Group #20276832 — subgroups

Subgroups — #20276832

Order 6784.

148 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 45 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	#16811039	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810890	C₂	1	2	yes	maximal, complemented
3	#16810453	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810436	C₂	1	2	yes	maximal, complemented
5	C₂×D₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#13133243	—	2	2	no	—
7	#13133223	C₂²	1	1	yes	characteristic
8	#13133157	C₂²	1	1	yes	characteristic
9	#13133156	—	2	1	no	—
10	C₁₆×D₅₃	—	2	1	no	—
11	#13133153	C₂²	1	2	yes	—
12	#13133134	—	2	2	no	—
13	C₂×D₈×C₅₃	C₂²	1	2	yes	—
14	D₁₆×C₅₃	—	2	1	no	—
15	D₁₆×C₅₃	—	2	1	no	—
16	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×D₄×C₅₃	—	2	2	no	—
18	#11719945	D₄	1	1	yes	characteristic
19	D₈×C₅₃	—	2	2	no	—
20	D₈×C₅₃	—	2	2	no	—
21	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
22	#11719923	—	4	2	no	—
23	#11719922	—	4	2	no	—
24	#11719917	—	2	2	no	—
25	#11719917	D₄	1	1	yes	characteristic
26	#11719912	—	2	1	no	—
27	C₈×D₅₃	—	2	1	no	—
28	C₁₆×C₅₃	D₄	1	1	yes	characteristic
29	C₁₆×C₅₃	D₄	1	1	yes	characteristic
30	#11719908	—	2	1	no	—
31	C₂³×C₅₃	—	4	2	no	—
32	D₄×C₅₃	—	4	2	no	—
33	D₄×C₅₃	—	4	2	no	—
34	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₂²⋊D₅₃	—	2	1	no	—
36	D₂₁₂	D₈	1	1	yes	characteristic
37	C₄×D₅₃	—	2	1	no	—
38	Dic₁₀₆	D₈	1	1	yes	characteristic
39	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₈×C₅₃	#41	1	1	yes	characteristic
41	#88322	—	2	1	no	—
42	#88322	—	4	2	no	—
43	C₂²×C₅₃	—	8	2	no	—
44	C₂²×C₅₃	—	16	2	no	—
45	C₂²×C₅₃	#135	1	1	yes	characteristic
46	C₂×D₅₃	—	2	1	no	—
47	C₄×C₅₃	#121	1	1	yes	characteristic
48	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
49	Dic₅₃	—	2	1	no	—
50	#2221	—	53	1	no	maximal, Sylow
51	C₂×C₅₃	—	2	1	no	—
52	C₂×C₅₃	—	16	2	no	—
53	C₂×C₅₃	#466	1	1	yes	characteristic
54	D₅₃	—	4	1	no	—
55	C₂×D₁₆	D₅₃	1	1	yes	characteristic, complemented
56	#504	—	53	1	no	—
57	#472	—	53	2	no	—
58	#368	—	53	1	no	—
59	#361	—	53	2	no	—
60	C₅₃	#2221	1	1	yes	characteristic, complemented, Sylow
61	AGL(1,ℤ/8)	—	106	2	no	—
62	C₂×D₈	C₂×D₅₃	1	2	yes	—
63	#131	—	53	1	no	—
64	D₁₆	—	2	1	no	—
65	D₁₆	—	2	1	no	—
66	M₃₂	—	53	1	no	—
67	M₃₂	—	106	1	no	—
68	C₂×C₁₆	—	106	1	no	—
69	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
70	#108	—	53	2	no	—
71	#100	—	106	2	no	—
72	#41	—	53	1	no	—
73	C₂×D₄	—	2	2	no	—
74	SD₁₆	—	212	2	no	—
75	D₈	—	2	2	no	—
76	D₈	—	2	2	no	—
77	D₈	—	212	2	no	—
78	M₁₆	—	53	1	no	—
79	M₁₆	—	106	1	no	—
80	M₁₆	—	106	2	no	—
81	C₂×C₈	—	106	1	no	—
82	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
83	C₁₆	—	106	1	no	—
84	C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
85	C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
86	C₂³	—	4	2	no	—
87	Q₈	—	53	1	no	—
88	D₄	—	4	2	no	—
89	D₄	—	4	2	no	—
90	D₄	—	53	1	no	—
91	D₄	—	106	1	no	—
92	C₂×C₄	—	106	1	no	—
93	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
94	C₈	—	106	1	no	—
95	C₈	—	212	2	no	—
96	C₈	#11719947	1	1	yes	characteristic
97	C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
98	C₂²	—	8	2	no	—
99	C₂²	—	16	2	no	—
100	C₂²	—	106	1	no	—
101	C₂²	#13133236	1	1	yes	characteristic
102	C₄	—	106	1	no	—
103	C₄	#13133188	1	1	yes	characteristic
104	C₄	D₈×D₅₃	1	1	yes	characteristic
105	C₂	—	2	1	no	—
106	C₂	—	16	2	no	—
107	C₂	—	212	1	no	—
108	C₂	#16810856	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811039

