Group #46782650 — subgroups

Subgroups — C₄×#22554405 #46782650

Order 33664.

228 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 121 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×#22554405	C₂	1	2	yes	maximal
2	C₄×#16862360	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₂₆₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#30681702	C₂	1	2	yes	maximal
5	C₄×M₁₆×C₂₆₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#16862360	C₂²	1	2	yes	—
7	C₂²×C₂₆₃⋊C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₂₆₃⋊C₈	C₂²	1	1	yes	characteristic
9	#22554405	C₄	1	8	yes	direct product factor
10	#22554381	—	2	1	no	—
11	C₄×C₂₆₃⋊C₈	—	2	1	no	—
12	C₄×C₂₆₃⋊C₈	C₄	1	4	yes	—
13	C₂×M₁₆×C₂₆₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₂₆₃	C₂²	1	1	yes	characteristic
15	#97×C₂₆₃	—	2	1	no	—
16	C₄×C₈×C₂₆₃	—	2	1	no	—
17	#16862360	—	2	2	no	—
18	#16862360	C₂×C₄	1	4	yes	—
19	C₂×C₂₆₃⋊C₈	—	2	2	no	—
20	C₂×C₂₆₃⋊C₈	—	2	2	no	—
21	C₂×C₂₆₃⋊C₈	C₂×C₄	1	2	yes	—
22	C₂×C₂₆₃⋊C₈	C₂×C₄	1	2	yes	—
23	C₂×C₂₆₃⋊C₈	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₂₆₃	Q₈	1	2	yes	—
25	C₂²×C₄×C₂₆₃	C₂³	1	1	yes	characteristic
26	M₁₆×C₂₆₃	—	2	2	no	—
27	M₁₆×C₂₆₃	C₂×C₄	1	4	yes	—
28	C₂×C₈×C₂₆₃	—	2	2	no	—
29	C₄²×C₂₆₃	C₂×C₄	1	2	yes	—
30	C₄²×C₂₆₃	D₄	1	1	yes	characteristic
31	C₄²×C₂₆₃	D₄	1	1	yes	characteristic
32	C₂₆₃⋊C₈	—	2	2	no	—
33	C₂₆₃⋊C₈	—	2	4	no	—
34	C₂₆₃⋊C₈	C₄²	1	4	yes	—
35	C₂³×C₂₆₃	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₄×C₂₆₃	—	2	4	no	—
37	C₂×C₄×C₂₆₃	C₄⋊C₄	1	2	yes	—
38	C₂×C₄×C₂₆₃	C₄⋊C₄	1	2	yes	—
39	C₂×C₄×C₂₆₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₂₆₃	C₂²×C₄	1	2	yes	—
41	C₂×C₄×C₂₆₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₆₃	#41	1	2	yes	—
43	C₈×C₂₆₃	—	2	4	no	—
44	C₂²×C₂₆₃	—	2	2	no	—
45	C₂²×C₂₆₃	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₂₆₃	C₄×Q₈	1	2	yes	—
47	C₄×C₂₆₃	—	2	2	no	—
48	C₄×C₂₆₃	#108	1	4	yes	complemented
49	C₄×C₂₆₃	C₂×C₄²	1	1	yes	characteristic
50	C₄×C₂₆₃	#117	1	1	yes	characteristic
51	C₄×C₂₆₃	C₄×D₄	1	2	yes	—
52	C₂×C₂₆₃	—	2	2	no	—
53	C₂×C₂₆₃	C₄×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₂₆₃	C₂×#108	1	2	yes	—
55	C₂₆₃	C₄×#108	1	1	yes	characteristic, complemented, Sylow
56	C₄×#108	—	263	1	no	maximal, Sylow
57	C₂×#108	—	263	2	no	—
58	C₄×M₁₆	—	263	1	no	—
59	C₄×M₁₆	D₂₆₃	1	1	yes	characteristic
60	C₂×C₄×C₈	—	263	1	no	—
61	#341	—	263	2	no	—
62	C₂×M₁₆	—	263	2	no	—
63	C₂×M₁₆	C₂×D₂₆₃	1	2	yes	—
64	C₂²×C₈	—	263	1	no	—
65	C₂²×C₈	—	263	1	no	—
66	C₂×C₄²	C₂×D₂₆₃	1	1	yes	characteristic
67	#108	—	263	8	no	—
68	#97	—	2	1	no	—
69	#97	—	526	1	no	—
70	C₄×C₈	—	2	1	no	—
71	C₄×C₈	—	263	4	no	—
72	C₄×C₈	—	526	1	no	—
73	C₂²×C₄	Dic₅₂₆	1	2	yes	—
74	C₂²×C₄	C₂²×D₂₆₃	1	1	yes	characteristic
75	M₁₆	—	2	2	no	—
76	M₁₆	—	263	4	no	—
77	M₁₆	—	526	2	no	—
78	M₁₆	C₄×D₂₆₃	1	4	yes	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	263	2	no	—
81	C₂×C₈	—	263	4	no	—
82	C₂×C₈	—	263	2	no	—
83	C₂×C₈	—	526	2	no	—
84	C₂×C₈	—	526	2	no	—
85	C₄²	C₄×D₂₆₃	1	2	yes	—
86	C₄²	C₂²⋊D₂₆₃	1	1	yes	characteristic
87	C₄²	C₂²⋊D₂₆₃	1	1	yes	characteristic
88	C₂³	C₂×Dic₅₂₆	1	1	yes	characteristic
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	Dic₂₆₃⋊C₄	1	2	yes	—
91	C₂×C₄	Dic₂₆₃⋊C₄	1	2	yes	—
92	C₂×C₄	C₂×C₄×D₂₆₃	1	1	yes	characteristic
93	C₂×C₄	C₂×C₄×D₂₆₃	1	2	yes	—
94	C₂×C₄	C₂×C₂²⋊D₂₆₃	1	1	yes	characteristic
95	C₂×C₄	#16862376	1	2	yes	—
96	C₈	—	2	4	no	—
97	C₈	—	263	4	no	—
98	C₈	—	526	2	no	—
99	C₈	—	526	4	no	—
100	C₂²	—	2	2	no	—
101	C₂²	C₂×Dic₂₆₃⋊C₄	1	1	yes	characteristic
102	C₂²	C₄×Dic₅₂₆	1	2	yes	—
103	C₄	—	2	2	no	—
104	C₄	#22554405	1	4	yes	direct product factor
105	C₄	C₄²×D₂₆₃	1	1	yes	characteristic
106	C₄	#22554427	1	1	yes	characteristic
107	C₄	C₄×C₂²⋊D₂₆₃	1	2	yes	—
108	C₂	—	2	2	no	—
109	C₂	C₄×Dic₂₆₃⋊C₄	1	1	yes	characteristic
110	C₂	C₂×#22554405	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22554405

