Group #44550498 — subgroups

Subgroups — #44550498

Order 32128.

224 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 101 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×#22548626	C₂	1	1	yes	maximal, characteristic
2	#30631557	C₂	1	1	yes	maximal, characteristic
3	#30631449	C₂	1	4	yes	maximal
4	#382×C₂₅₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₂₅₁⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₂₅₁	—	2	2	no	—
7	C₂×C₄⋊Dic₂₅₁	C₂²	1	2	yes	—
8	#22548626	C₄	1	4	yes	complemented
9	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	2	yes	—
10	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
11	#95×C₂₅₁	—	2	2	no	—
12	C₂²×Dic₂₅₁	—	2	2	no	—
13	C₂×C₂₅₁⋊C₈	—	2	2	no	—
14	C₂×C₂₅₁⋊C₈	—	2	2	no	—
15	C₂×C₂₅₁⋊C₈	C₂×C₄	1	4	yes	—
16	C₄⋊Dic₂₅₁	—	2	4	no	—
17	C₄⋊Dic₂₅₁	—	4	4	no	—
18	C₂²×C₄×C₂₅₁	—	2	2	no	—
19	C₂²×C₄×C₂₅₁	D₄	1	2	yes	—
20	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
21	C₄⋊C₄×C₂₅₁	—	2	4	no	—
22	C₄²×C₂₅₁	C₂×C₄	1	2	yes	—
23	C₄²×C₂₅₁	Q₈	1	2	yes	—
24	C₂×Dic₂₅₁	—	4	2	no	—
25	C₂×Dic₂₅₁	—	4	4	no	—
26	C₂₅₁⋊C₈	—	2	4	no	—
27	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
28	C₂×C₄×C₂₅₁	—	2	2	no	—
29	C₂×C₄×C₂₅₁	—	2	2	no	—
30	C₂×C₄×C₂₅₁	—	4	2	no	—
31	C₂×C₄×C₂₅₁	—	4	2	no	—
32	C₂×C₄×C₂₅₁	—	4	2	no	—
33	C₂×C₄×C₂₅₁	C₄⋊C₄	1	4	yes	—
34	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₂₅₁	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₄×C₂₅₁	#41	1	2	yes	—
37	C₂×C₄×C₂₅₁	#41	1	2	yes	—
38	C₂²×C₂₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₂₅₁	#124	1	1	yes	characteristic
40	C₂²×C₂₅₁	#124	1	1	yes	characteristic
41	C₂²×C₂₅₁	AGL(1,ℤ/8)	1	2	yes	—
42	C₂²×C₂₅₁	#137	1	2	yes	—
43	C₄×C₂₅₁	—	2	4	no	—
44	C₄×C₂₅₁	—	8	2	no	—
45	C₄×C₂₅₁	#117	1	2	yes	—
46	C₄×C₂₅₁	#125	1	2	yes	—
47	Dic₂₅₁	—	8	2	no	—
48	C₂×C₂₅₁	#382	1	1	yes	characteristic
49	C₂×C₂₅₁	#428	1	2	yes	—
50	C₂×C₂₅₁	#489	1	2	yes	—
51	C₂×C₂₅₁	#490	1	1	yes	characteristic
52	C₂×C₂₅₁	#491	1	1	yes	characteristic
53	C₂₅₁	#1836	1	1	yes	characteristic, complemented, Sylow
54	#1836	—	251	1	no	maximal, Sylow
55	C₂×#97	—	251	1	no	—
56	#384	—	251	1	no	—
57	#382	D₂₅₁	1	1	yes	characteristic
58	#340	—	251	4	no	—
59	C₂²×C₈	—	251	2	no	—
60	C₂×C₄⋊C₄	—	502	2	no	—
61	C₂×C₄⋊C₄	—	251	2	no	—
62	C₂×C₄⋊C₄	C₂×D₂₅₁	1	2	yes	—
63	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
64	#97	—	251	4	no	—
65	#95	—	2	2	no	—
66	C₂²×C₄	—	2	2	no	—
67	C₂²×C₄	—	502	2	no	—
68	C₂²×C₄	C₂²⋊D₂₅₁	1	2	yes	—
69	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
70	C₂×C₈	—	502	2	no	—
71	C₂×C₈	—	502	2	no	—
72	C₂×C₈	—	251	4	no	—
73	C₄⋊C₄	—	2	4	no	—
74	C₄⋊C₄	—	1004	4	no	—
75	C₄⋊C₄	—	502	4	no	—
76	C₄²	C₄×D₂₅₁	1	2	yes	—
77	C₄²	Dic₅₀₂	1	2	yes	—
78	C₂³	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	4	2	no	—
84	C₂×C₄	—	1004	4	no	—
85	C₂×C₄	—	1004	2	no	—
86	C₂×C₄	Dic₂₅₁⋊C₄	1	4	yes	—
87	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
88	C₂×C₄	C₂×Dic₅₀₂	1	1	yes	characteristic
89	C₂×C₄	#16860809	1	2	yes	—
90	C₂×C₄	#16860811	1	2	yes	—
91	C₈	—	502	4	no	—
92	C₂²	C₂×Dic₂₅₁⋊C₄	1	1	yes	characteristic
93	C₂²	#22548702	1	1	yes	characteristic
94	C₂²	#22548704	1	1	yes	characteristic
95	C₂²	#22548740	1	2	yes	—
96	C₂²	#22548745	1	2	yes	—
97	C₄	—	2	4	no	—
98	C₄	—	8	2	no	—
99	C₄	—	2008	2	no	—
100	C₄	#22548673	1	2	yes	—
101	C₄	#22548706	1	2	yes	—
102	C₂	#30631552	1	1	yes	characteristic
103	C₂	#30631730	1	2	yes	—
104	C₂	#30632046	1	2	yes	—
105	C₂	#30632053	1	1	yes	characteristic
106	C₂	#30632057	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22548626