C₂

yes

maximal, characteristic, complemented

#16810890

C₂

yes

maximal, complemented

#16810453

C₂

yes

maximal, characteristic, complemented

#16810436

C₂

yes

maximal, complemented

C₂×D₁₆×C₅₃

C₂

yes

maximal, characteristic, complemented

#13133243

—

#13133223

C₂²

yes

characteristic

#13133157

C₂²

yes

characteristic

#13133156

—

C₁₆×D₅₃

—

#13133153

C₂²

yes

—

#13133134

—

C₂×D₈×C₅₃

C₂²

yes

—

D₁₆×C₅₃

—

D₁₆×C₅₃

—

C₂×C₁₆×C₅₃

C₂²

yes

characteristic

C₂×D₄×C₅₃

—

#11719945

D₄

yes

characteristic

D₈×C₅₃

—

D₈×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

—

—

—

D₄

yes

characteristic

#11719912

—

C₈×D₅₃

—

C₁₆×C₅₃

D₄

yes

characteristic

C₁₆×C₅₃

D₄

yes

characteristic

#11719908

—

C₂³×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²⋊D₅₃

—

D₂₁₂

D₈

yes

characteristic

C₄×D₅₃

—

Dic₁₀₆

D₈

yes

characteristic

C₈×C₅₃

C₂×D₄

yes

C₈×C₅₃

yes

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#135

yes

characteristic

C₂×D₅₃

—

C₄×C₅₃

#121

yes

characteristic

C₄×C₅₃

C₂×D₈

yes

characteristic

Dic₅₃

—

#2221

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#466

yes

characteristic

D₅₃

—

C₂×D₁₆

D₅₃

yes

characteristic, complemented

—

—

—

—

C₅₃

#2221

yes

characteristic, complemented, Sylow

AGL(1,ℤ/8)

—

106

—

C₂×D₈

C₂×D₅₃

yes

—

#131

—

D₁₆

—

D₁₆

—

M₃₂

—

M₃₂

—

106

—

C₂×C₁₆

—

106

—

C₂×C₁₆

C₂×D₅₃

yes

characteristic

#108

—

#100

—

106

—

#41

—

C₂×D₄

—

SD₁₆

—

212

—

D₈

—

D₈

—

D₈

—

212

—

M₁₆

—

M₁₆

—

106

—

M₁₆

—

106

—

C₂×C₈

—

106

—

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₁₆

—

106

—

C₁₆

C₂²⋊D₅₃

yes

characteristic

C₁₆

C₂²⋊D₅₃

yes

characteristic

C₂³

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

D₄×D₅₃

yes

characteristic

C₈

—

106

—

C₈

—

212

—

C₈

#11719947

yes

characteristic

C₈

C₂×C₂²⋊D₅₃

yes

characteristic

C₂²

—

C₂²

—

100

C₂²

—

106

—

101

C₂²

#13133236

yes

characteristic

102

C₄

—

106

—

103

C₄

#13133188

yes

characteristic

104

C₄

D₈×D₅₃

yes

characteristic

105

C₂

—

106

C₂

—

107

C₂

—

212

—

108

C₂

#16810856

yes

characteristic

	subgroup	G/H
center	C₂	#16810856
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₂×D₁₆×C₅₃	C₂
socle	C₂×C₅₃	#466
hypercenter	C₂×D₁₆	D₅₃
hypocenter	C₅₃	#2221
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₁₆

subgroup

G/H

center

C₂

#16810856

commutator

C₂×C₈×C₅₃

C₂³

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₂×D₁₆×C₅₃

C₂

socle

C₂×C₅₃

#466

hypercenter

C₂×D₁₆

D₅₃

hypocenter

C₅₃

#2221

LCS

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₁₆

N	H ≅ G/N	type
1. #16811039	C₂	semidirect
2. #16810890	C₂	semidirect
3. #16810453	C₂	semidirect
4. #16810436	C₂	semidirect
5. C₂×D₁₆×C₅₃	C₂	semidirect
55. C₂×D₁₆	D₅₃	semidirect
60. C₅₃	#2221	semidirect

H ≅ G/N

type

1. #16811039

C₂

semidirect

2. #16810890

C₂

semidirect

3. #16810453

C₂

semidirect

4. #16810436

C₂

semidirect

5. C₂×D₁₆×C₅₃

C₂

semidirect

55. C₂×D₁₆

D₅₃

semidirect

60. C₅₃

#2221

semidirect

Subgroups — #20276832

Conjugacy classes of subgroups

Complements