Group #20274205 — subgroups

Subgroups — #20274205

Order 6784.

166 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 55 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, complemented
2	#16810904	C₂	1	2	yes	maximal
3	#16810702	C₂	1	1	yes	maximal, characteristic
4	#16810368	C₂	1	2	yes	maximal, complemented
5	C₅₃×#334	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₂₁₂	—	2	1	no	—
7	C₂×Dic₂₁₂	—	2	1	no	—
8	C₂×#11719912	—	2	1	no	—
9	#13133217	C₂²	1	2	yes	—
10	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
11	#13133151	—	2	2	no	—
12	C₄⋊#88322	C₂²	1	1	yes	characteristic
13	#13133139	—	2	2	no	—
14	#13133129	—	2	1	no	—
15	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
16	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×D₂₁₂	D₄	1	1	yes	characteristic
18	C₂×C₄×D₅₃	—	2	1	no	—
19	C₂×Dic₁₀₆	—	2	2	no	—
20	C₂×C₈×C₅₃	—	2	2	no	—
21	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
22	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
23	C₄²×C₅₃	C₂³	1	1	yes	characteristic
24	#11719921	—	2	1	no	—
25	C₄⋊Dic₅₃	—	2	2	no	—
26	C₄⋊Dic₅₃	—	2	2	no	—
27	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
28	C₂×#88322	—	2	1	no	—
29	Dic₂₁₂	—	4	2	no	—
30	Dic₂₁₂	—	4	2	no	—
31	#11719912	—	4	2	no	—
32	C₂²×D₅₃	—	2	1	no	—
33	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
36	C₂×Dic₅₃	—	2	1	no	—
37	C₂×Dic₅₃	—	4	2	no	—
38	D₂₁₂	—	2	2	no	—
39	C₄×D₅₃	—	4	1	no	—
40	C₄×D₅₃	—	4	1	no	—
41	Dic₁₀₆	—	4	2	no	—
42	Dic₁₀₆	—	4	2	no	—
43	C₈×C₅₃	—	2	1	no	—
44	C₈×C₅₃	—	4	2	no	—
45	C₈×C₅₃	D₈	1	2	yes	—
46	#88322	—	4	1	no	—
47	C₂²×C₅₃	#121	1	1	yes	characteristic
48	C₂×D₅₃	—	4	1	no	—
49	C₂×D₅₃	—	4	1	no	—
50	C₂×D₅₃	—	4	1	no	—
51	C₄×C₅₃	—	2	1	no	—
52	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
53	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
54	C₄×C₅₃	#137	1	2	yes	—
55	Dic₅₃	—	4	1	no	—
56	Dic₅₃	—	8	2	no	—
57	#1689	—	53	1	no	maximal, Sylow
58	C₂×C₅₃	#459	1	1	yes	characteristic
59	C₂×C₅₃	#470	1	1	yes	characteristic
60	C₂×C₅₃	#473	1	1	yes	characteristic
61	D₅₃	—	8	1	no	—
62	#494	—	53	1	no	—
63	#474	—	53	2	no	—
64	#436	—	53	1	no	—
65	#334	D₅₃	1	1	yes	characteristic, complemented
66	#332	—	53	2	no	—
67	C₅₃	#1689	1	1	yes	characteristic, complemented, Sylow
68	C₂×Q₁₆	—	106	1	no	—
69	C₂×Q₁₆	—	106	1	no	—
70	C₂×M₁₆	—	106	1	no	—
71	C₄⋊Q₈	—	53	2	no	—
72	C₄×D₄	—	53	1	no	—
73	#107	—	106	2	no	—
74	C₄⋊C₈	—	53	1	no	—
75	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
76	#102	—	106	2	no	—
77	C₂²⋊C₈	—	106	1	no	—
78	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
79	C₂×Q₈	—	106	2	no	—
80	C₂×D₄	—	53	1	no	—
81	C₂²×C₄	—	106	1	no	—
82	Q₁₆	—	212	2	no	—
83	Q₁₆	—	212	2	no	—
84	M₁₆	—	212	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	106	1	no	—
87	C₂×C₈	D₂₁₂	1	1	yes	characteristic
88	C₂×C₈	D₂₁₂	1	1	yes	characteristic
89	C₄⋊C₄	—	53	1	no	—
90	C₄⋊C₄	—	106	2	no	—
91	C₄⋊C₄	—	106	2	no	—
92	C₂²⋊C₄	—	106	1	no	—
93	C₄²	C₂²×D₅₃	1	1	yes	characteristic
94	C₂³	—	106	1	no	—
95	Q₈	—	212	2	no	—
96	Q₈	—	212	2	no	—
97	D₄	—	106	2	no	—
98	C₂×C₄	—	106	1	no	—
99	C₂×C₄	—	212	2	no	—
100	C₂×C₄	—	212	1	no	—
101	C₂×C₄	—	212	1	no	—
102	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
103	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
104	C₂×C₄	#11719949	1	1	yes	characteristic
105	C₈	—	2	1	no	—
106	C₈	—	4	2	no	—
107	C₈	—	212	1	no	—
108	C₈	D₄₂₄	1	2	yes	—
109	C₂²	—	212	1	no	—
110	C₂²	—	212	1	no	—
111	C₂²	—	212	1	no	—
112	C₂²	#13133192	1	1	yes	characteristic
113	C₄	—	2	1	no	—
114	C₄	—	424	2	no	—
115	C₄	—	212	1	no	—
116	C₄	C₂×D₄₂₄	1	1	yes	characteristic
117	C₄	#13133245	1	1	yes	characteristic
118	C₄	#13133246	1	2	yes	—
119	C₂	—	424	1	no	—
120	C₂	#16810819	1	1	yes	characteristic
121	C₂	#16810882	1	1	yes	characteristic
122	C₂	#16810899	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811003

