Group #20273928 — subgroups

Subgroups — #20273928

Order 6784.

208 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16811330	C₂	1	1	yes	maximal, characteristic
2	#16810975	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810830	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810371	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810353	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄⋊#88322	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#474	C₂	1	1	yes	maximal, characteristic
8	C₂×#11719925	—	2	1	no	—
9	#13133217	C₂²	1	1	yes	characteristic
10	#13133208	—	2	1	no	—
11	#13133196	—	2	1	no	—
12	#13133194	—	2	2	no	—
13	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
14	D₄×Dic₅₃	—	2	1	no	—
15	C₂×C₄⋊Dic₅₃	—	2	1	no	—
16	C₄⋊#88322	C₂²	1	1	yes	characteristic
17	#13133141	—	2	1	no	—
18	#13133140	—	2	2	no	—
19	C₄×#88322	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
22	#107×C₅₃	—	2	1	no	—
23	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
24	#102×C₅₃	—	2	1	no	—
25	C₂×Q₈×C₅₃	—	2	1	no	—
26	C₂×D₄×C₅₃	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₅₃	—	2	1	no	—
28	C₂²×Dic₅₃	—	2	2	no	—
29	C₂×Dic₁₀₆	D₄	1	1	yes	characteristic
30	C₂×Dic₁₀₆	D₄	1	2	yes	—
31	C₂×C₈×C₅₃	—	2	1	no	—
32	C₄⋊C₄×C₅₃	—	2	1	no	—
33	C₄⋊C₄×C₅₃	—	2	1	no	—
34	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
35	C₂²⋊C₄×C₅₃	—	2	1	no	—
36	C₄²×C₅₃	C₂³	1	1	yes	characteristic
37	C₂²⋊Dic₅₃	—	2	2	no	—
38	#11719925	—	4	2	no	—
39	C₄⋊Dic₅₃	—	2	2	no	—
40	C₄⋊Dic₅₃	—	2	1	no	—
41	C₄⋊Dic₅₃	—	2	1	no	—
42	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
43	#11719919	—	2	2	no	—
44	#11719919	—	2	1	no	—
45	C₄×Dic₅₃	—	2	1	no	—
46	C₂×#88322	—	2	1	no	—
47	C₂×#88322	—	2	1	no	—
48	C₂³×C₅₃	—	2	1	no	—
49	Q₈×C₅₃	—	4	1	no	—
50	Q₈×C₅₃	—	4	1	no	—
51	D₄×C₅₃	—	2	1	no	—
52	D₄×C₅₃	Q₁₆	1	2	yes	complemented
53	C₂×C₄×C₅₃	—	2	1	no	—
54	C₂×C₄×C₅₃	—	4	1	no	—
55	C₂×C₄×C₅₃	—	4	1	no	—
56	C₂×C₄×C₅₃	—	4	1	no	—
57	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂×Dic₅₃	—	2	2	no	—
61	C₂×Dic₅₃	—	2	1	no	—
62	C₂×Dic₅₃	—	2	1	no	—
63	C₂×Dic₅₃	—	8	2	no	—
64	Dic₁₀₆	—	2	1	no	—
65	Dic₁₀₆	—	2	1	no	—
66	Dic₁₀₆	—	4	2	no	—
67	C₈×C₅₃	—	4	1	no	—
68	#88322	—	4	1	no	—
69	#88322	—	4	1	no	—
70	C₂²×C₅₃	—	2	2	no	—
71	C₂²×C₅₃	—	4	1	no	—
72	C₂²×C₅₃	—	4	1	no	—
73	C₂²×C₅₃	#120	1	1	yes	characteristic
74	C₄×C₅₃	—	2	1	no	—
75	C₄×C₅₃	—	4	1	no	—
76	C₄×C₅₃	—	8	1	no	—
77	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
78	C₄×C₅₃	#135	1	1	yes	characteristic
79	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
80	C₄×C₅₃	#137	1	1	yes	characteristic
81	Dic₅₃	—	4	1	no	—
82	Dic₅₃	—	4	2	no	—
