Group #44558433 — subgroups

Subgroups — #44558433

Order 32128.

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

normal subgroups only

This group has 380 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	C₂×#22548707	C₂	1	2	yes	maximal
2	#30631627	C₂	1	4	yes	maximal
3	#30631619	C₂	1	4	yes	maximal
4	#30631617	C₂	1	4	yes	maximal
5	#382×C₂₅₁	C₂	1	1	yes	maximal, characteristic
6	#22548707	—	2	8	no	—
7	C₂×C₄⋊Dic₂₅₁	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₂₅₁	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₂₅₁	C₂²	1	4	yes	—
10	C₂×Dic₂₅₁⋊C₄	C₂²	1	4	yes	—
11	C₂×Dic₂₅₁⋊C₄	C₂²	1	4	yes	—
12	C₂₅₁⋊#95	C₂²	1	8	yes	—
13	C₂₅₁⋊#95	C₂²	1	4	yes	—
14	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	2	yes	—
15	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
16	#95×C₂₅₁	C₂²	1	4	yes	—
17	C₂²×Dic₂₅₁	C₂³	1	4	yes	—
18	C₂²×Dic₂₅₁	C₂³	1	4	yes	—
19	C₄⋊Dic₂₅₁	—	2	4	no	—
20	C₄⋊Dic₂₅₁	—	2	4	no	—
21	C₄⋊Dic₂₅₁	—	4	4	no	—
22	Dic₂₅₁⋊C₄	—	2	8	no	—
23	Dic₂₅₁⋊C₄	—	2	8	no	—
24	C₂²×C₄×C₂₅₁	C₂³	1	4	yes	—
25	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
26	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₂₅₁	—	4	2	no	—
28	C₄²×C₂₅₁	D₄	1	4	yes	—
29	C₂×Dic₂₅₁	—	2	4	no	—
30	C₂×Dic₂₅₁	—	2	4	no	—
31	C₂×Dic₂₅₁	—	2	4	no	—
32	C₂×Dic₂₅₁	—	2	8	no	—
33	C₂×Dic₂₅₁	—	2	4	no	—
34	C₂³×C₂₅₁	C₂⁴	1	1	yes	characteristic
35	C₂×C₄×C₂₅₁	—	2	1	no	—
36	C₂×C₄×C₂₅₁	—	2	4	no	—
37	C₂×C₄×C₂₅₁	—	2	1	no	—
38	C₂×C₄×C₂₅₁	—	2	4	no	—
39	C₂×C₄×C₂₅₁	—	2	8	no	—
40	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
41	C₂×C₄×C₂₅₁	C₂×D₄	1	4	yes	—
42	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
43	C₂²×C₂₅₁	H(5,𝔽₂)	1	1	yes	characteristic
44	C₂²×C₂₅₁	2₋¹⁺⁴	1	4	yes	—
45	C₂²×C₂₅₁	2₋¹⁺⁴	1	1	yes	characteristic
46	C₄×C₂₅₁	—	2	4	no	—
47	C₄×C₂₅₁	—	2	2	no	—
48	C₄×C₂₅₁	—	4	4	no	—
49	Dic₂₅₁	—	4	4	no	—
50	Dic₂₅₁	—	4	4	no	—
51	C₂×C₂₅₁	#536	1	2	yes	—
52	C₂×C₂₅₁	#537	1	1	yes	characteristic
53	C₂×C₂₅₁	#563	1	2	yes	—
54	C₂×C₂₅₁	#564	1	2	yes	—
55	C₂₅₁	#2822	1	1	yes	characteristic, complemented, Sylow
56	#2822	—	251	1	no	maximal, Sylow
57	C₂×#125	—	251	2	no	—
58	#400	—	251	4	no	—
59	#398	—	251	4	no	—
60	#398	—	251	4	no	—
61	#382	D₂₅₁	1	1	yes	characteristic
62	#125	—	502	8	no	—
63	C₂×C₄⋊C₄	—	251	4	no	—
64	C₂×C₄⋊C₄	—	251	4	no	—
65	C₂×C₄⋊C₄	—	251	2	no	—
66	C₂×C₄⋊C₄	—	251	4	no	—
67	C₂×C₄⋊C₄	—	251	2	no	—
68	C₂×C₄⋊C₄	C₂×D₂₅₁	1	2	yes	—
69	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
70	#95	—	251	8	no	—
71	#95	—	251	4	no	—
72	#95	C₂×D₂₅₁	1	4	yes	—
73	C₂²×C₄	—	251	4	no	—
74	C₂²×C₄	—	251	4	no	—
75	C₂²×C₄	C₂²×D₂₅₁	1	4	yes	—
76	C₂²×C₄	C₂²×D₂₅₁	1	2	yes	—
77	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
78	C₄⋊C₄	—	4	2	no	—
79	C₄⋊C₄	—	1004	4	no	—
80	C₄⋊C₄	—	502	8	no	—
81	C₄⋊C₄	—	502	8	no	—
82	C₄⋊C₄	—	502	4	no	—
83	C₄⋊C₄	—	502	4	no	—
84	C₄²	D₁₀₀₄	1	4	yes	—
85	C₂³	C₂³×D₂₅₁	1	1	yes	characteristic
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	2	8	no	—
91	C₂×C₄	—	502	4	no	—
92	C₂×C₄	—	502	4	no	—
93	C₂×C₄	—	502	4	no	—
94	C₂×C₄	—	502	8	no	—
95	C₂×C₄	—	502	4	no	—
96	C₂×C₄	C₂×D₁₀₀₄	1	4	yes	—
97	C₂×C₄	C₂×D₁₀₀₄	1	2	yes	—
98	C₂²	C₂²×D₁₀₀₄	1	1	yes	characteristic
99	C₂²	#22548760	1	1	yes	characteristic
100	C₂²	#22548762	1	4	yes	—
101	C₂²	#22548763	1	1	yes	characteristic
102	C₄	—	2	4	no	—
103	C₄	—	2	2	no	—
104	C₄	—	4	4	no	—
105	C₄	—	1004	4	no	—
106	C₄	—	1004	4	no	—
107	C₂	#30632263	1	2	yes	—
108	C₂	#30632268	1	1	yes	characteristic
109	C₂	#30632437	1	2	yes	—
110	C₂	C₂₅₁⋊#564	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22548707

