Group #44551357 — subgroups

Subgroups — #44551357

Order 32128.

228 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 89 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	1	yes	maximal, characteristic
2	C₄×#16860794	C₂	1	1	yes	maximal, characteristic
3	#30631455	C₂	1	4	yes	maximal
4	#423×C₂₅₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#16860794	C₂²	1	2	yes	—
6	#22548707	—	2	2	no	—
7	#22548707	—	2	1	no	—
8	#22548707	—	2	1	no	—
9	C₂×C₄⋊Dic₂₅₁	C₂²	1	2	yes	—
10	C₂×Dic₂₅₁⋊C₄	C₄	1	4	yes	—
11	#22548626	—	2	1	no	—
12	C₄×C₂₅₁⋊C₈	—	2	1	no	—
13	C₂×M₁₆×C₂₅₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₂₅₁	—	2	2	no	—
16	C₂²×Dic₂₅₁	C₂×C₄	1	4	yes	—
17	#16860794	—	2	2	no	—
18	#16860794	Q₈	1	4	yes	—
19	C₂×C₂₅₁⋊C₈	—	2	2	no	—
20	C₄⋊Dic₂₅₁	—	2	2	no	—
21	C₄⋊Dic₂₅₁	—	2	2	no	—
22	Dic₂₅₁⋊C₄	—	2	4	no	—
23	Dic₂₅₁⋊C₄	—	2	4	no	—
24	C₂²×C₄×C₂₅₁	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
26	M₁₆×C₂₅₁	—	2	4	no	—
27	C₂×C₈×C₂₅₁	—	2	2	no	—
28	C₄²×C₂₅₁	D₄	1	2	yes	—
29	C₄²×C₂₅₁	D₄	1	1	yes	characteristic
30	C₄²×C₂₅₁	D₄	1	1	yes	characteristic
31	C₂×Dic₂₅₁	—	2	4	no	—
32	C₂×Dic₂₅₁	—	4	4	no	—
33	C₂₅₁⋊C₈	—	2	4	no	—
34	C₂³×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₂₅₁	—	4	2	no	—
36	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₂₅₁	C₂×Q₈	1	2	yes	—
41	C₂×C₄×C₂₅₁	#41	1	2	yes	—
42	C₈×C₂₅₁	—	4	2	no	—
43	C₂²×C₂₅₁	—	2	2	no	—
44	C₂²×C₂₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₂₅₁	C₄×Q₈	1	2	yes	—
46	C₄×C₂₅₁	—	2	2	no	—
47	C₄×C₂₅₁	—	2	2	no	—
48	C₄×C₂₅₁	C₂²⋊Q₈	1	2	yes	—
49	C₄×C₂₅₁	#124	1	1	yes	characteristic
50	C₄×C₂₅₁	C₄⋊Q₈	1	1	yes	characteristic
51	Dic₂₅₁	—	4	4	no	—
52	C₂×C₂₅₁	—	2	2	no	—
53	C₂×C₂₅₁	#391	1	1	yes	characteristic
54	C₂×C₂₅₁	#413	1	2	yes	—
55	C₂₅₁	#1998	1	1	yes	characteristic, complemented, Sylow
56	#1998	—	251	1	no	maximal, Sylow
57	C₂×#125	—	251	1	no	—
58	#423	D₂₅₁	1	1	yes	characteristic
59	C₄×M₁₆	—	251	1	no	—
60	#343	—	251	4	no	—
61	C₂×M₁₆	—	251	2	no	—
62	C₂×M₁₆	C₂×D₂₅₁	1	2	yes	—
63	#125	—	502	2	no	—
64	#125	—	502	1	no	—
65	#125	—	502	1	no	—
66	C₂×C₄⋊C₄	—	251	4	no	—
67	C₂×C₄⋊C₄	—	251	2	no	—
68	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
69	C₄⋊C₈	—	2	2	no	—
70	#97	—	502	1	no	—
71	C₄×C₈	—	502	1	no	—
72	C₂²×C₄	—	251	4	no	—
73	C₂²×C₄	C₄×D₂₅₁	1	2	yes	—
74	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
75	M₁₆	—	2	4	no	—
76	M₁₆	—	502	2	no	—
77	M₁₆	—	251	4	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	502	2	no	—
80	C₄⋊C₄	—	502	4	no	—
81	C₄⋊C₄	—	502	4	no	—
82	C₄⋊C₄	—	502	2	no	—
83	C₄⋊C₄	—	502	2	no	—
84	C₄²	C₂²⋊D₂₅₁	1	2	yes	—
85	C₄²	D₁₀₀₄	1	1	yes	characteristic
86	C₄²	D₁₀₀₄	1	1	yes	characteristic
87	C₂³	C₂×C₄×D₂₅₁	1	1	yes	characteristic
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	—	1004	4	no	—
90	C₂×C₄	—	502	4	no	—
91	C₂×C₄	(C₂×D₂₅₁)⋊C₄	1	2	yes	—
92	C₂×C₄	(C₂×D₂₅₁)⋊C₄	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
94	C₂×C₄	C₂×D₁₀₀₄	1	1	yes	characteristic
95	C₂×C₄	Q₈×D₂₅₁	1	2	yes	—
96	C₂×C₄	#16860809	1	2	yes	—
97	C₈	—	4	2	no	—
98	C₈	—	502	4	no	—
99	C₂²	—	2	2	no	—
100	C₂²	C₂×(C₂×D₂₅₁)⋊C₄	1	1	yes	characteristic
101	C₂²	#22548680	1	2	yes	—
102	C₄	—	2	2	no	—
103	C₄	—	2	2	no	—
104	C₄	—	1004	4	no	—
105	C₄	#22548695	1	2	yes	—
106	C₄	#22548702	1	1	yes	characteristic
107	C₄	#22548713	1	1	yes	characteristic
108	C₂	—	2	2	no	—
109	C₂	#30631592	1	1	yes	characteristic
110	C₂	#30631669	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22548707

