Group #20275983 — subgroups

Subgroups — #20275983

Order 6784.

256 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₂²×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
2	#16810631	C₂	1	2	yes	maximal
3	C₂×#13133135	C₂	1	2	yes	maximal
4	#16810388	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#424	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₀₆	C₂²	1	2	yes	—
7	C₂×Dic₂₁₂	—	2	2	no	—
8	C₂×Dic₂₁₂	—	2	4	no	—
9	C₂×#11719917	C₂²	1	2	yes	—
10	#13133140	—	2	4	no	—
11	#13133135	—	2	4	no	—
12	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	#117×C₅₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₅₃	—	2	2	no	—
16	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
17	C₂²×Dic₅₃	—	2	2	no	—
18	C₂×Dic₁₀₆	—	2	2	no	—
19	C₂×Dic₁₀₆	—	2	2	no	—
20	C₂×Dic₁₀₆	—	2	2	no	—
21	C₂×Dic₁₀₆	—	2	2	no	—
22	C₂×Dic₁₀₆	—	2	2	no	—
23	C₂×Dic₁₀₆	D₄	1	4	yes	—
24	M₁₆×C₅₃	—	2	2	no	—
25	C₂×C₈×C₅₃	—	2	2	no	—
26	C₂×C₈×C₅₃	—	2	1	no	—
27	C₂×C₈×C₅₃	D₄	1	2	yes	—
28	C₄⋊C₄×C₅₃	D₄	1	2	yes	—
29	C₂²⋊C₄×C₅₃	—	2	1	no	—
30	C₄²×C₅₃	—	2	1	no	—
31	#11719917	—	2	2	no	—
32	#11719917	—	2	2	no	—
33	C₂×#88322	—	2	2	no	—
34	Dic₂₁₂	—	2	8	no	—
35	C₂³×C₅₃	#41	1	1	yes	characteristic
36	C₂×C₄×C₅₃	—	2	2	no	—
37	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₅₃	#41	1	2	yes	—
41	C₂×Dic₅₃	—	2	4	no	—
42	C₂×Dic₅₃	—	4	2	no	—
43	C₂×Dic₅₃	—	4	2	no	—
44	Dic₁₀₆	—	2	4	no	—
45	Dic₁₀₆	—	2	4	no	—
46	Dic₁₀₆	—	4	2	no	—
47	Dic₁₀₆	—	4	2	no	—
48	C₈×C₅₃	—	2	2	no	—
49	C₈×C₅₃	—	4	1	no	—
50	#88322	—	4	2	no	—
51	C₂²×C₅₃	—	2	2	no	—
52	C₂²×C₅₃	#121	1	1	yes	characteristic
53	C₂²×C₅₃	#123	1	2	yes	—
54	C₄×C₅₃	—	4	2	no	—
55	C₄×C₅₃	#120	1	1	yes	characteristic
56	C₄×C₅₃	#121	1	2	yes	—
57	C₄×C₅₃	#124	1	1	yes	characteristic
58	Dic₅₃	—	4	4	no	—
59	#2047	—	53	1	no	maximal, Sylow
60	C₂×C₅₃	—	2	2	no	—
61	C₂×C₅₃	#394	1	1	yes	characteristic
62	C₂×C₅₃	#473	1	2	yes	—
63	C₂²×Q₁₆	—	53	1	no	—
64	#424	D₅₃	1	1	yes	characteristic
65	#419	—	53	2	no	—
66	C₂×#101	—	53	2	no	—
67	#341	—	53	1	no	—
68	C₅₃	#2047	1	1	yes	characteristic, complemented, Sylow
69	C₂²×Q₈	—	53	2	no	—
70	C₂×Q₁₆	—	106	4	no	—
71	C₂×Q₁₆	—	106	2	no	—
72	C₂×M₁₆	—	53	2	no	—
73	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
74	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
75	#117	C₂×D₅₃	1	1	yes	characteristic
76	C₄⋊C₈	—	2	2	no	—
77	#103	—	106	4	no	—
78	#101	—	106	4	no	—
79	C₂×Q₈	—	53	4	no	—
80	C₂×Q₈	—	106	2	no	—
81	C₂×Q₈	—	106	2	no	—
82	C₂×Q₈	—	106	2	no	—
83	C₂×Q₈	—	106	2	no	—
84	C₂×Q₈	—	106	2	no	—
85	C₂²×C₄	—	106	2	no	—
86	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
87	Q₁₆	—	106	8	no	—
88	M₁₆	—	2	2	no	—
89	M₁₆	—	106	2	no	—
90	M₁₆	—	106	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	106	2	no	—
94	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
95	C₄⋊C₄	D₂₁₂	1	2	yes	—
96	C₂²⋊C₄	—	2	1	no	—
97	C₄²	—	2	1	no	—
98	C₂³	#11719945	1	1	yes	characteristic
99	Q₈	—	106	4	no	—
100	Q₈	—	106	4	no	—
101	Q₈	—	212	2	no	—
102	Q₈	—	212	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	106	4	no	—
105	C₂×C₄	—	212	2	no	—
106	C₂×C₄	—	212	2	no	—
107	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
108	C₂×C₄	D₄×D₅₃	1	2	yes	—
109	C₂×C₄	#11719949	1	2	yes	—
110	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
111	C₈	—	2	2	no	—
112	C₈	—	4	1	no	—
113	C₈	—	212	2	no	—
114	C₂²	—	2	2	no	—
115	C₂²	#13133191	1	1	yes	characteristic
116	C₂²	#13133202	1	2	yes	—
117	C₄	—	4	2	no	—
118	C₄	—	212	4	no	—
119	C₄	#13133185	1	1	yes	characteristic
120	C₄	#13133192	1	2	yes	—
121	C₄	#13133206	1	1	yes	characteristic
122	C₂	—	2	2	no	—
123	C₂	#16810540	1	1	yes	characteristic
124	C₂	#16810898	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₂₁₂

