Group #20276002 — subgroups

Subgroups — #20276002

Order 6784.

266 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 65 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	#16811423	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133145	C₂	1	2	yes	maximal, complemented
3	#16810605	C₂	1	2	yes	maximal, complemented
4	#16810381	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#423	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11719945	C₂²	1	2	yes	—
7	#13133250	—	2	2	no	—
8	#13133245	—	2	2	no	—
9	#13133239	—	2	2	no	—
10	C₂×#11719917	C₂²	1	2	yes	—
11	#13133145	—	2	2	no	—
12	#13133145	—	2	2	no	—
13	#13133135	—	2	2	no	—
14	#13133133	—	2	2	no	—
15	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
18	C₄⋊C₈×C₅₃	—	2	2	no	—
19	C₂²×C₄×C₅₃	—	2	1	no	—
20	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
21	C₂×C₂²⋊D₅₃	—	2	2	no	—
22	#11719945	—	2	2	no	—
23	#11719945	—	2	2	no	—
24	#11719945	—	2	2	no	—
25	#11719945	—	2	2	no	—
26	C₂×D₂₁₂	D₄	1	2	yes	—
27	C₂×C₄×D₅₃	—	2	2	no	—
28	C₂×Dic₁₀₆	D₄	1	2	yes	—
29	M₁₆×C₅₃	—	2	2	no	—
30	M₁₆×C₅₃	—	2	2	no	—
31	C₂×C₈×C₅₃	—	2	1	no	—
32	C₂×C₈×C₅₃	D₄	1	2	yes	—
33	C₄²×C₅₃	—	2	1	no	—
34	C₄²×C₅₃	D₄	1	1	yes	characteristic
35	C₄²×C₅₃	D₄	1	1	yes	characteristic
36	#11719917	—	2	2	no	—
37	#11719917	—	2	2	no	—
38	C₂×#88322	—	2	2	no	—
39	Dic₂₁₂	—	2	2	no	—
40	D₄₂₄	—	2	2	no	—
41	#11719913	—	2	4	no	—
42	C₂³×C₅₃	#41	1	1	yes	characteristic
43	C₂²×D₅₃	—	2	2	no	—
44	C₂×C₄×C₅₃	—	2	1	no	—
45	C₂×C₄×C₅₃	—	2	1	no	—
46	C₂×C₄×C₅₃	—	4	2	no	—
47	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₅₃	#41	1	2	yes	—
51	C₂²⋊D₅₃	—	4	2	no	—
52	C₂²⋊D₅₃	—	4	2	no	—
53	C₂×Dic₅₃	—	2	2	no	—
54	D₂₁₂	—	2	2	no	—
55	D₂₁₂	—	2	2	no	—
56	C₄×D₅₃	—	4	2	no	—
57	C₄×D₅₃	—	4	2	no	—
58	Dic₁₀₆	—	2	2	no	—
59	Dic₁₀₆	—	2	2	no	—
60	C₈×C₅₃	—	2	2	no	—
61	C₈×C₅₃	—	4	1	no	—
62	#88322	—	4	2	no	—
63	C₂²×C₅₃	—	4	1	no	—
64	C₂²×C₅₃	#121	1	1	yes	characteristic
65	C₂²×C₅₃	#123	1	2	yes	—
66	C₂×D₅₃	—	4	2	no	—
67	C₂×D₅₃	—	8	2	no	—
68	C₄×C₅₃	—	2	2	no	—
69	C₄×C₅₃	—	4	1	no	—
70	C₄×C₅₃	#120	1	1	yes	characteristic
71	C₄×C₅₃	#121	1	2	yes	—
72	C₄×C₅₃	#124	1	1	yes	characteristic
73	Dic₅₃	—	4	2	no	—
74	#2050	—	53	1	no	maximal, Sylow
75	C₂×C₅₃	—	2	1	no	—
76	C₂×C₅₃	—	2	2	no	—
