Group #20274181 — subgroups

Subgroups — #20274181

Order 6784.

182 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 55 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	#16810996	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810904	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810819	C₂	1	1	yes	maximal, characteristic
4	#16810702	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810368	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810366	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#335	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11719913	—	2	1	no	—
9	C₂×#11719913	—	2	1	no	—
10	C₂×D₄₂₄	—	2	1	no	—
11	C₂×#11719912	—	2	1	no	—
12	#13133217	C₂²	1	1	yes	characteristic
13	#13133214	C₂²	1	1	yes	characteristic
14	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
15	#13133151	—	2	1	no	—
16	C₄⋊#88322	C₂²	1	1	yes	characteristic
17	#13133139	—	2	1	no	—
18	#13133139	—	2	1	no	—
19	#13133129	—	2	1	no	—
20	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×D₂₁₂	—	2	1	no	—
24	C₂×D₂₁₂	—	2	1	no	—
25	C₂×D₂₁₂	—	2	1	no	—
26	C₂×D₂₁₂	D₄	1	1	yes	characteristic
27	C₂×C₄×D₅₃	—	2	1	no	—
28	C₂×Dic₁₀₆	—	2	1	no	—
29	C₂×C₈×C₅₃	—	2	1	no	—
30	C₂×C₈×C₅₃	—	2	1	no	—
31	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
32	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
33	C₄²×C₅₃	C₂³	1	1	yes	characteristic
34	#11719921	—	2	1	no	—
35	C₄⋊Dic₅₃	—	2	1	no	—
36	C₄⋊Dic₅₃	—	2	1	no	—
37	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
38	C₂×#88322	—	2	1	no	—
39	D₄₂₄	—	4	2	no	—
40	#11719913	—	4	2	no	—
41	#11719913	—	4	2	no	—
42	#11719912	—	4	2	no	—
43	C₂²×D₅₃	—	2	1	no	—
44	C₂²×D₅₃	—	4	1	no	—
45	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
48	C₂×Dic₅₃	—	2	1	no	—
49	C₂×Dic₅₃	—	4	1	no	—
50	D₂₁₂	—	2	1	no	—
51	D₂₁₂	—	2	1	no	—
52	D₂₁₂	—	4	1	no	—
53	D₂₁₂	—	4	1	no	—
54	D₂₁₂	—	4	1	no	—
55	D₂₁₂	—	4	1	no	—
56	D₂₁₂	—	8	1	no	—
57	C₄×D₅₃	—	4	1	no	—
58	C₄×D₅₃	—	4	1	no	—
59	Dic₁₀₆	—	4	1	no	—
60	Dic₁₀₆	—	4	1	no	—
61	C₈×C₅₃	—	2	1	no	—
62	C₈×C₅₃	—	4	1	no	—
63	C₈×C₅₃	—	4	1	no	—
64	C₈×C₅₃	D₈	1	2	yes	complemented
65	#88322	—	4	1	no	—
66	C₂²×C₅₃	#121	1	1	yes	characteristic
67	C₂×D₅₃	—	4	1	no	—
68	C₂×D₅₃	—	4	1	no	—
69	C₂×D₅₃	—	4	1	no	—
70	C₂×D₅₃	—	8	1	no	—
71	C₂×D₅₃	—	8	1	no	—
72	C₂×D₅₃	—	8	1	no	—
73	C₄×C₅₃	—	2	1	no	—
74	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
75	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
76	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₄×C₅₃	#137	1	1	yes	characteristic
78	Dic₅₃	—	4	1	no	—
79	Dic₅₃	—	8	1	no	—
