Group #20275542 — subgroups

Subgroups — #20275542

Order 6784.

360 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 97 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	C₄²×Dic₅₃	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	—	2	2	no	—
12	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
13	#13133143	—	2	8	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	8	no	—
35	C₄×Dic₅₃	—	2	2	no	—
36	C₄×Dic₅₃	D₄	1	4	yes	complemented
37	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×D₅₃	—	2	2	no	—
39	C₂×C₄×C₅₃	—	2	4	no	—
40	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	#41	1	2	yes	—
46	C₂²⋊D₅₃	—	4	4	no	—
47	C₂×Dic₅₃	—	2	8	no	—
48	C₂×Dic₅₃	—	2	4	no	—
49	C₂×Dic₅₃	—	2	2	no	—
50	D₂₁₂	—	2	4	no	—
51	D₂₁₂	—	4	4	no	—
52	C₄×D₅₃	—	4	2	no	—
53	C₄×D₅₃	—	4	2	no	—
54	Dic₁₀₆	—	2	4	no	—
55	C₈×C₅₃	—	4	2	no	—
56	C₂²×C₅₃	—	2	2	no	—
57	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₅₃	#121	1	2	yes	—
59	C₂×D₅₃	—	4	2	no	—
60	C₂×D₅₃	—	4	4	no	—
61	C₄×C₅₃	—	2	2	no	—
62	C₄×C₅₃	#104	1	4	yes	complemented
63	C₄×C₅₃	C₄×D₄	1	2	yes	—
64	C₄×C₅₃	#124	1	1	yes	characteristic
65	C₄×C₅₃	#127	1	1	yes	characteristic
66	Dic₅₃	—	2	8	no	—
67	Dic₅₃	—	4	2	no	—
68	#1962	—	53	1	no	maximal, Sylow
69	C₂×C₅₃	—	2	2	no	—
70	C₂×C₅₃	#390	1	1	yes	characteristic
71	C₂×C₅₃	C₂×#104	1	2	yes	—
72	D₅₃	—	8	2	no	—
73	#532	—	53	1	no	—
74	#423	D₅₃	1	1	yes	characteristic, complemented
75	C₂×#104	—	53	4	no	—
76	C₄³	—	53	1	no	—
77	C₅₃	#1962	1	1	yes	characteristic, complemented, Sylow
78	C₂×#41	—	53	2	no	—
79	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
80	C₄⋊Q₈	—	53	1	no	—
81	#127	—	53	1	no	—
82	#124	—	53	2	no	—
83	#121	—	106	2	no	—
84	C₄×D₄	—	106	2	no	—
85	C₂×C₄²	—	53	2	no	—
86	C₂×C₄²	—	106	2	no	—
87	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
88	C₄⋊C₈	—	2	2	no	—
89	#104	—	106	8	no	—
90	#41	—	106	4	no	—
91	#41	—	106	4	no	—
92	C₂×Q₈	—	53	2	no	—
93	C₂×D₄	—	53	2	no	—
94	C₂×D₄	—	106	2	no	—
95	C₂×D₄	—	106	2	no	—
96	C₂²×C₄	—	106	2	no	—
97	C₂²×C₄	—	106	2	no	—
98	C₂²×C₄	D₂₁₂	1	2	yes	—
99	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
100	M₁₆	—	2	4	no	—
101	C₂×C₈	—	2	2	no	—
102	C₄⋊C₄	—	106	2	no	—
103	C₂²⋊C₄	—	106	4	no	—
104	C₄²	—	53	4	no	—
105	C₄²	—	106	8	no	—
106	C₄²	—	106	2	no	—
107	C₄²	C₄×D₅₃	1	1	yes	characteristic
108	C₄²	C₄×D₅₃	1	1	yes	characteristic
109	C₄²	C₂²⋊D₅₃	1	2	yes	—
110	C₂³	—	106	2	no	—
111	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
112	Q₈	—	106	4	no	—
113	D₄	—	106	4	no	—
114	D₄	—	212	4	no	—
115	D₄	—	212	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	106	8	no	—
118	C₂×C₄	—	106	4	no	—
119	C₂×C₄	—	106	2	no	—
120	C₂×C₄	—	212	2	no	—
121	C₂×C₄	—	212	2	no	—
122	C₂×C₄	#11719921	1	2	yes	—
123	C₂×C₄	#11719921	1	2	yes	—
124	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
125	C₂×C₄	D₄×D₅₃	1	2	yes	—
126	C₂×C₄	#11719949	1	2	yes	—
127	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
128	C₈	—	4	2	no	—
129	C₂²	—	2	2	no	—
130	C₂²	—	212	2	no	—
131	C₂²	—	212	4	no	—
132	C₂²	C₂×#11719921	1	1	yes	characteristic
133	C₂²	#13133192	1	2	yes	—
134	C₄	—	2	2	no	—
135	C₄	—	106	8	no	—
136	C₄	—	212	2	no	—
137	C₄	#13133143	1	4	yes	complemented
138	C₄	#13133179	1	2	yes	—
139	C₄	#13133206	1	1	yes	characteristic
140	C₄	#13133213	1	1	yes	characteristic
141	C₂	—	2	2	no	—
142	C₂	—	424	2	no	—
143	C₂	#16810524	1	1	yes	characteristic
144	C₂	C₂×#13133143	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16811173