C₂

yes

maximal

C₄×#16862360

C₂

yes

maximal, characteristic

C₂×C₄×C₂₆₃⋊C₈

C₂

yes

maximal, characteristic

#30681702

C₂

yes

maximal

C₄×M₁₆×C₂₆₃

C₂

yes

maximal, characteristic

C₂×#16862360

C₂²

yes

—

C₂²×C₂₆₃⋊C₈

C₂²

yes

characteristic

C₂²×C₂₆₃⋊C₈

C₂²

yes

characteristic

#22554405

C₄

yes

direct product factor

#22554381

—

C₄×C₂₆₃⋊C₈

—

C₄×C₂₆₃⋊C₈

C₄

yes

—

C₂×M₁₆×C₂₆₃

C₂²

yes

—

C₂×C₄²×C₂₆₃

C₂²

yes

characteristic

#97×C₂₆₃

—

C₄×C₈×C₂₆₃

—

#16862360

—

#16862360

C₂×C₄

yes

—

C₂×C₂₆₃⋊C₈

—

C₂×C₂₆₃⋊C₈

—

C₂×C₂₆₃⋊C₈

C₂×C₄

yes

—

C₂×C₂₆₃⋊C₈

C₂×C₄

yes

—

C₂×C₂₆₃⋊C₈

C₂×C₄

yes

—

C₂²×C₄×C₂₆₃

Q₈

yes

—

C₂²×C₄×C₂₆₃

C₂³

yes

characteristic

M₁₆×C₂₆₃

—

M₁₆×C₂₆₃

C₂×C₄

yes

—

C₂×C₈×C₂₆₃

—

C₄²×C₂₆₃

C₂×C₄

yes

—

C₄²×C₂₆₃

D₄

yes

characteristic

C₄²×C₂₆₃

D₄

yes

characteristic

C₂₆₃⋊C₈

—

C₂₆₃⋊C₈

—

C₂₆₃⋊C₈

C₄²

yes

—

C₂³×C₂₆₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₆₃

—

C₂×C₄×C₂₆₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₆₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₆₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₆₃

