Group #20276793 — subgroups

Subgroups — #20276793

Order 6784.

150 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 47 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	#16811064	C₂	1	2	yes	maximal
2	#16810741	C₂	1	1	yes	maximal, characteristic
3	#16810444	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810439	C₂	1	2	yes	maximal, complemented
5	C₅₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
6	#13133239	C₂²	1	1	yes	characteristic
7	#13133223	—	2	1	no	—
8	#13133163	—	2	2	no	—
9	#13133160	—	2	2	no	—
10	#13133157	C₂²	1	2	yes	—
11	#13133153	C₂²	1	1	yes	characteristic
12	#13133145	—	2	1	no	—
13	C₂×Q₁₆×C₅₃	—	2	1	no	—
14	C₂×Q₁₆×C₅₃	C₂²	1	2	yes	—
15	C₄⋊Q₈×C₅₃	—	2	1	no	—
16	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₅₃	—	2	2	no	—
18	#11719945	—	2	1	no	—
19	Q₁₆×C₅₃	—	2	2	no	—
20	Q₁₆×C₅₃	—	2	2	no	—
21	Q₁₆×C₅₃	—	4	1	no	—
22	Q₁₆×C₅₃	—	4	1	no	—
23	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₅₃	—	4	2	no	—
26	C₄²×C₅₃	D₄	1	1	yes	characteristic
27	#11719917	—	2	1	no	—
28	Dic₂₁₂	D₄	1	1	yes	characteristic
29	D₄₂₄	D₄	1	1	yes	characteristic
30	#11719913	—	2	1	no	—
31	#11719912	—	4	1	no	—
32	C₈×D₅₃	—	4	1	no	—
33	#11719908	—	2	2	no	—
34	Q₈×C₅₃	—	4	2	no	—
35	Q₈×C₅₃	—	4	2	no	—
36	C₂×C₄×C₅₃	—	2	1	no	—
37	C₂×C₄×C₅₃	—	4	2	no	—
38	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂²⋊D₅₃	—	4	1	no	—
40	D₂₁₂	—	2	1	no	—
41	C₄×D₅₃	—	4	1	no	—
42	Dic₁₀₆	—	2	1	no	—
43	C₈×C₅₃	D₈	1	1	yes	characteristic
44	C₈×C₅₃	D₈	1	1	yes	characteristic
45	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₈×C₅₃	#41	1	1	yes	characteristic
47	#88322	—	4	1	no	—
48	C₂²×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
49	C₂×D₅₃	—	4	1	no	—
50	C₄×C₅₃	—	2	2	no	—
51	C₄×C₅₃	—	8	2	no	—
52	C₄×C₅₃	#121	1	1	yes	characteristic
53	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
54	Dic₅₃	—	4	1	no	—
55	#2214	—	53	1	no	maximal, Sylow
56	C₂×C₅₃	—	2	1	no	—
57	C₂×C₅₃	#459	1	1	yes	characteristic
58	D₅₃	—	8	1	no	—
59	#510	—	53	2	no	—
60	#494	D₅₃	1	1	yes	characteristic, complemented
61	#443	—	53	1	no	—
62	#364	—	53	1	no	—
63	#362	—	53	2	no	—
64	C₅₃	#2214	1	1	yes	characteristic, complemented, Sylow
65	#135	—	53	1	no	—
66	C₂×Q₁₆	—	2	1	no	—
67	C₂×Q₁₆	C₂×D₅₃	1	2	yes	—
68	#131	—	106	1	no	—
69	C₄⋊Q₈	—	2	1	no	—
70	Q₃₂	—	106	2	no	—
71	SD₃₂	—	106	2	no	—
72	M₃₂	—	53	2	no	—
73	#108	—	53	1	no	—
74	#104	—	106	1	no	—
75	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
76	#41	—	106	1	no	—
77	C₂×Q₈	—	2	2	no	—
78	Q₁₆	—	2	2	no	—
79	Q₁₆	—	2	2	no	—
80	Q₁₆	—	4	1	no	—
81	Q₁₆	—	4	1	no	—
82	Q₁₆	—	53	1	no	—
83	SD₁₆	—	106	1	no	—
84	D₈	—	53	1	no	—
85	M₁₆	—	106	1	no	—
86	M₁₆	—	212	1	no	—
87	C₂×C₈	—	212	1	no	—
88	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
89	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
90	C₄⋊C₄	—	4	2	no	—
91	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
92	C₁₆	—	106	2	no	—
93	Q₈	—	4	2	no	—
94	Q₈	—	4	2	no	—
95	Q₈	—	106	1	no	—
96	D₄	—	106	1	no	—
97	D₄	—	212	1	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	4	2	no	—
100	C₂×C₄	—	212	1	no	—
101	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
102	C₈	—	212	1	no	—
103	C₈	#11719922	1	1	yes	characteristic
104	C₈	#11719922	1	1	yes	characteristic
105	C₈	D₄×D₅₃	1	1	yes	characteristic
106	C₈	#11719947	1	1	yes	characteristic
107	C₂²	—	212	1	no	—
108	C₂²	#13133243	1	1	yes	characteristic
109	C₄	—	2	2	no	—
110	C₄	—	8	2	no	—
111	C₄	—	212	1	no	—
112	C₄	#13133188	1	1	yes	characteristic
113	C₄	C₂×#11719922	1	1	yes	characteristic
114	C₂	—	2	1	no	—
115	C₂	—	424	1	no	—
116	C₂	#16810815	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811064

