Group #20273931 — subgroups

Subgroups — #20273931

Order 6784.

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

normal subgroups only

This group has 208 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	#16811333	C₂	1	1	yes	maximal, characteristic
2	#16810974	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810904	C₂	1	1	yes	maximal, characteristic
4	#16810833	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810371	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810352	C₂	1	1	yes	maximal, characteristic
7	C₅₃×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11719925	—	2	1	no	—
9	#13133217	C₂²	1	1	yes	characteristic
10	#13133217	C₂²	1	1	yes	characteristic
11	#13133209	—	2	1	no	—
12	#13133195	—	2	2	no	—
13	#13133193	—	2	1	no	—
14	C₄×D₂₁₂	—	2	1	no	—
15	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×D₅₃	—	2	1	no	—
17	#13133151	—	2	1	no	—
18	C₄⋊#88322	C₂²	1	1	yes	characteristic
19	#13133142	—	2	2	no	—
20	#13133140	—	2	1	no	—
21	#13133139	—	2	1	no	—
22	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
26	C₂×D₂₁₂	D₄	1	1	yes	characteristic
27	C₂×C₄×D₅₃	—	2	1	no	—
28	C₂×C₄×D₅₃	—	2	2	no	—
29	C₂×Dic₁₀₆	—	2	1	no	—
30	C₂×Dic₁₀₆	D₄	1	2	yes	—
31	C₂×C₈×C₅₃	—	2	1	no	—
32	C₂×C₈×C₅₃	—	2	1	no	—
33	C₄⋊C₄×C₅₃	—	2	1	no	—
34	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
35	C₄²×C₅₃	—	2	1	no	—
36	C₄²×C₅₃	C₂³	1	1	yes	characteristic
37	#11719925	—	4	2	no	—
38	#11719921	—	2	2	no	—
39	#11719921	—	2	1	no	—
40	C₄⋊Dic₅₃	—	2	1	no	—
41	C₄⋊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	#11719919	—	2	2	no	—
46	#11719919	—	2	2	no	—
47	C₂×#88322	—	2	1	no	—
48	C₂²×D₅₃	—	2	1	no	—
49	Q₈×C₅₃	—	2	1	no	—
50	Q₈×C₅₃	—	2	1	no	—
51	C₂×C₄×C₅₃	—	2	1	no	—
52	C₂×C₄×C₅₃	—	2	1	no	—
53	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂×Dic₅₃	—	2	2	no	—
57	C₂×Dic₅₃	—	2	1	no	—
58	C₂×Dic₅₃	—	4	1	no	—
59	D₂₁₂	—	2	1	no	—
60	D₂₁₂	Q₁₆	1	2	yes	complemented
61	C₄×D₅₃	—	4	1	no	—
62	C₄×D₅₃	—	4	1	no	—
63	C₄×D₅₃	—	8	2	no	—
64	Dic₁₀₆	—	4	2	no	—
65	Dic₁₀₆	—	4	1	no	—
66	Dic₁₀₆	—	4	1	no	—
67	C₈×C₅₃	—	4	1	no	—
68	C₈×C₅₃	—	4	1	no	—
69	#88322	—	4	1	no	—
70	C₂²×C₅₃	#120	1	1	yes	characteristic
71	C₂×D₅₃	—	2	2	no	—
72	C₂×D₅₃	—	4	1	no	—
73	C₂×D₅₃	—	4	1	no	—
74	C₄×C₅₃	—	2	1	no	—
75	C₄×C₅₃	—	4	1	no	—
76	C₄×C₅₃	—	4	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	—	8	1	no	—
