Group #20273767 — subgroups

Subgroups — #20273767

Order 6784.

336 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16811258	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810815	C₂	1	2	yes	maximal, complemented
3	#16810367	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810352	C₂	1	2	yes	maximal, complemented
5	C₅₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄×D₅₃	—	2	2	no	—
7	C₂×#11719922	—	2	2	no	—
8	#13133188	—	2	2	no	—
9	#13133185	—	2	2	no	—
10	C₄×D₂₁₂	C₂²	1	2	yes	—
11	C₄⋊#88322	C₂²	1	2	yes	complemented
12	#13133138	—	2	2	no	—
13	C₂×D₈×C₅₃	—	2	2	no	—
14	#127×C₅₃	C₂²	1	1	yes	characteristic
15	#127×C₅₃	C₂²	1	1	yes	characteristic, complemented
16	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
17	C₂³×D₅₃	—	2	2	no	—
18	C₂×D₄×C₅₃	—	2	2	no	—
19	C₂×D₄×C₅₃	—	2	1	no	—
20	C₂×D₄×C₅₃	—	2	2	no	—
21	C₂×D₄×C₅₃	D₄	1	2	yes	—
22	C₂×C₂²⋊D₅₃	—	2	4	no	—
23	D₄×D₅₃	—	4	4	no	—
24	D₄×D₅₃	—	4	4	no	—
25	C₂×D₂₁₂	D₄	1	2	yes	—
26	C₂×C₄×D₅₃	—	2	2	no	—
27	D₈×C₅₃	—	4	4	no	—
28	C₂×C₈×C₅₃	—	2	1	no	—
29	C₄²×C₅₃	C₂³	1	1	yes	characteristic
30	C₂²⋊Dic₅₃	—	2	2	no	—
31	#11719922	—	4	4	no	—
32	#11719921	—	2	2	no	—
33	C₄⋊Dic₅₃	D₄	1	2	yes	—
34	C₂×#88322	—	2	2	no	—
35	C₂³×C₅₃	—	2	2	no	—
36	C₂³×C₅₃	—	4	1	no	—
37	C₂²×D₅₃	—	2	2	no	—
38	C₂²×D₅₃	—	4	8	no	—
39	C₂²×D₅₃	—	8	2	no	—
40	D₄×C₅₃	—	4	2	no	—
41	D₄×C₅₃	—	4	2	no	—
42	D₄×C₅₃	—	4	2	no	—
43	D₄×C₅₃	—	4	2	no	—
44	D₄×C₅₃	—	4	2	no	—
45	D₄×C₅₃	—	4	2	no	—
46	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂²⋊D₅₃	—	4	8	no	—
49	C₂×Dic₅₃	—	2	2	no	—
50	D₂₁₂	—	2	2	no	—
51	D₂₁₂	D₈	1	4	yes	complemented
52	C₄×D₅₃	—	4	2	no	—
53	C₄×D₅₃	—	4	2	no	—
54	C₈×C₅₃	—	2	2	no	—
55	#88322	—	4	2	no	—
56	C₂²×C₅₃	—	4	4	no	—
57	C₂²×C₅₃	—	4	2	no	—
58	C₂²×C₅₃	—	8	2	no	—
59	C₂²×C₅₃	—	8	1	no	—
60	C₂²×C₅₃	#120	1	1	yes	characteristic
61	C₂×D₅₃	—	2	4	no	—
62	C₂×D₅₃	—	4	2	no	—
63	C₂×D₅₃	—	4	2	no	—
64	C₂×D₅₃	—	8	8	no	—
65	C₄×C₅₃	—	2	1	no	—
66	C₄×C₅₃	C₂×D₈	1	2	yes	—
67	C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
68	Dic₅₃	—	4	2	no	—
69	#1619	—	53	1	no	maximal, Sylow
70	C₂×C₅₃	—	8	2	no	—
71	C₂×C₅₃	—	16	1	no	—
72	C₂×C₅₃	#447	1	2	yes	—
73	C₂×C₅₃	#453	1	1	yes	characteristic
74	D₅₃	—	4	4	no	—
75	D₄²	—	53	1	no	—
76	#493	D₅₃	1	1	yes	characteristic, complemented
77	#459	—	53	2	no	—
78	#331	—	53	1	no	—
79	D₄⋊C₈	—	53	2	no	—
80	C₅₃	#1619	1	1	yes	characteristic, complemented, Sylow
81	C₂²×D₄	—	106	2	no	—
82	C₂×D₈	—	2	2	no	—
83	C₂×D₈	—	106	2	no	—
84	#127	C₂×D₅₃	1	1	yes	characteristic
85	#127	C₂×D₅₃	1	1	yes	characteristic, complemented
86	#121	—	106	2	no	—
87	#120	—	106	2	no	—
88	C₄×D₄	—	53	2	no	—
89	C₄⋊C₈	—	53	2	no	—
90	#102	—	106	2	no	—
91	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic, complemented
92	C₂⁴	—	106	2	no	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	—	2	2	no	—
95	C₂×D₄	—	2	1	no	—
96	C₂×D₄	—	53	2	no	—
97	C₂×D₄	—	106	4	no	—
98	C₂×D₄	—	212	4	no	—
99	C₂×D₄	—	212	4	no	—
100	C₂×D₄	C₂²⋊D₅₃	1	2	yes	—
101	C₂²×C₄	—	106	2	no	—
102	D₈	—	4	4	no	—
103	D₈	—	212	4	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	106	2	no	—
106	C₄⋊C₄	—	53	2	no	—
107	C₂²⋊C₄	—	106	2	no	—
108	C₂²⋊C₄	—	106	2	no	—
109	C₄²	C₂²×D₅₃	1	1	yes	characteristic
110	C₂³	—	2	2	no	—
111	C₂³	—	4	1	no	—
112	C₂³	—	106	2	no	—
113	C₂³	—	424	2	no	—
114	C₂³	—	212	8	no	—
115	D₄	—	4	2	no	—
116	D₄	—	4	2	no	—
117	D₄	—	4	2	no	—
118	D₄	—	4	2	no	—
119	D₄	—	4	2	no	—
120	D₄	—	4	2	no	—
121	D₄	—	53	4	no	—
122	D₄	—	106	2	no	—
123	D₄	—	212	8	no	—
124	C₂×C₄	—	106	2	no	—
125	C₂×C₄	—	212	2	no	—
126	C₂×C₄	—	212	2	no	—
127	C₂×C₄	D₄×D₅₃	1	2	yes	—
128	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
129	C₈	—	2	2	no	—
130	C₈	—	212	2	no	—
131	C₂²	—	4	4	no	—
132	C₂²	—	4	2	no	—
133	C₂²	—	8	2	no	—
134	C₂²	—	8	1	no	—
135	C₂²	—	106	4	no	—
136	C₂²	—	424	8	no	—
137	C₂²	—	212	2	no	—
138	C₂²	—	212	2	no	—
139	C₂²	#13133185	1	1	yes	characteristic
140	C₄	—	2	1	no	—
141	C₄	—	212	2	no	—
142	C₄	D₈×D₅₃	1	2	yes	—
143	C₄	#13133241	1	2	yes	—
144	C₂	—	8	2	no	—
145	C₂	—	16	1	no	—
146	C₂	—	212	4	no	—
147	C₂	#16810755	1	2	yes	—
148	C₂	#16810790	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811258

