Group #45664221 — subgroups

Subgroups — #45664221

Order 32896.

228 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 109 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×#22551527	C₂	1	2	yes	maximal
2	M₁₆×Dic₂₅₇	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₂₅₇	C₂	1	1	yes	maximal, characteristic
4	#30656502	C₂	1	1	yes	maximal, characteristic
5	#341×C₂₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#16861602	C₂²	1	2	yes	—
7	C₂²×#14523339	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₂₅₇	C₂²	1	1	yes	characteristic
9	#22551527	C₄	1	8	yes	complemented
10	#22551502	—	2	2	no	—
11	C₈×Dic₂₅₇	—	2	2	no	—
12	C₈×Dic₂₅₇	C₄	1	4	yes	—
13	C₂×M₁₆×C₂₅₇	C₂²	1	2	yes	—
14	C₂²×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
15	C₂²×Dic₂₅₇	Q₈	1	2	yes	—
16	#16861602	—	2	2	no	—
17	#16861602	C₂×C₄	1	4	yes	—
18	C₂×#14523339	—	2	2	no	—
19	C₂×#14523339	C₂×C₄	1	2	yes	—
20	C₂×#14523339	C₂×C₄	1	4	yes	—
21	C₄×Dic₂₅₇	C₂×C₄	1	2	yes	—
22	C₄×Dic₂₅₇	D₄	1	1	yes	characteristic
23	C₄×Dic₂₅₇	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
25	M₁₆×C₂₅₇	—	2	2	no	—
26	M₁₆×C₂₅₇	C₂×C₄	1	4	yes	—
27	C₂×C₈×C₂₅₇	—	2	2	no	—
28	C₂×C₈×C₂₅₇	—	2	2	no	—
29	C₂×C₈×C₂₅₇	C₂×C₄	1	2	yes	—
30	C₂×Dic₂₅₇	—	2	4	no	—
31	C₂×Dic₂₅₇	C₄⋊C₄	1	2	yes	—
32	C₂×Dic₂₅₇	C₄⋊C₄	1	2	yes	—
33	#14523339	—	2	4	no	—
34	#14523339	C₄²	1	4	yes	—
35	C₂³×C₂₅₇	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₂₅₇	C₂²×C₄	1	2	yes	—
38	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₂₅₇	#41	1	2	yes	—
40	C₈×C₂₅₇	—	2	2	no	—
41	C₈×C₂₅₇	—	2	4	no	—
42	C₂²×C₂₅₇	—	2	2	no	—
43	C₂²×C₂₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₂₅₇	C₄×Q₈	1	2	yes	—
45	C₄×C₂₅₇	C₂×C₄²	1	1	yes	characteristic
46	C₄×C₂₅₇	#117	1	1	yes	characteristic
47	C₄×C₂₅₇	C₄×D₄	1	2	yes	—
48	Dic₂₅₇	—	2	2	no	—
49	Dic₂₅₇	#108	1	4	yes	complemented
50	C₂×C₂₅₇	—	2	2	no	—
51	C₂×C₂₅₇	C₄×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₂₅₇	C₂×#108	1	2	yes	—
53	C₂₅₇	C₄×#108	1	1	yes	characteristic, complemented, Sylow
54	C₄×#108	—	257	1	no	maximal, Sylow
55	C₂×#108	—	257	2	no	—
56	C₄×M₁₆	—	257	2	no	—
57	C₂×C₄×C₈	—	257	1	no	—
58	#341	—	257	1	no	—
59	#341	D₂₅₇	1	1	yes	characteristic
60	C₂×M₁₆	—	257	2	no	—
61	C₂×M₁₆	C₂×D₂₅₇	1	2	yes	—
62	C₂²×C₈	—	257	1	no	—
63	C₂²×C₈	C₂×D₂₅₇	1	1	yes	characteristic
64	C₂×C₄²	—	257	1	no	—
65	#108	—	257	8	no	—
66	#97	—	514	2	no	—
67	C₄×C₈	—	257	4	no	—
68	C₄×C₈	—	514	2	no	—
69	C₂²×C₄	—	257	2	no	—
70	C₂²×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
71	M₁₆	—	257	4	no	—
72	M₁₆	—	2	2	no	—
73	M₁₆	—	514	2	no	—
74	M₁₆	C₄×D₂₅₇	1	4	yes	—
75	C₂×C₈	—	257	2	no	—
76	C₂×C₈	—	257	4	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	514	2	no	—
80	C₂×C₈	C₄×D₂₅₇	1	2	yes	—
81	C₄²	—	257	2	no	—
82	C₄²	—	257	1	no	—
83	C₄²	—	257	1	no	—
84	C₂³	Q₈×D₂₅₇	1	1	yes	characteristic
85	C₂×C₄	—	257	2	no	—
86	C₂×C₄	—	257	2	no	—
87	C₂×C₄	—	514	4	no	—
88	C₂×C₄	C₂×C₄×D₂₅₇	1	2	yes	—
89	C₂×C₄	C₂×C₄×D₂₅₇	1	1	yes	characteristic
90	C₂×C₄	D₄×D₂₅₇	1	1	yes	characteristic
91	C₂×C₄	#16861617	1	2	yes	—
92	C₈	—	257	4	no	—
93	C₈	—	2	4	no	—
94	C₈	—	2	2	no	—
95	C₈	—	514	4	no	—
96	C₂²	—	2	2	no	—
97	C₂²	C₄⋊C₄×D₂₅₇	1	1	yes	characteristic
98	C₂²	#22551557	1	2	yes	—
99	C₄	—	257	4	no	—
100	C₄	—	514	2	no	—
101	C₄	C₄²×D₂₅₇	1	1	yes	characteristic
102	C₄	#22551548	1	1	yes	characteristic
103	C₄	#22551552	1	2	yes	—
104	C₂	—	2	2	no	—
105	C₂	#30656588	1	1	yes	characteristic
106	C₂	#108×D₂₅₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22551527