80	#1685	—	53	1	no	maximal, Sylow
81	C₂×C₅₃	#459	1	1	yes	characteristic
82	C₂×C₅₃	#468	1	1	yes	characteristic
83	C₂×C₅₃	#471	1	1	yes	characteristic
84	D₅₃	—	8	1	no	—
85	D₅₃	—	16	1	no	—
86	#492	—	53	1	no	—
87	#474	—	53	1	no	—
88	#459	—	53	1	no	—
89	#436	—	53	1	no	—
90	#335	D₅₃	1	1	yes	characteristic, complemented
91	#332	—	53	1	no	—
92	#331	—	53	1	no	—
93	C₅₃	#1685	1	1	yes	characteristic, complemented, Sylow
94	C₂×SD₁₆	—	106	1	no	—
95	C₂×SD₁₆	—	106	1	no	—
96	C₂×D₈	—	106	1	no	—
97	C₂×M₁₆	—	106	1	no	—
98	C₄⋊Q₈	—	53	1	no	—
99	#127	—	53	1	no	—
100	C₄×D₄	—	53	1	no	—
101	#107	—	106	1	no	—
102	C₄⋊C₈	—	53	1	no	—
103	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
104	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
105	#102	—	106	1	no	—
106	#102	—	106	1	no	—
107	C₂²⋊C₈	—	106	1	no	—
108	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
109	C₂×Q₈	—	106	1	no	—
110	C₂×D₄	—	53	1	no	—
111	C₂×D₄	—	106	1	no	—
112	C₂×D₄	—	106	1	no	—
113	C₂×D₄	—	106	1	no	—
114	C₂²×C₄	—	106	1	no	—
115	SD₁₆	—	212	2	no	—
116	SD₁₆	—	212	2	no	—
117	D₈	—	212	2	no	—
118	M₁₆	—	212	2	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	106	1	no	—
122	C₂×C₈	D₂₁₂	1	1	yes	characteristic
123	C₂×C₈	D₂₁₂	1	1	yes	characteristic
124	C₄⋊C₄	—	53	1	no	—
125	C₄⋊C₄	—	106	1	no	—
126	C₄⋊C₄	—	106	1	no	—
127	C₂²⋊C₄	—	106	1	no	—
128	C₄²	C₂²×D₅₃	1	1	yes	characteristic
129	C₂³	—	106	1	no	—
130	C₂³	—	212	1	no	—
131	Q₈	—	212	1	no	—
132	Q₈	—	212	1	no	—
133	D₄	—	106	1	no	—
134	D₄	—	106	1	no	—
135	D₄	—	424	1	no	—
136	D₄	—	212	1	no	—
137	D₄	—	212	1	no	—
138	D₄	—	212	1	no	—
139	D₄	—	212	1	no	—
140	C₂×C₄	—	106	1	no	—
141	C₂×C₄	—	212	1	no	—
142	C₂×C₄	—	212	1	no	—
143	C₂×C₄	—	212	1	no	—
144	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
145	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
146	C₂×C₄	#11719949	1	1	yes	characteristic
147	C₈	—	2	1	no	—
148	C₈	—	4	1	no	—
149	C₈	—	4	1	no	—
150	C₈	—	212	1	no	—
151	C₈	D₄₂₄	1	2	yes	complemented
152	C₂²	—	424	1	no	—
153	C₂²	—	424	1	no	—
154	C₂²	—	424	1	no	—
155	C₂²	—	212	1	no	—
156	C₂²	—	212	1	no	—
157	C₂²	—	212	1	no	—
158	C₂²	#13133192	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	424	1	no	—
161	C₄	—	212	1	no	—
162	C₄	C₂×D₄₂₄	1	1	yes	characteristic
163	C₄	#13133241	1	1	yes	characteristic
164	C₄	#13133245	1	1	yes	characteristic
165	C₄	#13133249	1	1	yes	characteristic
166	C₂	—	848	1	no	—
167	C₂	—	424	1	no	—
168	C₂	#16810819	1	1	yes	characteristic
169	C₂	#16810870	1	1	yes	characteristic
170	C₂	#16810887	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810996

