Group #20276077 — subgroups

Subgroups — #20276077

Order 6784.

208 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 69 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	#16810665	C₂	1	1	yes	maximal, characteristic
2	#16810649	C₂	1	1	yes	maximal, characteristic
3	#16810633	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133142	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133135	C₂	1	1	yes	maximal, characteristic
6	#16810391	C₂	1	1	yes	maximal, characteristic
7	C₂×#108×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₀₆	C₂²	1	1	yes	characteristic
9	C₂×#11719917	C₂²	1	1	yes	characteristic
10	#13133176	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
12	#13133151	—	2	1	no	—
13	#13133149	—	2	1	no	—
14	#13133146	—	2	2	no	—
15	#13133142	—	2	2	no	—
16	#13133142	—	2	1	no	—
17	#13133142	—	2	1	no	—
18	#13133135	—	2	2	no	—
19	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
21	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	#108×C₅₃	—	2	2	no	—
23	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
24	C₂²×Dic₅₃	—	2	1	no	—
25	C₂²×Dic₅₃	—	2	1	no	—
26	C₂×Dic₁₀₆	—	2	1	no	—
27	C₂×Dic₁₀₆	—	2	1	no	—
28	C₂×Dic₁₀₆	—	2	1	no	—
29	C₂×Dic₁₀₆	—	2	1	no	—
30	C₂×Dic₁₀₆	—	2	1	no	—
31	C₂×Dic₁₀₆	Q₈	1	2	yes	—
32	M₁₆×C₅₃	—	2	1	no	—
33	M₁₆×C₅₃	—	2	1	no	—
34	M₁₆×C₅₃	—	2	1	no	—
35	M₁₆×C₅₃	D₄	1	2	yes	—
36	C₂×C₈×C₅₃	—	2	1	no	—
37	C₂×C₈×C₅₃	—	2	1	no	—
38	C₂×C₈×C₅₃	—	2	1	no	—
39	C₂×C₈×C₅₃	—	2	1	no	—
40	C₂×C₈×C₅₃	D₄	1	2	yes	—
41	C₂²⋊Dic₅₃	—	2	1	no	—
42	C₄⋊Dic₅₃	—	2	1	no	—
43	C₄⋊Dic₅₃	—	2	1	no	—
44	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
45	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
46	C₄×Dic₅₃	—	2	1	no	—
47	#11719917	—	2	1	no	—
48	#11719917	—	2	1	no	—
49	C₂×#88322	—	2	1	no	—
50	C₂³×C₅₃	#41	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
56	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
57	C₂×Dic₅₃	—	2	1	no	—
58	C₂×Dic₅₃	—	2	1	no	—
59	C₂×Dic₅₃	—	2	2	no	—
60	C₂×Dic₅₃	—	4	1	no	—
61	C₂×Dic₅₃	—	4	1	no	—
62	C₂×Dic₅₃	—	4	1	no	—
63	C₂×Dic₅₃	—	8	1	no	—
64	Dic₁₀₆	—	2	2	no	—
65	Dic₁₀₆	—	2	2	no	—
66	Dic₁₀₆	—	4	1	no	—
67	Dic₁₀₆	—	4	1	no	—
68	C₈×C₅₃	—	2	2	no	—
69	C₈×C₅₃	—	2	2	no	—
70	C₈×C₅₃	—	4	1	no	—
71	#88322	—	4	1	no	—
72	C₂²×C₅₃	—	2	1	no	—
73	C₂²×C₅₃	—	2	1	no	—
74	C₂²×C₅₃	#121	1	1	yes	characteristic
75	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂²×C₅₃	#126	1	1	yes	characteristic
77	C₄×C₅₃	#121	1	1	yes	characteristic
78	C₄×C₅₃	#121	1	1	yes	characteristic
79	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
80	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
81	Dic₅₃	—	4	2	no	—
82	Dic₅₃	—	4	1	no	—
83	Dic₅₃	—	4	1	no	—
84	Dic₅₃	—	8	1	no	—
85	#2064	—	53	1	no	maximal, Sylow
