Group #20285454 — subgroups

Subgroups — #20285454

Order 6784.

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

normal subgroups only

This group has 446 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	#16811174	C₂	1	2	yes	maximal, complemented
2	#16811000	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810971	C₂	1	2	yes	maximal, complemented
4	#16810877	C₂	1	4	yes	maximal, complemented
5	#16810734	C₂	1	4	yes	maximal, complemented
6	C₄×#11719912	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#498	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11719949	—	2	2	no	—
9	C₂×D₄₂₄	C₂²	1	4	yes	—
10	C₂×#11719912	C₂²	1	2	yes	—
11	#13133214	C₂²	1	2	yes	—
12	#13133206	—	2	2	no	—
13	#13133192	—	2	2	no	—
14	#13133192	C₂²	1	4	yes	—
15	#13133179	—	2	2	no	—
16	#13133179	C₂²	1	4	yes	—
17	C₄²×D₅₃	C₂²	1	1	yes	characteristic
18	#13133137	C₂²	1	8	yes	—
19	#13133127	C₂²	1	2	yes	—
20	C₄×#88322	C₂²	1	1	yes	characteristic
21	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
22	#106×C₅₃	C₂²	1	4	yes	—
23	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×Q₈×C₅₃	—	2	2	no	—
25	#11719949	—	4	8	no	—
26	C₂×D₂₁₂	—	2	4	no	—
27	C₂×D₂₁₂	—	2	2	no	—
28	C₂×D₂₁₂	—	2	2	no	—
29	C₂×D₂₁₂	C₂³	1	4	yes	—
30	C₂×C₄×D₅₃	—	2	4	no	—
31	C₂×C₄×D₅₃	D₄	1	2	yes	—
32	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₅₃	C₂³	1	2	yes	—
34	C₄⋊C₄×C₅₃	—	2	2	no	—
35	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
36	C₄²×C₅₃	C₂³	1	1	yes	characteristic
37	#11719921	—	2	4	no	—
38	#11719921	—	2	4	no	—
39	C₄×Dic₅₃	D₄	1	2	yes	—
40	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
41	C₂×#88322	C₂³	1	2	yes	—
42	D₄₂₄	—	2	8	no	—
43	#11719912	—	2	4	no	—
44	C₂²×D₅₃	—	2	4	no	—
45	C₂²×D₅₃	C₂×D₄	1	1	yes	characteristic
46	Q₈×C₅₃	—	4	4	no	—
47	C₂×C₄×C₅₃	—	2	4	no	—
48	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
51	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
52	D₂₁₂	—	2	8	no	—
53	D₂₁₂	—	4	8	no	—
54	D₂₁₂	—	4	4	no	—
55	D₂₁₂	—	4	4	no	—
56	C₄×D₅₃	—	2	4	no	—
57	C₄×D₅₃	—	2	2	no	—
58	C₄×D₅₃	—	4	8	no	—
59	C₈×C₅₃	#41	1	4	yes	complemented
60	#88322	—	2	2	no	—
61	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
62	C₂×D₅₃	—	2	1	no	—
63	C₂×D₅₃	—	2	2	no	—
64	C₂×D₅₃	—	4	4	no	—
65	C₂×D₅₃	—	4	4	no	—
66	C₂×D₅₃	—	4	4	no	—
67	C₄×C₅₃	—	4	4	no	—
