Group #20275546 — subgroups

Subgroups — #20275546

Order 6784.

320 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 89 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	#16811173	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133143	C₂	1	4	yes	maximal, complemented
3	#16810469	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#423	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11719945	C₂²	1	2	yes	—
6	#13133217	C₄	1	1	yes	characteristic, complemented
7	#13133214	C₄	1	1	yes	characteristic, complemented
8	#13133205	C₄	1	2	yes	complemented
9	#13133191	—	2	2	no	—
10	C₄×D₂₁₂	—	2	2	no	—
11	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
12	#13133143	—	2	8	no	—
13	#13133124	—	2	2	no	—
14	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₅₃	—	2	2	no	—
17	C₂²×C₄×C₅₃	D₄	1	2	yes	—
18	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
19	C₂×C₂²⋊D₅₃	—	2	2	no	—
20	C₂²×Dic₅₃	—	2	2	no	—
21	#11719945	—	2	4	no	—
22	#11719945	—	2	4	no	—
23	C₂×D₂₁₂	—	2	2	no	—
24	C₂×D₂₁₂	C₂×C₄	1	2	yes	—
25	C₂×C₄×D₅₃	—	2	2	no	—
26	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
27	M₁₆×C₅₃	—	2	4	no	—
28	C₂×C₈×C₅₃	—	2	2	no	—
29	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₅₃	D₄	1	2	yes	—
32	#11719921	—	2	4	no	—
33	C₄⋊Dic₅₃	—	2	2	no	—
34	C₄×Dic₅₃	—	2	2	no	—
35	C₄×Dic₅₃	D₄	1	4	yes	complemented
36	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₂²×D₅₃	—	2	2	no	—
38	C₂×C₄×C₅₃	—	4	2	no	—
39	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	#41	1	2	yes	—
45	C₂²⋊D₅₃	—	4	4	no	—
46	C₂×Dic₅₃	—	2	2	no	—
47	C₂×Dic₅₃	—	2	4	no	—
48	C₂×Dic₅₃	—	4	4	no	—
49	D₂₁₂	—	2	4	no	—
50	D₂₁₂	—	4	4	no	—
51	C₄×D₅₃	—	4	2	no	—
52	C₄×D₅₃	—	4	2	no	—
53	Dic₁₀₆	—	2	4	no	—
54	C₈×C₅₃	—	4	2	no	—
55	C₂²×C₅₃	—	2	2	no	—
56	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₅₃	#121	1	2	yes	—
58	C₂×D₅₃	—	4	2	no	—
59	C₂×D₅₃	—	4	4	no	—
60	C₄×C₅₃	—	2	2	no	—
61	C₄×C₅₃	—	2	2	no	—
62	C₄×C₅₃	C₄×D₄	1	2	yes	—
63	C₄×C₅₃	#124	1	1	yes	characteristic
64	C₄×C₅₃	#127	1	1	yes	characteristic
65	Dic₅₃	—	4	4	no	—
66	Dic₅₃	—	4	2	no	—
67	#1963	—	53	1	no	maximal, Sylow
68	C₂×C₅₃	—	2	2	no	—
69	C₂×C₅₃	#390	1	1	yes	characteristic
70	C₂×C₅₃	#421	1	2	yes	—
71	D₅₃	—	8	2	no	—
72	#532	—	53	1	no	—
73	#423	D₅₃	1	1	yes	characteristic, complemented
74	C₂×#104	—	53	4	no	—
75	#376	—	53	1	no	—
76	C₅₃	#1963	1	1	yes	characteristic, complemented, Sylow
77	C₂×#41	—	53	2	no	—
78	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
79	C₄⋊Q₈	—	53	1	no	—
80	#127	—	53	1	no	—
81	#124	—	53	2	no	—
82	#121	—	106	2	no	—
83	C₄×D₄	—	106	2	no	—
84	C₂×C₄²	—	53	2	no	—
85	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
86	C₄⋊C₈	—	2	2	no	—
87	#104	—	106	8	no	—
88	#95	—	106	2	no	—
89	#41	—	106	4	no	—
90	#41	—	106	4	no	—
91	C₂×Q₈	—	53	2	no	—
92	C₂×D₄	—	53	2	no	—
93	C₂×D₄	—	106	2	no	—
94	C₂×D₄	—	106	2	no	—
95	C₂²×C₄	—	106	2	no	—
96	C₂²×C₄	—	106	2	no	—
97	C₂²×C₄	D₂₁₂	1	2	yes	—
98	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
99	M₁₆	—	2	4	no	—
100	C₂×C₈	—	2	2	no	—
101	C₄⋊C₄	—	106	2	no	—
102	C₂²⋊C₄	—	106	4	no	—
103	C₄²	—	53	4	no	—
104	C₄²	—	106	2	no	—
105	C₄²	C₄×D₅₃	1	1	yes	characteristic
106	C₄²	C₄×D₅₃	1	1	yes	characteristic
107	C₄²	C₂²⋊D₅₃	1	2	yes	—
108	C₂³	—	106	2	no	—
109	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
110	Q₈	—	106	4	no	—
111	D₄	—	106	4	no	—
112	D₄	—	212	4	no	—
113	D₄	—	212	4	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	106	2	no	—
116	C₂×C₄	—	106	4	no	—
117	C₂×C₄	—	212	4	no	—
118	C₂×C₄	—	212	2	no	—
119	C₂×C₄	—	212	2	no	—
120	C₂×C₄	#11719921	1	2	yes	—
121	C₂×C₄	#11719921	1	2	yes	—
122	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
123	C₂×C₄	D₄×D₅₃	1	2	yes	—
124	C₂×C₄	#11719949	1	2	yes	—
125	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
126	C₈	—	4	2	no	—
127	C₂²	—	2	2	no	—
128	C₂²	—	212	2	no	—
129	C₂²	—	212	4	no	—
130	C₂²	C₂×#11719921	1	1	yes	characteristic
131	C₂²	#13133192	1	2	yes	—
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	—	212	4	no	—
135	C₄	—	212	2	no	—
136	C₄	#13133179	1	2	yes	—
137	C₄	#13133206	1	1	yes	characteristic
138	C₄	#13133213	1	1	yes	characteristic
139	C₂	—	2	2	no	—
140	C₂	—	424	2	no	—
141	C₂	#16810524	1	1	yes	characteristic
142	C₂	#16810639	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16811173

