Group #24906219 — subgroups

Subgroups — #24906219

Order 10624.

184 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 109 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	#19127725	C₂	1	2	yes	maximal
2	#19127452	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×C₈×C₈₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#13050574	—	2	2	no	—
5	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
6	C₄⋊(C₈₃⋊C₈)	C₂²	1	4	yes	—
7	C₂²×C₈×C₈₃	C₄	1	2	yes	—
8	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
9	C₄×C₈×C₈₃	C₄	1	1	yes	characteristic
10	C₄×C₈×C₈₃	C₄	1	1	yes	characteristic
11	C₄×C₈×C₈₃	C₂²	1	2	yes	—
12	#13050574	—	4	4	no	—
13	C₂×C₈₃⋊C₈	—	2	4	no	—
14	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₈₃	Q₈	1	1	yes	characteristic
17	C₂×C₈×C₈₃	—	2	4	no	—
18	C₂×C₈×C₈₃	C₂×C₄	1	2	yes	—
19	C₂×C₈×C₈₃	C₂×C₄	1	2	yes	—
20	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
21	C₄²×C₈₃	D₄	1	1	yes	characteristic
22	C₄²×C₈₃	Q₈	1	1	yes	characteristic
23	C₄²×C₈₃	C₂³	1	1	yes	characteristic
24	C₂³×C₈₃	C₄⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₈₃	—	2	2	no	—
26	C₂×C₄×C₈₃	—	2	2	no	—
27	C₂×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₈₃	SD₁₆	1	4	yes	—
31	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
34	C₈×C₈₃	—	2	2	no	—
35	C₈×C₈₃	M₁₆	1	4	yes	complemented
36	C₈₃⋊C₈	—	4	4	no	—
37	C₂²×C₈₃	—	2	1	no	—
38	C₂²×C₈₃	—	2	1	no	—
39	C₂²×C₈₃	#106	1	2	yes	—
40	C₂²×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₄×C₈₃	—	2	1	no	—
42	C₄×C₈₃	—	2	1	no	—
43	C₄×C₈₃	#106	1	2	yes	—
44	C₄×C₈₃	#108	1	2	yes	—
45	C₄×C₈₃	C₂×M₁₆	1	2	yes	—
46	C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
47	C₂×C₈₃	—	2	2	no	—
48	C₂×C₈₃	#423	1	1	yes	characteristic
49	C₂×C₈₃	C₂×#106	1	1	yes	characteristic
50	C₂×C₈₃	C₂×#108	1	1	yes	characteristic
51	#1566	—	83	1	no	maximal, Sylow
52	C₈₃	#1566	1	1	yes	characteristic, complemented, Sylow
53	#423	—	83	2	no	—
54	C₂×C₄×C₈	D₈₃	1	1	yes	characteristic
55	#334	—	83	4	no	—
56	C₂×M₁₆	—	166	2	no	—
57	C₂²×C₈	Dic₈₃	1	2	yes	—
58	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
59	C₄⋊C₈	—	83	4	no	—
60	C₄⋊C₈	—	166	2	no	—
61	C₄×C₈	Dic₈₃	1	1	yes	characteristic
62	C₄×C₈	Dic₈₃	1	1	yes	characteristic
63	C₄×C₈	C₂×D₈₃	1	2	yes	—
64	C₂²×C₄	Dic₁₆₆	1	1	yes	characteristic
65	C₂²×C₄	D₃₃₂	1	1	yes	characteristic
66	C₂²×C₄	C₂×Dic₈₃	1	1	yes	characteristic
67	M₁₆	—	332	4	no	—
68	C₂×C₈	—	2	4	no	—
69	C₂×C₈	—	166	4	no	—
70	C₂×C₈	C₂×Dic₈₃	1	2	yes	—
71	C₂×C₈	C₂×Dic₈₃	1	2	yes	—
72	C₄²	Dic₁₆₆	1	1	yes	characteristic
73	C₄²	D₃₃₂	1	1	yes	characteristic
74	C₄²	C₂×Dic₈₃	1	1	yes	characteristic
75	C₄²	C₂²×D₈₃	1	1	yes	characteristic
76	C₂³	C₄⋊Dic₈₃	1	1	yes	characteristic
77	C₂×C₄	—	2	2	no	—
78	C₂×C₄	—	2	2	no	—
79	C₂×C₄	C₄⋊Dic₈₃	1	1	yes	characteristic
80	C₂×C₄	C₄⋊Dic₈₃	1	1	yes	characteristic
81	C₂×C₄	C₄⋊Dic₈₃	1	1	yes	characteristic
82	C₂×C₄	#13050579	1	4	yes	—
83	C₂×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
84	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
85	C₂×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
86	C₈	—	2	2	no	—
87	C₈	—	332	4	no	—
88	C₈	#13050574	1	4	yes	complemented
89	C₂²	—	2	1	no	—
90	C₂²	—	2	1	no	—
91	C₂²	#14581724	1	2	yes	—
92	C₂²	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	2	1	no	—
95	C₄	#14581724	1	2	yes	—
96	C₄	#14581728	1	2	yes	—
97	C₄	C₂×#13050574	1	2	yes	—
98	C₄	C₂×#13050579	1	2	yes	—
99	C₂	—	2	2	no	—
100	C₂	#19127725	1	1	yes	characteristic
101	C₂	C₂×#14581724	1	1	yes	characteristic
102	C₂	C₂×#14581728	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19127725