77	C₂×C₅₃	#394	1	1	yes	characteristic
78	D₅₃	—	8	2	no	—
79	#575	—	53	1	no	—
80	#423	D₅₃	1	1	yes	characteristic, complemented
81	C₂×#104	—	53	2	no	—
82	#413	—	53	2	no	—
83	#338	—	53	1	no	—
84	C₅₃	#2050	1	1	yes	characteristic, complemented, Sylow
85	C₂×#41	—	53	2	no	—
86	#137	—	106	2	no	—
87	AGL(1,ℤ/8)	—	106	2	no	—
88	#135	—	106	2	no	—
89	C₂×M₁₆	—	53	2	no	—
90	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
91	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
92	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
93	C₄⋊C₈	—	2	2	no	—
94	#104	—	106	2	no	—
95	#104	—	106	2	no	—
96	#101	—	106	2	no	—
97	#100	—	106	2	no	—
98	#41	—	106	2	no	—
99	#41	—	106	2	no	—
100	#41	—	106	2	no	—
101	#41	—	106	2	no	—
102	C₂×Q₈	—	53	2	no	—
103	C₂×D₄	—	53	2	no	—
104	C₂×D₄	—	106	2	no	—
105	C₂²×C₄	—	2	1	no	—
106	C₂²×C₄	—	106	2	no	—
107	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
108	Q₁₆	—	106	2	no	—
109	SD₁₆	—	106	4	no	—
110	D₈	—	106	2	no	—
111	M₁₆	—	2	2	no	—
112	M₁₆	—	2	2	no	—
113	M₁₆	—	106	2	no	—
114	M₁₆	—	106	2	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	106	2	no	—
117	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
118	C₄²	—	2	1	no	—
119	C₄²	D₂₁₂	1	1	yes	characteristic
120	C₄²	D₂₁₂	1	1	yes	characteristic
121	C₂³	—	106	2	no	—
122	C₂³	#11719945	1	1	yes	characteristic
123	Q₈	—	106	2	no	—
124	Q₈	—	106	2	no	—
125	D₄	—	106	2	no	—
126	D₄	—	106	2	no	—
127	D₄	—	212	2	no	—
128	D₄	—	212	2	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	106	2	no	—
133	C₂×C₄	—	212	2	no	—
134	C₂×C₄	—	212	2	no	—
135	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
136	C₂×C₄	D₄×D₅₃	1	2	yes	—
137	C₂×C₄	#11719949	1	2	yes	—
138	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
139	C₈	—	2	2	no	—
140	C₈	—	4	1	no	—
141	C₈	—	212	2	no	—
142	C₂²	—	4	1	no	—
143	C₂²	—	424	2	no	—
144	C₂²	—	212	2	no	—
145	C₂²	#13133191	1	1	yes	characteristic
146	C₂²	#13133202	1	2	yes	—
147	C₄	—	2	2	no	—
148	C₄	—	4	1	no	—
149	C₄	—	212	2	no	—
150	C₄	#13133185	1	1	yes	characteristic
151	C₄	#13133192	1	2	yes	—
152	C₄	#13133206	1	1	yes	characteristic
153	C₂	—	2	1	no	—
154	C₂	—	2	2	no	—
155	C₂	—	424	2	no	—
156	C₂	#16810540	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811423

