Group #20276021 — subgroups

Subgroups — #20276021

Order 6784.

264 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 65 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	#16811423	C₂	1	1	yes	maximal, characteristic
2	#16810620	C₂	1	2	yes	maximal, complemented
3	#16810596	C₂	1	2	yes	maximal, complemented
4	#16810380	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#424	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11719945	C₂²	1	2	yes	—
7	#13133250	—	2	2	no	—
8	#13133245	—	2	2	no	—
9	#13133239	—	2	2	no	—
10	#13133176	C₂²	1	2	yes	—
11	#13133142	—	2	2	no	—
12	#13133139	—	2	2	no	—
13	#13133131	—	2	4	no	—
14	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	#117×C₅₃	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₅₃	—	2	2	no	—
18	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
19	C₂×C₂²⋊D₅₃	—	2	2	no	—
20	#11719945	—	2	2	no	—
21	#11719945	—	2	2	no	—
22	#11719945	—	2	2	no	—
23	#11719945	—	2	2	no	—
24	C₂×D₂₁₂	D₄	1	2	yes	—
25	C₂×C₄×D₅₃	—	2	2	no	—
26	C₂×Dic₁₀₆	D₄	1	2	yes	—
27	M₁₆×C₅₃	—	2	2	no	—
28	C₂×C₈×C₅₃	—	2	1	no	—
29	C₂×C₈×C₅₃	—	2	2	no	—
30	C₂×C₈×C₅₃	D₄	1	2	yes	—
31	C₄⋊C₄×C₅₃	—	2	1	no	—
32	C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
33	C₄²×C₅₃	—	2	1	no	—
34	C₂²⋊Dic₅₃	—	2	2	no	—
35	C₄⋊Dic₅₃	—	2	2	no	—
36	C₄×Dic₅₃	—	2	2	no	—
37	Dic₂₁₂	—	2	2	no	—
38	D₄₂₄	—	2	2	no	—
39	#11719913	—	2	4	no	—
40	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×D₅₃	—	2	2	no	—
42	C₂×C₄×C₅₃	—	2	2	no	—
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	1	yes	characteristic
46	C₂×C₄×C₅₃	#41	1	2	yes	—
47	C₂²⋊D₅₃	—	4	2	no	—
48	C₂²⋊D₅₃	—	4	2	no	—
49	C₂×Dic₅₃	—	2	2	no	—
50	C₂×Dic₅₃	—	4	2	no	—
51	D₂₁₂	—	2	2	no	—
52	D₂₁₂	—	2	2	no	—
53	C₄×D₅₃	—	4	2	no	—
54	C₄×D₅₃	—	4	2	no	—
55	Dic₁₀₆	—	2	2	no	—
56	Dic₁₀₆	—	2	2	no	—
57	C₈×C₅₃	—	2	2	no	—
58	C₈×C₅₃	—	4	1	no	—
59	C₂²×C₅₃	—	4	1	no	—
60	C₂²×C₅₃	#121	1	1	yes	characteristic
61	C₂²×C₅₃	#121	1	2	yes	—
62	C₂×D₅₃	—	4	2	no	—
63	C₂×D₅₃	—	8	2	no	—
64	C₄×C₅₃	—	4	2	no	—
65	C₄×C₅₃	#120	1	1	yes	characteristic
66	C₄×C₅₃	#123	1	2	yes	—
67	C₄×C₅₃	#124	1	1	yes	characteristic
68	Dic₅₃	—	4	2	no	—
69	Dic₅₃	—	8	2	no	—
70	#2053	—	53	1	no	maximal, Sylow
71	C₂×C₅₃	—	2	2	no	—
72	C₂×C₅₃	—	2	1	no	—
73	C₂×C₅₃	#394	1	1	yes	characteristic
74	D₅₃	—	8	2	no	—
75	#575	—	53	1	no	—
76	#424	D₅₃	1	1	yes	characteristic, complemented
77	#416	—	53	2	no	—
78	#410	—	53	2	no	—
79	H(3,ℤ/4)	—	53	1	no	—
80	C₅₃	#2053	1	1	yes	characteristic, complemented, Sylow
81	C₂×#41	—	53	2	no	—
82	#137	—	106	2	no	—
83	AGL(1,ℤ/8)	—	106	2	no	—
84	#135	—	106	2	no	—
85	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
86	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
87	#117	—	53	2	no	—
88	#117	C₂×D₅₃	1	1	yes	characteristic
89	C₄⋊C₈	—	2	2	no	—
90	#103	—	106	2	no	—
91	#102	—	106	2	no	—
92	#99	—	106	4	no	—
93	#41	—	106	2	no	—
94	#41	—	106	2	no	—
95	#41	—	106	2	no	—
96	#41	—	106	2	no	—
97	C₂×Q₈	—	53	2	no	—
98	C₂×D₄	—	53	2	no	—
99	C₂×D₄	—	106	2	no	—
100	C₂²×C₄	—	106	2	no	—
101	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
102	Q₁₆	—	106	2	no	—
103	SD₁₆	—	106	4	no	—
104	D₈	—	106	2	no	—
105	M₁₆	—	2	2	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	106	2	no	—
111	C₂²⋊C₄	—	106	2	no	—
112	C₂²⋊C₄	D₂₁₂	1	2	yes	—
113	C₄²	—	2	1	no	—
114	C₄²	—	106	2	no	—
115	C₂³	—	106	2	no	—
116	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
117	Q₈	—	106	2	no	—
118	Q₈	—	106	2	no	—
119	D₄	—	106	2	no	—
120	D₄	—	106	2	no	—
121	D₄	—	212	2	no	—
122	D₄	—	212	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	106	2	no	—
125	C₂×C₄	—	212	2	no	—
126	C₂×C₄	—	212	2	no	—
127	C₂×C₄	—	212	2	no	—
128	C₂×C₄	#11719945	1	1	yes	characteristic
129	C₂×C₄	D₄×D₅₃	1	2	yes	—
130	C₂×C₄	#11719949	1	2	yes	—
131	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
132	C₈	—	2	2	no	—
133	C₈	—	4	1	no	—
134	C₂²	—	4	1	no	—
135	C₂²	—	424	2	no	—
136	C₂²	—	212	2	no	—
137	C₂²	#13133191	1	1	yes	characteristic
138	C₂²	#13133192	1	2	yes	—
139	C₄	—	4	2	no	—
140	C₄	—	424	2	no	—
141	C₄	—	212	2	no	—
142	C₄	#13133185	1	1	yes	characteristic
143	C₄	#13133202	1	2	yes	—
144	C₄	#13133206	1	1	yes	characteristic
145	C₂	—	2	1	no	—
146	C₂	—	2	2	no	—
147	C₂	—	424	2	no	—
148	C₂	#16810540	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811423

