Group #20286109 — subgroups

Subgroups — #20286109

Order 6784.

342 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 125 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	#16811180	C₂	1	2	yes	maximal
2	#16810909	C₂	1	4	yes	maximal, complemented
3	#16810881	C₂	1	4	yes	maximal
4	#16810834	C₂	1	4	yes	maximal, complemented
5	C₅₃×#423	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₁₂	C₂²	1	4	yes	—
7	#13133217	C₂²	1	2	yes	—
8	#13133217	C₂²	1	2	yes	—
9	#13133209	—	2	2	no	—
10	#13133196	—	2	2	no	—
11	#13133196	C₂²	1	4	yes	—
12	C₄×Dic₁₀₆	—	2	2	no	—
13	C₄×Dic₁₀₆	C₂²	1	4	yes	—
14	#13133176	—	2	2	no	—
15	#13133149	C₂²	1	4	yes	—
16	#13133142	C₂²	1	8	yes	—
17	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₅₃	C₂²	1	4	yes	—
20	C₂²×C₄×C₅₃	D₄	1	2	yes	—
21	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
22	C₂×Dic₁₀₆	—	2	2	no	—
23	C₂×Dic₁₀₆	C₂³	1	4	yes	—
24	M₁₆×C₅₃	—	4	2	no	—
25	C₂×C₈×C₅₃	C₂³	1	4	yes	—
26	C₄²×C₅₃	D₄	1	1	yes	characteristic
27	C₄²×C₅₃	D₄	1	1	yes	characteristic
28	C₄²×C₅₃	C₂³	1	2	yes	—
29	C₂²⋊Dic₅₃	—	2	4	no	—
30	C₄⋊Dic₅₃	—	2	2	no	—
31	C₄⋊Dic₅₃	—	2	2	no	—
32	C₄⋊Dic₅₃	—	2	2	no	—
33	C₄⋊Dic₅₃	C₂³	1	4	yes	—
34	#11719919	—	2	4	no	—
35	#11719919	—	2	4	no	—
36	C₄×Dic₅₃	—	2	4	no	—
37	Dic₂₁₂	—	4	4	no	—
38	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₅₃	—	2	4	no	—
40	C₂×C₄×C₅₃	—	2	1	no	—
41	C₂×C₄×C₅₃	—	2	1	no	—
42	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
46	C₂×Dic₅₃	—	2	4	no	—
47	C₂×Dic₅₃	—	2	4	no	—
48	Dic₁₀₆	—	2	4	no	—
49	Dic₁₀₆	—	2	4	no	—
50	Dic₁₀₆	—	4	4	no	—
51	C₈×C₅₃	—	2	4	no	—
52	C₂²×C₅₃	—	2	1	no	—
53	C₂²×C₅₃	—	2	2	no	—
54	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
55	C₄×C₅₃	—	2	1	no	—
56	C₄×C₅₃	—	2	2	no	—
57	C₄×C₅₃	#137	1	4	yes	—
58	C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
59	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
60	Dic₅₃	—	4	4	no	—
61	Dic₅₃	—	4	4	no	—
62	#3255	—	53	1	no	maximal, Sylow
63	C₂×C₅₃	—	4	1	no	—
64	C₂×C₅₃	#537	1	1	yes	characteristic
65	C₂×C₅₃	C₂×#137	1	2	yes	—
66	#533	—	53	2	no	—
67	#475	—	53	4	no	—
68	#470	—	53	4	no	—
69	#462	—	53	4	no	—
70	#423	D₅₃	1	1	yes	characteristic
71	C₅₃	#3255	1	1	yes	characteristic, complemented, Sylow
72	C₂×Q₁₆	—	53	4	no	—
73	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
74	C₄⋊Q₈	—	53	2	no	—
75	C₄⋊Q₈	—	53	2	no	—
76	#125	—	106	2	no	—
77	C₂²⋊Q₈	—	53	4	no	—
78	C₂²⋊Q₈	—	106	2	no	—
79	C₄×Q₈	—	53	4	no	—
80	C₄×Q₈	—	106	2	no	—
81	#117	—	106	2	no	—
82	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
83	#106	—	53	4	no	—
84	C₄⋊C₈	C₂×D₅₃	1	4	yes	—
85	#103	—	53	8	no	—
86	C₂×Q₈	—	53	4	no	—
87	C₂×Q₈	—	106	2	no	—
88	C₂²×C₄	D₂₁₂	1	2	yes	—
89	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
90	Q₁₆	—	212	4	no	—
91	M₁₆	—	4	2	no	—
92	C₂×C₈	C₂²×D₅₃	1	4	yes	—
93	C₄⋊C₄	—	53	4	no	—
94	C₄⋊C₄	—	106	2	no	—
95	C₄⋊C₄	—	106	2	no	—
96	C₄⋊C₄	—	106	4	no	—
97	C₄⋊C₄	—	106	4	no	—
98	C₄⋊C₄	—	106	2	no	—
99	C₂²⋊C₄	—	106	4	no	—
100	C₄²	—	106	4	no	—
101	C₄²	D₂₁₂	1	1	yes	characteristic
102	C₄²	D₂₁₂	1	1	yes	characteristic
103	C₄²	C₂²×D₅₃	1	2	yes	—
104	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
105	Q₈	—	106	4	no	—
106	Q₈	—	106	4	no	—
107	Q₈	—	212	4	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	106	4	no	—
112	C₂×C₄	—	106	4	no	—
113	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
114	C₂×C₄	C₂×D₂₁₂	1	2	yes	—
115	C₂×C₄	C₂×D₂₁₂	1	2	yes	—
116	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
117	C₈	—	2	4	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	2	1	no	—
120	C₂²	C₂²×D₂₁₂	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	2	1	no	—
123	C₄	—	212	4	no	—
124	C₄	—	212	4	no	—
125	C₄	#13133250	1	4	yes	—
126	C₄	#13133262	1	1	yes	characteristic
127	C₄	#13133265	1	1	yes	characteristic
128	C₂	—	4	1	no	—
129	C₂	#16811204	1	1	yes	characteristic
130	C₂	C₂×#13133250	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16811180

