Group #35212419 — subgroups

Subgroups — #35212419

Order 20864.

188 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 105 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	#384×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	complemented
6	C₂×C₄⋊C₄×C₁₆₃	—	2	2	no	—
7	C₂×C₄⋊C₄×C₁₆₃	C₄	1	2	yes	—
8	C₂×C₄²×C₁₆₃	C₂²	1	1	yes	characteristic
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	—	2	2	no	—
13	C₂²×C₄×C₁₆₃	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₁₆₃	D₄	1	2	yes	—
15	C₄⋊C₄×C₁₆₃	—	4	2	no	—
16	C₄⋊C₄×C₁₆₃	—	4	2	no	—
17	C₄⋊C₄×C₁₆₃	—	4	2	no	—
18	C₄²×C₁₆₃	C₂×C₄	1	2	yes	—
19	C₄²×C₁₆₃	D₄	1	1	yes	characteristic
20	C₄²×C₁₆₃	Q₈	1	1	yes	characteristic
21	C₁₆₃⋊C₈	—	4	4	no	—
22	C₂³×C₁₆₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×C₄×C₁₆₃	—	2	1	no	—
24	C₂×C₄×C₁₆₃	—	2	1	no	—
25	C₂×C₄×C₁₆₃	—	4	2	no	—
26	C₂×C₄×C₁₆₃	—	4	4	no	—
27	C₂×C₄×C₁₆₃	C₄²	1	1	yes	characteristic
28	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	2	yes	—
29	C₂×C₄×C₁₆₃	C₄⋊C₄	1	2	yes	—
30	C₂×C₄×C₁₆₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₆₃	D₈	1	2	yes	—
32	C₂×C₄×C₁₆₃	SD₁₆	1	2	yes	—
33	C₂×C₄×C₁₆₃	SD₁₆	1	2	yes	—
34	C₂×C₄×C₁₆₃	Q₁₆	1	2	yes	—
35	C₂²×C₁₆₃	#95	1	1	yes	characteristic
36	C₂²×C₁₆₃	#100	1	1	yes	characteristic
37	C₂²×C₁₆₃	#101	1	1	yes	characteristic
38	C₂²×C₁₆₃	#102	1	2	yes	—
39	C₂²×C₁₆₃	#103	1	2	yes	—
40	C₄×C₁₆₃	—	2	1	no	—
41	C₄×C₁₆₃	—	2	1	no	—
42	C₄×C₁₆₃	—	8	2	no	—
43	C₄×C₁₆₃	#102	1	2	yes	—
44	C₄×C₁₆₃	#103	1	2	yes	—
45	C₄×C₁₆₃	#106	1	2	yes	—
46	C₄×C₁₆₃	#107	1	2	yes	—
47	C₂×C₁₆₃	#331	1	1	yes	characteristic
48	C₂×C₁₆₃	#332	1	2	yes	—
49	C₂×C₁₆₃	#333	1	1	yes	characteristic
50	C₂×C₁₆₃	#340	1	2	yes	—