84	#1644	—	53	1	no	maximal, Sylow
85	C₂×C₅₃	#449	1	1	yes	characteristic
86	C₂×C₅₃	#451	1	1	yes	characteristic
87	C₂×C₅₃	#455	1	1	yes	characteristic
88	D₅₃	—	4	2	no	—
89	#554	—	53	1	no	—
90	#487	—	53	1	no	—
91	#474	—	53	1	no	—
92	#462	—	53	1	no	—
93	#333	—	53	1	no	—
94	Q₈⋊C₈	D₅₃	1	1	yes	characteristic, complemented
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	—	53	1	no	—
100	#125	—	106	1	no	—
101	C₂²⋊Q₈	—	106	2	no	—
102	C₂²⋊Q₈	—	106	1	no	—
103	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
104	C₄×D₄	—	53	1	no	—
105	C₄×D₄	—	106	1	no	—
106	C₂×C₄⋊C₄	—	106	1	no	—
107	#107	—	106	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	—	106	1	no	—
113	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
114	C₂×Q₈	—	53	2	no	—
115	C₂×Q₈	—	106	1	no	—
116	C₂×Q₈	D₂₁₂	1	1	yes	characteristic
117	C₂×D₄	—	53	1	no	—
118	C₂²×C₄	—	106	2	no	—
119	C₂²×C₄	—	106	1	no	—
120	Q₁₆	—	212	2	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	106	1	no	—
124	C₄⋊C₄	—	2	1	no	—
125	C₄⋊C₄	—	53	1	no	—
126	C₄⋊C₄	—	106	2	no	—
127	C₄⋊C₄	—	106	2	no	—
128	C₄⋊C₄	—	106	1	no	—
129	C₄⋊C₄	—	106	1	no	—
130	C₄⋊C₄	—	106	1	no	—
131	C₄⋊C₄	—	106	1	no	—
132	C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
133	C₂²⋊C₄	—	106	2	no	—
134	C₂²⋊C₄	—	106	1	no	—
135	C₄²	—	2	1	no	—
136	C₄²	C₂²×D₅₃	1	1	yes	characteristic
137	C₂³	—	106	1	no	—
138	Q₈	—	2	1	no	—
139	Q₈	—	2	1	no	—
140	Q₈	—	212	2	no	—
141	Q₈	—	212	1	no	—
142	Q₈	—	212	1	no	—
143	D₄	—	53	2	no	—
144	D₄	—	106	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	106	1	no	—
148	C₂×C₄	—	106	2	no	—
149	C₂×C₄	—	424	2	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	D₄×D₅₃	1	1	yes	characteristic
155	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
156	C₈	—	4	1	no	—
157	C₈	—	4	1	no	—
158	C₈	—	212	1	no	—
159	C₂²	—	106	2	no	—
160	C₂²	—	212	1	no	—
161	C₂²	—	212	1	no	—
162	C₂²	#13133187	1	1	yes	characteristic
163	C₄	—	2	1	no	—
164	C₄	—	4	1	no	—
165	C₄	—	4	1	no	—
166	C₄	—	424	1	no	—
167	C₄	—	212	1	no	—
168	C₄	—	212	2	no	—
169	C₄	Q₁₆×D₅₃	1	1	yes	characteristic
170	C₄	#13133239	1	1	yes	characteristic
171	C₄	#13133245	1	1	yes	characteristic
172	C₄	#13133246	1	1	yes	characteristic
173	C₂	—	212	2	no	—
174	C₂	#16810770	1	1	yes	characteristic
175	C₂	#16810780	1	1	yes	characteristic
176	C₂	#16810802	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811333