C₂

yes

maximal, characteristic

C₂×#13133143

C₂

yes

maximal, complemented

#16810469

C₂

yes

maximal, characteristic, complemented

C₅₃×#423

C₂

yes

maximal, characteristic, complemented

C₂×#11719945

C₂²

yes

—

#13133217

C₄

yes

characteristic, complemented

#13133214

C₄

yes

characteristic, complemented

#13133205

C₄

yes

complemented

#13133191

—

C₄×D₂₁₂

—

C₂×C₄×Dic₅₃

C₂²

yes

—

#13133143

—

#13133124

—

C₂×M₁₆×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₂²⋊D₅₃

—

C₂²×Dic₅₃

—

#11719945

—

#11719945

—

C₂×D₂₁₂

—

C₂×D₂₁₂

C₂×C₄

yes

—

C₂×C₄×D₅₃

—

C₂×Dic₁₀₆

C₂×C₄

yes

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

—

#11719921

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

D₄

yes

complemented

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂²×D₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

#121

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×D₄

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

#1963

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#390

yes

characteristic

C₂×C₅₃

#421

yes

—

D₅₃

—

#532

—

#423

D₅₃

yes

characteristic, complemented

C₂×#104

—

#376

—

C₅₃

#1963

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×M₁₆

C₂×D₅₃

yes

—

C₄⋊Q₈

—

#127

—

#124

—

#121

—

106

—

C₄×D₄

—

106

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

#104

—

106

—

#95

—

106

—

#41

—

106

—

#41

—

106

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

106

—

C₂×D₄

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

D₂₁₂

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

100

C₂×C₈

—

101

C₄⋊C₄

—

106

—

102

C₂²⋊C₄

—

106

—

103

C₄²

—

104

C₄²

—

106

—

105

C₄²

C₄×D₅₃

yes

characteristic

106

C₄²

C₄×D₅₃

yes

characteristic

107

C₄²

C₂²⋊D₅₃

yes

—

108

C₂³

—

106

—

109

C₂³

C₂×D₂₁₂

yes

characteristic

110

Q₈

—

106

—

111

D₄

—

106

—

112

D₄

—

212

—

113

D₄

—

212

—

114

C₂×C₄

—

115

C₂×C₄

—

106

—

116

C₂×C₄

—

106

—

117

C₂×C₄

—

212

—

118

C₂×C₄

—

212

—

119

C₂×C₄

—

212

—

120

C₂×C₄

#11719921

yes

—

121

C₂×C₄

#11719921

yes

—

122

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

123

C₂×C₄

D₄×D₅₃

yes

—

124

C₂×C₄

#11719949

yes

—

125

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

126

C₈

—

127

C₂²

—

128

C₂²

—

212

—

129

C₂²

—

212

—

130

C₂²

C₂×#11719921

yes

characteristic

131

C₂²

#13133192

yes

—

132

C₄

—

133

C₄

—

134

C₄

—

212

—

135

C₄

—

212

—

136

C₄

#13133179

yes

—

137

C₄

#13133206

yes

characteristic

138

C₄

#13133213

yes

characteristic

139

C₂

—

140

C₂

—

424

—

141

C₂

#16810524

yes

characteristic

142

C₂

#16810639

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#423	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	#423	D₅₃
hypocenter	C₅₃	#1963
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#423

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
2. C₂×#13133143	C₂	semidirect
3. #16810469	C₂	semidirect
4. C₅₃×#423	C₂	semidirect
6. #13133217	C₄	semidirect
7. #13133214	C₄	semidirect
8. #13133205	C₄	semidirect
35. C₄×Dic₅₃	D₄	semidirect
73. #423	D₅₃	semidirect
76. C₅₃	#1963	semidirect

H ≅ G/N

type

2. C₂×#13133143

C₂

semidirect

3. #16810469

C₂

semidirect

4. C₅₃×#423

C₂

semidirect

6. #13133217

C₄

semidirect

7. #13133214

C₄

semidirect

8. #13133205

C₄

semidirect

35. C₄×Dic₅₃

D₄

semidirect

73. #423

D₅₃

semidirect

76. C₅₃

#1963

semidirect

Subgroups — #20275546

Conjugacy classes of subgroups

Complements