68	C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
69	C₄×C₅₃	C₂×#41	1	2	yes	—
70	Dic₅₃	—	2	1	no	—
71	Dic₅₃	—	2	2	no	—
72	#3184	—	53	1	no	maximal, Sylow
73	C₂×C₅₃	#532	1	1	yes	characteristic
74	C₂×C₅₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
75	D₅₃	—	4	1	no	—
76	D₅₃	—	8	4	no	—
77	#532	—	53	2	no	—
78	#498	D₅₃	1	1	yes	characteristic, complemented
79	#493	—	53	1	no	—
80	#486	—	53	2	no	—
81	#469	—	53	4	no	—
82	#442	—	53	4	no	—
83	C₄×M₁₆	—	53	1	no	—
84	C₅₃	#3184	1	1	yes	characteristic, complemented, Sylow
85	C₂×#41	—	106	2	no	—
86	C₂×D₈	—	53	4	no	—
87	C₂×M₁₆	—	53	2	no	—
88	C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
89	#127	—	53	2	no	—
90	#124	—	106	2	no	—
91	#121	—	53	4	no	—
92	#121	—	106	2	no	—
93	C₄×D₄	—	53	4	no	—
94	C₄×D₄	—	106	2	no	—
95	C₂×C₄²	—	53	1	no	—
96	#106	C₂×D₅₃	1	4	yes	—
97	#102	—	53	8	no	—
98	#97	—	53	2	no	—
99	C₄×C₈	—	53	1	no	—
100	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
101	#41	—	212	8	no	—
102	C₂×Q₈	—	2	2	no	—
103	C₂×D₄	—	53	4	no	—
104	C₂×D₄	—	106	4	no	—
105	C₂×D₄	—	106	2	no	—
106	C₂×D₄	—	106	2	no	—
107	C₂²×C₄	—	53	2	no	—
108	C₂²×C₄	—	53	1	no	—
109	C₂²×C₄	—	106	4	no	—
110	D₈	—	106	8	no	—
111	M₁₆	—	106	4	no	—
112	C₂×C₈	—	53	2	no	—
113	C₂×C₈	C₂²×D₅₃	1	2	yes	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
116	C₂²⋊C₄	—	106	4	no	—
117	C₂²⋊C₄	—	106	4	no	—
118	C₄²	—	53	2	no	—
119	C₄²	—	53	1	no	—
120	C₄²	C₂²×D₅₃	1	1	yes	characteristic
121	C₂³	—	53	1	no	—
122	C₂³	—	106	4	no	—
123	Q₈	—	4	4	no	—
124	D₄	—	106	8	no	—
125	D₄	—	212	8	no	—
126	D₄	—	212	4	no	—
127	D₄	—	212	4	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	53	1	no	—
130	C₂×C₄	—	53	2	no	—
131	C₂×C₄	—	106	4	no	—
132	C₂×C₄	—	106	2	no	—
133	C₂×C₄	—	212	8	no	—
134	C₂×C₄	D₄×D₅₃	1	2	yes	—
135	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
136	C₈	—	106	2	no	—
137	C₈	#11719949	1	4	yes	complemented
138	C₂²	—	106	2	no	—
139	C₂²	—	106	1	no	—
140	C₂²	—	212	4	no	—
141	C₂²	—	212	4	no	—
142	C₂²	—	212	4	no	—
143	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
144	C₄	—	4	4	no	—
145	C₄	—	106	2	no	—
146	C₄	—	106	1	no	—
147	C₄	#13133244	1	4	yes	—
148	C₄	C₂×#11719949	1	2	yes	—
149	C₂	—	424	4	no	—
150	C₂	—	212	1	no	—
151	C₂	#16811174	1	1	yes	characteristic
152	C₂	C₂×#13133244	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16811174