C₂

yes

maximal, characteristic

C₄×#16860794

C₂

yes

maximal, characteristic

#30631455

C₂

yes

maximal

#423×C₂₅₁

C₂

yes

maximal, characteristic

C₂×#16860794

C₂²

yes

—

#22548707

—

#22548707

—

#22548707

—

C₂×C₄⋊Dic₂₅₁

C₂²

yes

—

C₂×Dic₂₅₁⋊C₄

C₄

yes

—

#22548626

—

C₄×C₂₅₁⋊C₈

—

C₂×M₁₆×C₂₅₁

C₂²

yes

—

C₂×C₄²×C₂₅₁

C₂²

yes

characteristic

C₄⋊C₈×C₂₅₁

—

C₂²×Dic₂₅₁

C₂×C₄

yes

—

#16860794

—

#16860794

Q₈

yes

—

C₂×C₂₅₁⋊C₈

—

C₄⋊Dic₂₅₁

—

C₄⋊Dic₂₅₁

—

Dic₂₅₁⋊C₄

—

Dic₂₅₁⋊C₄

—

C₂²×C₄×C₂₅₁

C₂×C₄

yes

—

C₂²×C₄×C₂₅₁

C₂³

yes

characteristic

M₁₆×C₂₅₁

—

C₂×C₈×C₂₅₁

—

C₄²×C₂₅₁

D₄

yes

—

C₄²×C₂₅₁

D₄

yes

characteristic

C₄²×C₂₅₁

D₄

yes

characteristic

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂₅₁⋊C₈

—

C₂³×C₂₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₅₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂×Q₈

yes

—

C₂×C₄×C₂₅₁

#41

yes

—

C₈×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₅₁

C₄×Q₈

yes

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

C₂²⋊Q₈

yes

—

C₄×C₂₅₁

#124

yes

characteristic

C₄×C₂₅₁

C₄⋊Q₈

yes

characteristic

Dic₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

#391

yes

characteristic

C₂×C₂₅₁

#413

yes

—

C₂₅₁

#1998

yes

characteristic, complemented, Sylow

#1998

—

251

maximal, Sylow

C₂×#125

—

251

—

#423

D₂₅₁

yes

characteristic

C₄×M₁₆

—

251

—

#343

—

251

—

C₂×M₁₆

—

251

—

C₂×M₁₆

C₂×D₂₅₁

yes

—

#125

—

502

—

#125

—

502

—

#125

—

502

—

C₂×C₄⋊C₄

—

251

—

C₂×C₄⋊C₄

—

251

—

C₂×C₄²

C₂×D₂₅₁

yes

characteristic

C₄⋊C₈

—

#97

—

502

—

C₄×C₈

—

502

—

C₂²×C₄

—

251

—

C₂²×C₄

C₄×D₂₅₁

yes

—

C₂²×C₄

C₂²×D₂₅₁

yes

characteristic

M₁₆

—

M₁₆

—

502

—

M₁₆

—

251

—

C₂×C₈

—

C₂×C₈

—

502

—

C₄⋊C₄

—

502

—

C₄⋊C₄

—

502

—

C₄⋊C₄

—

502

—

C₄⋊C₄

—

502

—

C₄²

C₂²⋊D₂₅₁

yes

—

C₄²

D₁₀₀₄

yes

characteristic

C₄²

D₁₀₀₄

yes

characteristic

C₂³

C₂×C₄×D₂₅₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

1004

—

C₂×C₄

—

502

—

C₂×C₄

(C₂×D₂₅₁)⋊C₄

yes

—

C₂×C₄

(C₂×D₂₅₁)⋊C₄

yes

—

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

characteristic

C₂×C₄

C₂×D₁₀₀₄

yes

characteristic

C₂×C₄

Q₈×D₂₅₁

yes

—

C₂×C₄

#16860809

yes

—

C₈

—

C₈

—

502

—

C₂²

—

100

C₂²

C₂×(C₂×D₂₅₁)⋊C₄

yes

characteristic

101

C₂²

#22548680

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

1004

—

105

C₄

#22548695

yes

—

106

C₄

#22548702

yes

characteristic

107

C₄

#22548713

yes

characteristic

108

C₂

—

109

C₂

#30631592

yes

characteristic

110

C₂

#30631669

yes

—

	subgroup	G/H
center	C₂²	C₂×(C₂×D₂₅₁)⋊C₄
commutator, prime radical	C₂³×C₂₅₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₅₁
Fitting	#423×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₂₅₁	C₂³
hypercenter	#423	D₂₅₁
hypocenter	C₂₅₁	#1998
LCS	G ▷ C₂³×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂×(C₂×D₂₅₁)⋊C₄

commutator, prime radical

C₂³×C₂₅₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₅₁

Fitting

#423×C₂₅₁

C₂

socle

C₂²×C₂₅₁

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₂₅₁

C₂³

D₂₅₁

C₂₅₁

G ▷ C₂³×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
55. C₂₅₁	#1998	semidirect

H ≅ G/N

type

55. C₂₅₁

#1998

semidirect

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

subgroup	index
C₂×#22548707	2
C₄×#16860794	2
#30631455	2
#423×C₂₅₁	2
#1998	251

subgroup

index

#423×C₂₅₁

251

Subgroups — #44551357

Conjugacy classes of subgroups

Complements

Maximal subgroups