Group #44558634 — subgroups

Subgroups — #44558634

Order 32128.

348 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 133 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	#30631626	C₂	1	3	yes	maximal
2	#30631620	C₂	1	6	yes	maximal
3	#30631607	C₂	1	3	yes	maximal
4	#30631555	C₂	1	2	yes	maximal
5	#384×C₂₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₅₀₂	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₂₅₁	C₂²	1	3	yes	—
8	C₂×Dic₂₅₁⋊C₄	C₂²	1	6	yes	—
9	C₂₅₁⋊#95	C₂²	1	6	yes	—
10	C₂₅₁⋊#95	C₂²	1	6	yes	—
11	C₂₅₁⋊#95	C₂²	1	3	yes	—
12	C₂₅₁⋊#95	C₂²	1	3	yes	—
13	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	3	yes	—
14	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	3	yes	—
15	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
16	C₂×Dic₅₀₂	—	4	3	no	—
17	C₂²×Dic₂₅₁	C₂³	1	2	yes	—
18	C₂²×Dic₂₅₁	C₂³	1	6	yes	—
19	C₄⋊Dic₂₅₁	—	4	3	no	—
20	Dic₂₅₁⋊C₄	—	4	6	no	—
21	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
22	C₂²×C₄×C₂₅₁	C₂³	1	3	yes	—
23	C₂²×C₄×C₂₅₁	C₂³	1	3	yes	—
24	C₄⋊C₄×C₂₅₁	—	4	3	no	—
25	C₄⋊C₄×C₂₅₁	—	4	3	no	—
26	C₄²×C₂₅₁	Q₈	1	4	yes	complemented
27	Dic₅₀₂	—	4	4	no	—
28	C₂×Dic₂₅₁	—	2	6	no	—
29	C₂×Dic₂₅₁	—	2	6	no	—
30	C₂×Dic₂₅₁	—	2	12	no	—
31	C₂³×C₂₅₁	C₂⁴	1	1	yes	characteristic
32	C₂×C₄×C₂₅₁	—	2	3	no	—
33	C₂×C₄×C₂₅₁	—	2	3	no	—
34	C₂×C₄×C₂₅₁	—	2	6	no	—
35	C₂×C₄×C₂₅₁	—	2	3	no	—
36	C₂×C₄×C₂₅₁	—	2	3	no	—
37	C₂×C₄×C₂₅₁	C₂×Q₈	1	6	yes	—
38	C₂²×C₂₅₁	C₂²×Q₈	1	1	yes	characteristic
39	C₂²×C₂₅₁	H(5,𝔽₂)	1	3	yes	—
40	C₂²×C₂₅₁	2₋¹⁺⁴	1	3	yes	—
41	C₄×C₂₅₁	—	2	6	no	—
42	C₄×C₂₅₁	—	4	3	no	—
43	C₄×C₂₅₁	—	4	1	no	—
44	Dic₂₅₁	—	4	2	no	—
45	Dic₂₅₁	—	4	6	no	—
46	C₂×C₂₅₁	#544	1	3	yes	—
47	C₂×C₂₅₁	#560	1	1	yes	characteristic
48	C₂×C₂₅₁	#563	1	3	yes	—
49	C₂₅₁	#2843	1	1	yes	characteristic, complemented, Sylow
50	#2843	—	251	1	no	maximal, Sylow
51	#400	—	251	3	no	—
52	#398	—	251	6	no	—
53	#395	—	251	3	no	—
54	#384	D₂₅₁	1	1	yes	characteristic
55	#383	—	251	2	no	—
56	C₂²×Q₈	—	251	1	no	—
57	C₂×C₄⋊C₄	—	251	6	no	—
58	C₂×C₄⋊C₄	—	251	3	no	—
59	C₂×C₄⋊C₄	C₂×D₂₅₁	1	3	yes	—
60	C₂×C₄⋊C₄	C₂×D₂₅₁	1	3	yes	—
61	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
62	#95	—	251	6	no	—
63	#95	—	251	6	no	—
64	#95	—	251	3	no	—
65	#95	—	251	3	no	—
66	C₂×Q₈	—	1004	3	no	—
67	C₂²×C₄	—	251	6	no	—
68	C₂²×C₄	—	251	2	no	—
69	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
70	C₂²×C₄	C₂²×D₂₅₁	1	3	yes	—
71	C₂²×C₄	C₂²×D₂₅₁	1	3	yes	—
72	C₄⋊C₄	—	4	3	no	—
73	C₄⋊C₄	—	4	3	no	—
74	C₄⋊C₄	—	1004	6	no	—
75	C₄⋊C₄	—	1004	3	no	—
76	C₄²	Dic₅₀₂	1	4	yes	complemented
77	C₂³	C₂³×D₂₅₁	1	1	yes	characteristic
78	Q₈	—	1004	4	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	—	2	6	no	—
82	C₂×C₄	—	2	3	no	—
83	C₂×C₄	—	2	3	no	—
84	C₂×C₄	—	502	6	no	—
85	C₂×C₄	—	502	6	no	—
86	C₂×C₄	—	502	12	no	—
87	C₂×C₄	C₂×Dic₅₀₂	1	6	yes	—
88	C₂²	C₂²×Dic₅₀₂	1	1	yes	characteristic
89	C₂²	#22548760	1	3	yes	—
90	C₂²	#22548763	1	3	yes	—
91	C₄	—	2	6	no	—
92	C₄	—	4	1	no	—
93	C₄	—	4	3	no	—
94	C₄	—	1004	2	no	—
95	C₄	—	1004	6	no	—
96	C₂	#30632318	1	3	yes	—
97	C₂	#30632424	1	1	yes	characteristic
98	C₂	#30632436	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#30631626