C₂

yes

maximal, complemented

#16811000

C₂

yes

maximal, characteristic, complemented

#16810971

C₂

yes

maximal, complemented

#16810877

C₂

yes

maximal, complemented

#16810734

C₂

yes

maximal, complemented

C₄×#11719912

C₂

yes

maximal, characteristic, complemented

C₅₃×#498

C₂

yes

maximal, characteristic, complemented

C₂×#11719949

—

C₂×D₄₂₄

C₂²

yes

—

C₂×#11719912

C₂²

yes

—

#13133214

C₂²

yes

—

#13133206

—

#13133192

—

#13133192

C₂²

yes

—

#13133179

—

#13133179

C₂²

yes

—

C₄²×D₅₃

C₂²

yes

characteristic

#13133137

C₂²

yes

—

#13133127

C₂²

yes

—

C₄×#88322

C₂²

yes

characteristic

C₄⋊Q₈×C₅₃

C₂²

yes

—

#106×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

#11719949

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

C₂³

yes

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

D₄

yes

—

C₂×C₄×D₅₃

C₂³

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₄²×C₅₃

C₂³

yes

characteristic

#11719921

—

#11719921

—

C₄×Dic₅₃

D₄

yes

—

C₄×Dic₅₃

C₂³

yes

characteristic

C₂×#88322

C₂³

yes

—

D₄₂₄

—

#11719912

—

C₂²×D₅₃

—

C₂²×D₅₃

C₂×D₄

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

C₂×D₄

yes

—

C₂×Dic₅₃

C₂×D₄

yes

characteristic

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

#41

yes

complemented

#88322

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₄×C₅₃

C₂×#41

yes

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

—

D₅₃

—

#532

—

#498

D₅₃

yes

characteristic, complemented

—

—

—

—

C₄×M₁₆

—

C₅₃

#3184

yes

characteristic, complemented, Sylow

C₂×#41

—

106

—

C₂×D₈

—

C₂×M₁₆

—

C₄⋊Q₈

C₂×D₅₃

yes

—

#127

—

#124

—

106

—

#121

—

#121

—

106

—

C₄×D₄

—

C₄×D₄

—

106

—

C₂×C₄²

—

#106

C₂×D₅₃

yes

—

#102

—

#97

—

C₄×C₈

—

100

C₄×C₈

C₂×D₅₃

yes

characteristic

101

#41

—

212

—

102

C₂×Q₈

—

103

C₂×D₄

—

104

C₂×D₄

—

106

—

105

C₂×D₄

—

106

—

106

C₂×D₄

—

106

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

106

—

110

D₈

—

106

—

111

M₁₆

—

106

—

112

C₂×C₈

—

113

C₂×C₈

C₂²×D₅₃

yes

—

114

C₄⋊C₄

—

115

C₄⋊C₄

C₂²×D₅₃

yes

—

116

C₂²⋊C₄

—

106

—

117

C₂²⋊C₄

—

106

—

118

C₄²

—

119

C₄²

—

120

C₄²

C₂²×D₅₃

yes

characteristic

121

C₂³

—

122

C₂³

—

106

—

123

Q₈

—

124

D₄

—

106

—

125

D₄

—

212

—

126

D₄

—

212

—

127

D₄

—

212

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

106

—

132

C₂×C₄

—

106

—

133

C₂×C₄

—

212

—

134

C₂×C₄

D₄×D₅₃

yes

—

135

C₂×C₄

C₂³×D₅₃

yes

characteristic

136

C₈

—

106

—

137

C₈

#11719949

yes

complemented

138

C₂²

—

106

—

139

C₂²

—

106

—

140

C₂²

—

212

—

141

C₂²

—

212

—

142

C₂²

—

212

—

143

C₂²

C₂×D₄×D₅₃

yes

characteristic

144

C₄

—

145

C₄

—

106

—

146

C₄

—

106

—

147

C₄

#13133244

yes

—

148

C₄

C₂×#11719949

yes

—

149

C₂

—

424

—

150

C₂

—

212

—

151

C₂

#16811174

yes

characteristic

152

C₂

C₂×#13133244

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#498	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#498	D₅₃
hypocenter	C₅₃	#3184
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

C₂×D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#498

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
1. #16811174	C₂	semidirect
2. #16811000	C₂	semidirect
3. #16810971	C₂	semidirect
4. #16810877	C₂	semidirect
5. #16810734	C₂	semidirect
6. C₄×#11719912	C₂	semidirect
7. C₅₃×#498	C₂	semidirect
59. C₈×C₅₃	#41	semidirect
78. #498	D₅₃	semidirect
84. C₅₃	#3184	semidirect
137. C₈	#11719949	semidirect

H ≅ G/N

type

1. #16811174

C₂

semidirect

2. #16811000

C₂

semidirect

3. #16810971

C₂

semidirect

4. #16810877

C₂

semidirect

5. #16810734

C₂

semidirect

6. C₄×#11719912

C₂

semidirect

7. C₅₃×#498

C₂

semidirect

59. C₈×C₅₃

#41

semidirect

78. #498

D₅₃

semidirect

84. C₅₃

#3184

semidirect

137. C₈

#11719949

semidirect

Subgroups — #20285454

Conjugacy classes of subgroups

Complements