C₂

yes

maximal, characteristic

#16810974

C₂

yes

maximal, characteristic, complemented

#16810904

C₂

yes

maximal, characteristic

#16810833

C₂

yes

maximal, characteristic, complemented

#16810371

C₂

yes

maximal, characteristic, complemented

#16810352

C₂

yes

maximal, characteristic

C₅₃×Q₈⋊C₈

C₂

yes

maximal, characteristic, complemented

C₂×#11719925

—

#13133217

C₂²

yes

characteristic

#13133217

C₂²

yes

—

—

—

C₄×D₂₁₂

—

C₄×D₂₁₂

C₂²

yes

characteristic

C₄⋊C₄×D₅₃

—

#13133151

—

C₄⋊#88322

C₂²

yes

—

—

—

C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

D₄

yes

characteristic

C₂×D₂₁₂

D₄

yes

characteristic

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

D₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

—

C₄²×C₅₃

C₂³

yes

—

—

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

—

—

—

C₂²×D₅₃

—

Q₈×C₅₃

—

Q₈×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

—

D₂₁₂

—

D₂₁₂

Q₁₆

yes

complemented

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

#120

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×Q₁₆

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

—

—

—

—

—

Q₈⋊C₈

D₅₃

yes

characteristic, complemented

D₄⋊C₈

—

C₅₃

#1644

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

106

—

C₄⋊Q₈

—

C₄⋊Q₈

—

100

#125

—

106

—

101

C₂²⋊Q₈

—

106

—

102

C₂²⋊Q₈

—

106

—

103

C₄×Q₈

C₂×D₅₃

yes

characteristic

104

C₄×D₄

—

105

C₄×D₄

—

106

—

106

C₂×C₄⋊C₄

—

106

—

107

#107

—

106

—

108

C₄⋊C₈

—

109

C₄⋊C₈

C₂×D₅₃

yes

characteristic

110

#103

—

106

—

111

#103

—

106

—

112

#102

—

106

—

113

C₄×C₈

C₂×D₅₃

yes

characteristic

114

C₂×Q₈

—

115

C₂×Q₈

—

106

—

116

C₂×Q₈

D₂₁₂

yes

characteristic

117

C₂×D₄

—

118

C₂²×C₄

—

106

—

119

C₂²×C₄

—

106

—

120

Q₁₆

—

212

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

106

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

106

—

127

C₄⋊C₄

—

106

—

128

C₄⋊C₄

—

106

—

129

C₄⋊C₄

—

106

—

130

C₄⋊C₄

—

106

—

131

C₄⋊C₄

—

106

—

132

C₄⋊C₄

D₂₁₂

yes

characteristic

133

C₂²⋊C₄

—

106

—

134

C₂²⋊C₄

—

106

—

135

C₄²

—

136

C₄²

C₂²×D₅₃

yes

characteristic

137

C₂³

—

106

—

138

Q₈

—

139

Q₈

—

140

Q₈

—

212

—

141

Q₈

—

212

—

142

Q₈

—

212

—

143

D₄

—

144

D₄

—

106

—

145

C₂×C₄

—

146

C₂×C₄

—

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

D₄×D₅₃

yes

characteristic

155

C₂×C₄

D₄×D₅₃

yes

characteristic

156

C₈

—

157

C₈

—

158

C₈

—

212

—

159

C₂²

—

106

—

160

C₂²

—

212

—

161

C₂²

—

212

—

162

C₂²

#13133187

yes

characteristic

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

424

—

167

C₄

—

212

—

168

C₄

—

212

—

169

C₄

Q₁₆×D₅₃

yes

characteristic

170

C₄

#13133239

yes

characteristic

171

C₄

#13133245

yes

characteristic

172

C₄

#13133246

yes

characteristic

173

C₂

—

212

—

174

C₂

#16810770

yes

characteristic

175

C₂

#16810780

yes

characteristic

176

C₂

#16810802

yes

characteristic

	subgroup	G/H
center	C₂²	#13133187
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×Q₈⋊C₈	C₂
socle	C₂²×C₅₃	#120
hypercenter	Q₈⋊C₈	D₅₃
hypocenter	C₅₃	#1644
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈⋊C₈

subgroup

G/H

center

C₂²

#13133187

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×Q₈⋊C₈

C₂

socle

C₂²×C₅₃

#120

hypercenter

Q₈⋊C₈

D₅₃

hypocenter

C₅₃

#1644

LCS

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈⋊C₈

N	H ≅ G/N	type
2. #16810974	C₂	semidirect
4. #16810833	C₂	semidirect
5. #16810371	C₂	semidirect
7. C₅₃×Q₈⋊C₈	C₂	semidirect
60. D₂₁₂	Q₁₆	semidirect
94. Q₈⋊C₈	D₅₃	semidirect
96. C₅₃	#1644	semidirect

H ≅ G/N

type

2. #16810974

C₂

semidirect

4. #16810833

C₂

semidirect

5. #16810371

C₂

semidirect

7. C₅₃×Q₈⋊C₈

C₂

semidirect

60. D₂₁₂

Q₁₆

semidirect

94. Q₈⋊C₈

D₅₃

semidirect

96. C₅₃

#1644

semidirect

Subgroups — #20273931

Conjugacy classes of subgroups

Complements