Group #24906270 — subgroups

Subgroups — Q₁₆×C₈₃⋊C₈ #24906270

Order 10624.

164 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 89 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	Q₈×C₈₃⋊C₈	C₂	1	2	yes	maximal
2	#19127454	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊(C₈₃⋊C₈)	C₂	1	2	yes	maximal
4	C₈×C₈₃⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₄×Q₁₆×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
7	C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
8	C₄×C₈₃⋊C₈	—	2	2	no	—
9	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
10	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×Q₁₆×C₈₃	C₄	1	1	yes	characteristic
12	C₄×Q₈×C₈₃	C₂²	1	2	yes	—
13	#107×C₈₃	C₄	1	1	yes	characteristic
14	#103×C₈₃	C₄	1	2	yes	—
15	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
16	C₂×C₈₃⋊C₈	—	2	1	no	—
17	C₂×C₈₃⋊C₈	—	2	2	no	—
18	C₂×C₈₃⋊C₈	D₄	1	1	yes	characteristic
19	C₂×C₈₃⋊C₈	D₄	1	1	yes	characteristic
20	C₂×Q₈×C₈₃	C₂×C₄	1	2	yes	—
21	Q₁₆×C₈₃	C₈	1	4	yes	complemented
22	C₂×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
24	C₄⋊C₄×C₈₃	—	2	2	no	—
25	C₄⋊C₄×C₈₃	C₂×C₄	1	2	yes	—
26	C₄²×C₈₃	—	2	2	no	—
27	C₄²×C₈₃	C₂³	1	1	yes	characteristic
28	Q₈×C₈₃	—	2	2	no	—
29	Q₈×C₈₃	C₂×C₈	1	4	yes	—
30	C₂×C₄×C₈₃	—	2	2	no	—
31	C₂×C₄×C₈₃	—	2	2	no	—
32	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
35	C₈×C₈₃	—	2	1	no	—
36	C₈×C₈₃	C₂×C₈	1	2	yes	—
37	C₈₃⋊C₈	—	2	2	no	—
38	C₈₃⋊C₈	—	2	1	no	—
39	C₈₃⋊C₈	—	4	2	no	—
40	C₈₃⋊C₈	Q₁₆	1	2	yes	direct product factor
41	C₂²×C₈₃	C₄×D₄	1	1	yes	characteristic
42	C₄×C₈₃	—	2	4	no	—
43	C₄×C₈₃	—	2	1	no	—
44	C₄×C₈₃	—	4	2	no	—
45	C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
46	C₄×C₈₃	#131	1	1	yes	characteristic
47	C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
48	C₄×C₈₃	#135	1	1	yes	characteristic
49	C₂×C₈₃	C₈×D₄	1	1	yes	characteristic
50	C₂×C₈₃	C₄×Q₁₆	1	1	yes	characteristic
51	C₂×C₈₃	#443	1	1	yes	characteristic
52	C₈×Q₁₆	—	83	1	no	maximal, Sylow
53	C₈₃	C₈×Q₁₆	1	1	yes	characteristic, complemented, Sylow
54	C₈×Q₈	—	83	2	no	—
55	C₄×Q₁₆	D₈₃	1	1	yes	characteristic
56	#335	—	83	1	no	—
57	Q₈⋊C₈	—	83	2	no	—
58	C₈²	—	83	1	no	—
59	C₂×Q₁₆	Dic₈₃	1	1	yes	characteristic
60	C₄×Q₈	C₂×D₈₃	1	2	yes	—
61	#107	Dic₈₃	1	1	yes	characteristic
62	C₄⋊C₈	—	83	2	no	—
63	C₄⋊C₈	—	166	2	no	—
64	#103	Dic₈₃	1	2	yes	—
65	C₄×C₈	—	83	1	no	—
66	C₄×C₈	—	83	1	no	—
67	C₄×C₈	—	166	2	no	—
68	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
69	C₂×Q₈	C₂×Dic₈₃	1	2	yes	—
70	Q₁₆	C₈₃⋊C₈	1	4	yes	direct product factor
71	C₂×C₈	—	83	1	no	—
72	C₂×C₈	—	83	1	no	—
73	C₂×C₈	—	166	2	no	—
74	C₂×C₈	—	166	1	no	—
75	C₂×C₈	C₂×Dic₈₃	1	1	yes	characteristic
76	C₂×C₈	C₂×Dic₈₃	1	1	yes	characteristic
77	C₄⋊C₄	—	2	2	no	—
78	C₄⋊C₄	C₂×Dic₈₃	1	2	yes	—
79	C₄²	—	2	2	no	—
80	C₄²	C₂²×D₈₃	1	1	yes	characteristic
81	Q₈	—	2	2	no	—
82	Q₈	C₂×C₈₃⋊C₈	1	4	yes	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
86	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
87	C₂×C₄	#13050589	1	1	yes	characteristic
88	C₈	—	2	1	no	—
89	C₈	—	332	2	no	—
90	C₈	—	83	2	no	—
91	C₈	—	166	2	no	—
92	C₈	—	166	1	no	—
93	C₈	C₂×C₈₃⋊C₈	1	2	yes	—
94	C₂²	D₄×Dic₈₃	1	1	yes	characteristic
95	C₄	—	2	1	no	—
96	C₄	—	2	4	no	—
97	C₄	—	4	2	no	—
98	C₄	C₂²×C₈₃⋊C₈	1	1	yes	characteristic
99	C₄	#14581799	1	1	yes	characteristic
100	C₄	Q₁₆×D₈₃	1	1	yes	characteristic
101	C₄	#14581815	1	1	yes	characteristic
102	C₂	D₄×C₈₃⋊C₈	1	1	yes	characteristic
103	C₂	Q₁₆×Dic₈₃	1	1	yes	characteristic
104	C₂	#19127818	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×C₈₃⋊C₈

