Group #24905608 — subgroups

Subgroups — #24905608

Order 10624.

122 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 122 conjugacy classes of subgroups — too many to draw. Only its 25 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	#19127494	C₂	1	1	yes	maximal, characteristic, complemented
2	#19127445	C₂	1	1	yes	maximal, characteristic, complemented
3	#453×C₈₃	C₂	1	1	yes	maximal, characteristic
4	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
5	#14581707	—	2	1	no	—
6	H(5,𝔽₂)×C₈₃	C₄	1	1	yes	characteristic, complemented
7	AGL(1,ℤ/8)×C₈₃	—	2	1	no	—
8	#127×C₈₃	C₂²	1	1	yes	characteristic
9	#104×C₈₃	—	2	1	no	—
10	#100×C₈₃	C₄	1	1	yes	characteristic, complemented
11	C₂×C₈₃⋊C₈	—	4	1	no	—
12	C₂²⋊Dic₈₃	—	4	1	no	—
13	#41×C₈₃	—	2	1	no	—
14	#41×C₈₃	—	4	1	no	—
15	C₂×D₄×C₈₃	—	4	1	no	—
16	C₂×D₄×C₈₃	—	4	1	no	—
17	C₂×D₄×C₈₃	—	4	1	no	—
18	C₂×D₄×C₈₃	C₂×C₄	1	1	yes	characteristic
19	C₂×D₄×C₈₃	D₄	1	1	yes	characteristic
20	SD₁₆×C₈₃	—	4	1	no	—
21	D₈×C₈₃	—	4	1	no	—
22	M₁₆×C₈₃	—	2	1	no	—
23	C₄²×C₈₃	D₄	1	1	yes	characteristic
24	C₂³×C₈₃	—	2	1	no	—
25	C₂³×C₈₃	—	2	1	no	—
26	C₂³×C₈₃	—	4	1	no	—
27	Q₈×C₈₃	—	2	1	no	—
28	D₄×C₈₃	—	2	1	no	—
29	D₄×C₈₃	—	4	1	no	—
30	D₄×C₈₃	—	4	1	no	—
31	D₄×C₈₃	—	4	2	no	—
32	D₄×C₈₃	—	4	1	no	—
33	D₄×C₈₃	—	8	1	no	—
34	D₄×C₈₃	—	8	1	no	—
35	C₂×C₄×C₈₃	—	2	1	no	—
36	C₂×C₄×C₈₃	—	4	1	no	—
37	C₂×C₄×C₈₃	—	4	1	no	—
38	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×Dic₈₃	—	8	1	no	—
40	C₈×C₈₃	—	4	1	no	—
41	C₈₃⋊C₈	—	8	1	no	—
42	C₂²×C₈₃	—	4	1	no	—
43	C₂²×C₈₃	—	4	1	no	—
44	C₂²×C₈₃	—	4	1	no	—
45	C₂²×C₈₃	—	8	1	no	—
46	C₂²×C₈₃	—	8	1	no	—
47	C₂²×C₈₃	—	16	1	no	—
48	C₂²×C₈₃	#99	1	1	yes	characteristic
49	C₄×C₈₃	—	2	1	no	—
50	C₄×C₈₃	—	2	1	no	—
51	C₄×C₈₃	—	2	1	no	—
52	C₄×C₈₃	—	4	1	no	—
53	Dic₈₃	—	16	1	no	—
54	C₂×C₈₃	—	2	1	no	—
55	C₂×C₈₃	—	8	1	no	—
56	C₂×C₈₃	—	8	1	no	—
57	C₂×C₈₃	—	8	1	no	—
58	C₂×C₈₃	AΣL(1,𝔽₁₆)	1	1	yes	characteristic
59	#1402	—	83	1	no	maximal, Sylow
60	C₈₃	#1402	1	1	yes	characteristic, complemented, Sylow
61	#453	D₈₃	1	1	yes	characteristic
62	#353	—	83	1	no	—
63	#331	—	83	1	no	—
64	H(5,𝔽₂)	Dic₈₃	1	1	yes	characteristic, complemented
65	AGL(1,ℤ/8)	—	2	1	no	—
66	#127	C₂×D₈₃	1	1	yes	characteristic
67	C₄⋊C₈	—	166	1	no	—
68	#104	—	2	1	no	—
69	#100	Dic₈₃	1	1	yes	characteristic, complemented
70	#99	—	166	1	no	—
71	#41	—	2	1	no	—
72	#41	—	4	1	no	—
73	C₂×D₄	—	4	1	no	—
74	C₂×D₄	—	4	1	no	—
75	C₂×D₄	—	4	1	no	—
76	C₂×D₄	C₂×Dic₈₃	1	1	yes	characteristic
77	C₂×D₄	C₂²⋊D₈₃	1	1	yes	characteristic
78	SD₁₆	—	4	1	no	—
79	D₈	—	4	1	no	—
80	M₁₆	—	2	1	no	—
81	C₂×C₈	—	332	1	no	—
82	C₂²⋊C₄	—	332	1	no	—
83	C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
84	C₂³	—	2	1	no	—
85	C₂³	—	2	1	no	—
86	C₂³	—	4	1	no	—
87	Q₈	—	2	1	no	—
88	D₄	—	2	1	no	—
89	D₄	—	4	1	no	—
90	D₄	—	4	1	no	—
91	D₄	—	4	2	no	—
92	D₄	—	4	1	no	—
93	D₄	—	8	1	no	—
94	D₄	—	8	1	no	—
95	C₂×C₄	—	2	1	no	—
96	C₂×C₄	—	4	1	no	—
97	C₂×C₄	—	4	1	no	—
98	C₂×C₄	—	664	1	no	—
99	C₂×C₄	C₂²⋊Dic₈₃	1	1	yes	characteristic
100	C₈	—	4	1	no	—
101	C₈	—	664	1	no	—
102	C₂²	—	4	1	no	—
103	C₂²	—	4	1	no	—
104	C₂²	—	4	1	no	—
105	C₂²	—	8	1	no	—
106	C₂²	—	8	1	no	—
107	C₂²	—	16	1	no	—
108	C₂²	#14581707	1	1	yes	characteristic
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	—	4	1	no	—
113	C₄	—	1328	1	no	—
114	C₂	—	2	1	no	—
115	C₂	—	8	1	no	—
116	C₂	—	8	1	no	—
117	C₂	—	8	1	no	—
118	C₂	#19127488	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19127494

