Group #45666397 — subgroups

Subgroups — #45666397

Order 32896.

170 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 47 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	#30657175	C₂	1	2	yes	maximal, complemented
2	#30657114	C₂	1	1	yes	maximal, characteristic, complemented
3	#30656855	C₂	1	1	yes	maximal, characteristic, complemented
4	#30656551	C₂	1	2	yes	maximal, complemented
5	#364×C₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#22551614	C₂²	1	1	yes	characteristic
7	C₂×D₂₀₅₆	—	2	1	no	—
8	C₂×D₂₀₅₆	C₂²	1	2	yes	—
9	#22551598	—	2	1	no	—
10	#22551589	—	2	1	no	—
11	#22551537	—	2	2	no	—
12	D₄₁₁₂	—	2	2	no	—
13	#22551528	C₂²	1	1	yes	characteristic
14	#22551520	—	2	1	no	—
15	M₃₂×C₂₅₇	C₂²	1	2	yes	—
16	C₄×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
17	#16861618	—	2	1	no	—
18	C₂×D₁₀₂₈	—	2	2	no	—
19	C₂×D₁₀₂₈	—	4	2	no	—
20	Dic₁₀₂₈	D₄	1	1	yes	characteristic, complemented
21	#16861607	—	2	1	no	—
22	D₂₀₅₆	—	2	2	no	—
23	D₂₀₅₆	—	2	2	no	—
24	D₂₀₅₆	—	4	1	no	—
25	D₂₀₅₆	—	4	1	no	—
26	D₂₀₅₆	D₄	1	1	yes	characteristic, complemented
27	#16861602	—	2	1	no	—
28	#16861601	—	4	1	no	—
29	C₈×D₂₅₇	—	4	1	no	—
30	C₂×C₈×C₂₅₇	D₄	1	1	yes	characteristic
31	C₂×C₈×C₂₅₇	C₂³	1	1	yes	characteristic
32	C₄²×C₂₅₇	D₄	1	1	yes	characteristic
33	C₁₆×C₂₅₇	—	2	2	no	—
34	C₂²×D₂₅₇	—	4	2	no	—
35	Dic₅₁₄	—	2	1	no	—
36	D₁₀₂₈	—	2	1	no	—
37	D₁₀₂₈	—	4	2	no	—
38	D₁₀₂₈	—	4	2	no	—
39	D₁₀₂₈	—	8	4	no	—
40	C₂²⋊D₂₅₇	—	4	1	no	—
41	C₄×D₂₅₇	—	4	1	no	—
42	#14523339	—	4	1	no	—
43	C₂×C₄×C₂₅₇	—	2	1	no	—
44	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
45	C₈×C₂₅₇	D₈	1	1	yes	characteristic
46	C₈×C₂₅₇	D₈	1	1	yes	characteristic
47	C₈×C₂₅₇	C₂×D₄	1	1	yes	characteristic
48	C₈×C₂₅₇	#41	1	1	yes	characteristic
49	C₂²×C₂₅₇	AGL(1,ℤ/8)	1	1	yes	characteristic
50	C₂×D₂₅₇	—	4	1	no	—
51	C₂×D₂₅₇	—	8	4	no	—
52	C₂×D₂₅₇	—	8	2	no	—
53	C₄×C₂₅₇	—	2	2	no	—
54	C₄×C₂₅₇	#121	1	1	yes	characteristic
55	C₄×C₂₅₇	C₂×D₈	1	1	yes	characteristic
56	Dic₂₅₇	—	4	1	no	—
57	C₂×C₂₅₇	—	2	1	no	—
58	C₂×C₂₅₇	#459	1	1	yes	characteristic
59	D₂₅₇	—	8	1	no	—
60	D₂₅₇	—	16	2	no	—
61	C₂₅₇	#2213	1	1	yes	characteristic, complemented, Sylow
62	#2213	—	257	1	no	maximal, Sylow
63	#509	—	257	2	no	—
64	#493	—	257	1	no	—
65	#443	—	257	1	no	—
66	#364	D₂₅₇	1	1	yes	characteristic, complemented
67	#361	—	257	2	no	—
68	#135	—	257	1	no	—
69	C₂×D₈	—	257	2	no	—
70	C₂×D₈	—	514	1	no	—
71	#131	—	514	1	no	—
72	#127	—	514	1	no	—
73	SD₃₂	—	514	2	no	—
74	D₁₆	—	514	2	no	—
75	M₃₂	C₂×D₂₅₇	1	2	yes	—
76	#108	—	257	1	no	—
77	#104	—	514	1	no	—
78	C₄×C₈	C₂×D₂₅₇	1	1	yes	characteristic
79	#41	—	514	1	no	—
80	C₂×D₄	—	514	2	no	—
81	C₂×D₄	—	1028	2	no	—
82	Q₁₆	—	257	1	no	—
83	SD₁₆	—	514	1	no	—
84	D₈	—	257	1	no	—
85	D₈	—	514	2	no	—
86	D₈	—	514	2	no	—
87	D₈	—	1028	1	no	—
88	D₈	—	1028	1	no	—
89	M₁₆	—	514	1	no	—
90	M₁₆	—	1028	1	no	—
91	C₂×C₈	—	1028	1	no	—
92	C₂×C₈	D₁₀₂₈	1	1	yes	characteristic
93	C₂×C₈	C₂²×D₂₅₇	1	1	yes	characteristic
94	C₄²	D₁₀₂₈	1	1	yes	characteristic
95	C₁₆	—	2	2	no	—
96	C₂³	—	1028	2	no	—
97	Q₈	—	514	1	no	—
98	D₄	—	514	1	no	—
99	D₄	—	1028	1	no	—
100	D₄	—	1028	2	no	—
101	D₄	—	1028	2	no	—
102	D₄	—	2056	4	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	1028	1	no	—
105	C₂×C₄	C₂×D₁₀₂₈	1	1	yes	characteristic
106	C₈	—	1028	1	no	—
107	C₈	D₂₀₅₆	1	1	yes	characteristic
108	C₈	D₂₀₅₆	1	1	yes	characteristic
109	C₈	D₄×D₂₅₇	1	1	yes	characteristic
110	C₈	#16861619	1	1	yes	characteristic
111	C₂²	—	1028	1	no	—
112	C₂²	—	2056	4	no	—
113	C₂²	—	2056	2	no	—
114	C₂²	#22551620	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	—	1028	1	no	—
117	C₄	#22551567	1	1	yes	characteristic
118	C₄	C₂×D₂₀₅₆	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2056	1	no	—
121	C₂	—	4112	2	no	—
122	C₂	#30656933	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30657175

