Group #35213864 — subgroups

Subgroups — #35213864

Order 20864.

172 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 49 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	#23782075	C₂	1	2	yes	maximal
2	#23782035	C₂	1	1	yes	maximal, characteristic
3	#23781407	C₂	1	1	yes	maximal, characteristic
4	#23781407	C₂	1	2	yes	maximal
5	#564×C₁₆₃	C₂	1	1	yes	maximal, characteristic
6	#19104735	C₂²	1	1	yes	characteristic
7	#19104668	—	2	2	no	—
8	#19104666	—	2	2	no	—
9	#19104658	—	2	2	no	—
10	#19104646	C₂²	1	1	yes	characteristic
11	#19104646	C₂²	1	2	yes	—
12	#125×C₁₆₃	—	2	2	no	—
13	#125×C₁₆₃	—	2	2	no	—
14	#125×C₁₆₃	—	2	2	no	—
15	#125×C₁₆₃	C₂²	1	1	yes	characteristic
16	#125×C₁₆₃	C₂²	1	2	yes	—
17	C₂×Dic₃₂₆	—	2	1	no	—
18	C₂×C₁₆₃⋊C₈	—	2	2	no	—
19	C₂×C₁₆₃⋊C₈	—	2	1	no	—
20	C₄⋊Dic₁₆₃	—	2	2	no	—
21	C₄⋊C₄×C₁₆₃	—	2	2	no	—
22	C₄⋊C₄×C₁₆₃	—	2	2	no	—
23	C₄⋊C₄×C₁₆₃	—	2	2	no	—
24	C₄⋊C₄×C₁₆₃	—	2	2	no	—
25	C₄⋊C₄×C₁₆₃	—	2	2	no	—
26	C₄⋊C₄×C₁₆₃	—	2	2	no	—
27	C₄⋊C₄×C₁₆₃	D₄	1	2	yes	—
28	C₄⋊C₄×C₁₆₃	D₄	1	2	yes	—
29	C₄⋊C₄×C₁₆₃	D₄	1	2	yes	—
30	C₄²×C₁₆₃	—	2	2	no	—
31	C₄²×C₁₆₃	C₂³	1	1	yes	characteristic
32	Dic₃₂₆	—	4	2	no	—
33	C₂×Dic₁₆₃	—	4	1	no	—
34	C₁₆₃⋊C₈	—	2	4	no	—
35	C₁₆₃⋊C₈	—	4	1	no	—
36	C₂×C₄×C₁₆₃	—	2	2	no	—
37	C₂×C₄×C₁₆₃	—	2	2	no	—
38	C₂×C₄×C₁₆₃	—	2	2	no	—
39	C₂×C₄×C₁₆₃	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₁₆₃	#120	1	1	yes	characteristic
42	C₄×C₁₆₃	—	2	1	no	—
43	C₄×C₁₆₃	—	2	2	no	—
44	C₄×C₁₆₃	—	4	2	no	—
45	C₄×C₁₆₃	—	4	2	no	—
46	C₄×C₁₆₃	—	4	2	no	—
47	Dic₁₆₃	—	8	1	no	—
48	C₂×C₁₆₃	#454	1	1	yes	characteristic
49	C₂×C₁₆₃	#456	1	2	yes	—
50	C₁₆₃	#1663	1	1	yes	characteristic, complemented, Sylow
51	#1663	—	163	1	no	maximal, Sylow
52	#564	D₁₆₃	1	1	yes	characteristic
53	#501	—	163	2	no	—
54	#491	—	163	1	no	—
55	#330	—	163	2	no	—
56	#330	—	163	1	no	—
57	C₄⋊Q₈	—	163	1	no	—
58	#125	—	2	2	no	—
59	#125	—	2	2	no	—
60	#125	—	2	2	no	—
61	#125	C₂×D₁₆₃	1	1	yes	characteristic
62	#125	C₂×D₁₆₃	1	2	yes	—
63	#107	—	326	2	no	—
64	#106	—	326	2	no	—
65	#103	—	326	2	no	—
66	#97	—	163	1	no	—
67	#97	—	163	2	no	—
68	C₂×Q₈	—	326	1	no	—
69	C₂×C₈	—	326	2	no	—
70	C₂×C₈	—	326	1	no	—
71	C₄⋊C₄	—	2	2	no	—
72	C₄⋊C₄	—	2	2	no	—
73	C₄⋊C₄	—	2	2	no	—
74	C₄⋊C₄	—	2	2	no	—
75	C₄⋊C₄	—	2	2	no	—
76	C₄⋊C₄	—	2	2	no	—
77	C₄⋊C₄	—	326	2	no	—
78	C₄⋊C₄	C₂²⋊D₁₆₃	1	2	yes	—
79	C₄⋊C₄	C₂²⋊D₁₆₃	1	2	yes	—
80	C₄⋊C₄	C₂²⋊D₁₆₃	1	2	yes	—
81	C₄²	—	2	2	no	—
82	C₄²	C₂²×D₁₆₃	1	1	yes	characteristic
83	Q₈	—	652	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	652	1	no	—
88	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	2	yes	—
89	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
90	C₈	—	326	4	no	—
91	C₈	—	652	1	no	—
92	C₂²	#19104704	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	2	2	no	—
95	C₄	—	4	2	no	—
96	C₄	—	4	2	no	—
97	C₄	—	4	2	no	—
98	C₄	—	1304	1	no	—
99	C₂	#23781838	1	1	yes	characteristic
100	C₂	#23781848	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23782075