C₂

yes

maximal, characteristic

#16810631

C₂

yes

maximal

C₂×#13133135

C₂

yes

maximal

#16810388

C₂

yes

maximal, characteristic

C₅₃×#424

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₂²

yes

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×#11719917

C₂²

yes

—

#13133140

—

#13133135

—

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

#117×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

D₄

yes

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

—

C₄⋊C₄×C₅₃

D₄

yes

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

#11719917

—

#11719917

—

C₂×#88322

—

Dic₂₁₂

—

C₂³×C₅₃

#41

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

#121

yes

characteristic

C₂²×C₅₃

#123

yes

—

C₄×C₅₃

—

C₄×C₅₃

#120

yes

characteristic

C₄×C₅₃

#121

yes

—

C₄×C₅₃

#124

yes

characteristic

Dic₅₃

—

#2047

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#394

yes

characteristic

C₂×C₅₃

#473

yes

—

C₂²×Q₁₆

—

#424

D₅₃

yes

—

—

—

C₅₃

#2047

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×Q₁₆

—

106

—

C₂×Q₁₆

—

106

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

characteristic

C₂²×C₈

C₂×D₅₃

yes

characteristic

#117

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

#103

—

106

—

#101

—

106

—

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

Q₁₆

—

106

—

M₁₆

—

M₁₆

—

106

—

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

C₂²⋊D₅₃

yes

—

C₄⋊C₄

D₂₁₂

yes

—

C₂²⋊C₄

—

C₄²

—

C₂³

#11719945

yes

characteristic

Q₈

—

106

—

100

Q₈

—

106

—

101

Q₈

—

212

—

102

Q₈

—

212

—

103

C₂×C₄

—

104

C₂×C₄

—

106

—

105

C₂×C₄

—

212

—

106

C₂×C₄

—

212

—

107

C₂×C₄

C₂×D₂₁₂

yes

characteristic

108

C₂×C₄

D₄×D₅₃

yes

—

109

C₂×C₄

#11719949

yes

—

110

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

—

212

—

114

C₂²

—

115

C₂²

#13133191

yes

characteristic

116

C₂²

#13133202

yes

—

117

C₄

—

118

C₄

—

212

—

119

C₄

#13133185

yes

characteristic

120

C₄

#13133192

yes

—

121

C₄

#13133206

yes

characteristic

122

C₂

—

123

C₂

#16810540

yes

characteristic

124

C₂

#16810898

yes

—

	subgroup	G/H
center	C₂²	#13133191
commutator	C₂²×C₄×C₅₃	C₂³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#424	C₂
socle	C₂²×C₅₃	#121
hypercenter	#424	D₅₃
hypocenter	C₅₃	#2047
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

subgroup

G/H

center

C₂²

#13133191

commutator

C₂²×C₄×C₅₃

C₂³

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#424

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

N	H ≅ G/N	type
68. C₅₃	#2047	semidirect

H ≅ G/N

type

68. C₅₃

#2047

semidirect

Subgroups — #20275983

Conjugacy classes of subgroups

Complements