C₂

yes

maximal, characteristic, complemented

#16810904

C₂

yes

maximal, characteristic, complemented

#16810819

C₂

yes

maximal, characteristic

#16810702

C₂

yes

maximal, characteristic, complemented

#16810368

C₂

yes

maximal, characteristic, complemented

#16810366

C₂

yes

maximal, characteristic, complemented

C₅₃×#335

C₂

yes

maximal, characteristic, complemented

C₂×#11719913

—

C₂×#11719913

—

C₂×D₄₂₄

—

C₂×#11719912

—

#13133217

C₂²

yes

characteristic

#13133214

C₂²

yes

characteristic

C₄×D₂₁₂

C₂²

yes

characteristic

#13133151

—

C₄⋊#88322

C₂²

yes

—

—

—

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

D₄

yes

characteristic

C₂×C₄×D₅₃

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

#11719921

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

characteristic

C₂×#88322

—

D₄₂₄

—

#11719913

—

#11719913

—

#11719912

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

D₈

yes

complemented

#88322

—

C₂²×C₅₃

#121

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×D₈

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

D₅₃

—

—

—

—

—

D₅₃

yes

characteristic, complemented

#332

—

#331

—

C₅₃

#1685

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

106

—

C₂×SD₁₆

—

106

—

C₂×D₈

—

106

—

C₂×M₁₆

—

106

—

C₄⋊Q₈

—

#127

—

100

C₄×D₄

—

101

#107

—

106

—

102

C₄⋊C₈

—

103

C₄⋊C₈

C₂×D₅₃

yes

characteristic

104

C₄⋊C₈

C₂×D₅₃

yes

characteristic

105

#102

—

106

—

106

#102

—

106

—

107

C₂²⋊C₈

—

106

—

108

C₄×C₈

C₂×D₅₃

yes

characteristic

109

C₂×Q₈

—

106

—

110

C₂×D₄

—

111

C₂×D₄

—

106

—

112

C₂×D₄

—

106

—

113

C₂×D₄

—

106

—

114

C₂²×C₄

—

106

—

115

SD₁₆

—

212

—

116

SD₁₆

—

212

—

117

D₈

—

212

—

118

M₁₆

—

212

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

106

—

122

C₂×C₈

D₂₁₂

yes

characteristic

123

C₂×C₈

D₂₁₂

yes

characteristic

124

C₄⋊C₄

—

125

C₄⋊C₄

—

106

—

126

C₄⋊C₄

—

106

—

127

C₂²⋊C₄

—

106

—

128

C₄²

C₂²×D₅₃

yes

characteristic

129

C₂³

—

106

—

130

C₂³

—

212

—

131

Q₈

—

212

—

132

Q₈

—

212

—

133

D₄

—

106

—

134

D₄

—

106

—

135

D₄

—

424

—

136

D₄

—

212

—

137

D₄

—

212

—

138

D₄

—

212

—

139

D₄

—

212

—

140

C₂×C₄

—

106

—

141

C₂×C₄

—

212

—

142

C₂×C₄

—

212

—

143

C₂×C₄

—

212

—

144

C₂×C₄

C₂×D₂₁₂

yes

characteristic

145

C₂×C₄

D₄×D₅₃

yes

characteristic

146

C₂×C₄

#11719949

yes

characteristic

147

C₈

—

148

C₈

—

149

C₈

—

150

C₈

—

212

—

151

C₈

D₄₂₄

yes

complemented

152

C₂²

—

424

—

153

C₂²

—

424

—

154

C₂²

—

424

—

155

C₂²

—

212

—

156

C₂²

—

212

—

157

C₂²

—

212

—

158

C₂²

#13133192

yes

characteristic

159

C₄

—

160

C₄

—

424

—

161

C₄

—

212

—

162

C₄

C₂×D₄₂₄

yes

characteristic

163

C₄

#13133241

yes

characteristic

164

C₄

#13133245

yes

characteristic

165

C₄

#13133249

yes

characteristic

166

C₂

—

848

—

167

C₂

—

424

—

168

C₂

#16810819

yes

characteristic

169

C₂

#16810870

yes

characteristic

170

C₂

#16810887

yes

characteristic

	subgroup	G/H
center	C₂²	#13133192
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#335	C₂
socle	C₂²×C₅₃	#121
hypercenter	#335	D₅₃
hypocenter	C₅₃	#1685
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #335

subgroup

G/H

center

C₂²

#13133192

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#335

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #335

N	H ≅ G/N	type
1. #16810996	C₂	semidirect
2. #16810904	C₂	semidirect
4. #16810702	C₂	semidirect
5. #16810368	C₂	semidirect
6. #16810366	C₂	semidirect
7. C₅₃×#335	C₂	semidirect
64. C₈×C₅₃	D₈	semidirect
90. #335	D₅₃	semidirect
93. C₅₃	#1685	semidirect
151. C₈	D₄₂₄	semidirect

H ≅ G/N

type

1. #16810996

C₂

semidirect

2. #16810904

C₂

semidirect

4. #16810702

C₂

semidirect

5. #16810368

C₂

semidirect

6. #16810366

C₂

semidirect

7. C₅₃×#335

C₂

semidirect

64. C₈×C₅₃

D₈

semidirect

90. #335

D₅₃

semidirect

93. C₅₃

#1685

semidirect

151. C₈

D₄₂₄

semidirect

Subgroups — #20274181

Conjugacy classes of subgroups

Complements