86	C₂×C₅₃	—	2	1	no	—
87	C₂×C₅₃	—	2	1	no	—
88	C₂×C₅₃	#396	1	1	yes	characteristic
89	C₂×C₅₃	#471	1	1	yes	characteristic
90	C₂×C₅₃	#473	1	1	yes	characteristic
91	C₂×#108	D₅₃	1	1	yes	characteristic
92	#428	—	53	1	no	—
93	#424	—	53	1	no	—
94	#419	—	53	1	no	—
95	C₂×#103	—	53	1	no	—
96	C₂×#101	—	53	1	no	—
97	#343	—	53	1	no	—
98	C₅₃	#2064	1	1	yes	characteristic, complemented, Sylow
99	C₂²×Q₈	—	53	1	no	—
100	C₂×M₁₆	—	53	1	no	—
101	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
102	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
103	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
104	#117	—	53	1	no	—
105	C₂×C₄⋊C₄	—	53	1	no	—
106	#108	—	2	2	no	—
107	#107	—	106	1	no	—
108	#106	—	106	1	no	—
109	C₄⋊C₈	—	106	2	no	—
110	#103	—	106	1	no	—
111	#103	—	106	1	no	—
112	#103	—	106	2	no	—
113	#101	—	106	2	no	—
114	C₂×Q₈	—	53	2	no	—
115	C₂×Q₈	—	106	1	no	—
116	C₂×Q₈	—	106	1	no	—
117	C₂×Q₈	—	106	1	no	—
118	C₂×Q₈	—	106	1	no	—
119	C₂×Q₈	—	106	1	no	—
120	C₂²×C₄	—	106	1	no	—
121	C₂²×C₄	—	106	1	no	—
122	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
123	M₁₆	—	2	1	no	—
124	M₁₆	—	2	1	no	—
125	M₁₆	—	2	1	no	—
126	M₁₆	—	106	1	no	—
127	M₁₆	—	106	1	no	—
128	M₁₆	C₂²⋊D₅₃	1	2	yes	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	106	1	no	—
134	C₂×C₈	D₂₁₂	1	2	yes	—
135	C₄⋊C₄	—	53	1	no	—
136	C₄⋊C₄	—	53	1	no	—
137	C₄⋊C₄	—	106	1	no	—
138	C₄⋊C₄	—	106	1	no	—
139	C₂²⋊C₄	—	106	1	no	—
140	C₄²	—	106	1	no	—
141	C₂³	#11719949	1	1	yes	characteristic
142	Q₈	—	106	2	no	—
143	Q₈	—	106	2	no	—
144	Q₈	—	212	1	no	—
145	Q₈	—	212	1	no	—
146	C₂×C₄	—	106	1	no	—
147	C₂×C₄	—	106	1	no	—
148	C₂×C₄	—	106	2	no	—
149	C₂×C₄	—	424	1	no	—
150	C₂×C₄	—	212	1	no	—
151	C₂×C₄	—	212	1	no	—
152	C₂×C₄	—	212	1	no	—
153	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
154	C₂×C₄	#11719945	1	1	yes	characteristic
155	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
156	C₂×C₄	#11719947	1	1	yes	characteristic
157	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
158	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
159	C₈	—	2	2	no	—
160	C₈	—	2	2	no	—
161	C₈	—	4	1	no	—
162	C₈	—	212	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	#13133192	1	1	yes	characteristic
166	C₂²	#13133193	1	1	yes	characteristic
167	C₂²	#13133212	1	1	yes	characteristic
168	C₄	—	424	1	no	—
169	C₄	—	212	2	no	—
170	C₄	—	212	1	no	—
171	C₄	—	212	1	no	—
172	C₄	#13133189	1	1	yes	characteristic
173	C₄	#13133191	1	1	yes	characteristic
174	C₄	#13133195	1	1	yes	characteristic
175	C₄	#13133197	1	1	yes	characteristic
176	C₂	—	2	1	no	—
177	C₂	—	2	1	no	—
178	C₂	#16810545	1	1	yes	characteristic
179	C₂	#16810887	1	1	yes	characteristic
180	C₂	#16810899	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810665

