Group #20283584 — subgroups

Subgroups — #20283584

Order 6784.

456 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 189 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	#16811100	C₂	1	2	yes	maximal
3	#16810731	C₂	1	4	yes	maximal, complemented
4	C₄×Dic₂₁₂	C₂	1	4	yes	maximal, complemented
5	#16810661	C₂	1	1	yes	maximal, characteristic
6	#16810633	C₂	1	2	yes	maximal
7	C₅₃×#405	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₀₆	C₂²	1	2	yes	—
9	C₂×Dic₂₁₂	C₄	1	8	yes	complemented
10	C₂×Dic₂₁₂	C₂²	1	2	yes	—
11	C₂×Dic₂₁₂	C₂²	1	2	yes	—
12	C₄×Dic₁₀₆	C₂²	1	8	yes	—
13	#13133182	—	2	4	no	—
14	#13133176	C₂²	1	2	yes	—
15	#13133173	—	2	4	no	—
16	#13133151	C₂²	1	2	yes	—
17	#13133149	C₂²	1	2	yes	—
18	#13133142	C₂²	1	8	yes	—
19	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	#117×C₅₃	C₂²	1	1	yes	characteristic
22	#97×C₅₃	C₂²	1	2	yes	—
23	C₄×C₈×C₅₃	C₂²	1	2	yes	—
24	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
25	C₂²×Dic₅₃	—	2	2	no	—
26	C₂×Dic₁₀₆	—	2	2	no	—
27	C₂×Dic₁₀₆	—	2	4	no	—
28	C₂×Dic₁₀₆	C₂×C₄	1	8	yes	—
29	C₂×Dic₁₀₆	C₂³	1	4	yes	—
30	M₁₆×C₅₃	—	2	2	no	—
31	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	—
32	C₂×C₈×C₅₃	C₂³	1	2	yes	—
33	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₅₃	D₄	1	2	yes	—
36	C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
37	C₄²×C₅₃	C₂³	1	2	yes	—
38	C₂²⋊Dic₅₃	—	2	2	no	—
39	C₄⋊Dic₅₃	C₂³	1	4	yes	—
40	#11719919	—	2	8	no	—
41	C₄×Dic₅₃	—	2	8	no	—
42	C₄×Dic₅₃	—	2	2	no	—
43	Dic₂₁₂	C₂×C₄	1	8	yes	—
44	Dic₂₁₂	C₂×C₄	1	8	yes	—
45	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	—
48	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₅₃	#41	1	2	yes	—
50	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
51	C₂×Dic₅₃	—	2	8	no	—
52	C₂×Dic₅₃	—	2	4	no	—
53	C₂×Dic₅₃	—	2	4	no	—
54	Dic₁₀₆	—	2	8	no	—
55	Dic₁₀₆	—	2	4	no	—
56	Dic₁₀₆	C₂²×C₄	1	8	yes	—
57	C₈×C₅₃	—	2	2	no	—
58	C₈×C₅₃	C₂²×C₄	1	2	yes	—
59	C₈×C₅₃	C₂²×C₄	1	2	yes	—
60	C₂²×C₅₃	—	2	2	no	—
61	C₂²×C₅₃	C₄×D₄	1	2	yes	—
62	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
63	C₄×C₅₃	—	2	4	no	—
64	C₄×C₅₃	C₄×D₄	1	2	yes	—
65	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
66	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
67	Dic₅₃	—	2	8	no	—
68	Dic₅₃	—	4	4	no	—
69	#2950	—	53	1	no	maximal, Sylow