C₂²×C₄

yes

—

C₂×C₄×C₂₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₃

#41

yes

—

C₈×C₂₆₃

—

C₂²×C₂₆₃

—

C₂²×C₂₆₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₆₃

C₄×Q₈

yes

—

C₄×C₂₆₃

—

C₄×C₂₆₃

#108

yes

complemented

C₄×C₂₆₃

C₂×C₄²

yes

characteristic

C₄×C₂₆₃

#117

yes

characteristic

C₄×C₂₆₃

C₄×D₄

yes

—

C₂×C₂₆₃

—

C₂×C₂₆₃

C₄×C₄⋊C₄

yes

characteristic

C₂×C₂₆₃

C₂×#108

yes

—

C₂₆₃

C₄×#108

yes

characteristic, complemented, Sylow

C₄×#108

—

263

maximal, Sylow

C₂×#108

—

263

—

C₄×M₁₆

—

263

—

C₄×M₁₆

D₂₆₃

yes

characteristic

C₂×C₄×C₈

—

263

—

#341

—

263

—

C₂×M₁₆

—

263

—

C₂×M₁₆

C₂×D₂₆₃

yes

—

C₂²×C₈

—

263

—

C₂²×C₈

—

263

—

C₂×C₄²

C₂×D₂₆₃

yes

characteristic

#108

—

263

—

#97

—

#97

—

526

—

C₄×C₈

—

C₄×C₈

—

263

—

C₄×C₈

—

526

—

C₂²×C₄

Dic₅₂₆

yes

—

C₂²×C₄

C₂²×D₂₆₃

yes

characteristic

M₁₆

—

M₁₆

—

263

—

M₁₆

—

526

—

M₁₆

C₄×D₂₆₃

yes

—

C₂×C₈

—

C₂×C₈

—

263

—

C₂×C₈

—

263

—

C₂×C₈

—

263

—

C₂×C₈

—

526

—

C₂×C₈

—

526

—

C₄²

C₄×D₂₆₃

yes

—

C₄²

C₂²⋊D₂₆₃

yes

characteristic

C₄²

C₂²⋊D₂₆₃

yes

characteristic

C₂³

C₂×Dic₅₂₆

yes

characteristic

C₂×C₄

—

C₂×C₄

Dic₂₆₃⋊C₄

yes

—

C₂×C₄

Dic₂₆₃⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₂₆₃

yes

characteristic

C₂×C₄

C₂×C₄×D₂₆₃

yes

—

C₂×C₄

C₂×C₂²⋊D₂₆₃

yes

characteristic

C₂×C₄

#16862376

yes

—

C₈

—

C₈

—

263

—

C₈

—

526

—

C₈

—

526

—

100

C₂²

—

101

C₂²

C₂×Dic₂₆₃⋊C₄

yes

characteristic

102

C₂²

C₄×Dic₅₂₆

yes

—

103

C₄

—

104

C₄

#22554405

yes

direct product factor

105

C₄

C₄²×D₂₆₃

yes

characteristic

106

C₄

#22554427

yes

characteristic

107

C₄

C₄×C₂²⋊D₂₆₃

yes

—

108

C₂

—

109

C₂

C₄×Dic₂₆₃⋊C₄

yes

characteristic

110

C₂

C₂×#22554405

yes

—

	subgroup	G/H
center	C₄²	C₂²⋊D₂₆₃
commutator	C₄×C₂₆₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₂₆₃
Fitting	C₄×M₁₆×C₂₆₃	C₂
socle	C₂²×C₂₆₃	C₂×C₄⋊C₄
prime radical	C₂³×C₂₆₃	C₂×Q₈
cosocle	C₂²×C₄×C₂₆₃	C₂³
hypercenter	C₄×M₁₆	D₂₆₃
hypocenter	C₂₆₃	C₄×#108
LCS	G ▷ C₄×C₂₆₃ ▷ C₂×C₂₆₃ ▷ C₂₆₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₄²

C₂²⋊D₂₆₃

commutator

C₄×C₂₆₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₂₆₃

Fitting

C₄×M₁₆×C₂₆₃

C₂

socle

C₂²×C₂₆₃

C₂×C₄⋊C₄

prime radical

C₂³×C₂₆₃

C₂×Q₈

cosocle

C₂²×C₄×C₂₆₃

C₂³

hypercenter

C₄×M₁₆

D₂₆₃

hypocenter

C₂₆₃

C₄×#108

LCS

G ▷ C₄×C₂₆₃ ▷ C₂×C₂₆₃ ▷ C₂₆₃

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
9. #22554405	C₄	direct (N × H)
48. C₄×C₂₆₃	#108	semidirect
55. C₂₆₃	C₄×#108	semidirect
104. C₄	#22554405	direct (N × H)

H ≅ G/N

type

9. #22554405

C₄

direct (N × H)

48. C₄×C₂₆₃

#108

semidirect

55. C₂₆₃

C₄×#108

semidirect

104. C₄

#22554405

direct (N × H)

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

subgroup	index
C₂×#22554405	2
C₄×#16862360	2
C₂×C₄×C₂₆₃⋊C₈	2
#30681702	2
C₄×M₁₆×C₂₆₃	2
C₄×#108	263

subgroup

index

C₂×C₄×C₂₆₃⋊C₈

C₄×M₁₆×C₂₆₃

263

Subgroups — C₄×#22554405 #46782650

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C4×#22554405 #46782650

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#22554405 #46782650