C₂

yes

maximal

#23782035

C₂

yes

maximal, characteristic

#23781407

C₂

yes

maximal, characteristic

#23781407

C₂

yes

maximal

#564×C₁₆₃

C₂

yes

maximal, characteristic

#19104735

C₂²

yes

—

—

—

C₂²

yes

characteristic

#19104646

C₂²

yes

—

#125×C₁₆₃

—

#125×C₁₆₃

—

#125×C₁₆₃

—

#125×C₁₆₃

C₂²

yes

characteristic

#125×C₁₆₃

C₂²

yes

—

C₂×Dic₃₂₆

—

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

—

C₄⋊Dic₁₆₃

—

C₄⋊C₄×C₁₆₃

—

C₄⋊C₄×C₁₆₃

—

C₄⋊C₄×C₁₆₃

—

C₄⋊C₄×C₁₆₃

—

C₄⋊C₄×C₁₆₃

—

C₄⋊C₄×C₁₆₃

—

C₄⋊C₄×C₁₆₃

D₄

yes

—

C₄⋊C₄×C₁₆₃

D₄

yes

—

C₄⋊C₄×C₁₆₃

D₄

yes

—

C₄²×C₁₆₃

—

C₄²×C₁₆₃

C₂³

yes

characteristic

Dic₃₂₆

—

C₂×Dic₁₆₃

—

C₁₆₃⋊C₈

—

C₁₆₃⋊C₈

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

C₂×D₄

yes

—

C₂×C₄×C₁₆₃

C₂×D₄

yes

characteristic

C₂²×C₁₆₃

#120

yes

characteristic

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

—

Dic₁₆₃

—

C₂×C₁₆₃

#454

yes

characteristic

C₂×C₁₆₃

#456

yes

—

C₁₆₃

#1663

yes

characteristic, complemented, Sylow

#1663

—

163

maximal, Sylow

#564

D₁₆₃

yes

characteristic

#501

—

163

—

#491

—

163

—

#330

—

163

—

#330

—

163

—

C₄⋊Q₈

—

163

—

—

—

—

C₂×D₁₆₃

yes

characteristic

#125

C₂×D₁₆₃

yes

—

#107

—

326

—

#106

—

326

—

#103

—

326

—

#97

—

163

—

#97

—

163

—

C₂×Q₈

—

326

—

C₂×C₈

—

326

—

C₂×C₈

—

326

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

326

—

C₄⋊C₄

C₂²⋊D₁₆₃

yes

—

C₄⋊C₄

C₂²⋊D₁₆₃

yes

—

C₄⋊C₄

C₂²⋊D₁₆₃

yes

—

C₄²

—

C₄²

C₂²×D₁₆₃

yes

characteristic

Q₈

—

652

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

652

—

C₂×C₄

C₂×C₂²⋊D₁₆₃

yes

—

C₂×C₄

C₂×C₂²⋊D₁₆₃

yes

characteristic

C₈

—

326

—

C₈

—

652

—

C₂²

#19104704

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

1304

—

C₂

#23781838

yes

characteristic

100

C₂

#23781848

yes

—

	subgroup	G/H
center	C₂²	#19104704
commutator, cosocle	C₄²×C₁₆₃	C₂³
Frattini	C₄²	C₂²×D₁₆₃
Fitting	#564×C₁₆₃	C₂
socle, prime radical	C₂²×C₁₆₃	#120
hypercenter	#564	D₁₆₃
hypocenter	C₁₆₃	#1663
LCS	G ▷ C₄²×C₁₆₃ ▷ C₂²×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #564

subgroup

G/H

center

C₂²

#19104704

commutator, cosocle

C₄²×C₁₆₃

C₂³

Frattini

C₄²

C₂²×D₁₆₃

Fitting

#564×C₁₆₃

C₂

C₂²×C₁₆₃

D₁₆₃

C₁₆₃

G ▷ C₄²×C₁₆₃ ▷ C₂²×C₁₆₃ ▷ C₁₆₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #564

N	H ≅ G/N	type
50. C₁₆₃	#1663	semidirect

H ≅ G/N

type

50. C₁₆₃

#1663

semidirect

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

subgroup	index
#23782075	2
#23782035	2
#23781407	2
#564×C₁₆₃	2
#1663	163

subgroup

index

#564×C₁₆₃

163

Subgroups — #35213864

Conjugacy classes of subgroups

Complements

Maximal subgroups