Group #46784016 — subgroups

Subgroups — #46784016

Order 33664.

256 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 67 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	C₂²×Dic₁₀₅₂	C₂	1	1	yes	maximal, characteristic
2	#30681964	C₂	1	1	yes	maximal, characteristic
3	#30681948	C₂	1	2	yes	maximal
4	C₂×#22554388	C₂	1	2	yes	maximal
5	#341×C₂₆₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₅₂₆	C₂²	1	2	yes	—
7	C₂×Dic₁₀₅₂	—	2	4	no	—
8	C₂×Dic₁₀₅₂	—	2	2	no	—
9	C₂×#16862360	C₂²	1	1	yes	characteristic
10	#22554429	C₂²	1	1	yes	characteristic
11	Dic₂₆₃⋊C₈	—	2	2	no	—
12	#22554395	—	2	4	no	—
13	#22554388	—	2	4	no	—
14	C₂×M₁₆×C₂₆₃	C₂²	1	2	yes	—
15	C₂²×C₈×C₂₆₃	C₂²	1	1	yes	characteristic
16	C₂×Dic₅₂₆	—	2	2	no	—
17	C₂×Dic₅₂₆	—	2	2	no	—
18	C₂×Dic₅₂₆	—	2	2	no	—
19	C₂×Dic₅₂₆	—	2	2	no	—
20	C₂×Dic₅₂₆	—	2	2	no	—
21	C₂×Dic₅₂₆	D₄	1	4	yes	—
22	C₂²×Dic₂₆₃	—	2	2	no	—
23	Dic₁₀₅₂	—	2	8	no	—
24	#16862360	—	2	2	no	—
25	C₂×C₂₆₃⋊C₈	—	2	1	no	—
26	C₄⋊Dic₂₆₃	D₄	1	2	yes	—
27	C₂²⋊Dic₂₆₃	—	2	1	no	—
28	C₄×Dic₂₆₃	—	2	1	no	—
29	C₂²×C₄×C₂₆₃	C₂³	1	1	yes	characteristic
30	M₁₆×C₂₆₃	—	2	2	no	—
31	M₁₆×C₂₆₃	—	2	2	no	—
32	C₂×C₈×C₂₆₃	—	2	2	no	—
33	C₂×C₈×C₂₆₃	—	2	2	no	—
34	C₂×C₈×C₂₆₃	D₄	1	2	yes	—
35	Dic₅₂₆	—	2	4	no	—
36	Dic₅₂₆	—	2	4	no	—
37	Dic₅₂₆	—	4	2	no	—
38	Dic₅₂₆	—	4	2	no	—
39	C₂×Dic₂₆₃	—	2	2	no	—
40	C₂×Dic₂₆₃	—	2	4	no	—
41	C₂×Dic₂₆₃	—	4	2	no	—
42	C₂×Dic₂₆₃	—	4	2	no	—
43	C₂₆₃⋊C₈	—	4	1	no	—
44	C₂³×C₂₆₃	#41	1	1	yes	characteristic
45	C₂×C₄×C₂₆₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₂₆₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₂₆₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₂₆₃	#41	1	2	yes	—
49	C₈×C₂₆₃	—	2	2	no	—
50	C₈×C₂₆₃	—	4	2	no	—
51	C₂²×C₂₆₃	—	2	2	no	—
52	C₂²×C₂₆₃	#121	1	1	yes	characteristic
53	C₂²×C₂₆₃	#123	1	2	yes	—
54	C₄×C₂₆₃	#120	1	1	yes	characteristic
55	C₄×C₂₆₃	#121	1	2	yes	—