C₂

yes

maximal

#30631627

C₂

yes

maximal

#30631619

C₂

yes

maximal

#30631617

C₂

yes

maximal

#382×C₂₅₁

C₂

yes

maximal, characteristic

#22548707

—

C₂×C₄⋊Dic₂₅₁

C₂²

yes

—

C₂×C₄⋊Dic₂₅₁

C₂²

yes

—

C₂×C₄⋊Dic₂₅₁

C₂²

yes

—

C₂×Dic₂₅₁⋊C₄

C₂²

yes

—

C₂×Dic₂₅₁⋊C₄

C₂²

yes

—

C₂₅₁⋊#95

C₂²

yes

—

C₂₅₁⋊#95

C₂²

yes

—

C₂×C₄⋊C₄×C₂₅₁

C₂²

yes

—

C₂×C₄²×C₂₅₁

C₂²

yes

characteristic

#95×C₂₅₁

C₂²

yes

—

C₂²×Dic₂₅₁

C₂³

yes

—

C₂²×Dic₂₅₁

C₂³

yes

—

C₄⋊Dic₂₅₁

—

C₄⋊Dic₂₅₁

—

C₄⋊Dic₂₅₁

—

Dic₂₅₁⋊C₄

—

Dic₂₅₁⋊C₄

—

C₂²×C₄×C₂₅₁

C₂³

yes

—

C₂²×C₄×C₂₅₁

C₂³

yes

—

C₂²×C₄×C₂₅₁

C₂³

yes

characteristic

C₄⋊C₄×C₂₅₁

—

C₄²×C₂₅₁

D₄

yes

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

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×D₄

yes

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

—

C₂²×C₂₅₁

C₂²×D₄

yes

characteristic

C₂²×C₂₅₁

H(5,𝔽₂)

