Group #46782413 — subgroups

Subgroups — #46782413

Order 33664.

242 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 69 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	#30682315	C₂	1	1	yes	maximal, characteristic, complemented
2	#30682315	C₂	1	2	yes	maximal, complemented
3	#30682311	C₂	1	1	yes	maximal, characteristic, complemented
4	#30682285	C₂	1	2	yes	maximal, complemented
5	#322×C₂₆₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16862365	—	2	2	no	—
7	C₂×D₂₁₀₄	—	2	4	no	—
8	C₂×D₂₁₀₄	—	2	2	no	—
9	#22554470	C₂²	1	1	yes	characteristic
10	#22554467	C₂²	1	2	yes	—
11	#22554467	C₂²	1	1	yes	characteristic
12	#22554392	—	2	4	no	—
13	C₄×C₈×C₂₆₃	C₂²	1	2	yes	—
14	C₄×C₈×C₂₆₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₅₂₆	—	2	1	no	—
16	C₂×D₁₀₅₂	—	2	2	no	—
17	C₂×D₁₀₅₂	—	2	2	no	—
18	C₂×D₁₀₅₂	—	2	2	no	—
19	C₂×D₁₀₅₂	—	2	2	no	—
20	C₂×D₁₀₅₂	—	2	1	no	—
21	#16862365	—	4	4	no	—
22	D₂₁₀₄	—	4	8	no	—
23	D₂₁₀₄	—	4	4	no	—
24	C₄⋊Dic₂₆₃	—	2	2	no	—
25	C₂×C₈×C₂₆₃	D₄	1	4	yes	—
26	C₂×C₈×C₂₆₃	D₄	1	2	yes	—
27	C₄²×C₂₆₃	C₂³	1	1	yes	characteristic
28	C₂²×D₂₆₃	—	4	2	no	—
29	C₂²×D₂₆₃	—	4	1	no	—
30	Dic₅₂₆	—	4	2	no	—
31	D₁₀₅₂	—	4	4	no	—
32	D₁₀₅₂	—	4	2	no	—
33	D₁₀₅₂	—	4	4	no	—
34	D₁₀₅₂	—	4	4	no	—
35	D₁₀₅₂	—	4	4	no	—
36	C₂×Dic₂₆₃	—	4	1	no	—
37	C₂×C₄×C₂₆₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₂₆₃	C₂×D₄	1	2	yes	—
39	C₈×C₂₆₃	—	2	4	no	—
40	C₈×C₂₆₃	D₈	1	4	yes	complemented
41	C₂²×C₂₆₃	#127	1	1	yes	characteristic
42	C₂×D₂₆₃	—	8	2	no	—
43	C₂×D₂₆₃	—	8	1	no	—
44	C₂×D₂₆₃	—	8	2	no	—
45	C₂×D₂₆₃	—	8	2	no	—
46	C₂×D₂₆₃	—	8	2	no	—
47	C₄×C₂₆₃	C₂×D₈	1	2	yes	—
48	C₄×C₂₆₃	AGL(1,ℤ/8)	1	4	yes	—
49	Dic₂₆₃	—	8	1	no	—
50	C₂×C₂₆₃	#493	1	1	yes	characteristic
51	C₂×C₂₆₃	#496	1	2	yes	—
52	D₂₆₃	—	16	2	no	—
53	D₂₆₃	—	16	1	no	—
54	C₂₆₃	#1721	1	1	yes	characteristic, complemented, Sylow
55	#1721	—	263	1	no	maximal, Sylow
56	#493	—	263	2	no	—
57	#493	—	263	1	no	—
58	#492	—	263	1	no	—
59	#486	—	263	2	no	—
60	#322	D₂₆₃	1	1	yes	characteristic, complemented
61	C₂×SD₁₆	—	526	2	no	—
62	C₂×D₈	—	526	4	no	—
63	C₂×D₈	—	526	2	no	—
64	C₄⋊Q₈	—	263	1	no	—
65	#127	—	263	2	no	—
66	#127	—	263	1	no	—
67	#102	—	526	4	no	—
68	C₄×C₈	C₂×D₂₆₃	1	2	yes	—
69	C₄×C₈	C₂×D₂₆₃	1	1	yes	characteristic
70	C₂×Q₈	—	526	1	no	—
71	C₂×D₄	—	526	2	no	—
72	C₂×D₄	—	526	2	no	—
73	C₂×D₄	—	526	2	no	—
74	C₂×D₄	—	526	2	no	—
75	C₂×D₄	—	526	1	no	—
76	SD₁₆	—	1052	4	no	—
77	D₈	—	1052	8	no	—
78	D₈	—	1052	4	no	—
79	C₂×C₈	D₁₀₅₂	1	4	yes	—
80	C₂×C₈	D₁₀₅₂	1	2	yes	—
81	C₄⋊C₄	—	526	2	no	—
82	C₄²	C₂²×D₂₆₃	1	1	yes	characteristic
83	C₂³	—	1052	2	no	—
84	C₂³	—	1052	1	no	—
85	Q₈	—	1052	2	no	—
86	D₄	—	1052	4	no	—
87	D₄	—	1052	2	no	—
88	D₄	—	1052	4	no	—
89	D₄	—	1052	4	no	—
90	D₄	—	1052	4	no	—
91	C₂×C₄	—	1052	1	no	—
92	C₂×C₄	C₂×D₁₀₅₂	1	1	yes	characteristic
93	C₂×C₄	C₂×D₁₀₅₂	1	2	yes	—
94	C₈	—	2	4	no	—
95	C₈	D₂₁₀₄	1	4	yes	complemented
96	C₂²	—	2104	2	no	—
97	C₂²	—	2104	1	no	—
98	C₂²	—	2104	2	no	—
99	C₂²	—	2104	2	no	—
100	C₂²	—	2104	2	no	—
101	C₂²	#22554467	1	1	yes	characteristic
102	C₄	—	2104	1	no	—
103	C₄	C₂×D₂₁₀₄	1	2	yes	—
104	C₄	#22554498	1	4	yes	—
105	C₂	—	4208	2	no	—
106	C₂	—	4208	1	no	—
107	C₂	#30682315	1	1	yes	characteristic
108	C₂	#30682327	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#30682315

