Group #35212371 — subgroups

Subgroups — #35212371

Order 20864.

180 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 113 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×C₄⋊(C₁₆₃⋊C₈)	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×C₁₆₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₁₆₃⋊C₈	—	2	2	no	—
4	C₄⋊(C₁₆₃⋊C₈)	—	2	2	no	—
5	C₄⋊(C₁₆₃⋊C₈)	C₄	1	4	yes	—
6	C₂²×C₈×C₁₆₃	C₄	1	2	yes	—
7	C₂×C₄²×C₁₆₃	C₂²	1	1	yes	characteristic
8	C₄×C₈×C₁₆₃	—	2	2	no	—
9	C₂×C₁₆₃⋊C₈	—	2	4	no	—
10	C₂×C₁₆₃⋊C₈	—	4	2	no	—
11	C₂×C₁₆₃⋊C₈	—	4	2	no	—
12	C₂²×C₄×C₁₆₃	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₄×C₁₆₃	D₄	1	1	yes	characteristic
14	C₂²×C₄×C₁₆₃	Q₈	1	1	yes	characteristic
15	C₂×C₈×C₁₆₃	—	2	2	no	—
16	C₂×C₈×C₁₆₃	—	2	2	no	—
17	C₂×C₈×C₁₆₃	C₈	1	4	yes	complemented
18	C₄²×C₁₆₃	C₂×C₄	1	2	yes	—
19	C₄²×C₁₆₃	D₄	1	2	yes	—
20	C₁₆₃⋊C₈	—	4	4	no	—
21	C₂³×C₁₆₃	C₄⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₁₆₃	—	2	2	no	—
23	C₂×C₄×C₁₆₃	C₄²	1	1	yes	characteristic
24	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	2	yes	—
25	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₄×C₁₆₃	C₄⋊C₄	1	2	yes	—
27	C₂×C₄×C₁₆₃	C₂×C₈	1	2	yes	—
28	C₂×C₄×C₁₆₃	M₁₆	1	2	yes	—
29	C₂×C₄×C₁₆₃	D₈	1	1	yes	characteristic
30	C₂×C₄×C₁₆₃	SD₁₆	1	2	yes	—
31	C₂×C₄×C₁₆₃	Q₁₆	1	1	yes	characteristic
32	C₈×C₁₆₃	—	2	4	no	—
33	C₂²×C₁₆₃	#95	1	1	yes	characteristic
34	C₂²×C₁₆₃	C₄⋊C₈	1	2	yes	—
35	C₂²×C₁₆₃	#106	1	1	yes	characteristic
36	C₂²×C₁₆₃	#107	1	1	yes	characteristic
37	C₂²×C₁₆₃	#108	1	2	yes	—
38	C₄×C₁₆₃	—	2	2	no	—
39	C₄×C₁₆₃	C₄×C₈	1	1	yes	characteristic
40	C₄×C₁₆₃	#97	1	1	yes	characteristic
41	C₄×C₁₆₃	C₂²⋊C₈	1	2	yes	—
42	C₄×C₁₆₃	#102	1	2	yes	—
43	C₄×C₁₆₃	#103	1	2	yes	—
44	C₂×C₁₆₃	#334	1	2	yes	—
45	C₂×C₁₆₃	#335	1	2	yes	—