56	C₄×C₂₆₃	#124	1	1	yes	characteristic
57	Dic₂₆₃	—	4	4	no	—
58	Dic₂₆₃	—	4	2	no	—
59	C₂×C₂₆₃	—	2	2	no	—
60	C₂×C₂₆₃	#394	1	1	yes	characteristic
61	C₂×C₂₆₃	#473	1	2	yes	—
62	C₂₆₃	#2047	1	1	yes	characteristic, complemented, Sylow
63	#2047	—	263	1	no	maximal, Sylow
64	C₂²×Q₁₆	—	263	1	no	—
65	#424	—	263	1	no	—
66	#419	—	263	2	no	—
67	C₂×#101	—	263	2	no	—
68	#341	D₂₆₃	1	1	yes	characteristic
69	C₂²×Q₈	—	263	2	no	—
70	C₂×Q₁₆	—	526	4	no	—
71	C₂×Q₁₆	—	526	2	no	—
72	C₂×M₁₆	—	263	1	no	—
73	C₂×M₁₆	C₂×D₂₆₃	1	2	yes	—
74	C₂²×C₈	C₂×D₂₆₃	1	1	yes	characteristic
75	#117	—	263	1	no	—
76	C₄⋊C₈	—	526	2	no	—
77	#103	—	526	4	no	—
78	#101	—	526	4	no	—
79	C₂×Q₈	—	263	4	no	—
80	C₂×Q₈	—	526	2	no	—
81	C₂×Q₈	—	526	2	no	—
82	C₂×Q₈	—	526	2	no	—
83	C₂×Q₈	—	526	2	no	—
84	C₂×Q₈	—	526	2	no	—
85	C₂²×C₄	—	526	2	no	—
86	C₂²×C₄	C₂²×D₂₆₃	1	1	yes	characteristic
87	Q₁₆	—	526	8	no	—
88	M₁₆	—	2	2	no	—
89	M₁₆	—	2	2	no	—
90	M₁₆	—	526	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	526	1	no	—
94	C₂×C₈	D₁₀₅₂	1	2	yes	—
95	C₄⋊C₄	—	263	2	no	—
96	C₂²⋊C₄	—	526	1	no	—
97	C₄²	—	526	1	no	—
98	C₂³	#16862377	1	1	yes	characteristic
99	Q₈	—	526	4	no	—
100	Q₈	—	526	4	no	—
101	Q₈	—	1052	2	no	—
102	Q₈	—	1052	2	no	—
103	C₂×C₄	—	526	2	no	—
104	C₂×C₄	—	526	4	no	—
105	C₂×C₄	—	1052	2	no	—
106	C₂×C₄	—	1052	2	no	—
107	C₂×C₄	D₄×D₂₆₃	1	2	yes	—
108	C₂×C₄	D₄×D₂₆₃	1	1	yes	characteristic
109	C₂×C₄	C₂×D₁₀₅₂	1	1	yes	characteristic
110	C₂×C₄	#16862376	1	2	yes	—
111	C₈	—	2	2	no	—
112	C₈	—	4	2	no	—
113	C₈	—	1052	1	no	—
114	C₂²	—	2	2	no	—
115	C₂²	#22554445	1	1	yes	characteristic
116	C₂²	#22554455	1	2	yes	—
117	C₄	—	1052	4	no	—
118	C₄	—	1052	2	no	—
119	C₄	#22554440	1	1	yes	characteristic
120	C₄	#22554443	1	2	yes	—
121	C₄	#22554458	1	1	yes	characteristic
122	C₂	—	2	2	no	—
123	C₂	#30681856	1	1	yes	characteristic
124	C₂	#30682214	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₁₀₅₂