83	Dic₅₃	—	4	1	no	—
84	#1644	—	53	1	no	maximal, Sylow
85	C₂×C₅₃	—	4	2	no	—
86	C₂×C₅₃	#449	1	1	yes	characteristic
87	C₂×C₅₃	#451	1	1	yes	characteristic
88	C₂×C₅₃	#455	1	1	yes	characteristic
89	#554	—	53	1	no	—
90	#487	—	53	1	no	—
91	#474	D₅₃	1	1	yes	characteristic
92	#462	—	53	1	no	—
93	#333	—	53	1	no	—
94	Q₈⋊C₈	—	53	1	no	—
95	D₄⋊C₈	—	53	1	no	—
96	C₅₃	#1644	1	1	yes	characteristic, complemented, Sylow
97	C₂×Q₁₆	—	106	1	no	—
98	C₄⋊Q₈	—	53	1	no	—
99	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
100	#125	—	106	1	no	—
101	C₂²⋊Q₈	—	106	2	no	—
102	C₂²⋊Q₈	—	106	1	no	—
103	C₄×Q₈	—	53	1	no	—
104	C₄×D₄	—	106	1	no	—
105	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
106	C₂×C₄⋊C₄	—	106	1	no	—
107	#107	—	2	1	no	—
108	C₄⋊C₈	—	53	1	no	—
109	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
110	#103	—	106	2	no	—
111	#103	—	106	1	no	—
112	#102	—	2	1	no	—
113	C₄×C₈	—	53	1	no	—
114	C₂×Q₈	—	2	1	no	—
115	C₂×Q₈	—	53	1	no	—
116	C₂×Q₈	—	53	2	no	—
117	C₂×D₄	C₂²⋊D₅₃	1	1	yes	characteristic
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	—	106	2	no	—
120	Q₁₆	—	212	2	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	106	1	no	—
123	C₂×C₈	—	106	1	no	—
124	C₄⋊C₄	—	2	1	no	—
125	C₄⋊C₄	—	2	1	no	—
126	C₄⋊C₄	—	53	1	no	—
127	C₄⋊C₄	—	106	2	no	—
128	C₄⋊C₄	—	106	2	no	—
129	C₄⋊C₄	—	106	1	no	—
130	C₄⋊C₄	—	106	1	no	—
131	C₄⋊C₄	—	106	1	no	—
132	C₄⋊C₄	C₂²⋊D₅₃	1	1	yes	characteristic
133	C₂²⋊C₄	—	2	1	no	—
134	C₂²⋊C₄	—	106	2	no	—
135	C₄²	—	106	1	no	—
136	C₄²	C₂²×D₅₃	1	1	yes	characteristic
137	C₂³	—	2	1	no	—
138	Q₈	—	4	1	no	—
139	Q₈	—	4	1	no	—
140	Q₈	—	106	1	no	—
141	Q₈	—	106	1	no	—
142	Q₈	—	212	2	no	—
143	D₄	—	2	1	no	—
144	D₄	#11719925	1	2	yes	complemented
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	106	1	no	—
150	C₂×C₄	—	106	1	no	—
151	C₂×C₄	—	106	2	no	—
152	C₂×C₄	—	424	2	no	—
153	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
154	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
155	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
156	C₈	—	4	1	no	—
157	C₈	—	212	1	no	—
158	C₈	—	212	1	no	—
159	C₂²	—	2	2	no	—
160	C₂²	—	4	1	no	—
161	C₂²	—	4	1	no	—
162	C₂²	#13133185	1	1	yes	characteristic
163	C₄	—	2	1	no	—
164	C₄	—	4	1	no	—
165	C₄	—	8	1	no	—
166	C₄	—	212	1	no	—
167	C₄	—	212	1	no	—
168	C₄	—	212	2	no	—
169	C₄	C₂×#11719925	1	1	yes	characteristic
170	C₄	#13133236	1	1	yes	characteristic
171	C₄	#13133241	1	1	yes	characteristic
172	C₄	#13133247	1	1	yes	characteristic
173	C₂	—	4	2	no	—
174	C₂	#16810765	1	1	yes	characteristic
175	C₂	#16810778	1	1	yes	characteristic
176	C₂	#16810799	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811330