C₂

yes

maximal, characteristic, complemented

#30682315

C₂

yes

maximal, complemented

#30682311

C₂

yes

maximal, characteristic, complemented

#30682285

C₂

yes

maximal, complemented

#322×C₂₆₃

C₂

yes

maximal, characteristic, complemented

C₂×#16862365

—

C₂×D₂₁₀₄

—

C₂×D₂₁₀₄

—

#22554470

C₂²

yes

characteristic

#22554467

C₂²

yes

—

#22554467

C₂²

yes

characteristic

#22554392

—

C₄×C₈×C₂₆₃

C₂²

yes

—

C₄×C₈×C₂₆₃

C₂²

yes

characteristic

C₂×Dic₅₂₆

—

C₂×D₁₀₅₂

—

C₂×D₁₀₅₂

—

C₂×D₁₀₅₂

—

C₂×D₁₀₅₂

—

C₂×D₁₀₅₂

—

#16862365

—

D₂₁₀₄

—

D₂₁₀₄

—

C₄⋊Dic₂₆₃

—

C₂×C₈×C₂₆₃

D₄

yes

—

C₂×C₈×C₂₆₃

D₄

yes

—

C₄²×C₂₆₃

C₂³

yes

characteristic

C₂²×D₂₆₃

—

C₂²×D₂₆₃

—

Dic₅₂₆

—

D₁₀₅₂

—

D₁₀₅₂

—

D₁₀₅₂

—

D₁₀₅₂

—

D₁₀₅₂

—

C₂×Dic₂₆₃

—

C₂×C₄×C₂₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₃

C₂×D₄

yes

—

C₈×C₂₆₃

—

C₈×C₂₆₃

D₈

yes

complemented

C₂²×C₂₆₃

#127

yes

characteristic

C₂×D₂₆₃

—

C₂×D₂₆₃

—

C₂×D₂₆₃

—

C₂×D₂₆₃

—

C₂×D₂₆₃

—

C₄×C₂₆₃

C₂×D₈

yes

—

C₄×C₂₆₃

AGL(1,ℤ/8)