C₂

yes

maximal

#30631620

C₂

yes

maximal

#30631607

C₂

yes

maximal

#30631555

C₂

yes

maximal

#384×C₂₅₁

C₂

yes

maximal, characteristic

C₂²×Dic₅₀₂

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₅₁

C₂²

yes

—

C₂×Dic₂₅₁⋊C₄

C₂²

yes

—

C₂₅₁⋊#95

C₂²

yes

—

C₂₅₁⋊#95

C₂²

yes

—

C₂₅₁⋊#95

C₂²

yes

—

C₂₅₁⋊#95

C₂²

yes

—

C₂×C₄⋊C₄×C₂₅₁

C₂²

yes

—

C₂×C₄⋊C₄×C₂₅₁

C₂²

yes

—

C₂×C₄²×C₂₅₁

C₂²

yes

characteristic

C₂×Dic₅₀₂

—

C₂²×Dic₂₅₁

C₂³

yes

—

C₂²×Dic₂₅₁

C₂³

yes

—

C₄⋊Dic₂₅₁

—

Dic₂₅₁⋊C₄

—

C₂²×C₄×C₂₅₁

C₂³

yes

characteristic

C₂²×C₄×C₂₅₁

C₂³

yes

—

C₂²×C₄×C₂₅₁

C₂³

yes

—

C₄⋊C₄×C₂₅₁

—

C₄⋊C₄×C₂₅₁

—

C₄²×C₂₅₁

Q₈

yes

complemented

Dic₅₀₂

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂³×C₂₅₁

C₂⁴

yes

characteristic

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂×Q₈

yes

—

C₂²×C₂₅₁

C₂²×Q₈

yes

characteristic

C₂²×C₂₅₁

H(5,𝔽₂)

yes

—

C₂²×C₂₅₁

2₋¹⁺⁴

yes

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

Dic₂₅₁

—

Dic₂₅₁

—

C₂×C₂₅₁

#544

yes

—

C₂×C₂₅₁

#560

yes

characteristic

C₂×C₂₅₁

#563

yes

—

C₂₅₁

#2843

yes

characteristic, complemented, Sylow

#2843

—

251

maximal, Sylow

#400

—

251

—

#398

—

251

—

#395

—

251

—

#384

D₂₅₁

yes

characteristic

#383

—

251

—

C₂²×Q₈

—

251

—

C₂×C₄⋊C₄

—

251

—

C₂×C₄⋊C₄

—

251

—

C₂×C₄⋊C₄

C₂×D₂₅₁

yes

—

C₂×C₄⋊C₄

C₂×D₂₅₁

yes

—

C₂×C₄²

C₂×D₂₅₁

yes

characteristic

#95

—

251

—

#95

—

251

—

#95

—

251

—

#95

—

251

—

C₂×Q₈

—

1004

—

C₂²×C₄

—

251

—

C₂²×C₄

—

251

—

C₂²×C₄

C₂²×D₂₅₁

yes

characteristic

C₂²×C₄

C₂²×D₂₅₁

yes

—

C₂²×C₄

C₂²×D₂₅₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

1004

—

C₄⋊C₄

—

1004

—

C₄²

Dic₅₀₂

yes

complemented

C₂³

C₂³×D₂₅₁

yes

characteristic

Q₈

—

1004

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

502

—

C₂×C₄

—

502

—

C₂×C₄

—

502

—

C₂×C₄

C₂×Dic₅₀₂

yes

—

C₂²

C₂²×Dic₅₀₂

yes

characteristic

C₂²

#22548760

yes

—

C₂²

#22548763

yes

—

C₄

—

C₄

—

C₄

—

C₄

—

1004

—

C₄

—

1004

—

C₂

#30632318

yes

—

C₂

#30632424

yes

characteristic

C₂

#30632436

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₅₁
commutator, socle, prime radical, cosocle	C₂³×C₂₅₁	C₂⁴
Fitting	#384×C₂₅₁	C₂
hypercenter	#384	D₂₅₁
hypocenter	C₂₅₁	#2843
LCS	G ▷ C₂³×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₂₅₁

commutator, socle, prime radical, cosocle

C₂³×C₂₅₁

C₂⁴

Fitting

#384×C₂₅₁

C₂

D₂₅₁

C₂₅₁

G ▷ C₂³×C₂₅₁ ▷ C₂₅₁

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
26. C₄²×C₂₅₁	Q₈	semidirect
49. C₂₅₁	#2843	semidirect
76. C₄²	Dic₅₀₂	semidirect

H ≅ G/N

type

26. C₄²×C₂₅₁

Q₈

semidirect

49. C₂₅₁

#2843

semidirect

76. C₄²

Dic₅₀₂

semidirect

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

subgroup	index
#30631626	2
#30631620	2
#30631607	2
#30631555	2
#384×C₂₅₁	2
#2843	251

subgroup

index

#384×C₂₅₁

251

Subgroups — #44558634

Conjugacy classes of subgroups

Complements

Maximal subgroups