Group #45664401 — subgroups

Subgroups — #45664401

Order 32896.

272 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 85 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×#22551551	C₂	1	1	yes	maximal, characteristic
2	#30656701	C₂	1	1	yes	maximal, characteristic
3	#30656505	C₂	1	4	yes	maximal, complemented
4	#407×C₂₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#16861602	C₂²	1	2	yes	—
6	#22551551	—	2	2	no	—
7	#22551551	—	2	2	no	—
8	C₂×C₄⋊Dic₂₅₇	C₂²	1	2	yes	—
9	C₂×C₂²⋊Dic₂₅₇	C₄	1	2	yes	—
10	C₂×C₄×Dic₂₅₇	C₄	1	2	yes	—
11	C₂²⋊#14523339	—	2	2	no	—
12	C₂³×C₄×C₂₅₇	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₂₅₇	C₂²	1	2	yes	—
14	C₂²⋊C₈×C₂₅₇	—	2	2	no	—
15	C₂²×Dic₂₅₇	C₂×C₄	1	4	yes	—
16	#16861602	—	2	4	no	—
17	C₂×#14523339	—	2	2	no	—
18	C₄⋊Dic₂₅₇	—	2	2	no	—
19	C₄⋊Dic₂₅₇	—	2	2	no	—
20	C₂²⋊Dic₂₅₇	—	2	4	no	—
21	C₄×Dic₂₅₇	—	2	4	no	—
22	C₂⁴×C₂₅₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₂₅₇	—	2	4	no	—
24	C₂²×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₂₅₇	D₄	1	2	yes	—
26	C₂²×C₄×C₂₅₇	D₄	1	2	yes	—
27	C₂²×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
28	M₁₆×C₂₅₇	—	2	4	no	—
29	C₂×C₈×C₂₅₇	—	2	2	no	—
30	C₂×Dic₂₅₇	—	2	4	no	—
31	C₂×Dic₂₅₇	—	4	4	no	—
32	#14523339	—	4	2	no	—
33	C₂³×C₂₅₇	—	2	2	no	—
34	C₂³×C₂₅₇	—	4	2	no	—
35	C₂³×C₂₅₇	C₂²⋊C₄	1	2	yes	—
36	C₂³×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₂₅₇	—	2	2	no	—
38	C₂×C₄×C₂₅₇	—	2	4	no	—
39	C₂×C₄×C₂₅₇	—	2	4	no	—
40	C₂×C₄×C₂₅₇	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₂₅₇	#41	1	4	yes	—
44	C₈×C₂₅₇	—	4	2	no	—
45	C₂²×C₂₅₇	—	2	2	no	—
46	C₂²×C₂₅₇	—	2	2	no	—
47	C₂²×C₂₅₇	—	2	4	no	—
48	C₂²×C₂₅₇	—	4	4	no	—
49	C₂²×C₂₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₂₅₇	#117	1	2	yes	—
51	C₄×C₂₅₇	—	2	2	no	—
52	C₄×C₂₅₇	#123	1	2	yes	—
53	C₄×C₂₅₇	#123	1	2	yes	—
