Group #20276790 — subgroups

Subgroups — #20276790

Order 6784.

170 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 43 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	#16811061	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811059	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810998	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810739	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810443	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810435	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#361	C₂	1	1	yes	maximal, characteristic, complemented
8	#13133239	C₂²	1	1	yes	characteristic
9	C₂×D₄₂₄	C₂²	1	1	yes	characteristic
10	C₂×#11719922	—	2	1	no	—
11	#13133225	—	2	1	no	—
12	#13133213	—	2	1	no	—
13	#13133162	—	2	1	no	—
14	#13133161	—	2	1	no	—
15	D₈₄₈	—	2	1	no	—
16	#13133158	—	2	1	no	—
17	#13133157	C₂²	1	1	yes	characteristic
18	#13133143	—	2	1	no	—
19	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
20	C₂×D₈×C₅₃	C₂²	1	1	yes	characteristic
21	M₃₂×C₅₃	C₂²	1	1	yes	characteristic
22	#108×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×D₄×C₅₃	—	2	1	no	—
24	C₂×C₂²⋊D₅₃	—	4	2	no	—
25	#11719945	—	2	1	no	—
26	C₂×D₂₁₂	—	2	1	no	—
27	D₈×C₅₃	—	2	1	no	—
28	D₈×C₅₃	—	2	1	no	—
29	M₁₆×C₅₃	—	2	1	no	—
30	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
31	#11719922	—	4	1	no	—
32	#11719922	—	4	1	no	—
33	C₄×Dic₅₃	D₄	1	1	yes	characteristic
34	C₂×#88322	D₄	1	1	yes	characteristic
35	Dic₂₁₂	D₄	1	1	yes	characteristic, complemented
36	D₄₂₄	—	2	1	no	—
37	D₄₂₄	—	2	1	no	—
38	D₄₂₄	D₄	1	1	yes	characteristic, complemented
39	#11719913	—	2	1	no	—
40	#11719912	—	4	1	no	—
41	C₈×D₅₃	—	4	1	no	—
42	C₁₆×C₅₃	—	2	1	no	—
43	#11719908	—	2	1	no	—
44	C₂³×C₅₃	—	4	1	no	—
45	C₂²×D₅₃	—	4	1	no	—
46	D₄×C₅₃	—	4	1	no	—
47	D₄×C₅₃	—	4	1	no	—
48	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂²⋊D₅₃	—	4	1	no	—
50	C₂²⋊D₅₃	—	8	4	no	—
51	C₂×Dic₅₃	—	2	1	no	—
52	D₂₁₂	—	2	1	no	—
53	D₂₁₂	—	4	1	no	—
54	D₂₁₂	—	4	1	no	—
55	C₄×D₅₃	—	4	1	no	—
56	Dic₁₀₆	—	2	1	no	—
57	C₈×C₅₃	—	4	1	no	—
58	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₈×C₅₃	#41	1	1	yes	characteristic
60	#88322	D₈	1	1	yes	characteristic
61	#88322	D₈	1	1	yes	characteristic
62	C₂²×C₅₃	—	8	2	no	—
63	C₂²×C₅₃	—	8	1	no	—
64	C₂²×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₂×D₅₃	—	4	1	no	—
66	C₂×D₅₃	—	8	2	no	—
67	C₂×D₅₃	—	8	1	no	—
68	C₄×C₅₃	#121	1	1	yes	characteristic
69	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
70	Dic₅₃	—	2	2	no	—
71	Dic₅₃	—	4	1	no	—
72	#2213	—	53	1	no	maximal, Sylow
73	C₂×C₅₃	—	2	1	no	—
74	C₂×C₅₃	—	16	1	no	—
75	C₂×C₅₃	#459	1	1	yes	characteristic
76	D₅₃	—	8	1	no	—
77	D₅₃	—	16	1	no	—
78	#509	—	53	1	no	—
79	#509	—	53	1	no	—
80	#493	—	53	1	no	—
81	#443	—	53	1	no	—
82	#364	—	53	1	no	—
83	#361	—	53	1	no	—
84	#361	D₅₃	1	1	yes	characteristic, complemented
85	C₅₃	#2213	1	1	yes	characteristic, complemented, Sylow
86	#135	—	53	1	no	—
87	C₂×D₈	—	53	1	no	—
88	C₂×D₈	—	106	1	no	—
89	C₂×D₈	C₂×D₅₃	1	1	yes	characteristic
90	#131	—	106	1	no	—
91	#127	—	106	1	no	—
92	SD₃₂	—	106	1	no	—
93	SD₃₂	—	106	1	no	—
94	D₁₆	—	106	1	no	—
95	D₁₆	—	106	1	no	—
96	M₃₂	—	53	1	no	—
97	M₃₂	C₂×D₅₃	1	1	yes	characteristic
98	#108	C₂×D₅₃	1	1	yes	characteristic
99	#104	—	106	1	no	—
100	C₄×C₈	—	53	1	no	—
101	#41	—	106	1	no	—
102	C₂×D₄	—	2	1	no	—
103	C₂×D₄	—	106	1	no	—
104	C₂×D₄	—	212	2	no	—
105	Q₁₆	—	53	1	no	—
106	SD₁₆	—	106	1	no	—
107	D₈	—	2	1	no	—
108	D₈	—	2	1	no	—
109	D₈	—	53	1	no	—
110	D₈	—	106	1	no	—
111	D₈	—	106	1	no	—
112	D₈	—	212	1	no	—
113	D₈	—	212	1	no	—
114	M₁₆	—	2	1	no	—
115	M₁₆	—	212	1	no	—
116	C₂×C₈	—	53	1	no	—
117	C₂×C₈	—	212	1	no	—
118	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
119	C₄²	—	53	1	no	—
120	C₁₆	—	2	1	no	—
121	C₁₆	—	106	1	no	—
122	C₂³	—	4	1	no	—
123	C₂³	—	212	1	no	—
124	Q₈	—	106	1	no	—
125	D₄	—	4	1	no	—
126	D₄	—	4	1	no	—
127	D₄	—	106	1	no	—
128	D₄	—	424	4	no	—
129	D₄	—	212	1	no	—
130	D₄	—	212	1	no	—
131	D₄	—	212	1	no	—
132	C₂×C₄	—	106	1	no	—
133	C₂×C₄	—	212	1	no	—
134	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
135	C₈	—	4	1	no	—
136	C₈	—	53	1	no	—
137	C₈	—	53	1	no	—
138	C₈	#11719945	1	1	yes	characteristic
139	C₈	D₄×D₅₃	1	1	yes	characteristic
140	C₂²	—	8	2	no	—
141	C₂²	—	8	1	no	—
142	C₂²	—	424	2	no	—
143	C₂²	—	424	1	no	—
144	C₂²	—	212	1	no	—
145	C₂²	#13133244	1	1	yes	characteristic
146	C₄	—	106	2	no	—
147	C₄	—	212	1	no	—
148	C₄	#13133190	1	1	yes	characteristic
149	C₄	D₈×D₅₃	1	1	yes	characteristic
150	C₂	—	2	1	no	—
151	C₂	—	16	1	no	—
152	C₂	—	848	1	no	—
153	C₂	—	424	1	no	—
154	C₂	#16810817	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811061

