Group #44550225 — subgroups

Subgroups — #44550225

Order 32128.

260 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 129 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×Dic₂₅₁⋊C₈	C₂	1	2	yes	maximal
2	C₂×C₄×C₂₅₁⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#30631533	C₂	1	1	yes	maximal, characteristic
4	#30631441	C₂	1	2	yes	maximal
5	C₂×#97×C₂₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₂₅₁⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
9	Dic₂₅₁⋊C₈	C₄	1	8	yes	complemented
10	C₄×C₂₅₁⋊C₈	—	2	2	no	—
11	C₂₅₁⋊#95	C₄	1	4	yes	—
12	C₂²×C₈×C₂₅₁	C₂²	1	2	yes	—
13	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
14	#97×C₂₅₁	—	2	2	no	—
15	C₂²×Dic₂₅₁	C₂×C₄	1	2	yes	—
16	C₂²×Dic₂₅₁	C₂×C₄	1	2	yes	—
17	C₂×C₂₅₁⋊C₈	—	2	2	no	—
18	C₂×C₂₅₁⋊C₈	—	2	2	no	—
19	C₂×C₂₅₁⋊C₈	C₂×C₄	1	4	yes	—
20	C₄×Dic₂₅₁	—	2	2	no	—
21	C₄×Dic₂₅₁	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₂₅₁	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
24	C₂×C₈×C₂₅₁	—	2	2	no	—
25	C₂×C₈×C₂₅₁	—	2	2	no	—
26	C₂×C₈×C₂₅₁	C₂×C₄	1	4	yes	—
27	C₄²×C₂₅₁	D₄	1	2	yes	—
28	C₄²×C₂₅₁	Q₈	1	2	yes	—
29	C₂×Dic₂₅₁	—	2	4	no	—
30	C₂×Dic₂₅₁	—	2	2	no	—
31	C₂×Dic₂₅₁	—	2	4	no	—
32	C₂×Dic₂₅₁	C₄²	1	4	yes	—
33	C₂₅₁⋊C₈	—	2	4	no	—
34	C₂³×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₂₅₁	—	2	4	no	—
36	C₂×C₄×C₂₅₁	C₄⋊C₄	1	4	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×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₂₅₁	#41	1	2	yes	—
41	C₈×C₂₅₁	—	2	4	no	—
42	C₂²×C₂₅₁	C₂×C₄²	1	1	yes	characteristic
43	C₂²×C₂₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₂₅₁	#117	1	1	yes	characteristic
45	C₂²×C₂₅₁	#131	1	4	yes	—
46	C₄×C₂₅₁	—	2	4	no	—
47	C₄×C₂₅₁	C₄×D₄	1	2	yes	—
48	C₄×C₂₅₁	C₄×Q₈	1	2	yes	—
49	Dic₂₅₁	—	2	4	no	—
50	Dic₂₅₁	—	4	2	no	—
51	C₂×C₂₅₁	C₄×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₂₅₁	#405	1	2	yes	—
53	C₂×C₂₅₁	#424	1	2	yes	—
54	C₂×C₂₅₁	#433	1	2	yes	—
55	C₂₅₁	#1768	1	1	yes	characteristic, complemented, Sylow
56	#1768	—	251	1	no	maximal, Sylow
57	C₂×C₄⋊C₈	—	251	2	no	—
58	C₂×#97	D₂₅₁	1	1	yes	characteristic
59	C₂×C₄×C₈	—	251	1	no	—
60	#376	—	251	1	no	—
61	#336	—	251	2	no	—
62	C₂²×C₈	—	251	2	no	—
63	C₂²×C₈	C₂×D₂₅₁	1	2	yes	—
64	C₂×C₄²	—	251	1	no	—
65	C₂×C₄²	—	251	1	no	—
66	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
67	C₄⋊C₈	—	251	8	no	—
68	#97	—	2	2	no	—
69	C₄×C₈	—	502	2	no	—
70	#95	—	251	4	no	—
71	C₂²×C₄	—	251	2	no	—
72	C₂²×C₄	—	251	2	no	—
73	C₂²×C₄	C₄×D₂₅₁	1	2	yes	—
74	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	502	2	no	—
78	C₂×C₈	—	502	2	no	—
79	C₂×C₈	—	251	4	no	—
80	C₂×C₈	C₄×D₂₅₁	1	4	yes	—
81	C₄²	—	502	2	no	—
82	C₄²	—	251	4	no	—
83	C₄²	C₂²⋊D₂₅₁	1	2	yes	—
84	C₄²	Dic₅₀₂	1	2	yes	—
85	C₂³	C₂×C₄×D₂₅₁	1	1	yes	characteristic
86	C₂×C₄	—	2	4	no	—
87	C₂×C₄	—	502	2	no	—
88	C₂×C₄	—	502	4	no	—
89	C₂×C₄	—	502	4	no	—
90	C₂×C₄	—	251	4	no	—
91	C₂×C₄	Dic₂₅₁⋊C₄	1	4	yes	—
92	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
94	C₂×C₄	C₂×Dic₅₀₂	1	1	yes	characteristic
95	C₂×C₄	#16860810	1	2	yes	—
96	C₈	—	2	4	no	—
97	C₈	—	502	4	no	—
98	C₂²	C₄²×D₂₅₁	1	1	yes	characteristic
99	C₂²	C₂×Dic₂₅₁⋊C₄	1	1	yes	characteristic
100	C₂²	#22548672	1	1	yes	characteristic
101	C₂²	#22548723	1	4	yes	—
102	C₄	—	2	4	no	—
103	C₄	—	1004	2	no	—
104	C₄	—	502	4	no	—
105	C₄	C₄×C₂²⋊D₂₅₁	1	2	yes	—
106	C₄	C₄×Dic₅₀₂	1	2	yes	—
107	C₂	C₄×Dic₂₅₁⋊C₄	1	1	yes	characteristic
108	C₂	#30631644	1	2	yes	—
109	C₂	#30631715	1	2	yes	—
110	C₂	#30631749	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₂₅₁⋊C₈