C₂

yes

maximal, characteristic

#16810620

C₂

yes

maximal, complemented

#16810596

C₂

yes

maximal, complemented

#16810380

C₂

yes

maximal, characteristic, complemented

C₅₃×#424

C₂

yes

maximal, characteristic, complemented

C₂×#11719945

C₂²

yes

—

—

—

—

C₂²

yes

—

#13133142

—

#13133139

—

#13133131

—

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

#117×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₂²⋊D₅₃

—

—

—

—

—

C₂×D₂₁₂

D₄

yes

—

C₂×C₄×D₅₃

—

C₂×Dic₁₀₆

D₄

yes

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

D₄

yes

—

C₄²×C₅₃

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

Dic₂₁₂

—

D₄₂₄

—

#11719913

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂²×D₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂²⋊D₅₃

—

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#121

yes

characteristic

C₂²×C₅₃

#121

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#120

yes

characteristic

C₄×C₅₃

#123

yes

—

C₄×C₅₃

#124

yes

characteristic

Dic₅₃

—

Dic₅₃

—

#2053

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#394

yes

characteristic

D₅₃

—

#575

—

#424

D₅₃

yes

characteristic, complemented

#416

—

#410

—

H(3,ℤ/4)

—

C₅₃

#2053

yes

characteristic, complemented, Sylow

C₂×#41

—

#137

—

106

—

AGL(1,ℤ/8)

—

106

—

#135

—

106

—

C₂×M₁₆

C₂×D₅₃

yes

characteristic

C₂²×C₈

C₂×D₅₃

yes

characteristic

#117

—

#117

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

#103

—

106

—

#102

—

106

—

#99

—

106

—

#41

—

106

—

#41

—

106

—

#41

—

106

—

#41

—

106

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

106

—

100

C₂²×C₄

—

106

—

101

C₂²×C₄

C₂²×D₅₃

yes

characteristic

102

Q₁₆

—

106

—

103

SD₁₆

—

106

—

104

D₈

—

106

—

105

M₁₆

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

C₂²⋊D₅₃

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

106

—

111

C₂²⋊C₄

—

106

—

112

C₂²⋊C₄

D₂₁₂

yes

—

113

C₄²

—

114

C₄²

—

106

—

115

C₂³

—

106

—

116

C₂³

C₂×D₂₁₂

yes

characteristic

117

Q₈

—

106

—

118

Q₈

—

106

—

119

D₄

—

106

—

120

D₄

—

106

—

121

D₄

—

212

—

122

D₄

—

212

—

123

C₂×C₄

—

124

C₂×C₄

—

106

—

125

C₂×C₄

—

212

—

126

C₂×C₄

—

212

—

127

C₂×C₄

—

212

—

128

C₂×C₄

#11719945

yes

characteristic

129

C₂×C₄

D₄×D₅₃

yes

—

130

C₂×C₄

#11719949

yes

—

131

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

132

C₈

—

133

C₈

—

134

C₂²

—

135

C₂²

—

424

—

136

C₂²

—

212

—

137

C₂²

#13133191

yes

characteristic

138

C₂²

#13133192

yes

—

139

C₄

—

140

C₄

—

424

—

141

C₄

—

212

—

142

C₄

#13133185

yes

characteristic

143

C₄

#13133202

yes

—

144

C₄

#13133206

yes

characteristic

145

C₂

—

146

C₂

—

147

C₂

—

424

—

148

C₂

#16810540

yes

characteristic

	subgroup	G/H
center	C₄	#13133206
commutator	C₂²×C₄×C₅₃	C₂³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#424	C₂
socle	C₂×C₅₃	#394
hypercenter	#424	D₅₃
hypocenter	C₅₃	#2053
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #424

subgroup

G/H

center

C₄

#13133206

commutator

C₂²×C₄×C₅₃

C₂³

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#424

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #424

N	H ≅ G/N	type
2. #16810620	C₂	semidirect
3. #16810596	C₂	semidirect
4. #16810380	C₂	semidirect
5. C₅₃×#424	C₂	semidirect
76. #424	D₅₃	semidirect
80. C₅₃	#2053	semidirect

H ≅ G/N

type

2. #16810620

C₂

semidirect

3. #16810596

C₂

semidirect

4. #16810380

C₂

semidirect

5. C₅₃×#424

C₂

semidirect

76. #424

D₅₃

semidirect

80. C₅₃

#2053

semidirect

Subgroups — #20276021

Conjugacy classes of subgroups

Complements