C₂

yes

maximal, characteristic

C₂×#13133145

C₂

yes

maximal, complemented

#16810605

C₂

yes

maximal, complemented

#16810381

C₂

yes

maximal, characteristic, complemented

C₅₃×#423

C₂

yes

maximal, characteristic, complemented

C₂×#11719945

C₂²

yes

—

—

—

—

C₂²

yes

—

—

—

—

—

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₂²⋊D₅₃

—

—

—

—

—

C₂×D₂₁₂

D₄

yes

—

C₂×C₄×D₅₃

—

C₂×Dic₁₀₆

D₄

yes

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

—

—

—

Dic₂₁₂

—

D₄₂₄

—

#11719913

—

C₂³×C₅₃

#41

yes

characteristic

C₂²×D₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

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

—

C₂²⋊D₅₃

—

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

#121

yes

characteristic

C₂²×C₅₃

#123

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#120

yes

characteristic

C₄×C₅₃

#121

yes

—

C₄×C₅₃

#124

yes

characteristic

Dic₅₃

—

#2050

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#394

yes

characteristic

D₅₃

—

#575

—

#423

D₅₃

yes

characteristic, complemented

C₂×#104

—

#413

—

#338

—

C₅₃

#2050

yes

characteristic, complemented, Sylow

C₂×#41

—

#137

—

106

—

AGL(1,ℤ/8)

—

106

—

#135

—

106

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

characteristic

C₂×M₁₆

C₂×D₅₃

yes

characteristic

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

#104

—

106

—

#104

—

106

—

#101

—

106

—

#100

—

106

—

#41

—

106

—

#41

—

106

—

100

#41

—

106

—

101

#41

—

106

—

102

C₂×Q₈

—

103

C₂×D₄

—

104

C₂×D₄

—

106

—

105

C₂²×C₄

—

106

C₂²×C₄

—

106

—

107

C₂²×C₄

C₂²×D₅₃

yes

characteristic

108

Q₁₆

—

106

—

109

SD₁₆

—

106

—

110

D₈

—

106

—

111

M₁₆

—

112

M₁₆

—

113

M₁₆

—

106

—

114

M₁₆

—

106

—

115

C₂×C₈

—

116

C₂×C₈

—

106

—

117

C₂×C₈

C₂²⋊D₅₃

yes

—

118

C₄²

—

119

C₄²

D₂₁₂

yes

characteristic

120

C₄²

D₂₁₂

yes

characteristic

121

C₂³

—

106

—

122

C₂³

#11719945

yes

characteristic

123

Q₈

—

106

—

124

Q₈

—

106

—

125

D₄

—

106

—

126

D₄

—

106

—

127

D₄

—

212

—

128

D₄

—

212

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

106

—

133

C₂×C₄

—

212

—

134

C₂×C₄

—

212

—

135

C₂×C₄

C₂×D₂₁₂

yes

characteristic

136

C₂×C₄

D₄×D₅₃

yes

—

137

C₂×C₄

#11719949

yes

—

138

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

139

C₈

—

140

C₈

—

141

C₈

—

212

—

142

C₂²

—

143

C₂²

—

424

—

144

C₂²

—

212

—

145

C₂²

#13133191

yes

characteristic

146

C₂²

#13133202

yes

—

147

C₄

—

148

C₄

—

149

C₄

—

212

—

150

C₄

#13133185

yes

characteristic

151

C₄

#13133192

yes

—

152

C₄

#13133206

yes

characteristic

153

C₂

—

154

C₂

—

155

C₂

—

424

—

156

C₂

#16810540

yes

characteristic

	subgroup	G/H
center	C₄	#13133206
commutator	C₂²×C₄×C₅₃	C₂³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#423	C₂
socle	C₂×C₅₃	#394
hypercenter	#423	D₅₃
hypocenter	C₅₃	#2050
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #423

subgroup

G/H

center

C₄

#13133206

commutator

C₂²×C₄×C₅₃

C₂³

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#423

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #423

N	H ≅ G/N	type
2. C₂×#13133145	C₂	semidirect
3. #16810605	C₂	semidirect
4. #16810381	C₂	semidirect
5. C₅₃×#423	C₂	semidirect
80. #423	D₅₃	semidirect
84. C₅₃	#2050	semidirect

H ≅ G/N

type

2. C₂×#13133145

C₂

semidirect

3. #16810605

C₂

semidirect

4. #16810381

C₂

semidirect

5. C₅₃×#423

C₂

semidirect

80. #423

D₅₃

semidirect

84. C₅₃

#2050

semidirect

Subgroups — #20276002

Conjugacy classes of subgroups

Complements