C₂

yes

maximal

#19127452

C₂

yes

maximal, complemented

C₂×C₄×C₈×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050574

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

C₂²

yes

—

C₂²×C₈×C₈₃

C₄

yes

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₄×C₈×C₈₃

C₄

yes

characteristic

C₄×C₈×C₈₃

C₄

yes

characteristic

C₄×C₈×C₈₃

C₂²

yes

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

D₄

yes

characteristic

C₂²×C₄×C₈₃

Q₈

yes

characteristic

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂×C₄

yes

—

C₂×C₈×C₈₃

C₂×C₄

yes

—

C₄²×C₈₃

C₂×C₄

yes

characteristic

C₄²×C₈₃

D₄

yes

characteristic

C₄²×C₈₃

Q₈

yes

characteristic

C₄²×C₈₃

C₂³

yes

characteristic

C₂³×C₈₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

SD₁₆

yes

—

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×Q₈

yes

characteristic

C₈×C₈₃

—

C₈×C₈₃

M₁₆

yes

complemented

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

#106

yes

—

C₂²×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#106

yes

—

C₄×C₈₃

#108

yes

—

C₄×C₈₃

C₂×M₁₆

yes

—

C₄×C₈₃

C₂×SD₁₆

yes

—

C₂×C₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

#423

—

C₂×C₄×C₈

D₈₃

yes

characteristic

#334

—

C₂×M₁₆

—

166

—

C₂²×C₈

Dic₈₃

yes

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

166

—

C₄×C₈

Dic₈₃

yes

characteristic

C₄×C₈

Dic₈₃

yes

characteristic

C₄×C₈

C₂×D₈₃

yes

—

C₂²×C₄

Dic₁₆₆

yes

characteristic

C₂²×C₄

D₃₃₂

yes

characteristic

C₂²×C₄

C₂×Dic₈₃

yes

characteristic

M₁₆

—

332

—

C₂×C₈

—

C₂×C₈

—

166

—

C₂×C₈

C₂×Dic₈₃

yes

—

C₂×C₈

C₂×Dic₈₃

yes

—

C₄²

Dic₁₆₆

yes

characteristic

C₄²

D₃₃₂

yes

characteristic

C₄²

C₂×Dic₈₃

yes

characteristic

C₄²

C₂²×D₈₃

yes

characteristic

C₂³

C₄⋊Dic₈₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊Dic₈₃

yes

characteristic

C₂×C₄

C₄⋊Dic₈₃

yes

characteristic

C₂×C₄

C₄⋊Dic₈₃

yes

characteristic

C₂×C₄

#13050579

yes

—

C₂×C₄

C₂²×Dic₈₃

yes

characteristic

C₂×C₄

C₂×D₃₃₂

yes

characteristic

C₂×C₄

C₂×Dic₁₆₆

yes

characteristic

C₈

—

C₈

—

332

—

C₈

#13050574

yes

complemented

C₂²

—

C₂²

—

C₂²

#14581724

yes

—

C₂²

C₂×C₄⋊Dic₈₃

yes

characteristic

C₄

—

C₄

—

C₄

#14581724

yes

—

C₄

#14581728

yes

—

C₄

C₂×#13050574

yes

—

C₄

C₂×#13050579

yes

—

C₂

—

100

C₂

#19127725

yes

characteristic

101

C₂

C₂×#14581724

yes

characteristic

102

C₂

C₂×#14581728

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃₃₂
commutator	C₂×C₄×C₈₃	C₂²×C₄
Frattini	C₄²	C₂²×D₈₃
Fitting	C₂×C₄×C₈×C₈₃	C₂
socle	C₂²×C₈₃	C₂×C₄⋊C₄
prime radical	C₂³×C₈₃	C₄⋊C₄
cosocle	C₄²×C₈₃	C₂³
hypercenter	C₂×C₄×C₈	D₈₃
hypocenter	C₈₃	#1566
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

C₂×D₃₃₂

commutator

C₂×C₄×C₈₃

C₂²×C₄

Frattini

C₄²

C₂²×D₈₃

Fitting

C₂×C₄×C₈×C₈₃

C₂

socle

C₂²×C₈₃

C₂×C₄⋊C₄

prime radical

C₂³×C₈₃

C₄⋊C₄

cosocle

C₄²×C₈₃

C₂³

hypercenter

C₂×C₄×C₈

D₈₃

hypocenter

C₈₃

#1566

LCS

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #19127452	C₂	semidirect
35. C₈×C₈₃	M₁₆	semidirect
52. C₈₃	#1566	semidirect
88. C₈	#13050574	semidirect

H ≅ G/N

type

2. #19127452

C₂

semidirect

35. C₈×C₈₃

M₁₆

semidirect

52. C₈₃

#1566

semidirect

88. C₈

#13050574

semidirect

Subgroups — #24906219

Conjugacy classes of subgroups

Complements