Group #20286154 — subgroups

Subgroups — #20286154

Order 6784.

370 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 117 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	#16811228	C₂	1	2	yes	maximal, complemented
2	#16810924	C₂	1	4	yes	maximal, complemented
3	#16810876	C₂	1	2	yes	maximal, complemented
4	#16810859	C₂	1	2	yes	maximal, complemented
5	#16810819	C₂	1	4	yes	maximal, complemented
6	C₅₃×#424	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×D₄₂₄	C₂²	1	2	yes	—
8	C₂×D₄₂₄	C₂²	1	2	yes	—
9	#13133214	C₂²	1	2	yes	—
10	#13133209	C₂²	1	2	yes	—
11	#13133202	—	2	4	no	—
12	#13133191	C₂²	1	4	yes	—
13	#13133190	—	2	4	no	—
14	C₄×D₂₁₂	C₂²	1	4	yes	—
15	#13133151	C₂²	1	2	yes	—
16	#13133149	C₂²	1	2	yes	—
17	#13133139	C₂²	1	4	yes	—
18	#13133139	C₂²	1	4	yes	—
19	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	#117×C₅₃	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₅₃	C₂²	1	4	yes	—
23	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
24	C₂×C₂²⋊D₅₃	—	2	4	no	—
25	C₂×D₂₁₂	—	2	4	no	—
26	C₂×D₂₁₂	C₂³	1	4	yes	—
27	C₂×C₄×D₅₃	—	2	4	no	—
28	M₁₆×C₅₃	—	4	1	no	—
29	C₂×C₈×C₅₃	—	4	1	no	—
30	C₂×C₈×C₅₃	C₂³	1	2	yes	—
31	C₂×C₈×C₅₃	C₂³	1	2	yes	—
32	C₄⋊C₄×C₅₃	D₄	1	2	yes	—
33	C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
34	C₄²×C₅₃	C₂³	1	2	yes	—
35	#11719921	—	2	4	no	—
36	#11719921	—	2	4	no	—
37	C₄⋊Dic₅₃	C₂³	1	4	yes	—
38	#11719919	—	2	4	no	—
39	D₄₂₄	—	4	2	no	—
40	D₄₂₄	—	4	2	no	—
41	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×D₅₃	—	2	4	no	—
43	C₂×C₄×C₅₃	—	2	1	no	—
44	C₂×C₄×C₅₃	—	2	1	no	—
45	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	—
46	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
48	C₂²⋊D₅₃	—	8	4	no	—
49	C₂×Dic₅₃	—	2	4	no	—
50	D₂₁₂	—	2	4	no	—
51	D₂₁₂	—	2	4	no	—
52	D₂₁₂	—	8	4	no	—
53	C₄×D₅₃	—	8	4	no	—
54	C₈×C₅₃	—	2	2	no	—
55	C₈×C₅₃	—	2	2	no	—
56	C₂²×C₅₃	—	2	1	no	—
57	C₂²×C₅₃	—	2	2	no	—
58	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
59	C₂×D₅₃	—	4	4	no	—
60	C₂×D₅₃	—	4	4	no	—
61	C₂×D₅₃	—	4	4	no	—
62	C₄×C₅₃	—	2	1	no	—
63	C₄×C₅₃	—	2	4	no	—
64	C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
65	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
66	Dic₅₃	—	4	4	no	—
67	#3259	—	53	1	no	maximal, Sylow
68	C₂×C₅₃	—	4	1	no	—
69	C₂×C₅₃	#537	1	1	yes	characteristic
70	C₂×C₅₃	#576	1	2	yes	—
71	D₅₃	—	8	4	no	—
72	#540	—	53	2	no	—
73	#478	—	53	4	no	—
74	#469	—	53	2	no	—
75	#466	—	53	2	no	—
76	#459	—	53	4	no	—
77	#424	D₅₃	1	1	yes	characteristic, complemented
78	C₅₃	#3259	1	1	yes	characteristic, complemented, Sylow
79	C₂×D₈	—	53	2	no	—
80	C₂×D₈	—	53	2	no	—
81	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
82	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
83	#127	—	53	2	no	—
84	#125	—	53	2	no	—
85	#123	—	106	4	no	—
86	#121	—	53	4	no	—
87	#121	—	106	4	no	—
88	C₄×D₄	—	53	4	no	—
89	#117	C₂×D₅₃	1	1	yes	characteristic
90	#107	—	53	2	no	—
91	#106	—	53	2	no	—
92	C₄⋊C₈	C₂×D₅₃	1	4	yes	—
93	#102	—	53	4	no	—
94	#102	—	53	4	no	—
95	C₂×D₄	—	53	4	no	—
96	C₂×D₄	—	106	4	no	—
97	C₂×D₄	—	106	4	no	—
98	C₂²×C₄	—	106	4	no	—
99	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
100	D₈	—	212	2	no	—
101	D₈	—	212	2	no	—
102	M₁₆	—	4	1	no	—
103	C₂×C₈	—	4	1	no	—
104	C₂×C₈	C₂²×D₅₃	1	2	yes	—
105	C₂×C₈	C₂²×D₅₃	1	2	yes	—
106	C₄⋊C₄	—	53	4	no	—
107	C₄⋊C₄	—	106	4	no	—
108	C₄⋊C₄	D₂₁₂	1	2	yes	—
109	C₂²⋊C₄	—	106	4	no	—
110	C₂²⋊C₄	—	106	4	no	—
111	C₂²⋊C₄	D₂₁₂	1	2	yes	—
112	C₄²	C₂²×D₅₃	1	2	yes	—
113	C₂³	—	106	4	no	—
114	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
115	D₄	—	106	4	no	—
116	D₄	—	106	4	no	—
117	D₄	—	424	4	no	—
118	D₄	—	424	4	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	106	4	no	—
122	C₂×C₄	—	424	4	no	—
123	C₂×C₄	C₂×D₂₁₂	1	4	yes	—
124	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
125	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
126	C₈	—	2	2	no	—
127	C₈	—	2	2	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	212	4	no	—
131	C₂²	—	212	4	no	—
132	C₂²	—	212	4	no	—
133	C₂²	C₂²×D₂₁₂	1	1	yes	characteristic
134	C₄	—	2	4	no	—
135	C₄	—	2	1	no	—
136	C₄	—	212	4	no	—
137	C₄	#13133262	1	1	yes	characteristic
138	C₄	#13133265	1	1	yes	characteristic
139	C₂	—	4	1	no	—
140	C₂	—	424	4	no	—
141	C₂	#16811204	1	1	yes	characteristic
142	C₂	#16811428	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16811228