54	Dic₂₅₇	—	4	4	no	—
55	C₂×C₂₅₇	—	2	2	no	—
56	C₂×C₂₅₇	—	4	2	no	—
57	C₂×C₂₅₇	#381	1	1	yes	characteristic
58	C₂×C₂₅₇	#413	1	2	yes	—
59	C₂₅₇	#1822	1	1	yes	characteristic, complemented, Sylow
60	#1822	—	257	1	no	maximal, Sylow
61	C₂×#117	—	257	1	no	—
62	#407	—	257	1	no	—
63	#407	D₂₅₇	1	1	yes	characteristic
64	#343	—	257	4	no	—
65	C₂³×C₄	C₂×D₂₅₇	1	1	yes	characteristic
66	C₂×M₁₆	—	257	2	no	—
67	C₂×M₁₆	C₂×D₂₅₇	1	2	yes	—
68	#117	—	514	2	no	—
69	#117	—	514	2	no	—
70	C₂×C₄⋊C₄	—	257	2	no	—
71	C₂×C₂²⋊C₄	—	257	2	no	—
72	C₂×C₄²	—	257	2	no	—
73	C₂²⋊C₈	—	2	2	no	—
74	C₂²⋊C₈	—	514	2	no	—
75	C₂⁴	C₄×D₂₅₇	1	1	yes	characteristic
76	C₂²×C₄	—	257	4	no	—
77	C₂²×C₄	—	2	4	no	—
78	C₂²×C₄	C₄×D₂₅₇	1	1	yes	characteristic
79	C₂²×C₄	C₂²⋊D₂₅₇	1	2	yes	—
80	C₂²×C₄	D₁₀₂₈	1	2	yes	—
81	C₂²×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
82	M₁₆	—	2	4	no	—
83	M₁₆	—	514	4	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	514	2	no	—
86	C₄⋊C₄	—	514	2	no	—
87	C₄⋊C₄	—	514	2	no	—
88	C₂²⋊C₄	—	514	4	no	—
89	C₄²	—	514	4	no	—
90	C₂³	—	2	2	no	—
91	C₂³	—	4	2	no	—
92	C₂³	#16861595	1	2	yes	—
93	C₂³	C₂×C₄×D₂₅₇	1	1	yes	characteristic
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	514	4	no	—
98	C₂×C₄	—	1028	4	no	—
99	C₂×C₄	#16861595	1	2	yes	—
100	C₂×C₄	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
101	C₂×C₄	C₂×D₁₀₂₈	1	1	yes	characteristic
102	C₂×C₄	#16861617	1	4	yes	—
103	C₈	—	4	2	no	—
104	C₈	—	1028	2	no	—
105	C₂²	—	2	2	no	—
106	C₂²	—	2	2	no	—
107	C₂²	—	2	4	no	—
108	C₂²	—	4	4	no	—
109	C₂²	C₂×#16861595	1	1	yes	characteristic
110	C₂²	#22551548	1	2	yes	—
111	C₄	—	2	2	no	—
112	C₄	—	1028	4	no	—
113	C₄	#22551573	1	2	yes	—
114	C₄	#22551576	1	2	yes	—
115	C₂	—	2	2	no	—
116	C₂	—	4	2	no	—
117	C₂	#30656598	1	1	yes	characteristic
118	C₂	#30656719	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22551551