46	C₂×C₁₆₃	#336	1	1	yes	characteristic
47	C₂×C₁₆₃	#340	1	1	yes	characteristic
48	C₂×C₁₆₃	#341	1	1	yes	characteristic
49	C₁₆₃	#1277	1	1	yes	characteristic, complemented, Sylow
50	#1277	—	163	1	no	maximal, Sylow
51	C₂×C₄⋊C₈	—	163	2	no	—
52	C₂×C₄×C₈	D₁₆₃	1	1	yes	characteristic
53	C₂²×C₈	—	326	2	no	—
54	C₂²×C₈	Dic₁₆₃	1	2	yes	—
55	C₂×C₄²	C₂×D₁₆₃	1	1	yes	characteristic
56	C₄⋊C₈	—	326	2	no	—
57	C₄⋊C₈	—	163	4	no	—
58	C₄×C₈	—	2	2	no	—
59	C₂²×C₄	C₂×Dic₁₆₃	1	1	yes	characteristic
60	C₂²×C₄	D₆₅₂	1	1	yes	characteristic
61	C₂²×C₄	Dic₃₂₆	1	1	yes	characteristic
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	—	326	4	no	—
65	C₂×C₈	—	652	2	no	—
66	C₂×C₈	—	652	2	no	—
67	C₂×C₈	C₁₆₃⋊C₈	1	4	yes	complemented
68	C₄²	C₄×D₁₆₃	1	2	yes	—
69	C₄²	C₂²⋊D₁₆₃	1	2	yes	—
70	C₂³	C₄⋊Dic₁₆₃	1	1	yes	characteristic
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	C₄×Dic₁₆₃	1	1	yes	characteristic
73	C₂×C₄	(C₂×D₁₆₃)⋊C₄	1	2	yes	—
74	C₂×C₄	C₂²⋊Dic₁₆₃	1	1	yes	characteristic
75	C₂×C₄	Dic₁₆₃⋊C₄	1	2	yes	—
76	C₂×C₄	C₂×C₁₆₃⋊C₈	1	2	yes	—
77	C₂×C₄	#14579683	1	2	yes	—
78	C₂×C₄	D₁₃₀₄	1	1	yes	characteristic
79	C₂×C₄	#14579688	1	2	yes	—
80	C₂×C₄	Dic₆₅₂	1	1	yes	characteristic
81	C₈	—	2	4	no	—
82	C₈	—	652	4	no	—
83	C₂²	C₁₆₃⋊#95	1	1	yes	characteristic
84	C₂²	C₄⋊(C₁₆₃⋊C₈)	1	2	yes	—
85	C₂²	#19104667	1	1	yes	characteristic
86	C₂²	#19104669	1	1	yes	characteristic
87	C₂²	#19104671	1	2	yes	—
88	C₄	—	2	2	no	—
89	C₄	C₄×C₁₆₃⋊C₈	1	1	yes	characteristic
90	C₄	#19104646	1	1	yes	characteristic
91	C₄	C₂²⋊(C₁₆₃⋊C₈)	1	2	yes	—
92	C₄	#19104657	1	2	yes	—
93	C₄	#19104660	1	2	yes	—
94	C₂	#23781416	1	2	yes	—
95	C₂	#23781418	1	2	yes	—
96	C₂	#23781420	1	1	yes	characteristic
97	C₂	#23781428	1	1	yes	characteristic
98	C₂	#23781430	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₁₆₃⋊C₈)

