Group #46781485 — subgroups

Subgroups — #46781485

Order 33664.

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

normal subgroups only

This group has 210 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	#30681962	C₂	1	2	yes	maximal
2	#30681684	C₂	1	4	yes	maximal, complemented
3	#533×C₂₆₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#16862360	—	2	2	no	—
5	C₄⋊(C₂₆₃⋊C₈)	—	2	2	no	—
6	C₄⋊(C₂₆₃⋊C₈)	C₂²	1	4	yes	—
7	C₄⋊Q₈×C₂₆₃	C₂²	1	2	yes	—
8	#125×C₂₆₃	C₄	1	2	yes	—
9	C₂²⋊Q₈×C₂₆₃	—	2	2	no	—
10	C₄×Q₈×C₂₆₃	—	2	2	no	—
11	#117×C₂₆₃	C₄	1	2	yes	—
12	C₂×C₄²×C₂₆₃	C₂²	1	1	yes	characteristic
13	#16862360	—	4	4	no	—
14	C₂×C₂₆₃⋊C₈	—	2	4	no	—
15	C₂×Q₈×C₂₆₃	—	2	2	no	—
16	C₂²×C₄×C₂₆₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₂₆₃	D₄	1	2	yes	—
18	C₄⋊C₄×C₂₆₃	—	2	4	no	—
19	C₄⋊C₄×C₂₆₃	—	2	2	no	—
20	C₄⋊C₄×C₂₆₃	C₂×C₄	1	4	yes	—
21	C₂²⋊C₄×C₂₆₃	—	2	2	no	—
22	C₄²×C₂₆₃	—	2	2	no	—
23	C₄²×C₂₆₃	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₂₆₃	D₄	1	2	yes	—
25	C₄²×C₂₆₃	C₂³	1	1	yes	characteristic
26	C₂₆₃⋊C₈	—	4	4	no	—
27	C₂³×C₂₆₃	C₂²⋊C₄	1	1	yes	characteristic
28	Q₈×C₂₆₃	—	4	2	no	—
29	Q₈×C₂₆₃	—	4	2	no	—
30	C₂×C₄×C₂₆₃	—	2	2	no	—
31	C₂×C₄×C₂₆₃	—	2	2	no	—
32	C₂×C₄×C₂₆₃	—	2	2	no	—
33	C₂×C₄×C₂₆₃	—	2	4	no	—
34	C₂×C₄×C₂₆₃	C₂²⋊C₄	1	2	yes	—
35	C₂×C₄×C₂₆₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₂₆₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₂₆₃	C₂×D₄	1	2	yes	—
38	C₂²×C₂₆₃	—	2	1	no	—
39	C₂²×C₂₆₃	—	2	2	no	—
40	C₂²×C₂₆₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
41	C₄×C₂₆₃	—	2	1	no	—
42	C₄×C₂₆₃	—	2	2	no	—
43	C₄×C₂₆₃	—	4	4	no	—
44	C₄×C₂₆₃	#101	1	2	yes	—
45	C₄×C₂₆₃	AGL(1,ℤ/8)	1	2	yes	—
46	C₄×C₂₆₃	#137	1	2	yes	—
47	C₂×C₂₆₃	—	4	1	no	—
48	C₂×C₂₆₃	C₂×#101	1	1	yes	characteristic
49	C₂×C₂₆₃	#417	1	2	yes	—
50	C₂₆₃	#1561	1	1	yes	characteristic, complemented, Sylow
51	#1561	—	263	1	no	maximal, Sylow
52	#533	D₂₆₃	1	1	yes	characteristic
53	#423	—	263	2	no	—
54	#332	—	263	4	no	—
55	C₂×M₁₆	—	526	2	no	—
56	C₄⋊Q₈	C₂×D₂₆₃	1	2	yes	—
57	#125	Dic₂₆₃	1	2	yes	—
58	C₂²⋊Q₈	—	2	2	no	—
59	C₄×Q₈	—	2	2	no	—
60	#117	Dic₂₆₃	1	2	yes	—
61	C₂×C₄²	C₂×D₂₆₃	1	1	yes	characteristic
62	C₄⋊C₈	—	263	4	no	—
63	C₄⋊C₈	—	526	2	no	—
64	C₂×Q₈	—	2	2	no	—
65	C₂²×C₄	C₂×Dic₂₆₃	1	1	yes	characteristic
66	C₂²×C₄	C₂²⋊D₂₆₃	1	2	yes	—
67	M₁₆	—	1052	4	no	—
68	C₂×C₈	—	526	4	no	—
69	C₄⋊C₄	—	2	4	no	—
70	C₄⋊C₄	—	2	2	no	—
71	C₄⋊C₄	C₂×Dic₂₆₃	1	4	yes	—
72	C₂²⋊C₄	—	2	2	no	—
73	C₄²	—	2	2	no	—
74	C₄²	C₂×Dic₂₆₃	1	1	yes	characteristic
75	C₄²	C₂²⋊D₂₆₃	1	2	yes	—
76	C₄²	C₂²×D₂₆₃	1	1	yes	characteristic
77	C₂³	C₂²⋊Dic₂₆₃	1	1	yes	characteristic
78	Q₈	—	4	2	no	—
79	Q₈	—	4	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	4	no	—
84	C₂×C₄	C₂²⋊Dic₂₆₃	1	2	yes	—
85	C₂×C₄	C₂²⋊Dic₂₆₃	1	1	yes	characteristic
86	C₂×C₄	C₂²×Dic₂₆₃	1	1	yes	characteristic
87	C₂×C₄	C₂×C₂²⋊D₂₆₃	1	2	yes	—
88	C₈	—	1052	4	no	—
89	C₂²	—	2	1	no	—
90	C₂²	—	2	2	no	—
91	C₂²	C₂×C₂²⋊Dic₂₆₃	1	1	yes	characteristic
92	C₄	—	2	1	no	—
93	C₄	—	2	2	no	—
94	C₄	—	4	4	no	—
95	C₄	#22554389	1	2	yes	—
96	C₄	#22554495	1	2	yes	—
97	C₄	#22554500	1	2	yes	—
98	C₂	—	4	1	no	—
99	C₂	C₂×#22554389	1	1	yes	characteristic
100	C₂	#30681939	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#30681962