C₂

yes

maximal

C₂×C₄×C₂₅₁⋊C₈

C₂

yes

maximal, characteristic

#30631533

C₂

yes

maximal, characteristic

#30631441

C₂

yes

maximal

C₂×#97×C₂₅₁

C₂

yes

maximal, characteristic

C₂²×C₂₅₁⋊C₈

C₂²

yes

—

C₂×C₄×Dic₂₅₁

C₂²

yes

characteristic

C₂×C₄×Dic₂₅₁

C₂²

yes

characteristic

Dic₂₅₁⋊C₈

C₄

yes

complemented

C₄×C₂₅₁⋊C₈

—

C₂₅₁⋊#95

C₄

yes

—

C₂²×C₈×C₂₅₁

C₂²

yes

—

C₂×C₄²×C₂₅₁

C₂²

yes

characteristic

#97×C₂₅₁

—

C₂²×Dic₂₅₁

C₂×C₄

yes

—

C₂²×Dic₂₅₁

C₂×C₄

yes

—

C₂×C₂₅₁⋊C₈

—

C₂×C₂₅₁⋊C₈

—

C₂×C₂₅₁⋊C₈

C₂×C₄

yes

—

C₄×Dic₂₅₁

—

C₄×Dic₂₅₁

C₂×C₄

yes

—

C₂²×C₄×C₂₅₁

C₂×C₄

yes

—

C₂²×C₄×C₂₅₁

C₂³

yes

characteristic

C₂×C₈×C₂₅₁

—

C₂×C₈×C₂₅₁

—

C₂×C₈×C₂₅₁

C₂×C₄

yes

—

C₄²×C₂₅₁

D₄

yes

—

C₄²×C₂₅₁

Q₈

yes

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

C₄²

yes

—

C₂₅₁⋊C₈

—

C₂³×C₂₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₄⋊C₄

yes

—

C₂×C₄×C₂₅₁

C₂²×C₄

yes

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₅₁

#41

yes

—

C₈×C₂₅₁

—

C₂²×C₂₅₁

C₂×C₄²

yes

characteristic

C₂²×C₂₅₁

C₂×C₄⋊C₄

yes

C₂²×C₂₅₁

yes

C₂²×C₂₅₁

yes

—

C₄×C₂₅₁

—

C₄×C₂₅₁

C₄×D₄

yes

—

C₄×C₂₅₁

C₄×Q₈

yes

—

Dic₂₅₁

—

Dic₂₅₁

—

C₂×C₂₅₁

C₄×C₄⋊C₄

yes

characteristic

C₂×C₂₅₁

#405

yes

—

C₂×C₂₅₁

#424

yes

—

C₂×C₂₅₁

#433

yes

—

C₂₅₁

#1768

yes

characteristic, complemented, Sylow

#1768

—

251

maximal, Sylow

C₂×C₄⋊C₈

—

251

—

C₂×#97

D₂₅₁

yes

characteristic

C₂×C₄×C₈

—

251

—

#376

—

251

—

#336

—

251

—

C₂²×C₈

—

251

—

C₂²×C₈

C₂×D₂₅₁

yes

—

C₂×C₄²

—

251

—

C₂×C₄²

—

251

—

C₂×C₄²

C₂×D₂₅₁

yes

characteristic

C₄⋊C₈

—

251

—

#97

—

C₄×C₈

—

502

—

#95

—

251

—

C₂²×C₄

—

251

—

C₂²×C₄

—

251

—

C₂²×C₄

C₄×D₂₅₁

yes

—

C₂²×C₄

C₂²×D₂₅₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

502