C₂

yes

maximal

C₂×C₄×C₈×C₁₆₃

C₂

yes

maximal, characteristic

C₂²×C₁₆₃⋊C₈

—

C₄⋊(C₁₆₃⋊C₈)

—

C₄⋊(C₁₆₃⋊C₈)

C₄

yes

—

C₂²×C₈×C₁₆₃

C₄

yes

—

C₂×C₄²×C₁₆₃

C₂²

yes

characteristic

C₄×C₈×C₁₆₃

—

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

—

C₂²×C₄×C₁₆₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₆₃

D₄

yes

characteristic

C₂²×C₄×C₁₆₃

Q₈

yes

characteristic

C₂×C₈×C₁₆₃

—

C₂×C₈×C₁₆₃

—

C₂×C₈×C₁₆₃

C₈

yes

complemented

C₄²×C₁₆₃

C₂×C₄

yes

—

C₄²×C₁₆₃

D₄

yes

—

C₁₆₃⋊C₈

—

C₂³×C₁₆₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

C₄²

yes

characteristic

C₂×C₄×C₁₆₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₆₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₆₃

C₄⋊C₄

yes

—

C₂×C₄×C₁₆₃

C₂×C₈

yes

—

C₂×C₄×C₁₆₃

M₁₆

yes

—

C₂×C₄×C₁₆₃

D₈

yes

characteristic

C₂×C₄×C₁₆₃

SD₁₆

yes

—

C₂×C₄×C₁₆₃

Q₁₆

yes

characteristic

C₈×C₁₆₃

—

C₂²×C₁₆₃

#95

yes

characteristic

C₂²×C₁₆₃

C₄⋊C₈

yes

—

C₂²×C₁₆₃

yes

C₂²×C₁₆₃

yes

C₂²×C₁₆₃

yes

—

C₄×C₁₆₃

—

C₄×C₁₆₃

C₄×C₈

yes

characteristic

C₄×C₁₆₃

#97

yes

characteristic

C₄×C₁₆₃

C₂²⋊C₈

yes

—

C₄×C₁₆₃

#102

yes

—

C₄×C₁₆₃

#103

yes

—

C₂×C₁₆₃

#334

yes

—

C₂×C₁₆₃

#335

yes

—

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

C₁₆₃

yes

characteristic, complemented, Sylow

#1277

—

163

maximal, Sylow

C₂×C₄⋊C₈

—

163

—

C₂×C₄×C₈

D₁₆₃

yes

characteristic

C₂²×C₈

—

326

—

C₂²×C₈

Dic₁₆₃

yes

—

C₂×C₄²

C₂×D₁₆₃

yes

characteristic

C₄⋊C₈

—

326

—

C₄⋊C₈

—

163

—

C₄×C₈

—

C₂²×C₄

C₂×Dic₁₆₃

yes

characteristic

C₂²×C₄

D₆₅₂

yes

characteristic

C₂²×C₄

Dic₃₂₆

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

326

—

C₂×C₈

—

652

—

C₂×C₈

—

652

—

C₂×C₈

C₁₆₃⋊C₈

yes

complemented

C₄²

C₄×D₁₆₃

yes

—

C₄²

C₂²⋊D₁₆₃

yes

—

C₂³

C₄⋊Dic₁₆₃

yes

characteristic

C₂×C₄

—

C₂×C₄

C₄×Dic₁₆₃

yes

characteristic

C₂×C₄

(C₂×D₁₆₃)⋊C₄

yes

—

C₂×C₄

C₂²⋊Dic₁₆₃

yes

characteristic

C₂×C₄

Dic₁₆₃⋊C₄

yes

—

C₂×C₄

C₂×C₁₆₃⋊C₈

yes

—

C₂×C₄

#14579683

yes

—

C₂×C₄

D₁₃₀₄

yes

characteristic

C₂×C₄

#14579688

yes

—

C₂×C₄

Dic₆₅₂

yes

characteristic

C₈

—

C₈

—

652

—

C₂²

C₁₆₃⋊#95

yes

characteristic

C₂²

C₄⋊(C₁₆₃⋊C₈)

yes

—

C₂²

yes

C₂²

yes

C₂²

yes

—

C₄

—

C₄

C₄×C₁₆₃⋊C₈

yes

characteristic

C₄

#19104646

yes

characteristic

C₄

C₂²⋊(C₁₆₃⋊C₈)

yes

—

C₄

#19104657

yes

—

C₄

#19104660

yes

—

C₂

#23781416

yes

—

C₂

#23781418

yes

—

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂²×C₄	D₆₅₂
commutator	C₄×C₁₆₃	C₄×C₈
Frattini	C₂×C₄²	C₂×D₁₆₃
Fitting	C₂×C₄×C₈×C₁₆₃	C₂
socle, prime radical	C₂³×C₁₆₃	C₄⋊C₄
cosocle	C₂×C₄²×C₁₆₃	C₂²
hypercenter	C₂×C₄×C₈	D₁₆₃
hypocenter	C₁₆₃	#1277
LCS	G ▷ C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂²×C₄

D₆₅₂

commutator

C₄×C₁₆₃

C₄×C₈

Frattini

C₂×C₄²

C₂×D₁₆₃

Fitting

C₂×C₄×C₈×C₁₆₃

C₂

socle, prime radical

C₂³×C₁₆₃

C₄⋊C₄

cosocle

C₂×C₄²×C₁₆₃

C₂²

hypercenter

C₂×C₄×C₈

D₁₆₃

hypocenter

C₁₆₃

#1277

LCS

G ▷ C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
17. C₂×C₈×C₁₆₃	C₈	semidirect
49. C₁₆₃	#1277	semidirect
67. C₂×C₈	C₁₆₃⋊C₈	semidirect

H ≅ G/N

type

17. C₂×C₈×C₁₆₃

C₈

semidirect

49. C₁₆₃

#1277

semidirect

67. C₂×C₈

C₁₆₃⋊C₈

semidirect

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

subgroup	index
C₂×C₄⋊(C₁₆₃⋊C₈)	2
C₂×C₄×C₈×C₁₆₃	2
#1277	163

subgroup

index

C₂×C₄⋊(C₁₆₃⋊C₈)

C₂×C₄×C₈×C₁₆₃

#1277

163

Subgroups — #35212371

Conjugacy classes of subgroups

Complements

Maximal subgroups