Group #24905506 — subgroups

Subgroups — #24905506

Order 10624.

102 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 102 conjugacy classes of subgroups — too many to draw. Only its 41 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	C₄⋊(C₈₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
2	#19127452	C₂	1	1	yes	maximal, characteristic, complemented
3	#495×C₈₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈₃⋊C₁₆	—	2	1	no	—
5	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
6	C₂×Q₁₆×C₈₃	C₄	1	1	yes	characteristic
7	C₂×SD₁₆×C₈₃	—	2	1	no	—
8	C₂×D₈×C₈₃	C₄	1	1	yes	characteristic
9	#124×C₈₃	—	2	1	no	—
10	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
11	C₂×C₈₃⋊C₈	—	4	1	no	—
12	C₈₃⋊C₁₆	—	4	1	no	—
13	C₂×Q₈×C₈₃	—	2	1	no	—
14	C₂×D₄×C₈₃	—	2	1	no	—
15	Q₁₆×C₈₃	—	4	1	no	—
16	SD₁₆×C₈₃	—	4	1	no	—
17	SD₁₆×C₈₃	—	4	1	no	—
18	D₈×C₈₃	—	4	1	no	—
19	C₂×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₈₃	D₄	1	1	yes	characteristic
21	C₂²⋊C₄×C₈₃	—	4	1	no	—
22	C₂²⋊C₄×C₈₃	—	4	1	no	—
23	C₄²×C₈₃	D₄	1	1	yes	characteristic
24	C₂³×C₈₃	—	4	1	no	—
25	Q₈×C₈₃	—	4	1	no	—
26	Q₈×C₈₃	—	4	1	no	—
27	D₄×C₈₃	—	4	1	no	—
28	D₄×C₈₃	—	4	1	no	—
29	C₂×C₄×C₈₃	—	4	1	no	—
30	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₈₃	D₈	1	1	yes	characteristic
32	C₂×C₄×C₈₃	SD₁₆	1	1	yes	characteristic
33	C₈×C₈₃	—	2	1	no	—
34	C₈×C₈₃	D₈	1	1	yes	characteristic
35	C₈×C₈₃	SD₁₆	1	1	yes	characteristic
36	C₈₃⋊C₈	—	8	1	no	—
37	C₂²×C₈₃	—	8	1	no	—
38	C₂²×C₈₃	—	8	1	no	—
39	C₂²×C₈₃	—	8	1	no	—
40	C₂²×C₈₃	#102	1	1	yes	characteristic
41	C₄×C₈₃	—	2	1	no	—
42	C₄×C₈₃	—	2	1	no	—
43	C₄×C₈₃	—	8	1	no	—
44	C₄×C₈₃	#100	1	1	yes	characteristic
45	C₄×C₈₃	#102	1	1	yes	characteristic
46	C₂×C₈₃	—	16	1	no	—
47	C₂×C₈₃	#331	1	1	yes	characteristic
48	C₂×C₈₃	#359	1	1	yes	characteristic
49	C₂×C₈₃	#360	1	1	yes	characteristic
50	#1360	—	83	1	no	maximal, Sylow
51	C₈₃	#1360	1	1	yes	characteristic, complemented, Sylow
52	#495	D₈₃	1	1	yes	characteristic
53	C₄⋊C₁₆	—	83	1	no	—
54	#334	—	83	1	no	—
55	C₂×Q₁₆	Dic₈₃	1	1	yes	characteristic
56	C₂×SD₁₆	—	2	1	no	—
57	C₂×D₈	Dic₈₃	1	1	yes	characteristic
58	#124	—	2	1	no	—
59	C₂×C₁₆	—	166	1	no	—
60	C₄⋊C₈	—	166	1	no	—
61	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
62	C₂×Q₈	—	2	1	no	—
63	C₂×D₄	—	2	1	no	—
64	Q₁₆	—	4	1	no	—
65	SD₁₆	—	4	1	no	—
66	SD₁₆	—	4	1	no	—
67	D₈	—	4	1	no	—
68	C₂×C₈	—	332	1	no	—
69	C₂×C₈	C₂×Dic₈₃	1	1	yes	characteristic
70	C₂×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
71	C₂²⋊C₄	—	4	1	no	—
72	C₂²⋊C₄	—	4	1	no	—
73	C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
74	C₁₆	—	332	1	no	—
75	C₂³	—	4	1	no	—
76	Q₈	—	4	1	no	—
77	Q₈	—	4	1	no	—
78	D₄	—	4	1	no	—
79	D₄	—	4	1	no	—
80	C₂×C₄	—	4	1	no	—
81	C₂×C₄	C₂²⋊Dic₈₃	1	1	yes	characteristic
82	C₂×C₄	#13050575	1	1	yes	characteristic
83	C₂×C₄	#13050578	1	1	yes	characteristic
84	C₈	—	2	1	no	—
85	C₈	—	664	1	no	—
86	C₈	#13050575	1	1	yes	characteristic
87	C₈	#13050578	1	1	yes	characteristic
88	C₂²	—	8	1	no	—
89	C₂²	—	8	1	no	—
90	C₂²	—	8	1	no	—
91	C₂²	#14581713	1	1	yes	characteristic
92	C₄	—	2	1	no	—
93	C₄	—	2	1	no	—
94	C₄	—	8	1	no	—
95	C₄	#14581709	1	1	yes	characteristic
96	C₄	#14581713	1	1	yes	characteristic
97	C₂	—	16	1	no	—
98	C₂	#19127445	1	1	yes	characteristic
99	C₂	#19127508	1	1	yes	characteristic
100	C₂	#19127511	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊(C₈₃⋊C₁₆)