C₂

yes

maximal

#16810909

C₂

yes

maximal, complemented

#16810881

C₂

yes

maximal

#16810834

C₂

yes

maximal, complemented

C₅₃×#423

C₂

yes

maximal, characteristic

C₂×Dic₂₁₂

C₂²

yes

—

#13133217

C₂²

yes

—

#13133217

C₂²

yes

—

#13133209

—

#13133196

—

#13133196

C₂²

yes

—

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

C₂²

yes

—

#13133176

—

#13133149

C₂²

yes

—

#13133142

C₂²

yes

—

C₂×M₁₆×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂³

yes

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

—

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

#11719919

—

C₄×Dic₅₃

—

Dic₂₁₂

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#137

yes

—

C₄×C₅₃

H(5,𝔽₂)

yes

characteristic

C₄×C₅₃

2₋¹⁺⁴

yes

characteristic

Dic₅₃

—

Dic₅₃

—

#3255

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#537

yes

characteristic

C₂×C₅₃

C₂×#137

yes

—

—

—

—

—

D₅₃

yes

characteristic

C₅₃

#3255

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

106

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

106

—

C₄×Q₈

—

C₄×Q₈

—

106

—

#117

—

106

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#106

—

C₄⋊C₈

C₂×D₅₃

yes

—

#103

—

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂²×C₄

D₂₁₂

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

Q₁₆

—

212

—

M₁₆

—

C₂×C₈

C₂²×D₅₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₂²⋊C₄

—

106

—

100

C₄²

—

106

—

101

C₄²

D₂₁₂

yes

characteristic

102

C₄²

D₂₁₂

yes

characteristic

103

C₄²

C₂²×D₅₃

yes

—

104

C₂³

C₂×D₂₁₂

yes

characteristic

105

Q₈

—

106

—

106

Q₈

—

106

—

107

Q₈

—

212

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

106

—

112

C₂×C₄

—

106

—

113

C₂×C₄

C₂×D₂₁₂

yes

characteristic

114

C₂×C₄

C₂×D₂₁₂

yes

—

115

C₂×C₄

C₂×D₂₁₂

yes

—

116

C₂×C₄

C₂³×D₅₃

yes

characteristic

117

C₈

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂²×D₂₁₂

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

212

—

124

C₄

—

212

—

125

C₄

#13133250

yes

—

126

C₄

#13133262

yes

characteristic

127

C₄

#13133265

yes

characteristic

128

C₂

—

129

C₂

#16811204

yes

characteristic

130

C₂

C₂×#13133250

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#423	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#423	D₅₃
hypocenter	C₅₃	#3255
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂²×D₂₁₂

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#423

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
2. #16810909	C₂	semidirect
4. #16810834	C₂	semidirect
71. C₅₃	#3255	semidirect

H ≅ G/N

type

2. #16810909

C₂

semidirect

4. #16810834

C₂

semidirect

71. C₅₃

#3255

semidirect

Subgroups — #20286109

Conjugacy classes of subgroups

Complements