C₂

yes

maximal

M₁₆×Dic₂₅₇

C₂

yes

maximal

C₂×C₈×Dic₂₅₇

C₂

yes

maximal, characteristic

#30656502

C₂

yes

maximal, characteristic

#341×C₂₅₇

C₂

yes

maximal, characteristic

C₂×#16861602

C₂²

yes

—

C₂²×#14523339

C₂²

yes

characteristic

C₂×C₄×Dic₂₅₇

C₂²

yes

characteristic

#22551527

C₄

yes

complemented

#22551502

—

C₈×Dic₂₅₇

—

C₈×Dic₂₅₇

C₄

yes

—

C₂×M₁₆×C₂₅₇

C₂²

yes

—

C₂²×C₈×C₂₅₇

C₂²

yes

characteristic

C₂²×Dic₂₅₇

Q₈

yes

—

#16861602

—

#16861602

C₂×C₄

yes

—

C₂×#14523339

—

C₂×#14523339

C₂×C₄

yes

—

C₂×#14523339

C₂×C₄

yes

—

C₄×Dic₂₅₇

C₂×C₄

yes

—

C₄×Dic₂₅₇

D₄

yes

characteristic

C₄×Dic₂₅₇

D₄

yes

characteristic

C₂²×C₄×C₂₅₇

C₂³

yes

characteristic

M₁₆×C₂₅₇

—

M₁₆×C₂₅₇

C₂×C₄

yes

—

C₂×C₈×C₂₅₇

—

C₂×C₈×C₂₅₇

—

C₂×C₈×C₂₅₇

C₂×C₄

yes

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

C₄⋊C₄

yes

—

C₂×Dic₂₅₇

C₄⋊C₄

yes

—

#14523339

—

#14523339

C₄²

yes

—

C₂³×C₂₅₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₇

C₂²×C₄

yes

—

C₂×C₄×C₂₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₇

#41

yes

—

C₈×C₂₅₇

—

C₈×C₂₅₇

—

C₂²×C₂₅₇

—

C₂²×C₂₅₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₅₇

C₄×Q₈

yes

—

C₄×C₂₅₇

C₂×C₄²

yes

characteristic

C₄×C₂₅₇

#117

yes

characteristic

C₄×C₂₅₇

C₄×D₄

yes

—

Dic₂₅₇

—

Dic₂₅₇

#108

yes

complemented

C₂×C₂₅₇

—

C₂×C₂₅₇

C₄×C₄⋊C₄

yes

characteristic

C₂×C₂₅₇

C₂×#108

yes

—

C₂₅₇

C₄×#108

yes

characteristic, complemented, Sylow

C₄×#108

—

257

maximal, Sylow

C₂×#108

—

257

—

C₄×M₁₆

—

257

—

C₂×C₄×C₈

—

257

—

#341

—

257

—

#341

D₂₅₇

yes

characteristic

C₂×M₁₆

—

257

—

C₂×M₁₆

C₂×D₂₅₇

yes

—

C₂²×C₈

—

257

—

C₂²×C₈

C₂×D₂₅₇

yes

characteristic

C₂×C₄²

—

257

—

#108

—

257

—

#97

—

514

—

C₄×C₈

—

257

—

C₄×C₈

—

514

—

C₂²×C₄

—

257

—

C₂²×C₄

C₂²×D₂₅₇

yes

characteristic

M₁₆