70	C₂×C₅₃	—	2	2	no	—
71	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
72	C₂×C₅₃	#578	1	2	yes	—
73	C₂²×Q₁₆	—	53	1	no	—
74	#519	—	53	2	no	—
75	#441	—	53	4	no	—
76	C₄×Q₁₆	—	53	4	no	—
77	#427	—	53	1	no	—
78	#419	—	53	2	no	—
79	#405	D₅₃	1	1	yes	characteristic
80	C₅₃	#2950	1	1	yes	characteristic, complemented, Sylow
81	C₂²×Q₈	—	53	2	no	—
82	C₂×Q₁₆	—	53	8	no	—
83	C₂×Q₁₆	—	53	2	no	—
84	C₂×Q₁₆	—	53	2	no	—
85	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
86	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
87	C₄×Q₈	—	53	8	no	—
88	C₄×Q₈	—	106	4	no	—
89	#117	—	53	2	no	—
90	#117	—	106	4	no	—
91	#117	C₂×D₅₃	1	1	yes	characteristic
92	#107	—	53	2	no	—
93	#106	—	53	2	no	—
94	#103	—	53	8	no	—
95	#97	C₂×D₅₃	1	2	yes	—
96	C₄×C₈	C₂×D₅₃	1	2	yes	—
97	C₂×Q₈	—	53	8	no	—
98	C₂×Q₈	—	53	4	no	—
99	C₂×Q₈	—	106	2	no	—
100	C₂×Q₈	—	106	4	no	—
101	C₂²×C₄	—	106	2	no	—
102	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
103	Q₁₆	—	53	8	no	—
104	Q₁₆	—	53	8	no	—
105	M₁₆	—	2	2	no	—
106	C₂×C₈	C₄×D₅₃	1	4	yes	—
107	C₂×C₈	C₂²×D₅₃	1	2	yes	—
108	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
109	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
110	C₄⋊C₄	—	53	4	no	—
111	C₄⋊C₄	—	106	8	no	—
112	C₄⋊C₄	D₂₁₂	1	2	yes	—
113	C₂²⋊C₄	—	106	2	no	—
114	C₂²⋊C₄	D₂₁₂	1	2	yes	—
115	C₄²	—	106	2	no	—
116	C₄²	—	106	8	no	—
117	C₄²	C₂²×D₅₃	1	2	yes	—
118	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
119	Q₈	—	53	8	no	—
120	Q₈	—	106	8	no	—
121	Q₈	—	106	4	no	—
122	C₂×C₄	—	106	8	no	—
123	C₂×C₄	—	106	4	no	—
124	C₂×C₄	—	106	4	no	—
125	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
126	C₂×C₄	C₂×D₂₁₂	1	4	yes	—
127	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
128	C₂×C₄	#11719945	1	2	yes	—
129	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
130	C₈	—	2	2	no	—
131	C₈	C₂×C₄×D₅₃	1	2	yes	—
132	C₈	C₂×C₄×D₅₃	1	2	yes	—
133	C₂²	—	2	2	no	—
134	C₂²	C₄×D₂₁₂	1	2	yes	—
135	C₂²	C₂²×D₂₁₂	1	1	yes	characteristic
136	C₄	—	2	4	no	—
137	C₄	—	106	8	no	—
138	C₄	—	212	4	no	—
139	C₄	C₄×D₂₁₂	1	2	yes	—
140	C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
141	C₄	C₂×#11719945	1	1	yes	characteristic
142	C₂	—	2	2	no	—
143	C₂	C₂×C₄×D₂₁₂	1	1	yes	characteristic
144	C₂	#16811440	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₂₁₂