51	C₂×C₁₆₃	#343	1	1	yes	characteristic
52	C₁₆₃	#1296	1	1	yes	characteristic, complemented, Sylow
53	#1296	—	163	1	no	maximal, Sylow
54	C₂×C₄⋊C₈	—	163	2	no	—
55	#384	D₁₆₃	1	1	yes	characteristic
56	C₂²×C₈	—	326	2	no	—
57	C₂×C₄⋊C₄	—	2	2	no	—
58	C₂×C₄⋊C₄	Dic₁₆₃	1	2	yes	—
59	C₂×C₄²	C₂×D₁₆₃	1	1	yes	characteristic
60	C₄⋊C₈	—	326	2	no	—
61	C₄⋊C₈	—	163	4	no	—
62	C₂²×C₄	—	2	2	no	—
63	C₂²×C₄	C₂×Dic₁₆₃	1	1	yes	characteristic
64	C₂²×C₄	C₂²⋊D₁₆₃	1	2	yes	—
65	C₂×C₈	—	326	4	no	—
66	C₂×C₈	—	652	2	no	—
67	C₂×C₈	—	652	2	no	—
68	C₄⋊C₄	—	4	2	no	—
69	C₄⋊C₄	—	4	2	no	—
70	C₄⋊C₄	—	4	2	no	—
71	C₄²	C₄×D₁₆₃	1	2	yes	—
72	C₄²	D₆₅₂	1	1	yes	characteristic
73	C₄²	Dic₃₂₆	1	1	yes	characteristic
74	C₂³	C₂²⋊Dic₁₆₃	1	1	yes	characteristic
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	4	2	no	—
78	C₂×C₄	—	4	4	no	—
79	C₂×C₄	C₄×Dic₁₆₃	1	1	yes	characteristic
80	C₂×C₄	(C₂×D₁₆₃)⋊C₄	1	2	yes	—
81	C₂×C₄	Dic₁₆₃⋊C₄	1	2	yes	—
82	C₂×C₄	C₄⋊Dic₁₆₃	1	1	yes	characteristic
83	C₂×C₄	#14579684	1	2	yes	—
84	C₂×C₄	#14579686	1	2	yes	—
85	C₂×C₄	#14579687	1	2	yes	—
86	C₂×C₄	#14579689	1	2	yes	—
87	C₈	—	652	4	no	—
88	C₂²	C₁₆₃⋊#95	1	1	yes	characteristic
89	C₂²	#19104652	1	1	yes	characteristic
90	C₂²	#19104654	1	1	yes	characteristic
91	C₂²	#19104656	1	2	yes	—
92	C₂²	#19104659	1	2	yes	—
93	C₄	—	2	1	no	—
94	C₄	—	2	1	no	—
95	C₄	—	8	2	no	—
96	C₄	#19104655	1	2	yes	—
97	C₄	#19104658	1	2	yes	—
98	C₄	#19104666	1	2	yes	—
99	C₄	#19104668	1	2	yes	—
100	C₂	#23781409	1	1	yes	characteristic
101	C₂	#23781411	1	2	yes	—
102	C₂	#23781414	1	1	yes	characteristic
103	C₂	#23781427	1	2	yes	—
104	C₂	#23781434	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₁₆₃⋊C₈)