C₂

yes

maximal, characteristic, complemented

#16811059

C₂

yes

maximal, characteristic, complemented

#16810998

C₂

yes

maximal, characteristic, complemented

#16810739

C₂

yes

maximal, characteristic, complemented

#16810443

C₂

yes

maximal, characteristic, complemented

#16810435

C₂

yes

maximal, characteristic, complemented

C₅₃×#361

C₂

yes

maximal, characteristic, complemented

#13133239

C₂²

yes

characteristic

C₂×D₄₂₄

C₂²

yes

—

—

—

—

—

D₈₄₈

—

#13133158

—

#13133157

C₂²

yes

characteristic

#13133143

—

C₈×Dic₅₃

C₂²

yes

characteristic

C₂×D₈×C₅₃

C₂²

yes

characteristic

M₃₂×C₅₃

C₂²

yes

characteristic

#108×C₅₃

C₂²

yes

characteristic

C₂×D₄×C₅₃

—

C₂×C₂²⋊D₅₃

—

#11719945

—

C₂×D₂₁₂

—

D₈×C₅₃

—

D₈×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

#11719922

—

#11719922

—

C₄×Dic₅₃

D₄

yes

characteristic

C₂×#88322

D₄

yes

characteristic

Dic₂₁₂

D₄

yes

characteristic, complemented

D₄₂₄

—

D₄₂₄

—

D₄₂₄

D₄

yes

characteristic, complemented

#11719913

—

#11719912

—

C₈×D₅₃

—

C₁₆×C₅₃

—

#11719908

—

C₂³×C₅₃

—

C₂²×D₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²⋊D₅₃

—

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

C₂×D₄

yes

C₈×C₅₃

yes

D₈

yes

characteristic

#88322

D₈

yes

characteristic

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

AGL(1,ℤ/8)

