Group #20274086 — subgroups

Subgroups — #20274086

Order 6784.

156 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16811003	C₂	1	1	yes	maximal, characteristic
2	#16810909	C₂	1	2	yes	maximal
3	C₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
4	#16810368	C₂	1	2	yes	maximal
5	C₅₃×#335	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₁₂	—	2	1	no	—
7	C₂×Dic₂₁₂	—	2	1	no	—
8	#13133217	C₂²	1	2	yes	—
9	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
10	#13133149	—	2	2	no	—
11	C₄⋊#88322	C₂²	1	1	yes	characteristic
12	#13133146	—	2	1	no	—
13	#13133142	—	2	2	no	—
14	C₈×Dic₅₃	—	2	1	no	—
15	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
16	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₀₆	—	2	2	no	—
18	C₂×Dic₁₀₆	D₄	1	1	yes	characteristic
19	C₂×C₈×C₅₃	—	2	2	no	—
20	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
21	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
22	C₄²×C₅₃	C₂³	1	1	yes	characteristic
23	C₄⋊Dic₅₃	—	2	2	no	—
24	C₄⋊Dic₅₃	—	2	2	no	—
25	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
26	#11719919	—	2	1	no	—
27	C₄×Dic₅₃	—	2	1	no	—
28	C₂×#88322	—	2	1	no	—
29	Dic₂₁₂	—	4	2	no	—
30	Dic₂₁₂	—	4	2	no	—
31	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
34	C₂×Dic₅₃	—	2	1	no	—
35	C₂×Dic₅₃	—	2	1	no	—
36	C₂×Dic₅₃	—	4	2	no	—
37	Dic₁₀₆	—	2	1	no	—
38	Dic₁₀₆	—	4	2	no	—
39	Dic₁₀₆	—	4	2	no	—
40	Dic₁₀₆	Q₁₆	1	2	yes	complemented
41	C₈×C₅₃	—	2	1	no	—
42	C₈×C₅₃	—	4	2	no	—
43	C₈×C₅₃	SD₁₆	1	2	yes	—
44	#88322	—	4	1	no	—
45	C₂²×C₅₃	#121	1	1	yes	characteristic
46	C₄×C₅₃	—	2	1	no	—
47	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
48	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
49	C₄×C₅₃	#135	1	1	yes	characteristic
50	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
51	Dic₅₃	—	2	2	no	—
52	Dic₅₃	—	4	1	no	—
53	Dic₅₃	—	8	2	no	—
54	#1670	—	53	1	no	maximal, Sylow
55	C₂×C₅₃	#460	1	1	yes	characteristic
56	C₂×C₅₃	#467	1	1	yes	characteristic
57	C₂×C₅₃	#473	1	1	yes	characteristic
58	#494	—	53	1	no	—
59	#475	—	53	2	no	—
60	C₈×Q₈	—	53	1	no	—
61	#335	D₅₃	1	1	yes	characteristic
62	#332	—	53	2	no	—
63	C₅₃	#1670	1	1	yes	characteristic, complemented, Sylow
64	C₂×Q₁₆	—	106	1	no	—
65	C₂×Q₁₆	—	106	1	no	—
66	C₄⋊Q₈	—	53	2	no	—
67	C₄×Q₈	—	53	1	no	—
68	#106	—	106	2	no	—
69	C₄⋊C₈	—	53	1	no	—
70	C₄⋊C₈	—	106	1	no	—
71	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
72	#103	—	106	2	no	—
73	C₄×C₈	—	106	1	no	—
74	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
75	C₂×Q₈	—	53	1	no	—
76	C₂×Q₈	—	106	2	no	—
77	Q₁₆	—	212	2	no	—
78	Q₁₆	—	212	2	no	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	106	1	no	—
81	C₂×C₈	D₂₁₂	1	1	yes	characteristic
82	C₂×C₈	D₂₁₂	1	1	yes	characteristic
83	C₄⋊C₄	—	53	1	no	—
84	C₄⋊C₄	—	106	1	no	—
85	C₄⋊C₄	—	106	2	no	—
86	C₄⋊C₄	—	106	2	no	—
87	C₄²	—	106	1	no	—
88	C₄²	C₂²×D₅₃	1	1	yes	characteristic
89	Q₈	—	53	2	no	—
90	Q₈	—	106	1	no	—
91	Q₈	—	212	2	no	—
92	Q₈	—	212	2	no	—
93	C₂×C₄	—	106	1	no	—
94	C₂×C₄	—	106	1	no	—
95	C₂×C₄	—	212	2	no	—
96	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
97	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
98	C₂×C₄	#11719949	1	1	yes	characteristic
99	C₈	—	2	1	no	—
100	C₈	—	4	2	no	—
101	C₈	—	212	1	no	—
102	C₈	#11719913	1	2	yes	—
103	C₂²	#13133192	1	1	yes	characteristic
104	C₄	—	2	1	no	—
105	C₄	—	106	2	no	—
106	C₄	—	424	2	no	—
107	C₄	—	212	1	no	—
108	C₄	C₂×#11719913	1	1	yes	characteristic
109	C₄	Q₁₆×D₅₃	1	1	yes	characteristic
110	C₄	#13133236	1	1	yes	characteristic
111	C₄	#13133245	1	1	yes	characteristic
112	C₂	#16810824	1	1	yes	characteristic
113	C₂	#16810865	1	1	yes	characteristic
114	C₂	#16810899	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811003