C₂

yes

maximal, characteristic

#30656701

C₂

yes

maximal, characteristic

#30656505

C₂

yes

maximal, complemented

#407×C₂₅₇

C₂

yes

maximal, characteristic

C₂×#16861602

C₂²

yes

—

#22551551

—

#22551551

—

C₂×C₄⋊Dic₂₅₇

C₂²

yes

—

C₂×C₂²⋊Dic₂₅₇

C₄

yes

—

C₂×C₄×Dic₂₅₇

C₄

yes

—

C₂²⋊#14523339

—

C₂³×C₄×C₂₅₇

C₂²

yes

characteristic

C₂×M₁₆×C₂₅₇

C₂²

yes

—

C₂²⋊C₈×C₂₅₇

—

C₂²×Dic₂₅₇

C₂×C₄

yes

—

#16861602

—

C₂×#14523339

—

C₄⋊Dic₂₅₇

—

C₄⋊Dic₂₅₇

—

C₂²⋊Dic₂₅₇

—

C₄×Dic₂₅₇

—

C₂⁴×C₂₅₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₅₇

—

C₂²×C₄×C₂₅₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₅₇

D₄

yes

—

C₂²×C₄×C₂₅₇

D₄

yes

—

C₂²×C₄×C₂₅₇

C₂³

yes

characteristic

M₁₆×C₂₅₇

—

C₂×C₈×C₂₅₇

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

#14523339

—

C₂³×C₂₅₇

—

C₂³×C₂₅₇

—

C₂³×C₂₅₇

C₂²⋊C₄

yes

—

C₂³×C₂₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

C₂²⋊C₄

yes

—

C₂×C₄×C₂₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₇

#41

yes

—

C₈×C₂₅₇

—

C₂²×C₂₅₇

—

C₂²×C₂₅₇

—

C₂²×C₂₅₇

—

C₂²×C₂₅₇

—

C₂²×C₂₅₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₅₇

#117

yes

—

C₄×C₂₅₇

—

C₄×C₂₅₇

#123

yes

—

C₄×C₂₅₇

#123

yes

—

Dic₂₅₇

—

C₂×C₂₅₇

—

C₂×C₂₅₇

—

C₂×C₂₅₇

#381

yes

characteristic

C₂×C₂₅₇

#413

yes

—

C₂₅₇

#1822

yes

characteristic, complemented, Sylow

#1822

—

257

maximal, Sylow

C₂×#117

—

257

—

#407

—

257

—

#407

D₂₅₇

yes

characteristic

#343

—

257

—

C₂³×C₄

C₂×D₂₅₇

yes

characteristic

C₂×M₁₆

—

257

—

C₂×M₁₆

C₂×D₂₅₇

yes

—

#117

—

514

—

#117

—

514

—

C₂×C₄⋊C₄

—

257

—

C₂×C₂²⋊C₄

—

257

—

C₂×C₄²

—

257

—

C₂²⋊C₈

—

C₂²⋊C₈

—

514

—

C₂⁴

C₄×D₂₅₇

yes

characteristic

C₂²×C₄

—

257

—

C₂²×C₄

—

C₂²×C₄

C₄×D₂₅₇

yes

characteristic

C₂²×C₄

C₂²⋊D₂₅₇

yes

—

C₂²×C₄

D₁₀₂₈

yes

—

C₂²×C₄

C₂²×D₂₅₇

yes

characteristic

M₁₆

—

M₁₆

—

514

—

C₂×C₈

—

C₂×C₈

—

514

—

C₄⋊C₄

—

514

—

C₄⋊C₄

—

514

—

C₂²⋊C₄

—

514

—

C₄²

—

514

—

C₂³

—

C₂³

—

C₂³

#16861595

yes

—

C₂³

C₂×C₄×D₂₅₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

514

—

C₂×C₄

—

1028

—

C₂×C₄

#16861595

yes

—

100

C₂×C₄

C₂×C₂²⋊D₂₅₇

yes

characteristic

101

C₂×C₄

C₂×D₁₀₂₈

yes

characteristic

102

C₂×C₄

#16861617

yes

—

103

C₈

—

104

C₈

—

1028

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

C₂×#16861595

yes

characteristic

110

C₂²

#22551548

yes

—

111

C₄

—

112

C₄

—

1028

—

113

C₄

#22551573

yes

—

114

C₄

#22551576

yes

—

115

C₂

—

116

C₂

—

117

C₂

#30656598

yes

characteristic

118

C₂

#30656719

yes

—

	subgroup	G/H
center	C₂²	C₂×#16861595
commutator	C₂³×C₂₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₅₇
Fitting	#407×C₂₅₇	C₂
socle	C₂²×C₂₅₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₂₅₇	C₂×C₄
cosocle	C₂²×C₄×C₂₅₇	C₂³
hypercenter	#407	D₂₅₇
hypocenter	C₂₅₇	#1822
LCS	G ▷ C₂³×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

subgroup

G/H

center

C₂²

C₂×#16861595

commutator

C₂³×C₂₅₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₅₇

Fitting

#407×C₂₅₇

C₂

socle

C₂²×C₂₅₇

C₂×C₂²⋊C₄

prime radical

C₂⁴×C₂₅₇

C₂×C₄

cosocle

C₂²×C₄×C₂₅₇

C₂³

D₂₅₇

C₂₅₇

G ▷ C₂³×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

N	H ≅ G/N	type
3. #30656505	C₂	semidirect
59. C₂₅₇	#1822	semidirect

H ≅ G/N

type

3. #30656505

C₂

semidirect

59. C₂₅₇

#1822

semidirect

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

subgroup	index
C₂×#22551551	2
#30656701	2
#30656505	2
#407×C₂₅₇	2
#1822	257

subgroup

index

#407×C₂₅₇

257

Subgroups — #45664401

Conjugacy classes of subgroups

Complements

Maximal subgroups