C₂

yes

maximal, complemented

#16810924

C₂

yes

maximal, complemented

#16810876

C₂

yes

maximal, complemented

#16810859

C₂

yes

maximal, complemented

#16810819

C₂

yes

maximal, complemented

C₅₃×#424

C₂

yes

maximal, characteristic, complemented

C₂×D₄₂₄

C₂²

yes

—

C₂×D₄₂₄

C₂²

yes

—

#13133214

C₂²

yes

—

#13133209

C₂²

yes

—

#13133202

—

#13133191

C₂²

yes

—

#13133190

—

C₄×D₂₁₂

C₂²

yes

—

#13133151

C₂²

yes

—

#13133149

C₂²

yes

—

#13133139

C₂²

yes

—

#13133139

C₂²

yes

—

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

#117×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₂²⋊D₅₃

—

C₂×D₂₁₂

—

C₂×D₂₁₂

C₂³

yes

—

C₂×C₄×D₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

—

C₂×C₈×C₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

D₄

yes

—

C₂²⋊C₄×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

—

#11719921

—

#11719921

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

D₄₂₄

—

D₄₂₄

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂²×D₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

H(5,𝔽₂)

yes

characteristic

C₄×C₅₃

2₋¹⁺⁴

yes

characteristic

Dic₅₃

—

#3259

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#537

yes

characteristic

C₂×C₅₃

#576

yes

—

D₅₃

—

—

—

—

—

—

D₅₃

yes

characteristic, complemented

C₅₃

#3259

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂×D₈

—

C₂×M₁₆

C₂×D₅₃

yes

characteristic

C₂²×C₈

C₂×D₅₃

yes

—

—

—

106

—

#121

—

#121

—

106

—

C₄×D₄

—

#117

C₂×D₅₃

yes

characteristic

#107

—

#106

—

C₄⋊C₈

C₂×D₅₃

yes

—

#102

—

#102

—

C₂×D₄

—

C₂×D₄

—

106

—

C₂×D₄

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

100

D₈

—

212

—

101

D₈

—

212

—

102

M₁₆

—

103

C₂×C₈

—

104

C₂×C₈

C₂²×D₅₃

yes

—

105

C₂×C₈

C₂²×D₅₃

yes

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

106

—

108

C₄⋊C₄

D₂₁₂

yes

—

109

C₂²⋊C₄

—

106

—

110

C₂²⋊C₄

—

106

—

111

C₂²⋊C₄

D₂₁₂

yes

—

112

C₄²

C₂²×D₅₃

yes

—

113

C₂³

—

106

—

114

C₂³

C₂×D₂₁₂

yes

characteristic

115

D₄

—

106

—

116

D₄

—

106

—

117

D₄

—

424

—

118

D₄

—

424

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

106

—

122

C₂×C₄

—

424

—

123

C₂×C₄

C₂×D₂₁₂

yes

—

124

C₂×C₄

C₂×D₂₁₂

yes

characteristic

125

C₂×C₄

C₂³×D₅₃

yes

characteristic

126

C₈

—

127

C₈

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

212

—

131

C₂²

—

212

—

132

C₂²

—

212

—

133

C₂²

C₂²×D₂₁₂

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

212

—

137

C₄

#13133262

yes

characteristic

138

C₄

#13133265

yes

characteristic

139

C₂

—

140

C₂

—

424

—

141

C₂

#16811204

yes

characteristic

142

C₂

#16811428

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#424	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#424	D₅₃
hypocenter	C₅₃	#3259
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #424

subgroup

G/H

center

C₂²

C₂²×D₂₁₂

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#424

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ #117 ◁ #424

N	H ≅ G/N	type
1. #16811228	C₂	semidirect
2. #16810924	C₂	semidirect
3. #16810876	C₂	semidirect
4. #16810859	C₂	semidirect
5. #16810819	C₂	semidirect
6. C₅₃×#424	C₂	semidirect
77. #424	D₅₃	semidirect
78. C₅₃	#3259	semidirect

H ≅ G/N

type

1. #16811228

C₂

semidirect

2. #16810924

C₂

semidirect

3. #16810876

C₂

semidirect

4. #16810859

C₂

semidirect

5. #16810819

C₂

semidirect

6. C₅₃×#424

C₂

semidirect

77. #424

D₅₃

semidirect

78. C₅₃

#3259

semidirect

Subgroups — #20286154

Conjugacy classes of subgroups

Complements