C₂

yes

maximal

#16810741

C₂

yes

maximal, characteristic

#16810444

C₂

yes

maximal, characteristic, complemented

#16810439

C₂

yes

maximal, complemented

C₅₃×#494

C₂

yes

maximal, characteristic, complemented

#13133239

C₂²

yes

—

—

—

C₂²

yes

—

#13133153

C₂²

yes

characteristic

#13133145

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

C₂²

yes

—

C₄⋊Q₈×C₅₃

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

#11719945

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

D₄

yes

characteristic

#11719917

—

Dic₂₁₂

D₄

yes

characteristic

D₄₂₄

D₄

yes

characteristic

#11719913

—

#11719912

—

C₈×D₅₃

—

#11719908

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²⋊D₅₃

—

D₂₁₂

—

C₄×D₅₃

—

Dic₁₀₆

—

C₈×C₅₃

D₈

yes

characteristic

C₈×C₅₃

D₈

yes

characteristic

C₈×C₅₃

C₂×D₄

yes

C₈×C₅₃

yes

—

C₂²×C₅₃

AGL(1,ℤ/8)

yes

characteristic

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#121

yes

characteristic

C₄×C₅₃

C₂×D₈

yes

characteristic

Dic₅₃

—

#2214

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#459

yes

characteristic

D₅₃

—

#510

—

#494

D₅₃

yes

characteristic, complemented

#443

—

#364

—

#362

—

C₅₃

#2214

yes

characteristic, complemented, Sylow

#135

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×D₅₃

yes

—

#131

—

106

—

C₄⋊Q₈

—

Q₃₂

—

106

—

SD₃₂

—

106

—

M₃₂

—

#108

—

#104

—

106

—

C₄×C₈

C₂×D₅₃

yes

characteristic

#41

—

106

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

SD₁₆

—

106

—

D₈

—

M₁₆

—

106

—

M₁₆

—

212

—

C₂×C₈

—

212

—

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

C₄²

C₂²⋊D₅₃

yes

characteristic

C₁₆

—

106

—

Q₈

—

Q₈

—

Q₈

—

106

—

D₄

—

106

—

D₄

—

212

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

212

—

101

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

102

C₈

—

212

—

103

C₈

#11719922

yes

characteristic

104

C₈

#11719922

yes

characteristic

105

C₈

D₄×D₅₃

yes

characteristic

106

C₈

#11719947

yes

characteristic

107

C₂²

—

212

—

108

C₂²

#13133243

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

212

—

112

C₄

#13133188

yes

characteristic

113

C₄

C₂×#11719922

yes

characteristic

114

C₂

—

115

C₂

—

424

—

116

C₂

#16810815

yes

characteristic

	subgroup	G/H
center	C₂	#16810815
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#494	C₂
socle	C₂×C₅₃	#459
hypercenter	#494	D₅₃
hypocenter	C₅₃	#2214
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #494

subgroup

G/H

center

C₂

#16810815

commutator

C₂×C₈×C₅₃

C₂³

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₅₃×#494

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #494

N	H ≅ G/N	type
3. #16810444	C₂	semidirect
4. #16810439	C₂	semidirect
5. C₅₃×#494	C₂	semidirect
60. #494	D₅₃	semidirect
64. C₅₃	#2214	semidirect

H ≅ G/N

type

3. #16810444

C₂

semidirect

4. #16810439

C₂

semidirect

5. C₅₃×#494

C₂

semidirect

60. #494

D₅₃

semidirect

64. C₅₃

#2214

semidirect

Subgroups — #20276793

Conjugacy classes of subgroups

Complements