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

#121

yes

characteristic

C₄×C₅₃

C₂×D₈

yes

characteristic

Dic₅₃

—

Dic₅₃

—

#2213

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#459

yes

characteristic

D₅₃

—

D₅₃

—

—

—

—

—

—

—

D₅₃

yes

characteristic, complemented

C₅₃

#2213

yes

characteristic, complemented, Sylow

#135

—

C₂×D₈

—

C₂×D₈

—

106

—

C₂×D₈

C₂×D₅₃

yes

characteristic

#131

—

106

—

#127

—

106

—

SD₃₂

—

106

—

SD₃₂

—

106

—

D₁₆

—

106

—

D₁₆

—

106

—

M₃₂

—

M₃₂

C₂×D₅₃

yes

characteristic

#108

C₂×D₅₃

yes

characteristic

#104

—

106

—

100

C₄×C₈

—

101

#41

—

106

—

102

C₂×D₄

—

103

C₂×D₄

—

106

—

104

C₂×D₄

—

212

—

105

Q₁₆

—

106

SD₁₆

—

106

—

107

D₈

—

108

D₈

—

109

D₈

—

110

D₈

—

106

—

111

D₈

—

106

—

112

D₈

—

212

—

113

D₈

—

212

—

114

M₁₆

—

115

M₁₆

—

212

—

116

C₂×C₈

—

117

C₂×C₈

—

212

—

118

C₂×C₈

C₂²×D₅₃

yes

characteristic

119

C₄²

—

120

C₁₆

—

121

C₁₆

—

106

—

122

C₂³

—

123

C₂³

—

212

—

124

Q₈

—

106

—

125

D₄

—

126

D₄

—

127

D₄

—

106

—

128

D₄

—

424

—

129

D₄

—

212

—

130

D₄

—

212

—

131

D₄

—

212

—

132

C₂×C₄

—

106

—

133

C₂×C₄

—

212

—

134

C₂×C₄

D₄×D₅₃

yes

characteristic

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

#11719945

yes

characteristic

139

C₈

D₄×D₅₃

yes

characteristic

140

C₂²

—

141

C₂²

—

142

C₂²

—

424

—

143

C₂²

—

424

—

144

C₂²

—

212

—

145

C₂²

#13133244

yes

characteristic

146

C₄

—

106

—

147

C₄

—

212

—

148

C₄

#13133190

yes

characteristic

149

C₄

D₈×D₅₃

yes

characteristic

150

C₂

—

151

C₂

—

152

C₂

—

848

—

153

C₂

—

424

—

154

C₂

#16810817

yes

characteristic

	subgroup	G/H
center	C₂	#16810817
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#361	C₂
socle	C₂×C₅₃	#459
hypercenter	#361	D₅₃
hypocenter	C₅₃	#2213
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #361

subgroup

G/H

center

C₂

#16810817

commutator

C₂×C₈×C₅₃

C₂³

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₅₃×#361

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #361

N	H ≅ G/N	type
1. #16811061	C₂	semidirect
2. #16811059	C₂	semidirect
3. #16810998	C₂	semidirect
4. #16810739	C₂	semidirect
5. #16810443	C₂	semidirect
6. #16810435	C₂	semidirect
7. C₅₃×#361	C₂	semidirect
35. Dic₂₁₂	D₄	semidirect
38. D₄₂₄	D₄	semidirect
84. #361	D₅₃	semidirect
85. C₅₃	#2213	semidirect

H ≅ G/N

type

1. #16811061

C₂

semidirect

2. #16811059

C₂

semidirect

3. #16810998

C₂

semidirect

4. #16810739

C₂

semidirect

5. #16810443

C₂

semidirect

6. #16810435

C₂

semidirect

7. C₅₃×#361

C₂

semidirect

35. Dic₂₁₂

D₄

semidirect

38. D₄₂₄

D₄

semidirect

84. #361

D₅₃

semidirect

85. C₅₃

#2213

semidirect

Subgroups — #20276790

Conjugacy classes of subgroups

Complements