁⋊#564

C₂

yes

maximal, characteristic

#30632096

C₂

yes

maximal

#30632057

C₂

yes

maximal, characteristic

#30631428

C₂

yes

maximal

#330×C₂₅₁

C₂

yes

maximal, characteristic

#22548715

C₂²

yes

characteristic

#22548707

C₂²

yes

—

—

—

—

—

—

—

C₂²

yes

characteristic

#125×C₂₅₁

C₂²

yes

characteristic

#97×C₂₅₁

C₂²

yes

—

C₂×Dic₅₀₂

—

C₂×C₂₅₁⋊C₈

—

C₄⋊Dic₂₅₁

—

C₄⋊Dic₂₅₁

—

C₄⋊Dic₂₅₁

D₄

yes

—

C₄⋊Dic₂₅₁

D₄

yes

—

Dic₂₅₁⋊C₄

—

Dic₂₅₁⋊C₄

—

Dic₂₅₁⋊C₄

—

Dic₂₅₁⋊C₄

—

C₄×Dic₂₅₁

—

C₂×C₈×C₂₅₁

—

C₄⋊C₄×C₂₅₁

—

C₄⋊C₄×C₂₅₁

D₄

yes

—

C₄²×C₂₅₁

C₂³

yes

characteristic

Dic₅₀₂

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂₅₁⋊C₈

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₈×C₂₅₁

—

C₂²×C₂₅₁

#120

yes

characteristic

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

Dic₂₅₁

—

Dic₂₅₁

—

Dic₂₅₁

—

C₂×C₂₅₁

#454

yes

characteristic

C₂×C₂₅₁

#456

yes

—

C₂₅₁

#1663

yes

characteristic, complemented, Sylow

#1663

—

251

maximal, Sylow

#564

—

251

—

#501

—

251

—

#491

—

251

—

#330

—

251

—

#330

D₂₅₁

yes

characteristic

C₄⋊Q₈

—

251

—

#125

—

502

—

#125

—

502

—

#125

—

502

—

#125

—

251

—

#125

C₂×D₂₅₁

yes

characteristic

#107

—

502

—

#106

—

502

—

#103

—

502

—

#97

—

251

—

#97

C₂×D₂₅₁

yes

—

C₂×Q₈

—

502

—

C₂×C₈

—

C₂×C₈

—

502

—

C₄⋊C₄

—

C₄⋊C₄

—

502

—

C₄⋊C₄

—

502

—

C₄⋊C₄

—

502

—

C₄⋊C₄

—

502

—

C₄⋊C₄

—

502

—

C₄⋊C₄

—

502

—

C₄⋊C₄

—

251

—

C₄⋊C₄

—

251

—

C₄⋊C₄

D₁₀₀₄

yes

—

C₄²

—

502

—

C₄²

C₂²×D₂₅₁

yes

characteristic

Q₈

—

1004

—

C₂×C₄

—

C₂×C₄

—

1004

—

C₂×C₄

—

502

—

C₂×C₄

—

502

—

C₂×C₄

D₄×D₂₅₁

yes

—

C₂×C₄

C₂×D₁₀₀₄

yes

characteristic

C₈

—

C₈

—

1004

—

C₂²

#22548685

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

2008

—

C₄

—

1004

—

C₄

—

1004

—

C₂

#30631862

yes

characteristic

100

C₂

#30631871

yes

—

	subgroup	G/H
center	C₂²	#22548685
commutator, cosocle	C₄²×C₂₅₁	C₂³
Frattini	C₄²	C₂²×D₂₅₁
Fitting	#330×C₂₅₁	C₂
socle, prime radical	C₂²×C₂₅₁	#120
hypercenter	#330	D₂₅₁
hypocenter	C₂₅₁	#1663
LCS	G ▷ C₄²×C₂₅₁ ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #330

subgroup

G/H

center

C₂²

#22548685

commutator, cosocle

C₄²×C₂₅₁

C₂³

Frattini

C₄²

C₂²×D₂₅₁

Fitting

#330×C₂₅₁

C₂

C₂²×C₂₅₁

D₂₅₁

C₂₅₁

G ▷ C₄²×C₂₅₁ ▷ C₂²×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #330

N	H ≅ G/N	type
50. C₂₅₁	#1663	semidirect

H ≅ G/N

type

50. C₂₅₁

#1663

semidirect

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

subgroup	index
C₂₅₁⋊#564	2
#30632096	2
#30632057	2
#30631428	2
#330×C₂₅₁	2
#1663	251

subgroup

index

C₂₅₁⋊#564

#330×C₂₅₁

251

Subgroups — #44549694

Conjugacy classes of subgroups

Complements

Maximal subgroups