C₂

yes

maximal

#19127454

C₂

yes

maximal, characteristic

Q₈⋊(C₈₃⋊C₈)

C₂

yes

maximal

C₈×C₈₃⋊C₈

C₂

yes

maximal, characteristic

C₄×Q₁₆×C₈₃

C₂

yes

maximal, characteristic

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

C₂²

yes

—

C₄×C₈₃⋊C₈

—

C₄×C₈₃⋊C₈

C₂²

yes

characteristic

C₄×C₈₃⋊C₈

C₂²

yes

characteristic

C₂×Q₁₆×C₈₃

C₄

yes

characteristic

C₄×Q₈×C₈₃

C₂²

yes

—

#107×C₈₃

C₄

yes

characteristic

#103×C₈₃

C₄

yes

—

C₄×C₈×C₈₃

C₂²

yes

characteristic

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

D₄

yes

characteristic

C₂×C₈₃⋊C₈

D₄

yes

characteristic

C₂×Q₈×C₈₃

C₂×C₄

yes

—

Q₁₆×C₈₃

C₈

yes

complemented

C₂×C₈×C₈₃

C₂×C₄

yes

characteristic

C₂×C₈×C₈₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

C₂×C₄

yes

—

C₄²×C₈₃

—

C₄²×C₈₃

C₂³

yes

characteristic

Q₈×C₈₃

—

Q₈×C₈₃

C₂×C₈

yes

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

#41

yes

characteristic

C₈×C₈₃

—

C₈×C₈₃

C₂×C₈

yes

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

Q₁₆

yes

direct product factor

C₂²×C₈₃

C₄×D₄

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂²×C₈

yes

characteristic

C₄×C₈₃

#131

yes

characteristic

C₄×C₈₃

C₂×Q₁₆

yes

characteristic

C₄×C₈₃

#135

yes

characteristic

C₂×C₈₃

C₈×D₄

yes

characteristic

C₂×C₈₃

C₄×Q₁₆

yes

characteristic

C₂×C₈₃

#443

yes

characteristic

C₈×Q₁₆

—

maximal, Sylow

C₈₃

C₈×Q₁₆

yes

characteristic, complemented, Sylow

C₈×Q₈

—

C₄×Q₁₆

D₈₃

yes

characteristic

#335

—

Q₈⋊C₈

—

C₈²

—

C₂×Q₁₆

Dic₈₃

yes

characteristic

C₄×Q₈

C₂×D₈₃

yes

—

#107

Dic₈₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

166

—

#103

Dic₈₃

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

166

—

C₄×C₈

C₂×D₈₃

yes

characteristic

C₂×Q₈

C₂×Dic₈₃

yes

—

Q₁₆

C₈₃⋊C₈

yes

direct product factor

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

166

—

C₂×C₈

—

166

—

C₂×C₈

C₂×Dic₈₃

yes

characteristic

C₂×C₈

C₂×Dic₈₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₈₃

yes

—

C₄²

—

C₄²

C₂²×D₈₃

yes

characteristic

Q₈

—

Q₈

C₂×C₈₃⋊C₈

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×Dic₈₃

yes

characteristic

C₂×C₄

D₄×D₈₃

yes

characteristic

C₂×C₄

#13050589

yes

characteristic

C₈

—

C₈

—

332

—

C₈

—

C₈

—

166

—

C₈

—

166

—

C₈

C₂×C₈₃⋊C₈

yes

—

C₂²

D₄×Dic₈₃

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

C₂²×C₈₃⋊C₈

yes

characteristic

C₄

#14581799

yes

characteristic

100

C₄

Q₁₆×D₈₃

yes

characteristic

101

C₄

#14581815

yes

characteristic

102

C₂

D₄×C₈₃⋊C₈

yes

characteristic

103

C₂

Q₁₆×Dic₈₃

yes

characteristic

104

C₂

#19127818

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₈₃
commutator	C₄×C₈₃	C₂²×C₈
Frattini	C₄²	C₂²×D₈₃
Fitting	C₄×Q₁₆×C₈₃	C₂
socle, prime radical	C₂²×C₈₃	C₄×D₄
cosocle	C₄²×C₈₃	C₂³
hypercenter	C₄×Q₁₆	D₈₃
hypocenter	C₈₃	C₈×Q₁₆
LCS	G ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

subgroup

G/H

center

C₂×C₄

D₄×D₈₃

commutator

C₄×C₈₃

C₂²×C₈

Frattini

C₄²

C₂²×D₈₃

Fitting

C₄×Q₁₆×C₈₃

C₂

socle, prime radical

C₂²×C₈₃

C₄×D₄

cosocle

C₄²×C₈₃

C₂³

hypercenter

C₄×Q₁₆

D₈₃

hypocenter

C₈₃

C₈×Q₁₆

LCS

G ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

N	H ≅ G/N	type
21. Q₁₆×C₈₃	C₈	semidirect
40. C₈₃⋊C₈	Q₁₆	direct (N × H)
53. C₈₃	C₈×Q₁₆	semidirect
70. Q₁₆	C₈₃⋊C₈	direct (N × H)

H ≅ G/N

type

21. Q₁₆×C₈₃

C₈

semidirect

40. C₈₃⋊C₈

Q₁₆

direct (N × H)

53. C₈₃

C₈×Q₁₆

semidirect

70. Q₁₆

C₈₃⋊C₈

direct (N × H)

Subgroups — Q₁₆×C₈₃⋊C₈ #24906270

Conjugacy classes of subgroups

Complements

Subgroups — Q16×C83⋊C8 #24906270

Conjugacy classes of subgroups

Complements

Subgroups — Q₁₆×C₈₃⋊C₈ #24906270