C₂

yes

maximal, characteristic

#30631557

C₂

yes

maximal, characteristic

#30631449

C₂

yes

maximal

#382×C₂₅₁

C₂

yes

maximal, characteristic

C₂²×C₂₅₁⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₂₅₁

—

C₂×C₄⋊Dic₂₅₁

C₂²

yes

—

#22548626

C₄

yes

complemented

C₂×C₄⋊C₄×C₂₅₁

C₂²

yes

—

C₂×C₄²×C₂₅₁

C₂²

yes

characteristic

#95×C₂₅₁

—

C₂²×Dic₂₅₁

—

C₂×C₂₅₁⋊C₈

—

C₂×C₂₅₁⋊C₈

—

C₂×C₂₅₁⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₂₅₁

—

C₄⋊Dic₂₅₁

—

C₂²×C₄×C₂₅₁

—

C₂²×C₄×C₂₅₁

D₄

yes

—

C₂²×C₄×C₂₅₁

C₂³

yes

characteristic

C₄⋊C₄×C₂₅₁

—

C₄²×C₂₅₁

C₂×C₄

yes

—

C₄²×C₂₅₁

Q₈

yes

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂₅₁⋊C₈

—

C₂³×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₄⋊C₄

yes

—

C₂×C₄×C₂₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₅₁

#41

yes

—

C₂×C₄×C₂₅₁

#41

yes

—

C₂²×C₂₅₁

C₂×C₄⋊C₄

yes

C₂²×C₂₅₁

yes

C₂²×C₂₅₁

yes

C₂²×C₂₅₁

yes

—

C₂²×C₂₅₁

#137

yes

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

#117

yes

—

C₄×C₂₅₁

#125

yes

—

Dic₂₅₁

—

C₂×C₂₅₁

#382

yes

characteristic

C₂×C₂₅₁

#428

yes

—

C₂×C₂₅₁

#489

yes

—

C₂×C₂₅₁

yes

C₂×C₂₅₁

yes

C₂₅₁

yes

characteristic, complemented, Sylow

#1836

—

251

maximal, Sylow

C₂×#97

—

251

—

#384

—

251

—

#382

D₂₅₁

yes

characteristic

#340

—

251

—

C₂²×C₈

—

251

—

C₂×C₄⋊C₄

—

502

—

C₂×C₄⋊C₄

—

251

—

C₂×C₄⋊C₄

C₂×D₂₅₁

yes

—

C₂×C₄²

C₂×D₂₅₁

yes

characteristic

#97

—

251

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

502

—

C₂²×C₄

C₂²⋊D₂₅₁

yes

—

C₂²×C₄

C₂²×D₂₅₁

yes

characteristic

C₂×C₈

—

502

—

C₂×C₈

—

502

—

C₂×C₈

—

251

—

C₄⋊C₄

—

C₄⋊C₄

—

1004

—

C₄⋊C₄

—

502

—

C₄²

C₄×D₂₅₁

yes

—

C₄²

Dic₅₀₂

yes

—

C₂³

C₂×C₂²⋊D₂₅₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

1004

—

C₂×C₄

—

1004

—

C₂×C₄

Dic₂₅₁⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₂₅₁

yes

characteristic

C₂×C₄

C₂×Dic₅₀₂

yes

characteristic

C₂×C₄

#16860809

yes

—

C₂×C₄

#16860811

yes

—

C₈

—

502

—

C₂²

C₂×Dic₂₅₁⋊C₄

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#22548745

yes

—

C₄

—

C₄

—

C₄

—

2008

—

100

C₄

#22548673

yes

—

101

C₄

#22548706

yes

—

102

C₂

#30631552

yes

characteristic

103

C₂

#30631730

yes

—

104

C₂

#30632046

yes

—

105

C₂

#30632053

yes

characteristic

106

C₂

#30632057

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₅₁
commutator	C₂×C₄×C₂₅₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₅₁
Fitting	#382×C₂₅₁	C₂
socle, prime radical	C₂³×C₂₅₁	C₂×D₄
cosocle	C₂²×C₄×C₂₅₁	C₂³
hypercenter	#382	D₂₅₁
hypocenter	C₂₅₁	#1836
LCS	G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₅₁

commutator

C₂×C₄×C₂₅₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₅₁

Fitting

#382×C₂₅₁

C₂

socle, prime radical

C₂³×C₂₅₁

C₂×D₄

cosocle

C₂²×C₄×C₂₅₁

C₂³

D₂₅₁

C₂₅₁

G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

N	H ≅ G/N	type
8. #22548626	C₄	semidirect
53. C₂₅₁	#1836	semidirect

H ≅ G/N

type

8. #22548626

C₄

semidirect

53. C₂₅₁

#1836

semidirect

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

subgroup	index
C₂×#22548626	2
#30631557	2
#30631449	2
#382×C₂₅₁	2
#1836	251

subgroup

index

#382×C₂₅₁

251

Subgroups — #44550498

Conjugacy classes of subgroups

Complements

Maximal subgroups