C₂

yes

maximal, characteristic

#16810649

C₂

yes

maximal, characteristic

#16810633

C₂

yes

maximal, characteristic

C₂×#13133142

C₂

yes

maximal, characteristic

C₂×#13133135

C₂

yes

maximal, characteristic

#16810391

C₂

yes

maximal, characteristic

C₂×#108×C₅₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₂²

yes

characteristic

C₂×#11719917

C₂²

yes

characteristic

#13133176

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

—

—

—

—

—

—

—

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

#108×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

Q₈

yes

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

characteristic

C₄⋊Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

—

#11719917

—

#11719917

—

C₂×#88322

—

C₂³×C₅₃

#41

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#121

yes

characteristic

C₂²×C₅₃

C₂²⋊Q₈

yes

C₂²×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

#2064

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

yes

—

—

—

—

—

—

C₅₃

#2064

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

100

C₂×M₁₆

—

101

C₂×M₁₆

C₂×D₅₃

yes

characteristic

102

C₂×M₁₆

C₂×D₅₃

yes

characteristic

103

C₂²×C₈

C₂×D₅₃

yes

characteristic

104

#117

—

105

C₂×C₄⋊C₄

—

106

#108

—

107

#107

—

106

—

108

#106

—

106

—

109

C₄⋊C₈

—

106

—

110

#103

—

106

—

111

#103

—

106

—

112

#103

—

106

—

113

#101

—

106

—

114

C₂×Q₈

—

115

C₂×Q₈

—

106

—

116

C₂×Q₈

—

106

—

117

C₂×Q₈

—

106

—

118

C₂×Q₈

—

106

—

119

C₂×Q₈

—

106

—

120

C₂²×C₄

—

106

—

121

C₂²×C₄

—

106

—

122

C₂²×C₄

C₂²×D₅₃

yes

characteristic

123

M₁₆

—

124

M₁₆

—

125

M₁₆

—

126

M₁₆

—

106

—

127

M₁₆

—

106

—

128

M₁₆

C₂²⋊D₅₃

yes

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

106

—

134

C₂×C₈

D₂₁₂

yes

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

106

—

138

C₄⋊C₄

—

106

—

139

C₂²⋊C₄

—

106

—

140

C₄²

—

106

—

141

C₂³

#11719949

yes

characteristic

142

Q₈

—

106

—

143

Q₈

—

106

—

144

Q₈

—

212

—

145

Q₈

—

212

—

146

C₂×C₄

—

106

—

147

C₂×C₄

—

106

—

148

C₂×C₄

—

106

—

149

C₂×C₄

—

424

—

150

C₂×C₄

—

212

—

151

C₂×C₄

—

212

—

152

C₂×C₄

—

212

—

153

C₂×C₄

C₂×D₂₁₂

yes

characteristic

154

C₂×C₄

#11719945

yes

characteristic

155

C₂×C₄

D₄×D₅₃

yes

characteristic

156

C₂×C₄

#11719947

yes

characteristic

157

C₂×C₄

Q₈×D₅₃

yes

characteristic

158

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

159

C₈

—

160

C₈

—

161

C₈

—

162

C₈

—

212

—

163

C₂²

—

164

C₂²

—

165

C₂²

#13133192

yes

characteristic

166

C₂²

#13133193

yes

characteristic

167

C₂²

#13133212

yes

characteristic

168

C₄

—

424

—

169

C₄

—

212

—

170

C₄

—

212

—

171

C₄

—

212

—

172

C₄

#13133189

yes

characteristic

173

C₄

#13133191

yes

characteristic

174

C₄

#13133195

yes

characteristic

175

C₄

#13133197

yes

characteristic

176

C₂

—

177

C₂

—

178

C₂

#16810545

yes

characteristic

179

C₂

#16810887

yes

characteristic

180

C₂

#16810899

yes

characteristic

	subgroup	G/H
center	C₂²	#13133192
commutator	C₂²×C₄×C₅₃	C₂³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×#108×C₅₃	C₂
socle	C₂²×C₅₃	#121
hypercenter	C₂×#108	D₅₃
hypocenter	C₅₃	#2064
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂²

#13133192

commutator

C₂²×C₄×C₅₃

C₂³

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×#108×C₅₃

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
98. C₅₃	#2064	semidirect

H ≅ G/N

type

98. C₅₃

#2064

semidirect

Subgroups — #20276077

Conjugacy classes of subgroups

Complements