Group #20274280 — subgroups

Subgroups — #20274280

Order 6784.

162 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 51 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	#16810830	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810820	C₂	1	1	yes	maximal, characteristic
3	#16810753	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810374	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810369	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810367	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#486	C₂	1	1	yes	maximal, characteristic
8	C₂×#11719925	—	2	1	no	—
9	C₂×#11719923	—	2	1	no	—
10	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
11	C₄⋊#88322	C₂²	1	1	yes	characteristic
12	C₄⋊#88322	C₂²	1	1	yes	characteristic
13	C₄⋊#88322	C₂²	1	1	yes	characteristic
14	#13133146	—	2	1	no	—
15	#13133141	—	2	1	no	—
16	#13133138	—	2	1	no	—
17	#13133127	—	2	1	no	—
18	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
19	#127×C₅₃	C₂²	1	1	yes	characteristic
20	#102×C₅₃	—	2	1	no	—
21	#102×C₅₃	—	2	1	no	—
22	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₅₃	—	2	1	no	—
24	C₂×D₄×C₅₃	—	2	1	no	—
25	C₂×D₄×C₅₃	—	2	1	no	—
26	C₂×D₄×C₅₃	—	2	1	no	—
27	C₂×Dic₁₀₆	D₄	1	1	yes	characteristic
28	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
29	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
30	C₄⋊C₄×C₅₃	—	2	1	no	—
31	C₄⋊C₄×C₅₃	—	2	1	no	—
32	C₄²×C₅₃	C₂³	1	1	yes	characteristic
33	#11719925	—	4	2	no	—
34	#11719923	—	4	2	no	—
35	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
36	#11719919	—	2	1	no	—
37	C₄×Dic₅₃	—	2	1	no	—
38	C₂×#88322	—	2	1	no	—
39	C₂×#88322	—	2	1	no	—
40	C₂×#88322	—	2	1	no	—
41	C₂³×C₅₃	—	4	1	no	—
42	Q₈×C₅₃	—	4	1	no	—
43	Q₈×C₅₃	—	4	1	no	—
44	D₄×C₅₃	—	4	1	no	—
45	D₄×C₅₃	—	4	1	no	—
46	D₄×C₅₃	—	4	1	no	—
47	D₄×C₅₃	—	4	1	no	—
48	D₄×C₅₃	—	8	1	no	—
49	C₂×C₄×C₅₃	—	4	1	no	—
50	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
53	C₂×Dic₅₃	—	2	1	no	—
54	C₂×Dic₅₃	—	2	1	no	—
55	Dic₁₀₆	—	2	1	no	—
56	Dic₁₀₆	—	2	1	no	—
57	C₈×C₅₃	—	2	1	no	—
58	C₈×C₅₃	—	2	1	no	—
59	#88322	—	4	1	no	—
60	#88322	—	4	1	no	—
61	#88322	—	4	1	no	—
62	C₂²×C₅₃	—	8	1	no	—
63	C₂²×C₅₃	—	8	1	no	—
64	C₂²×C₅₃	—	8	1	no	—
65	C₂²×C₅₃	#121	1	1	yes	characteristic
66	C₄×C₅₃	—	2	1	no	—
67	C₄×C₅₃	—	8	1	no	—
68	C₄×C₅₃	#135	1	1	yes	characteristic
69	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
70	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
71	C₄×C₅₃	#137	1	1	yes	characteristic
72	Dic₅₃	—	4	1	no	—
73	Dic₅₃	—	4	1	no	—
74	#1702	—	53	1	no	maximal, Sylow
75	C₂×C₅₃	—	16	1	no	—
76	C₂×C₅₃	#464	1	1	yes	characteristic
77	C₂×C₅₃	#469	1	1	yes	characteristic
78	C₂×C₅₃	#471	1	1	yes	characteristic
79	#486	D₅₃	1	1	yes	characteristic
80	#462	—	53	1	no	—
81	#460	—	53	1	no	—
82	#446	—	53	1	no	—
83	#334	—	53	1	no	—
84	#332	—	53	1	no	—
85	#331	—	53	1	no	—
86	C₅₃	#1702	1	1	yes	characteristic, complemented, Sylow
87	C₂×Q₁₆	—	106	1	no	—
88	C₂×SD₁₆	—	106	1	no	—
89	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
90	#127	C₂×D₅₃	1	1	yes	characteristic
91	C₄×Q₈	—	53	1	no	—
92	C₄⋊C₈	—	53	1	no	—
93	C₄⋊C₈	—	53	1	no	—
94	C₄⋊C₈	—	53	1	no	—
95	C₄⋊C₈	—	106	1	no	—
96	#103	—	106	1	no	—
97	#102	—	2	1	no	—
98	#102	—	2	1	no	—
99	#102	—	106	1	no	—
100	#97	—	106	1	no	—
101	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
102	C₂×Q₈	—	2	1	no	—
103	C₂×Q₈	—	53	1	no	—
104	C₂×D₄	—	2	1	no	—
105	C₂×D₄	—	2	1	no	—
106	C₂×D₄	—	2	1	no	—
107	Q₁₆	—	212	2	no	—
108	SD₁₆	—	212	2	no	—
109	C₂×C₈	—	106	1	no	—
110	C₂×C₈	—	106	1	no	—
111	C₂×C₈	—	106	1	no	—
112	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
113	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	2	1	no	—
116	C₄⋊C₄	—	53	1	no	—
117	C₄⋊C₄	—	106	1	no	—
118	C₄²	—	106	1	no	—
119	C₄²	C₂²×D₅₃	1	1	yes	characteristic
120	C₂³	—	4	1	no	—
121	Q₈	—	4	1	no	—
122	Q₈	—	4	1	no	—
123	Q₈	—	106	1	no	—
124	Q₈	—	106	1	no	—
125	D₄	—	4	1	no	—
126	D₄	—	4	1	no	—
127	D₄	—	4	1	no	—
128	D₄	—	4	1	no	—
129	D₄	—	8	1	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	106	1	no	—
132	C₂×C₄	—	106	1	no	—
133	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
134	C₂×C₄	#11719947	1	1	yes	characteristic
135	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
136	C₈	—	2	1	no	—
137	C₈	—	2	1	no	—
138	C₈	—	212	1	no	—
139	C₈	—	212	1	no	—
140	C₈	—	212	1	no	—
141	C₂²	—	8	1	no	—
142	C₂²	—	8	1	no	—
143	C₂²	—	8	1	no	—
144	C₂²	#13133188	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	8	1	no	—
147	C₄	—	212	1	no	—
148	C₄	—	212	1	no	—
149	C₄	#13133237	1	1	yes	characteristic
150	C₄	#13133241	1	1	yes	characteristic
151	C₄	#13133241	1	1	yes	characteristic
152	C₄	#13133247	1	1	yes	characteristic
153	C₂	—	16	1	no	—
154	C₂	#16810842	1	1	yes	characteristic
155	C₂	#16810873	1	1	yes	characteristic
156	C₂	#16810883	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810830