—

C₂×C₈

—

502

—

C₂×C₈

—

251

—

C₂×C₈

C₄×D₂₅₁

yes

—

C₄²

—

502

—

C₄²

—

251

—

C₄²

C₂²⋊D₂₅₁

yes

—

C₄²

Dic₅₀₂

yes

—

C₂³

C₂×C₄×D₂₅₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

502

—

C₂×C₄

—

502

—

C₂×C₄

—

502

—

C₂×C₄

—

251

—

C₂×C₄

Dic₂₅₁⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₂₅₁

yes

—

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

characteristic

C₂×C₄

C₂×Dic₅₀₂

yes

characteristic

C₂×C₄

#16860810

yes

—

C₈

—

C₈

—

502

—

C₂²

C₄²×D₂₅₁

yes

characteristic

C₂²

C₂×Dic₂₅₁⋊C₄

yes

characteristic

100

C₂²

#22548672

yes

characteristic

101

C₂²

#22548723

yes

—

102

C₄

—

103

C₄

—

1004

—

104

C₄

—

502

—

105

C₄

C₄×C₂²⋊D₂₅₁

yes

—

106

C₄

C₄×Dic₅₀₂

yes

—

107

C₂

C₄×Dic₂₅₁⋊C₄

yes

characteristic

108

C₂

#30631644

yes

—

109

C₂

#30631715

yes

—

110

C₂

#30631749

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₅₁
commutator	C₂²×C₂₅₁	C₂×C₄²
Fitting	C₂×#97×C₂₅₁	C₂
socle, prime radical	C₂³×C₂₅₁	C₂²×C₄
cosocle	C₂²×C₄×C₂₅₁	C₂³
hypercenter	C₂×#97	D₂₅₁
hypocenter	C₂₅₁	#1768
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₅₁

commutator

C₂²×C₂₅₁

C₂×C₄²

Fitting

C₂×#97×C₂₅₁

C₂

socle, prime radical

C₂³×C₂₅₁

C₂²×C₄

cosocle

C₂²×C₄×C₂₅₁

C₂³

hypercenter

C₂×#97

D₂₅₁

C₂₅₁

G ▷ C₂²×C₂₅₁ ▷ C₂₅₁

C₁ ◁ C₂²×C₄ ◁ C₂×#97

N	H ≅ G/N	type
9. Dic₂₅₁⋊C₈	C₄	semidirect
55. C₂₅₁	#1768	semidirect

H ≅ G/N

type

9. Dic₂₅₁⋊C₈

C₄

semidirect

55. C₂₅₁

#1768

semidirect

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

subgroup	index
C₂×Dic₂₅₁⋊C₈	2
C₂×C₄×C₂₅₁⋊C₈	2
#30631533	2
#30631441	2
C₂×#97×C₂₅₁	2
#1768	251

subgroup

index

C₂×Dic₂₅₁⋊C₈

C₂×C₄×C₂₅₁⋊C₈

#30631533

#30631441

C₂×#97×C₂₅₁

#1768

251

Subgroups — #44550225

Conjugacy classes of subgroups

Complements

Maximal subgroups