—

257

—

M₁₆

—

M₁₆

—

514

—

M₁₆

C₄×D₂₅₇

yes

—

C₂×C₈

—

257

—

C₂×C₈

—

257

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

514

—

C₂×C₈

C₄×D₂₅₇

yes

—

C₄²

—

257

—

C₄²

—

257

—

C₄²

—

257

—

C₂³

Q₈×D₂₅₇

yes

characteristic

C₂×C₄

—

257

—

C₂×C₄

—

257

—

C₂×C₄

—

514

—

C₂×C₄

C₂×C₄×D₂₅₇

yes

—

C₂×C₄

C₂×C₄×D₂₅₇

yes

characteristic

C₂×C₄

D₄×D₂₅₇

yes

characteristic

C₂×C₄

#16861617

yes

—

C₈

—

257

—

C₈

—

C₈

—

C₈

—

514

—

C₂²

—

C₂²

C₄⋊C₄×D₂₅₇

yes

characteristic

C₂²

#22551557

yes

—

C₄

—

257

—

100

C₄

—

514

—

101

C₄

C₄²×D₂₅₇

yes

characteristic

102

C₄

#22551548

yes

characteristic

103

C₄

#22551552

yes

—

104

C₂

—

105

C₂

#30656588

yes

characteristic

106

C₂

#108×D₂₅₇

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₂₅₇
commutator	C₄×C₂₅₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₂₅₇
Fitting	#341×C₂₅₇	C₂
socle	C₂²×C₂₅₇	C₂×C₄⋊C₄
prime radical	C₂³×C₂₅₇	C₂×Q₈
cosocle	C₂²×C₄×C₂₅₇	C₂³
hypercenter	#341	D₂₅₇
hypocenter	C₂₅₇	C₄×#108
LCS	G ▷ C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₂₅₇

commutator

C₄×C₂₅₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₂₅₇

Fitting

#341×C₂₅₇

C₂

socle

C₂²×C₂₅₇

C₂×C₄⋊C₄

prime radical

C₂³×C₂₅₇

C₂×Q₈

cosocle

C₂²×C₄×C₂₅₇

C₂³

D₂₅₇

C₂₅₇

G ▷ C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
9. #22551527	C₄	semidirect
49. Dic₂₅₇	#108	semidirect
53. C₂₅₇	C₄×#108	semidirect

H ≅ G/N

type

9. #22551527

C₄

semidirect

49. Dic₂₅₇

#108

semidirect

53. C₂₅₇

C₄×#108

semidirect

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

subgroup	index
C₂×#22551527	2
M₁₆×Dic₂₅₇	2
C₂×C₈×Dic₂₅₇	2
#30656502	2
#341×C₂₅₇	2
C₄×#108	257

subgroup

index

C₂×#22551527

M₁₆×Dic₂₅₇

C₂×C₈×Dic₂₅₇

#30656502

#341×C₂₅₇

C₄×#108

257

Subgroups — #45664221

Conjugacy classes of subgroups

Complements

Maximal subgroups