C₂

yes

maximal, characteristic

#16811100

C₂

yes

maximal

#16810731

C₂

yes

maximal, complemented

C₄×Dic₂₁₂

C₂

yes

maximal, complemented

#16810661

C₂

yes

maximal, characteristic

#16810633

C₂

yes

maximal

C₅₃×#405

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₂²

yes

—

C₂×Dic₂₁₂

C₄

yes

complemented

C₂×Dic₂₁₂

C₂²

yes

—

C₂×Dic₂₁₂

C₂²

yes

—

C₄×Dic₁₀₆

C₂²

yes

—

#13133182

—

#13133176

C₂²

yes

—

#13133173

—

#13133151

C₂²

yes

—

#13133149

C₂²

yes

—

#13133142

C₂²

yes

—

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

#117×C₅₃

C₂²

yes

characteristic

#97×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂×C₄

yes

—

C₂×Dic₁₀₆

C₂³

yes

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂³

yes

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

D₄

yes

—

C₂²⋊C₄×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

Dic₂₁₂

C₂×C₄

yes

—

Dic₂₁₂

C₂×C₄

yes

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

C₂²×C₄

yes

—

C₈×C₅₃

—

C₈×C₅₃

C₂²×C₄

yes

—

C₈×C₅₃

C₂²×C₄

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

C₄×D₄

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₄×D₄

yes

—

C₄×C₅₃

C₂³×C₄

yes

characteristic

C₄×C₅₃

C₂×#41

yes

characteristic

Dic₅₃

—

Dic₅₃

—

#2950

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×C₄×D₄

yes

characteristic

C₂×C₅₃

#578

yes

—

C₂²×Q₁₆

—

#519

—

#441

—

C₄×Q₁₆

—

#427

—

#419

—

#405

D₅₃

yes

characteristic

C₅₃

#2950

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

characteristic

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

106

—

#117

—

#117

—

106

—

#117

C₂×D₅₃

yes

—

—

—

C₂×D₅₃

yes

—

C₄×C₈

C₂×D₅₃

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

106

—

100

C₂×Q₈

—

106

—

101

C₂²×C₄

—

106

—

102

C₂²×C₄

C₂²×D₅₃

yes

characteristic

103

Q₁₆

—

104

Q₁₆

—

105

M₁₆

—

106

C₂×C₈

C₄×D₅₃

yes

—

107

C₂×C₈

C₂²×D₅₃

yes

—

108

C₂×C₈

C₂²×D₅₃

yes

characteristic

109

C₂×C₈

C₂²×D₅₃

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

106

—

112

C₄⋊C₄

D₂₁₂

yes

—

113

C₂²⋊C₄

—

106

—

114

C₂²⋊C₄

D₂₁₂

yes

—

115

C₄²

—

106

—

116

C₄²

—

106

—

117

C₄²

C₂²×D₅₃

yes

—

118

C₂³

C₂×D₂₁₂

yes

characteristic

119

Q₈

—

120

Q₈

—

106

—

121

Q₈

—

106

—

122

C₂×C₄

—

106

—

123

C₂×C₄

—

106

—

124

C₂×C₄

—

106

—

125

C₂×C₄

C₂×C₄×D₅₃

yes

—

126

C₂×C₄

C₂×D₂₁₂

yes

—

127

C₂×C₄

C₂×D₂₁₂

yes

characteristic

128

C₂×C₄

#11719945

yes

—

129

C₂×C₄

C₂³×D₅₃

yes

characteristic

130

C₈

—

131

C₈

C₂×C₄×D₅₃

yes

—

132

C₈

C₂×C₄×D₅₃

yes

—

133

C₂²

—

134

C₂²

C₄×D₂₁₂

yes

—

135

C₂²

C₂²×D₂₁₂

yes

characteristic

136

C₄

—

137

C₄

—

106

—

138

C₄

—

212

—

139

C₄

C₄×D₂₁₂

yes

—

140

C₄

C₂²×C₄×D₅₃

yes

characteristic

141

C₄

C₂×#11719945

yes

characteristic

142

C₂

—

143

C₂

C₂×C₄×D₂₁₂

yes

characteristic

144

C₂

#16811440

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₂₁₂
commutator	C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#405	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#405	D₅₃
hypocenter	C₅₃	#2950
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

subgroup

G/H

center

C₂²

C₂²×D₂₁₂

commutator

C₄×C₅₃

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#405

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ #117 ◁ #405

N	H ≅ G/N	type
3. #16810731	C₂	semidirect
4. C₄×Dic₂₁₂	C₂	semidirect
9. C₂×Dic₂₁₂	C₄	semidirect
80. C₅₃	#2950	semidirect

H ≅ G/N

type

3. #16810731

C₂

semidirect

4. C₄×Dic₂₁₂

C₂

semidirect

9. C₂×Dic₂₁₂

C₄

semidirect

80. C₅₃

#2950

semidirect

Subgroups — #20283584

Conjugacy classes of subgroups

Complements