C₂

yes

maximal, characteristic, complemented

#19127452

C₂

yes

maximal, characteristic, complemented

#495×C₈₃

C₂

yes

maximal, characteristic

C₂×C₈₃⋊C₁₆

—

C₄⋊(C₈₃⋊C₈)

—

C₂×Q₁₆×C₈₃

C₄

yes

characteristic

C₂×SD₁₆×C₈₃

—

C₂×D₈×C₈₃

C₄

yes

characteristic

#124×C₈₃

—

C₄×C₈×C₈₃

C₂²

yes

characteristic

C₂×C₈₃⋊C₈

—

C₈₃⋊C₁₆

—

C₂×Q₈×C₈₃

—

C₂×D₄×C₈₃

—

Q₁₆×C₈₃

—

SD₁₆×C₈₃

—

SD₁₆×C₈₃

—

D₈×C₈₃

—

C₂×C₈×C₈₃

C₂×C₄

yes

characteristic

C₂×C₈×C₈₃

D₄

yes

characteristic

C₂²⋊C₄×C₈₃

—

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

D₄

yes

characteristic

C₂³×C₈₃

—

Q₈×C₈₃

—

Q₈×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

D₈

yes

characteristic

C₂×C₄×C₈₃

SD₁₆

yes

characteristic

C₈×C₈₃

—

C₈×C₈₃

D₈

yes

characteristic

C₈×C₈₃

SD₁₆

yes

characteristic

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

#102

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₂×C₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

#495

D₈₃

yes

characteristic

C₄⋊C₁₆

—

#334

—

C₂×Q₁₆

Dic₈₃

yes

characteristic

C₂×SD₁₆

—

C₂×D₈

Dic₈₃

yes

characteristic

#124

—

C₂×C₁₆

—

166

—

C₄⋊C₈

—

166

—

C₄×C₈

C₂×D₈₃

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

Q₁₆

—

SD₁₆

—

SD₁₆

—

D₈

—

C₂×C₈

—

332

—

C₂×C₈

C₂×Dic₈₃

yes

characteristic

C₂×C₈

C₂²⋊D₈₃

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂²⋊D₈₃

yes

characteristic

C₁₆

—

332

—

C₂³

—

Q₈

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₈₃

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

664

—

C₈

yes

C₈

yes

C₂²

—

C₂²

—

C₂²

—

C₂²

#14581713

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

yes

C₄

yes

C₂

—

C₂

yes

C₂

yes

100

C₂

#19127511

yes

characteristic

	subgroup	G/H
center	C₂²	#14581713
commutator	C₂×C₈×C₈₃	C₂×C₄
Frattini	C₄×C₈	C₂×D₈₃
Fitting	#495×C₈₃	C₂
socle	C₂²×C₈₃	#102
prime radical	C₂×D₈×C₈₃	C₄
cosocle	C₄×C₈×C₈₃	C₂²
hypercenter	#495	D₈₃
hypocenter	C₈₃	#1360
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #495

subgroup

G/H

center

C₂²

#14581713

commutator

C₂×C₈×C₈₃

C₂×C₄

Frattini

C₄×C₈

C₂×D₈₃

Fitting

#495×C₈₃

C₂

socle

C₂²×C₈₃

#102

prime radical

C₂×D₈×C₈₃

C₄

cosocle

C₄×C₈×C₈₃

C₂²

D₈₃

C₈₃

G ▷ C₂×C₈×C₈₃ ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #495

N	H ≅ G/N	type
1. C₄⋊(C₈₃⋊C₁₆)	C₂	semidirect
2. #19127452	C₂	semidirect
51. C₈₃	#1360	semidirect

H ≅ G/N

type

1. C₄⋊(C₈₃⋊C₁₆)

C₂

semidirect

2. #19127452

C₂

semidirect

51. C₈₃

#1360

semidirect

Subgroups — #24905506

Conjugacy classes of subgroups

Complements