C₂

yes

maximal, characteristic

C₂×#13133143

C₂

yes

maximal, complemented

C₄²×Dic₅₃

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

—

C₂×C₄×Dic₅₃

C₂²

yes

—

#13133143

—

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

—

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

#104

yes

complemented

C₄×C₅₃

C₄×D₄

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

#1962

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#390

yes

characteristic

C₂×C₅₃

C₂×#104

yes

—

D₅₃

—

#532

—

#423

D₅₃

yes

characteristic, complemented

C₂×#104

—

C₄³

—

C₅₃

#1962

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×M₁₆

C₂×D₅₃

yes

—

C₄⋊Q₈

—

#127

—

#124

—

#121

—

106

—

C₄×D₄

—

106

—

C₂×C₄²

—

C₂×C₄²

—

106

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

#104

—

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

100

M₁₆

—

101

C₂×C₈

—

102

C₄⋊C₄

—

106

—

103

C₂²⋊C₄

—

106

—

104

C₄²

—

105

C₄²

—

106

—

106

C₄²

—

106

—

107

C₄²

C₄×D₅₃

yes

characteristic

108

C₄²

C₄×D₅₃

yes

characteristic

109

C₄²

C₂²⋊D₅₃

yes

—

110

C₂³

—

106

—

111

C₂³

C₂×D₂₁₂

yes

characteristic

112

Q₈

—

106

—

113

D₄

—

106

—

114

D₄

—

212

—

115

D₄

—

212

—

116

C₂×C₄

—

117

C₂×C₄

—

106

—

118

C₂×C₄

—

106

—

119

C₂×C₄

—

106

—

120

C₂×C₄

—

212

—

121

C₂×C₄

—

212

—

122

C₂×C₄

#11719921

yes

—

123

C₂×C₄

#11719921

yes

—

124

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

125

C₂×C₄

D₄×D₅₃

yes

—

126

C₂×C₄

#11719949

yes

—

127

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

128

C₈

—

129

C₂²

—

130

C₂²

—

212

—

131

C₂²

—

212

—

132

C₂²

C₂×#11719921

yes

characteristic

133

C₂²

#13133192

yes

—

134

C₄

—

135

C₄

—

106

—

136

C₄

—

212

—

137

C₄

#13133143

yes

complemented

138

C₄

#13133179

yes

—

139

C₄

#13133206

yes

characteristic

140

C₄

#13133213

yes

characteristic

141

C₂

—

142

C₂

—

424

—

143

C₂

#16810524

yes

characteristic

144

C₂

C₂×#13133143

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	#1962
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. C₄²×Dic₅₃	C₂	semidirect
4. C₅₃×#423	C₂	semidirect
6. #13133217	C₄	semidirect
7. #13133214	C₄	semidirect
8. #13133205	C₄	semidirect
36. C₄×Dic₅₃	D₄	semidirect
62. C₄×C₅₃	#104	semidirect
74. #423	D₅₃	semidirect
77. C₅₃	#1962	semidirect
137. C₄	#13133143	semidirect

H ≅ G/N

type

2. C₂×#13133143

C₂

semidirect

3. C₄²×Dic₅₃

C₂

semidirect

4. C₅₃×#423

C₂

semidirect

6. #13133217

C₄

semidirect

7. #13133214

C₄

semidirect

8. #13133205

C₄

semidirect

36. C₄×Dic₅₃

D₄

semidirect

62. C₄×C₅₃

#104

semidirect

74. #423

D₅₃

semidirect

77. C₅₃

#1962

semidirect

137. C₄

#13133143

semidirect

Subgroups — #20275542

Conjugacy classes of subgroups

Complements