C₂

yes

maximal

#384×C₁₆₃

C₂

yes

maximal, characteristic

C₂²×C₁₆₃⋊C₈

—

C₄⋊(C₁₆₃⋊C₈)

—

C₄⋊(C₁₆₃⋊C₈)

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₆₃

—

C₂×C₄⋊C₄×C₁₆₃

C₄

yes

—

C₂×C₄²×C₁₆₃

C₂²

yes

characteristic

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

—

C₂²×C₄×C₁₆₃

—

C₂²×C₄×C₁₆₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₆₃

D₄

yes

—

C₄⋊C₄×C₁₆₃

—

C₄⋊C₄×C₁₆₃

—

C₄⋊C₄×C₁₆₃

—

C₄²×C₁₆₃

C₂×C₄

yes

—

C₄²×C₁₆₃

D₄

yes

characteristic

C₄²×C₁₆₃

Q₈

yes

characteristic

C₁₆₃⋊C₈

—

C₂³×C₁₆₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

C₄²

yes

characteristic

C₂×C₄×C₁₆₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₆₃

C₄⋊C₄

yes

—

C₂×C₄×C₁₆₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₆₃

D₈

yes

—

C₂×C₄×C₁₆₃

SD₁₆

yes

—

C₂×C₄×C₁₆₃

SD₁₆

yes

—

C₂×C₄×C₁₆₃

Q₁₆

yes

—

C₂²×C₁₆₃

yes

C₂²×C₁₆₃

yes

C₂²×C₁₆₃

yes

C₂²×C₁₆₃

yes

—

C₂²×C₁₆₃

#103

yes

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

#102

yes

—

C₄×C₁₆₃

#103

yes

—

C₄×C₁₆₃

#106

yes

—

C₄×C₁₆₃

#107

yes

—

C₂×C₁₆₃

#331

yes

characteristic

C₂×C₁₆₃

#332

yes

—

C₂×C₁₆₃

#333

yes

characteristic

C₂×C₁₆₃

#340

yes

—

C₂×C₁₆₃

#343

yes

characteristic

C₁₆₃

#1296

yes

characteristic, complemented, Sylow

#1296

—

163

maximal, Sylow

C₂×C₄⋊C₈

—

163

—

#384

D₁₆₃

yes

characteristic

C₂²×C₈

—

326

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Dic₁₆₃

yes

—

C₂×C₄²

C₂×D₁₆₃

yes

characteristic

C₄⋊C₈

—

326

—

C₄⋊C₈

—

163

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₁₆₃

yes

characteristic

C₂²×C₄

C₂²⋊D₁₆₃

yes

—

C₂×C₈

—

326

—

C₂×C₈

—

652

—

C₂×C₈

—

652

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₄×D₁₆₃

yes

—

C₄²

D₆₅₂

yes

characteristic

C₄²

Dic₃₂₆

yes

characteristic

C₂³

C₂²⋊Dic₁₆₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₁₆₃

yes

characteristic

C₂×C₄

(C₂×D₁₆₃)⋊C₄

yes

—

C₂×C₄

Dic₁₆₃⋊C₄

yes

—

C₂×C₄

C₄⋊Dic₁₆₃

yes

characteristic

C₂×C₄

#14579684

yes

—

C₂×C₄

#14579686

yes

—

C₂×C₄

#14579687

yes

—

C₂×C₄

#14579689

yes

—

C₈

—

652

—

C₂²

C₁₆₃⋊#95

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#19104659

yes

—

C₄

—

C₄

—

C₄

—

C₄

#19104655

yes

—

C₄

#19104658

yes

—

C₄

#19104666

yes

—

C₄

#19104668

yes

—

100

C₂

#23781409

yes

characteristic

101

C₂

#23781411

yes

—

102

C₂

#23781414

yes

characteristic

103

C₂

#23781427

yes

—

104

C₂

#23781434

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²⋊Dic₁₆₃
commutator	C₂×C₄×C₁₆₃	C₄²
Frattini	C₂×C₄²	C₂×D₁₆₃
Fitting	#384×C₁₆₃	C₂
socle, prime radical	C₂³×C₁₆₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₆₃	C₂²
hypercenter	#384	D₁₆₃
hypocenter	C₁₆₃	#1296
LCS	G ▷ C₂×C₄×C₁₆₃ ▷ C₂²×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂²⋊Dic₁₆₃

commutator

C₂×C₄×C₁₆₃

C₄²

Frattini

C₂×C₄²

C₂×D₁₆₃

Fitting

#384×C₁₆₃

C₂

socle, prime radical

C₂³×C₁₆₃

C₂²⋊C₄

cosocle

C₂×C₄²×C₁₆₃

C₂²

D₁₆₃

C₁₆₃

G ▷ C₂×C₄×C₁₆₃ ▷ C₂²×C₁₆₃ ▷ C₁₆₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
5. C₄⋊(C₁₆₃⋊C₈)	C₄	semidirect
52. C₁₆₃	#1296	semidirect

H ≅ G/N

type

5. C₄⋊(C₁₆₃⋊C₈)

C₄

semidirect

52. C₁₆₃

#1296

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
#384×C₁₆₃	2
#1296	163

subgroup

index

C₂×C₄⋊(C₁₆₃⋊C₈)

#384×C₁₆₃

#1296

163

Subgroups — #35212419

Conjugacy classes of subgroups

Complements

Maximal subgroups