C₂

yes

maximal, characteristic

#16810909

C₂

yes

maximal

C₈×Dic₁₀₆

C₂

yes

maximal, characteristic

#16810368

C₂

yes

maximal

C₅₃×#335

C₂

yes

maximal, characteristic

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

#13133217

C₂²

yes

—

C₄×Dic₁₀₆

C₂²

yes

characteristic

#13133149

—

C₄⋊#88322

C₂²

yes

characteristic

#13133146

—

#13133142

—

C₈×Dic₅₃

—

C₄⋊C₈×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

D₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

characteristic

#11719919

—

C₄×Dic₅₃

—

C₂×#88322

—

Dic₂₁₂

—

Dic₂₁₂

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

Q₁₆

yes

complemented

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

SD₁₆

yes

—

#88322

—

C₂²×C₅₃

#121

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₄×C₅₃

C₂×Q₁₆

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

C₈×Q₈

—

#335

D₅₃

yes

characteristic

#332

—

C₅₃

#1670

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

106

—

C₂×Q₁₆

—

106

—

C₄⋊Q₈

—

C₄×Q₈

—

#106

—

106

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₄⋊C₈

C₂×D₅₃

yes

—

#103

—

106

—

C₄×C₈

—

106

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

106

—

Q₁₆

—

212

—

Q₁₆

—

212

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

D₂₁₂

yes

characteristic

C₂×C₈

D₂₁₂

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄²

—

106

—

C₄²

C₂²×D₅₃

yes

characteristic

Q₈

—

Q₈

—

106

—

Q₈

—

212

—

Q₈

—

212

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

212

—

C₂×C₄

C₂×D₂₁₂

yes

characteristic

C₂×C₄

D₄×D₅₃

yes

characteristic

C₂×C₄

#11719949

yes

characteristic

C₈

—

100

C₈

—

101

C₈

—

212

—

102

C₈

#11719913

yes

—

103

C₂²

#13133192

yes

characteristic

104

C₄

—

105

C₄

—

106

—

106

C₄

—

424

—

107

C₄

—

212

—

108

C₄

C₂×#11719913

yes

characteristic

109

C₄

Q₁₆×D₅₃

yes

characteristic

110

C₄

#13133236

yes

characteristic

111

C₄

#13133245

yes

characteristic

112

C₂

#16810824

yes

characteristic

113

C₂

#16810865

yes

characteristic

114

C₂

#16810899

yes

characteristic

	subgroup	G/H
center	C₂²	#13133192
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#335	C₂
socle	C₂²×C₅₃	#121
hypercenter	#335	D₅₃
hypocenter	C₅₃	#1670
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #335

subgroup

G/H

center

C₂²

#13133192

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#335

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #335

N	H ≅ G/N	type
40. Dic₁₀₆	Q₁₆	semidirect
63. C₅₃	#1670	semidirect

H ≅ G/N

type

40. Dic₁₀₆

Q₁₆

semidirect

63. C₅₃

#1670

semidirect

Subgroups — #20274086

Conjugacy classes of subgroups

Complements