yes

—

Dic₂₆₃

—

C₂×C₂₆₃

#493

yes

characteristic

C₂×C₂₆₃

#496

yes

—

D₂₆₃

—

D₂₆₃

—

C₂₆₃

#1721

yes

characteristic, complemented, Sylow

#1721

—

263

maximal, Sylow

#493

—

263

—

#493

—

263

—

#492

—

263

—

#486

—

263

—

#322

D₂₆₃

yes

characteristic, complemented

C₂×SD₁₆

—

526

—

C₂×D₈

—

526

—

C₂×D₈

—

526

—

C₄⋊Q₈

—

263

—

#127

—

263

—

#127

—

263

—

#102

—

526

—

C₄×C₈

C₂×D₂₆₃

yes

—

C₄×C₈

C₂×D₂₆₃

yes

characteristic

C₂×Q₈

—

526

—

C₂×D₄

—

526

—

C₂×D₄

—

526

—

C₂×D₄

—

526

—

C₂×D₄

—

526

—

C₂×D₄

—

526

—

SD₁₆

—

1052

—

D₈

—

1052

—

D₈

—

1052

—

C₂×C₈

D₁₀₅₂

yes

—

C₂×C₈

D₁₀₅₂

yes

—

C₄⋊C₄

—

526

—

C₄²

C₂²×D₂₆₃

yes

characteristic

C₂³

—

1052

—

C₂³

—

1052

—

Q₈

—

1052

—

D₄

—

1052

—

D₄

—

1052

—

D₄

—

1052

—

D₄

—

1052

—

D₄

—

1052

—

C₂×C₄

—

1052

—

C₂×C₄

C₂×D₁₀₅₂

yes

characteristic

C₂×C₄

C₂×D₁₀₅₂

yes

—

C₈

—

C₈

D₂₁₀₄

yes

complemented

C₂²

—

2104

—

C₂²

—

2104

—

C₂²

—

2104

—

C₂²

—

2104

—

100

C₂²

—

2104

—

101

C₂²

#22554467

yes

characteristic

102

C₄

—

2104

—

103

C₄

C₂×D₂₁₀₄

yes

—

104

C₄

#22554498

yes

—

105

C₂

—

4208

—

106

C₂

—

4208

—

107

C₂

#30682315

yes

characteristic

108

C₂

#30682327

yes

—

	subgroup	G/H
center	C₂²	#22554467
commutator, cosocle	C₄²×C₂₆₃	C₂³
Frattini	C₄²	C₂²×D₂₆₃
Fitting	#322×C₂₆₃	C₂
socle	C₂²×C₂₆₃	#127
hypercenter	#322	D₂₆₃
hypocenter	C₂₆₃	#1721
LCS	G ▷ C₄²×C₂₆₃ ▷ C₂²×C₂₆₃ ▷ C₂₆₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#22554467

commutator, cosocle

C₄²×C₂₆₃

C₂³

Frattini

C₄²

C₂²×D₂₆₃

Fitting

#322×C₂₆₃

C₂

C₂²×C₂₆₃

D₂₆₃

C₂₆₃

G ▷ C₄²×C₂₆₃ ▷ C₂²×C₂₆₃ ▷ C₂₆₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
1. #30682315	C₂	semidirect
2. #30682315	C₂	semidirect
3. #30682311	C₂	semidirect
4. #30682285	C₂	semidirect
5. #322×C₂₆₃	C₂	semidirect
40. C₈×C₂₆₃	D₈	semidirect
54. C₂₆₃	#1721	semidirect
60. #322	D₂₆₃	semidirect
95. C₈	D₂₁₀₄	semidirect

H ≅ G/N

type

1. #30682315

C₂

semidirect

2. #30682315

C₂

semidirect

3. #30682311

C₂

semidirect

4. #30682285

C₂

semidirect

5. #322×C₂₆₃

C₂

semidirect

40. C₈×C₂₆₃

D₈

semidirect

54. C₂₆₃

#1721

semidirect

60. #322

D₂₆₃

semidirect

95. C₈

D₂₁₀₄

semidirect

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

subgroup	index
#30682315	2
#30682311	2
#30682285	2
#322×C₂₆₃	2
#1721	263

subgroup

index

#322×C₂₆₃

263

Subgroups — #46782413

Conjugacy classes of subgroups

Complements

Maximal subgroups