C₂

yes

maximal, characteristic, complemented

#16810815

C₂

yes

maximal, complemented

#16810367

C₂

yes

maximal, characteristic, complemented

#16810352

C₂

yes

maximal, complemented

C₅₃×#493

C₂

yes

maximal, characteristic, complemented

C₂×D₄×D₅₃

—

C₂×#11719922

—

#13133188

—

#13133185

—

C₄×D₂₁₂

C₂²

yes

—

C₄⋊#88322

C₂²

yes

complemented

#13133138

—

C₂×D₈×C₅₃

—

#127×C₅₃

C₂²

yes

characteristic

#127×C₅₃

C₂²

yes

characteristic, complemented

C₄×C₈×C₅₃

C₂²

yes

characteristic, complemented

C₂³×D₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

D₄

yes

—

C₂×C₂²⋊D₅₃

—

D₄×D₅₃

—

D₄×D₅₃

—

C₂×D₂₁₂

D₄

yes

—

C₂×C₄×D₅₃

—

D₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

C₂²⋊Dic₅₃

—

#11719922

—

#11719921

—

C₄⋊Dic₅₃

D₄

yes

—

C₂×#88322

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

D₈

yes

complemented

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#120

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×D₈

yes

—

C₄×C₅₃

AGL(1,ℤ/8)