C₂

yes

maximal, characteristic

#16810975

C₂

yes

maximal, characteristic, complemented

#16810830

C₂

yes

maximal, characteristic, complemented

#16810371

C₂

yes

maximal, characteristic, complemented

#16810353

C₂

yes

maximal, characteristic, complemented

D₄⋊#88322

C₂

yes

maximal, characteristic

C₅₃×#474

C₂

yes

maximal, characteristic

C₂×#11719925

—

#13133217

C₂²

yes

—

—

—

C₄×Dic₁₀₆

C₂²

yes

characteristic

D₄×Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

C₄⋊#88322

C₂²

yes

—

—

C₂²

yes

characteristic

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

C₄×D₄×C₅₃

C₂²

yes

characteristic

#107×C₅₃

—

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

#102×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

D₄

yes

characteristic

C₂×Dic₁₀₆

D₄

yes

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

C₂²⋊Dic₅₃

—

#11719925

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

characteristic

#11719919

—

#11719919

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

Q₁₆

yes

complemented

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#120

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×Q₁₆

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

#1644

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

D₅₃

yes

characteristic

#462

—

#333

—

Q₈⋊C₈

—

D₄⋊C₈

—

C₅₃

#1644

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

106

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

100

#125

—

106

—

101

C₂²⋊Q₈

—

106

—

102

C₂²⋊Q₈

—

106

—

103

C₄×Q₈

—

104

C₄×D₄

—

106

—

105

C₄×D₄

C₂×D₅₃

yes

characteristic

106

C₂×C₄⋊C₄

—

106

—

107

#107

—

108

C₄⋊C₈

—

109

C₄⋊C₈

C₂×D₅₃

yes

characteristic

110

#103

—

106

—

111

#103

—

106

—

112

#102

—

113

C₄×C₈

—

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂×Q₈

—

117

C₂×D₄

C₂²⋊D₅₃

yes

characteristic

118

C₂²×C₄

—

119

C₂²×C₄

—

106

—

120

Q₁₆

—

212

—

121

C₂×C₈

—

122

C₂×C₈

—

106

—

123

C₂×C₈

—

106

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

106

—

128

C₄⋊C₄

—

106

—

129

C₄⋊C₄

—

106

—

130

C₄⋊C₄

—

106

—

131

C₄⋊C₄

—

106

—

132

C₄⋊C₄

C₂²⋊D₅₃

yes

characteristic

133

C₂²⋊C₄

—

134

C₂²⋊C₄

—

106

—

135

C₄²

—

106

—

136

C₄²

C₂²×D₅₃

yes

characteristic

137

C₂³

—

138

Q₈

—

139

Q₈

—

140

Q₈

—

106

—

141

Q₈

—

106

—

142

Q₈

—

212

—

143

D₄

—

144

D₄

#11719925

yes

complemented

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

106

—

150

C₂×C₄

—

106

—

151

C₂×C₄

—

106

—

152

C₂×C₄

—

424

—

153

C₂×C₄

D₄×D₅₃

yes

characteristic

154

C₂×C₄

D₄×D₅₃

yes

characteristic

155

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

156

C₈

—

157

C₈

—

212

—

158

C₈

—

212

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

162

C₂²

#13133185

yes

characteristic

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

212

—

167

C₄

—

212

—

168

C₄

—

212

—

169

C₄

C₂×#11719925

yes

characteristic

170

C₄

#13133236

yes

characteristic

171

C₄

#13133241

yes

characteristic

172

C₄

#13133247

yes

characteristic

173

C₂

—

174

C₂

#16810765

yes

characteristic

175

C₂

#16810778

yes

characteristic

176

C₂

#16810799

yes

characteristic

	subgroup	G/H
center	C₂²	#13133185
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#474	C₂
socle	C₂²×C₅₃	#120
hypercenter	#474	D₅₃
hypocenter	C₅₃	#1644
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #474

subgroup

G/H

center

C₂²

#13133185

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#474

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #474

N	H ≅ G/N	type
2. #16810975	C₂	semidirect
3. #16810830	C₂	semidirect
4. #16810371	C₂	semidirect
5. #16810353	C₂	semidirect
52. D₄×C₅₃	Q₁₆	semidirect
96. C₅₃	#1644	semidirect
144. D₄	#11719925	semidirect

H ≅ G/N

type

2. #16810975

C₂

semidirect

3. #16810830

C₂

semidirect

4. #16810371

C₂

semidirect

5. #16810353

C₂

semidirect

52. D₄×C₅₃

Q₁₆

semidirect

96. C₅₃

#1644

semidirect

144. D₄

#11719925

semidirect

Subgroups — #20273928

Conjugacy classes of subgroups

Complements