C₂

yes

maximal, characteristic

#30681964

C₂

yes

maximal, characteristic

#30681948

C₂

yes

maximal

C₂×#22554388

C₂

yes

maximal

#341×C₂₆₃

C₂

yes

maximal, characteristic

C₂²×Dic₅₂₆

C₂²

yes

—

C₂×Dic₁₀₅₂

—

C₂×Dic₁₀₅₂

—

C₂×#16862360

C₂²

yes

characteristic

#22554429

C₂²

yes

characteristic

Dic₂₆₃⋊C₈

—

#22554395

—

#22554388

—

C₂×M₁₆×C₂₆₃

C₂²

yes

—

C₂²×C₈×C₂₆₃

C₂²

yes

characteristic

C₂×Dic₅₂₆

—

C₂×Dic₅₂₆

—

C₂×Dic₅₂₆

—

C₂×Dic₅₂₆

—

C₂×Dic₅₂₆

—

C₂×Dic₅₂₆

D₄

yes

—

C₂²×Dic₂₆₃

—

Dic₁₀₅₂

—

#16862360

—

C₂×C₂₆₃⋊C₈

—

C₄⋊Dic₂₆₃

D₄

yes

—

C₂²⋊Dic₂₆₃

—

C₄×Dic₂₆₃

—

C₂²×C₄×C₂₆₃

C₂³

yes

characteristic

M₁₆×C₂₆₃

—

M₁₆×C₂₆₃

—

C₂×C₈×C₂₆₃

—

C₂×C₈×C₂₆₃

—

C₂×C₈×C₂₆₃

D₄

yes

—

Dic₅₂₆

—

Dic₅₂₆

—

Dic₅₂₆

—

Dic₅₂₆

—

C₂×Dic₂₆₃

—

C₂×Dic₂₆₃

—

C₂×Dic₂₆₃

—

C₂×Dic₂₆₃

—

C₂₆₃⋊C₈

—

C₂³×C₂₆₃

#41

yes

characteristic

C₂×C₄×C₂₆₃

C₂×D₄

yes

—

C₂×C₄×C₂₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₃

#41

yes

—

C₈×C₂₆₃

—

C₈×C₂₆₃

—

C₂²×C₂₆₃

—

C₂²×C₂₆₃

#121

yes

characteristic

C₂²×C₂₆₃

#123

yes

—

C₄×C₂₆₃

#120

yes

characteristic

C₄×C₂₆₃

#121

yes

—

C₄×C₂₆₃

#124

yes

characteristic

Dic₂₆₃

—

Dic₂₆₃

—

C₂×C₂₆₃

—

C₂×C₂₆₃

#394

yes

characteristic

C₂×C₂₆₃

#473

yes

—

C₂₆₃

#2047

yes

characteristic, complemented, Sylow

#2047

—

263

maximal, Sylow

C₂²×Q₁₆

—

263

—

#424

—

263

—

#419

—

263

—

C₂×#101

—

263

—

#341

D₂₆₃

yes

characteristic

C₂²×Q₈

—

263

—

C₂×Q₁₆

—

526

—

C₂×Q₁₆

—

526

—

C₂×M₁₆

—

263

—

C₂×M₁₆

C₂×D₂₆₃

yes

—

C₂²×C₈

C₂×D₂₆₃

yes

characteristic

#117

—

263

—

C₄⋊C₈

—

526

—

#103

—

526

—

#101

—

526

—

C₂×Q₈

—

263

—

C₂×Q₈

—

526

—

C₂×Q₈

—

526

—

C₂×Q₈

—

526

—

C₂×Q₈

—

526

—

C₂×Q₈

—

526

—

C₂²×C₄

—

526

—

C₂²×C₄

C₂²×D₂₆₃

yes

characteristic

Q₁₆

—

526

—

M₁₆

—

M₁₆

—

M₁₆

—

526

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

526

—

C₂×C₈

D₁₀₅₂

yes

—

C₄⋊C₄

—

263

—

C₂²⋊C₄

—

526

—

C₄²

—

526

—

C₂³

#16862377

yes

characteristic

Q₈

—

526

—

100

Q₈

—

526

—

101

Q₈

—

1052

—

102

Q₈

—

1052

—

103

C₂×C₄

—

526

—

104

C₂×C₄

—

526

—

105

C₂×C₄

—

1052

—

106

C₂×C₄

—

1052

—

107

C₂×C₄

D₄×D₂₆₃

yes

—

108

C₂×C₄

D₄×D₂₆₃

yes

characteristic

109

C₂×C₄

C₂×D₁₀₅₂

yes

characteristic

110

C₂×C₄

#16862376

yes

—

111

C₈

—

112

C₈

—

113

C₈

—

1052

—

114

C₂²

—

115

C₂²

#22554445

yes

characteristic

116

C₂²

#22554455

yes

—

117

C₄

—

1052

—

118

C₄

—

1052

—

119

C₄

#22554440

yes

characteristic

120

C₄

#22554443

yes

—

121

C₄

#22554458

yes

characteristic

122

C₂

—

123

C₂

#30681856

yes

characteristic

124

C₂

#30682214

yes

—

	subgroup	G/H
center	C₂²	#22554445
commutator, cosocle	C₂²×C₄×C₂₆₃	C₂³
Frattini	C₂²×C₄	C₂²×D₂₆₃
Fitting	#341×C₂₆₃	C₂
socle	C₂²×C₂₆₃	#121
prime radical	C₂³×C₂₆₃	#41
hypercenter	#341	D₂₆₃
hypocenter	C₂₆₃	#2047
LCS	G ▷ C₂²×C₄×C₂₆₃ ▷ C₂×C₂₆₃ ▷ C₂₆₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂²

#22554445

commutator, cosocle

C₂²×C₄×C₂₆₃

C₂³

Frattini

C₂²×C₄

C₂²×D₂₆₃

Fitting

#341×C₂₆₃

C₂

C₂²×C₂₆₃

C₂³×C₂₆₃

D₂₆₃

C₂₆₃

G ▷ C₂²×C₄×C₂₆₃ ▷ C₂×C₂₆₃ ▷ C₂₆₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
62. C₂₆₃	#2047	semidirect

H ≅ G/N

type

62. C₂₆₃

#2047

semidirect

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

subgroup	index
C₂²×Dic₁₀₅₂	2
#30681964	2
#30681948	2
C₂×#22554388	2
#341×C₂₆₃	2
#2047	263

subgroup

index

C₂²×Dic₁₀₅₂

#341×C₂₆₃

263

Subgroups — #46784016

Conjugacy classes of subgroups

Complements

Maximal subgroups