yes

—

Dic₅₃

—

#1619

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#447

yes

—

C₂×C₅₃

#453

yes

characteristic

D₅₃

—

D₄²

—

#493

D₅₃

yes

characteristic, complemented

#459

—

#331

—

D₄⋊C₈

—

C₅₃

#1619

yes

characteristic, complemented, Sylow

C₂²×D₄

—

106

—

C₂×D₈

—

C₂×D₈

—

106

—

#127

C₂×D₅₃

yes

characteristic

#127

C₂×D₅₃

yes

characteristic, complemented

#121

—

106

—

#120

—

106

—

C₄×D₄

—

C₄⋊C₈

—

#102

—

106

—

C₄×C₈

C₂×D₅₃

yes

characteristic, complemented

C₂⁴

—

106

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

106

—

C₂×D₄

—

212

—

C₂×D₄

—

212

—

100

C₂×D₄

C₂²⋊D₅₃

yes

—

101

C₂²×C₄

—

106

—

102

D₈

—

103

D₈

—

212

—

104

C₂×C₈

—

105

C₂×C₈

—

106

—

106

C₄⋊C₄

—

107

C₂²⋊C₄

—

106

—

108

C₂²⋊C₄

—

106

—

109

C₄²

C₂²×D₅₃

yes

characteristic

110

C₂³

—

111

C₂³

—

112

C₂³

—

106

—

113

C₂³

—

424

—

114

C₂³

—

212

—

115

D₄

—

116

D₄

—

117

D₄

—

118

D₄

—

119

D₄

—

120

D₄

—

121

D₄

—

122

D₄

—

106

—

123

D₄

—

212

—

124

C₂×C₄

—

106

—

125

C₂×C₄

—

212

—

126

C₂×C₄

—

212

—

127

C₂×C₄

D₄×D₅₃

yes

—

128

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

129

C₈

—

130

C₈

—

212

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

106

—

136

C₂²

—

424

—

137

C₂²

—

212

—

138

C₂²

—

212

—

139

C₂²

#13133185

yes

characteristic

140

C₄

—

141

C₄

—

212

—

142

C₄

D₈×D₅₃

yes

—

143

C₄

#13133241

yes

—

144

C₂

—

145

C₂

—

146

C₂

—

212

—

147

C₂

#16810755

yes

—

148

C₂

#16810790

yes

characteristic

	subgroup	G/H
center	C₂²	#13133185
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#493	C₂
socle	C₂²×C₅₃	#120
hypercenter	#493	D₅₃
hypocenter	C₅₃	#1619
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#13133185

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#493

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #16811258	C₂	semidirect
2. #16810815	C₂	semidirect
3. #16810367	C₂	semidirect
4. #16810352	C₂	semidirect
5. C₅₃×#493	C₂	semidirect
11. C₄⋊#88322	C₂²	semidirect
15. #127×C₅₃	C₂²	semidirect
16. C₄×C₈×C₅₃	C₂²	semidirect
51. D₂₁₂	D₈	semidirect
76. #493	D₅₃	semidirect
80. C₅₃	#1619	semidirect
85. #127	C₂×D₅₃	semidirect
91. C₄×C₈	C₂×D₅₃	semidirect

H ≅ G/N

type

1. #16811258

C₂

semidirect

2. #16810815

C₂

semidirect

3. #16810367

C₂

semidirect

4. #16810352

C₂

semidirect

5. C₅₃×#493

C₂

semidirect

11. C₄⋊#88322

C₂²

semidirect

15. #127×C₅₃

C₂²

semidirect

16. C₄×C₈×C₅₃

C₂²

semidirect

51. D₂₁₂

D₈

semidirect

76. #493

D₅₃

semidirect

80. C₅₃

#1619

semidirect

85. #127

C₂×D₅₃

semidirect

91. C₄×C₈

C₂×D₅₃

semidirect

Subgroups — #20273767

Conjugacy classes of subgroups

Complements