C₂

yes

maximal, complemented

#30657114

C₂

yes

maximal, characteristic, complemented

#30656855

C₂

yes

maximal, characteristic, complemented

#30656551

C₂

yes

maximal, complemented

#364×C₂₅₇

C₂

yes

maximal, characteristic, complemented

#22551614

C₂²

yes

characteristic

C₂×D₂₀₅₆

—

C₂×D₂₀₅₆

C₂²

yes

—

#22551598

—

#22551589

—

#22551537

—

D₄₁₁₂

—

#22551528

C₂²

yes

characteristic

#22551520

—

M₃₂×C₂₅₇

C₂²

yes

—

C₄×C₈×C₂₅₇

C₂²

yes

characteristic

#16861618

—

C₂×D₁₀₂₈

—

C₂×D₁₀₂₈

—

Dic₁₀₂₈

D₄

yes

characteristic, complemented

#16861607

—

D₂₀₅₆

—

D₂₀₅₆

—

D₂₀₅₆

—

D₂₀₅₆

—

D₂₀₅₆

D₄

yes

characteristic, complemented

#16861602

—

#16861601

—

C₈×D₂₅₇

—

C₂×C₈×C₂₅₇

D₄

yes

characteristic

C₂×C₈×C₂₅₇

C₂³

yes

characteristic

C₄²×C₂₅₇

D₄

yes

characteristic

C₁₆×C₂₅₇

—

C₂²×D₂₅₇

—

Dic₅₁₄

—

D₁₀₂₈

—

D₁₀₂₈

—

D₁₀₂₈

—

D₁₀₂₈

—

C₂²⋊D₂₅₇

—

C₄×D₂₅₇

—

#14523339

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

C₂×D₄

yes

characteristic

C₈×C₂₅₇

D₈

yes

characteristic

C₈×C₂₅₇

D₈

yes

characteristic

C₈×C₂₅₇

C₂×D₄

yes

C₈×C₂₅₇

yes

C₂²×C₂₅₇

yes

C₂×D₂₅₇

—

C₂×D₂₅₇

—

C₂×D₂₅₇

—

C₄×C₂₅₇

—

C₄×C₂₅₇

#121

yes

characteristic

C₄×C₂₅₇

C₂×D₈

yes

characteristic

Dic₂₅₇

—

C₂×C₂₅₇

—

C₂×C₂₅₇

#459

yes

characteristic

D₂₅₇

—

D₂₅₇

—

C₂₅₇

#2213

yes

characteristic, complemented, Sylow

#2213

—

257

maximal, Sylow

#509

—

257

—

#493

—

257

—

#443

—

257

—

#364

D₂₅₇

yes

characteristic, complemented

#361

—

257

—

#135

—

257

—

C₂×D₈

—

257

—

C₂×D₈

—

514

—

#131

—

514

—

#127

—

514

—

SD₃₂

—

514

—

D₁₆

—

514

—

M₃₂

C₂×D₂₅₇

yes

—

#108

—

257

—

#104

—

514

—

C₄×C₈

C₂×D₂₅₇

yes

characteristic

#41

—

514

—

C₂×D₄

—

514

—

C₂×D₄

—

1028

—

Q₁₆

—

257

—

SD₁₆

—

514

—

D₈

—

257

—

D₈

—

514

—

D₈

—

514

—

D₈

—

1028

—

D₈

—

1028

—

M₁₆

—

514

—

M₁₆

—

1028

—

C₂×C₈

—

1028

—

C₂×C₈

D₁₀₂₈

yes

characteristic

C₂×C₈

C₂²×D₂₅₇

yes