C₂

yes

maximal

#30681684

C₂

yes

maximal, complemented

#533×C₂₆₃

C₂

yes

maximal, characteristic

C₂×#16862360

—

C₄⋊(C₂₆₃⋊C₈)

—

C₄⋊(C₂₆₃⋊C₈)

C₂²

yes

—

C₄⋊Q₈×C₂₆₃

C₂²

yes

—

#125×C₂₆₃

C₄

yes

—

C₂²⋊Q₈×C₂₆₃

—

C₄×Q₈×C₂₆₃

—

#117×C₂₆₃

C₄

yes

—

C₂×C₄²×C₂₆₃

C₂²

yes

characteristic

#16862360

—

C₂×C₂₆₃⋊C₈

—

C₂×Q₈×C₂₆₃

—

C₂²×C₄×C₂₆₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₆₃

D₄

yes

—

C₄⋊C₄×C₂₆₃

—

C₄⋊C₄×C₂₆₃

—

C₄⋊C₄×C₂₆₃

C₂×C₄

yes

—

C₂²⋊C₄×C₂₆₃

—

C₄²×C₂₆₃

—

C₄²×C₂₆₃

C₂×C₄

yes

characteristic

C₄²×C₂₆₃

D₄

yes

—

C₄²×C₂₆₃

C₂³

yes

characteristic

C₂₆₃⋊C₈

—

C₂³×C₂₆₃

C₂²⋊C₄

yes

characteristic

Q₈×C₂₆₃

—

Q₈×C₂₆₃

—

C₂×C₄×C₂₆₃

—

C₂×C₄×C₂₆₃

—

C₂×C₄×C₂₆₃

—

C₂×C₄×C₂₆₃

—

C₂×C₄×C₂₆₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₆₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₆₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₆₃