C₂

yes

maximal, characteristic, complemented

#19127445

C₂

yes

maximal, characteristic, complemented

#453×C₈₃

C₂

yes

maximal, characteristic

C₄⋊(C₈₃⋊C₈)

—

#14581707

—

H(5,𝔽₂)×C₈₃

C₄

yes

characteristic, complemented

AGL(1,ℤ/8)×C₈₃

—

#127×C₈₃

C₂²

yes

characteristic

#104×C₈₃

—

#100×C₈₃

C₄

yes

characteristic, complemented

C₂×C₈₃⋊C₈

—

C₂²⋊Dic₈₃

—

#41×C₈₃

—

#41×C₈₃

—

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

C₂×C₄

yes

characteristic

C₂×D₄×C₈₃

D₄

yes

characteristic

SD₁₆×C₈₃

—

D₈×C₈₃

—

M₁₆×C₈₃

—

C₄²×C₈₃

D₄

yes

characteristic

C₂³×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

—

Q₈×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

#99

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

AΣL(1,𝔽₁₆)

yes

—

C₈₃

yes

characteristic, complemented, Sylow

#453

D₈₃

yes

—

—

Dic₈₃

yes

characteristic, complemented

AGL(1,ℤ/8)

—

#127

C₂×D₈₃

yes

characteristic

C₄⋊C₈

—

166

—

#104

—

#100

Dic₈₃

yes

characteristic, complemented

#99

—

166

—

#41

—

#41

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₈₃

yes

characteristic

C₂×D₄

C₂²⋊D₈₃

yes

characteristic

SD₁₆

—

D₈

—

M₁₆

—

C₂×C₈

—

332

—

C₂²⋊C₄

—

332

—

C₄²

C₂²⋊D₈₃

yes

characteristic

C₂³

—

C₂³

—

C₂³

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

664

—

C₂×C₄

C₂²⋊Dic₈₃

yes

characteristic

100

C₈

—

101

C₈

—

664

—

102

C₂²

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

#14581707

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

1328

—

114

C₂

—

115

C₂

—

116

C₂

—

117

C₂

—

118

C₂

#19127488

yes

characteristic

	subgroup	G/H
center	C₂	#19127488
commutator	C₂×D₄×C₈₃	C₂×C₄
Frattini	#127	C₂×D₈₃
Fitting, prime radical	#453×C₈₃	C₂
socle	C₂×C₈₃	AΣL(1,𝔽₁₆)
cosocle	#127×C₈₃	C₂²
hypercenter	#453	D₈₃
hypocenter	C₈₃	#1402
LCS	G ▷ C₂×D₄×C₈₃ ▷ C₂×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #127 ◁ #453

subgroup

G/H

center

C₂

#19127488

commutator

C₂×D₄×C₈₃

C₂×C₄

Frattini

#127

C₂×D₈₃

Fitting, prime radical

#453×C₈₃

C₂

socle

C₂×C₈₃

AΣL(1,𝔽₁₆)

cosocle

#127×C₈₃

C₂²

D₈₃

C₈₃

G ▷ C₂×D₄×C₈₃ ▷ C₂×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #127 ◁ #453

N	H ≅ G/N	type
1. #19127494	C₂	semidirect
2. #19127445	C₂	semidirect
6. H(5,𝔽₂)×C₈₃	C₄	semidirect
10. #100×C₈₃	C₄	semidirect
60. C₈₃	#1402	semidirect
64. H(5,𝔽₂)	Dic₈₃	semidirect
69. #100	Dic₈₃	semidirect

H ≅ G/N

type

1. #19127494

C₂

semidirect

2. #19127445

C₂

semidirect

6. H(5,𝔽₂)×C₈₃

C₄

semidirect

10. #100×C₈₃

C₄

semidirect

60. C₈₃

#1402

semidirect

64. H(5,𝔽₂)

Dic₈₃

semidirect

69. #100

Dic₈₃

semidirect

Subgroups — #24905608

Conjugacy classes of subgroups

Complements