characteristic

C₄²

D₁₀₂₈

yes

characteristic

C₁₆

—

C₂³

—

1028

—

Q₈

—

514

—

D₄

—

514

—

D₄

—

1028

—

100

D₄

—

1028

—

101

D₄

—

1028

—

102

D₄

—

2056

—

103

C₂×C₄

—

104

C₂×C₄

—

1028

—

105

C₂×C₄

C₂×D₁₀₂₈

yes

characteristic

106

C₈

—

1028

—

107

C₈

D₂₀₅₆

yes

characteristic

108

C₈

D₂₀₅₆

yes

characteristic

109

C₈

D₄×D₂₅₇

yes

characteristic

110

C₈

#16861619

yes

characteristic

111

C₂²

—

1028

—

112

C₂²

—

2056

—

113

C₂²

—

2056

—

114

C₂²

#22551620

yes

characteristic

115

C₄

—

116

C₄

—

1028

—

117

C₄

#22551567

yes

characteristic

118

C₄

C₂×D₂₀₅₆

yes

characteristic

119

C₂

—

120

C₂

—

2056

—

121

C₂

—

4112

—

122

C₂

#30656933

yes

characteristic

	subgroup	G/H
center	C₂	#30656933
commutator, cosocle	C₂×C₈×C₂₅₇	C₂³
Frattini	C₂×C₈	C₂²×D₂₅₇
Fitting	#364×C₂₅₇	C₂
socle	C₂×C₂₅₇	#459
hypercenter	#364	D₂₅₇
hypocenter	C₂₅₇	#2213
LCS	G ▷ C₂×C₈×C₂₅₇ ▷ C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

subgroup

G/H

center

C₂

#30656933

commutator, cosocle

C₂×C₈×C₂₅₇

C₂³

Frattini

C₂×C₈

C₂²×D₂₅₇

Fitting

#364×C₂₅₇

C₂

C₂×C₂₅₇

D₂₅₇

C₂₅₇

G ▷ C₂×C₈×C₂₅₇ ▷ C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

N	H ≅ G/N	type
1. #30657175	C₂	semidirect
2. #30657114	C₂	semidirect
3. #30656855	C₂	semidirect
4. #30656551	C₂	semidirect
5. #364×C₂₅₇	C₂	semidirect
20. Dic₁₀₂₈	D₄	semidirect
26. D₂₀₅₆	D₄	semidirect
61. C₂₅₇	#2213	semidirect
66. #364	D₂₅₇	semidirect

H ≅ G/N

type

1. #30657175

C₂

semidirect

2. #30657114

C₂

semidirect

3. #30656855

C₂

semidirect

4. #30656551

C₂

semidirect

5. #364×C₂₅₇

C₂

semidirect

20. Dic₁₀₂₈

D₄

semidirect

26. D₂₀₅₆

D₄

semidirect

61. C₂₅₇

#2213

semidirect

66. #364

D₂₅₇

semidirect

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

subgroup	index
#30657175	2
#30657114	2
#30656855	2
#30656551	2
#364×C₂₅₇	2
#2213	257

subgroup

index

#364×C₂₅₇

257

Subgroups — #45666397

Conjugacy classes of subgroups

Complements

Maximal subgroups