yes

characteristic

C₂²×C₂₅₁

2₋¹⁺⁴

yes

—

C₂²×C₂₅₁

2₋¹⁺⁴

yes

characteristic

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

Dic₂₅₁

—

Dic₂₅₁

—

C₂×C₂₅₁

#536

yes

—

C₂×C₂₅₁

#537

yes

characteristic

C₂×C₂₅₁

#563

yes

—

C₂×C₂₅₁

#564

yes

—

C₂₅₁

#2822

yes

characteristic, complemented, Sylow

#2822

—

251

maximal, Sylow

C₂×#125

—

251

—

#400

—

251

—

#398

—

251

—

#398

—

251

—

#382

D₂₅₁

yes

characteristic

#125

—

502

—

C₂×C₄⋊C₄

—

251

—

C₂×C₄⋊C₄

—

251

—

C₂×C₄⋊C₄

—

251

—

C₂×C₄⋊C₄

—

251

—

C₂×C₄⋊C₄

—

251

—

C₂×C₄⋊C₄

C₂×D₂₅₁

yes

—

C₂×C₄²

C₂×D₂₅₁

yes

characteristic

#95

—

251

—

#95

—

251

—

#95

C₂×D₂₅₁

yes

—

C₂²×C₄

—

251

—

C₂²×C₄

—

251

—

C₂²×C₄

C₂²×D₂₅₁

yes

—

C₂²×C₄

C₂²×D₂₅₁

yes

—

C₂²×C₄

C₂²×D₂₅₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

1004

—

C₄⋊C₄

—

502

—

C₄⋊C₄

—

502

—

C₄⋊C₄

—

502

—

C₄⋊C₄

—

502

—

C₄²

D₁₀₀₄

yes

—

C₂³

C₂³×D₂₅₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

502

—

C₂×C₄

—

502

—

C₂×C₄

—

502

—

C₂×C₄

—

502

—

C₂×C₄

—

502

—

C₂×C₄

C₂×D₁₀₀₄

yes

—

C₂×C₄

C₂×D₁₀₀₄

yes

—

C₂²

C₂²×D₁₀₀₄

yes

characteristic

C₂²

#22548760

yes

characteristic

100

C₂²

#22548762

yes

—

101

C₂²

#22548763

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

1004

—

106

C₄

—

1004

—

107

C₂

#30632263

yes

—

108

C₂

#30632268

yes

characteristic

109

C₂

#30632437

yes

—

110

C₂

C₂₅₁⋊#564

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₅₁
commutator, socle, prime radical, cosocle	C₂³×C₂₅₁	C₂⁴
Fitting	#382×C₂₅₁	C₂
hypercenter	#382	D₂₅₁
hypocenter	C₂₅₁	#2822
LCS	G ▷ C₂³×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂³ ◁ #382

subgroup

G/H

center, Frattini

C₂³

C₂³×D₂₅₁

commutator, socle, prime radical, cosocle

C₂³×C₂₅₁

C₂⁴

Fitting

#382×C₂₅₁

C₂

D₂₅₁

C₂₅₁

G ▷ C₂³×C₂₅₁ ▷ C₂₅₁

C₁ ◁ C₂³ ◁ #382

N	H ≅ G/N	type
55. C₂₅₁	#2822	semidirect

H ≅ G/N

type

55. C₂₅₁

#2822

semidirect

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

subgroup	index
C₂×#22548707	2
#30631627	2
#30631619	2
#30631617	2
#382×C₂₅₁	2
#2822	251

subgroup

index

#382×C₂₅₁

251

Subgroups — #44558433

Conjugacy classes of subgroups

Complements

Maximal subgroups