C₂×D₄

yes

—

C₂²×C₂₆₃

—

C₂²×C₂₆₃

—

C₂²×C₂₆₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₆₃

—

C₄×C₂₆₃

—

C₄×C₂₆₃

—

C₄×C₂₆₃

#101

yes

—

C₄×C₂₆₃

AGL(1,ℤ/8)

yes

—

C₄×C₂₆₃

#137

yes

—

C₂×C₂₆₃

—

C₂×C₂₆₃

C₂×#101

yes

characteristic

C₂×C₂₆₃

#417

yes

—

C₂₆₃

#1561

yes

characteristic, complemented, Sylow

#1561

—

263

maximal, Sylow

#533

D₂₆₃

yes

characteristic

#423

—

263

—

#332

—

263

—

C₂×M₁₆

—

526

—

C₄⋊Q₈

C₂×D₂₆₃

yes

—

#125

Dic₂₆₃

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

Dic₂₆₃

yes

—

C₂×C₄²

C₂×D₂₆₃

yes

characteristic

C₄⋊C₈

—

263

—

C₄⋊C₈

—

526

—

C₂×Q₈

—

C₂²×C₄

C₂×Dic₂₆₃

yes

characteristic

C₂²×C₄

C₂²⋊D₂₆₃

yes

—

M₁₆

—

1052

—

C₂×C₈

—

526

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₂₆₃

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×Dic₂₆₃

yes

characteristic

C₄²

C₂²⋊D₂₆₃

yes

—

C₄²

C₂²×D₂₆₃

yes

characteristic

C₂³

C₂²⋊Dic₂₆₃

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₂₆₃

yes

—

C₂×C₄

C₂²⋊Dic₂₆₃

yes

characteristic

C₂×C₄

C₂²×Dic₂₆₃

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₂₆₃

yes

—

C₈

—

1052

—

C₂²

—

C₂²

—

C₂²

C₂×C₂²⋊Dic₂₆₃

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#22554389

yes

—

C₄

#22554495

yes

—

C₄

#22554500

yes

—

C₂

—

C₂

C₂×#22554389

yes

characteristic

100

C₂

#30681939

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₂₆₃
commutator	C₂×C₄×C₂₆₃	C₂²×C₄
Frattini	C₄²	C₂²×D₂₆₃
Fitting	#533×C₂₆₃	C₂
socle	C₂²×C₂₆₃	C₂×C₂²⋊C₄
prime radical	C₂³×C₂₆₃	C₂²⋊C₄
cosocle	C₄²×C₂₆₃	C₂³
hypercenter	#533	D₂₆₃
hypocenter	C₂₆₃	#1561
LCS	G ▷ C₂×C₄×C₂₆₃ ▷ C₂²×C₂₆₃ ▷ C₂₆₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₂₆₃

commutator

C₂×C₄×C₂₆₃

C₂²×C₄

Frattini

C₄²

C₂²×D₂₆₃

Fitting

#533×C₂₆₃

C₂

socle

C₂²×C₂₆₃

C₂×C₂²⋊C₄

prime radical

C₂³×C₂₆₃

C₂²⋊C₄

cosocle

C₄²×C₂₆₃

C₂³

D₂₆₃

C₂₆₃

G ▷ C₂×C₄×C₂₆₃ ▷ C₂²×C₂₆₃ ▷ C₂₆₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #30681684	C₂	semidirect
50. C₂₆₃	#1561	semidirect

H ≅ G/N

type

2. #30681684

C₂

semidirect

50. C₂₆₃

#1561

semidirect

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

subgroup	index
#30681962	2
#30681684	2
#533×C₂₆₃	2
#1561	263

subgroup

index

#30681962

#30681684

#533×C₂₆₃

#1561

263

Subgroups — #46781485

Conjugacy classes of subgroups

Complements

Maximal subgroups