C₂

yes

maximal, characteristic, complemented

#16810904

C₂

yes

maximal

#16810702

C₂

yes

maximal, characteristic

#16810368

C₂

yes

maximal, complemented

C₅₃×#334

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×#11719912

—

#13133217

C₂²

yes

—

C₄×D₂₁₂

C₂²

yes

characteristic

#13133151

—

C₄⋊#88322

C₂²

yes

characteristic

#13133139

—

#13133129

—

C₄⋊C₈×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×D₂₁₂

D₄

yes

characteristic

C₂×C₄×D₅₃

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

#11719921

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

characteristic

C₂×#88322

—

Dic₂₁₂

—

Dic₂₁₂

—

#11719912

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

D₈

yes

—

#88322

—

C₂²×C₅₃

#121

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×D₈

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

—

—

—

D₅₃

yes

characteristic, complemented

#332

—

C₅₃

#1689

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

106

—

C₂×Q₁₆

—

106

—

C₂×M₁₆

—

106

—

C₄⋊Q₈

—

C₄×D₄

—

#107

—

106

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₅₃

yes

—

#102

—

106

—

C₂²⋊C₈

—

106

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×Q₈

—

106

—

C₂×D₄

—

C₂²×C₄

—

106

—

Q₁₆

—

212

—

Q₁₆

—

212

—

M₁₆

—

212

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

D₂₁₂

yes

characteristic

C₂×C₈

D₂₁₂

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₂²⋊C₄

—

106

—

C₄²

C₂²×D₅₃

yes

characteristic

C₂³

—

106

—

Q₈

—

212

—

Q₈

—

212

—

D₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

212

—

100

C₂×C₄

—

212

—

101

C₂×C₄

—

212

—

102

C₂×C₄

C₂×D₂₁₂

yes

characteristic

103

C₂×C₄

D₄×D₅₃

yes

characteristic

104

C₂×C₄

#11719949

yes

characteristic

105

C₈

—

106

C₈

—

107

C₈

—

212

—

108

C₈

D₄₂₄

yes

—

109

C₂²

—

212

—

110

C₂²

—

212

—

111

C₂²

—

212

—

112

C₂²

#13133192

yes

characteristic

113

C₄

—

114

C₄

—

424

—

115

C₄

—

212

—

116

C₄

C₂×D₄₂₄

yes

characteristic

117

C₄

#13133245

yes

characteristic

118

C₄

#13133246

yes

—

119

C₂

—

424

—

120

C₂

#16810819

yes

characteristic

121

C₂

#16810882

yes

characteristic

122

C₂

#16810899

yes

characteristic

	subgroup	G/H
center	C₂²	#13133192
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#334	C₂
socle	C₂²×C₅₃	#121
hypercenter	#334	D₅₃
hypocenter	C₅₃	#1689
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #334

subgroup

G/H

center

C₂²

#13133192

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#334

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #334

N	H ≅ G/N	type
1. #16811003	C₂	semidirect
4. #16810368	C₂	semidirect
5. C₅₃×#334	C₂	semidirect
65. #334	D₅₃	semidirect
67. C₅₃	#1689	semidirect

H ≅ G/N

type

1. #16811003

C₂

semidirect

4. #16810368

C₂

semidirect

5. C₅₃×#334

C₂

semidirect

65. #334

D₅₃

semidirect

67. C₅₃

#1689

semidirect

Subgroups — #20274205

Conjugacy classes of subgroups

Complements