C₂

yes

maximal, characteristic, complemented

#16810820

C₂

yes

maximal, characteristic

#16810753

C₂

yes

maximal, characteristic, complemented

#16810374

C₂

yes

maximal, characteristic, complemented

#16810369

C₂

yes

maximal, characteristic, complemented

#16810367

C₂

yes

maximal, characteristic

C₅₃×#486

C₂

yes

maximal, characteristic

C₂×#11719925

—

C₂×#11719923

—

C₄×Dic₁₀₆

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

—

—

—

—

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

#127×C₅₃

C₂²

yes

characteristic

#102×C₅₃

—

#102×C₅₃

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×Dic₁₀₆

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

#11719925

—

#11719923

—

C₄⋊Dic₅₃

D₄

yes

characteristic

#11719919

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#121

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

#1702

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

yes

—

—

—

—

—

—

C₅₃

#1702

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

106

—

C₂×SD₁₆

—

106

—

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

#127

C₂×D₅₃

yes

characteristic

C₄×Q₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

#103

—

106

—

#102

—

#102

—

#102

—

106

—

100

#97

—

106

—

101

C₄×C₈

C₂×D₅₃

yes

characteristic

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂×D₄

—

107

Q₁₆

—

212

—

108

SD₁₆

—

212

—

109

C₂×C₈

—

106

—

110

C₂×C₈

—

106

—

111

C₂×C₈

—

106

—

112

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

113

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

106

—

118

C₄²

—

106

—

119

C₄²

C₂²×D₅₃

yes

characteristic

120

C₂³

—

121

Q₈

—

122

Q₈

—

123

Q₈

—

106

—

124

Q₈

—

106

—

125

D₄

—

126

D₄

—

127

D₄

—

128

D₄

—

129

D₄

—

130

C₂×C₄

—

131

C₂×C₄

—

106

—

132

C₂×C₄

—

106

—

133

C₂×C₄

D₄×D₅₃

yes

characteristic

134

C₂×C₄

#11719947

yes

characteristic

135

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

136

C₈

—

137

C₈

—

138

C₈

—

212

—

139

C₈

—

212

—

140

C₈

—

212

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

#13133188

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

212

—

148

C₄

—

212

—

149

C₄

#13133237

yes

characteristic

150

C₄

#13133241

yes

characteristic

151

C₄

#13133241

yes

characteristic

152

C₄

#13133247

yes

characteristic

153

C₂

—

154

C₂

#16810842

yes

characteristic

155

C₂

#16810873

yes

characteristic

156

C₂

#16810883

yes

characteristic

	subgroup	G/H
center	C₂²	#13133188
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#486	C₂
socle	C₂²×C₅₃	#121
hypercenter	#486	D₅₃
hypocenter	C₅₃	#1702
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #486

subgroup

G/H

center

C₂²

#13133188

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#486

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #486

N	H ≅ G/N	type
1. #16810830	C₂	semidirect
3. #16810753	C₂	semidirect
4. #16810374	C₂	semidirect
5. #16810369	C₂	semidirect
86. C₅₃	#1702	semidirect

H ≅ G/N

type

1. #16810830

C₂

semidirect

3. #16810753

C₂

semidirect

4. #16810374

C₂

semidirect

5. #16810369

C₂

semidirect

86. C₅₃

#1702

semidirect

Subgroups — #20274280

Conjugacy classes of subgroups

Complements