Group #27220289 — subgroups

Subgroups — C₅₃×#12777 #27220289

Order 13568.

784 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 142 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	C₂×Q₈²×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#356	C₂	1	3	yes	maximal
3	C₅₃×#1886	C₂	1	3	yes	maximal
4	Q₈²×C₅₃	—	2	6	no	—
5	Q₈²×C₅₃	C₄	1	4	yes	complemented
6	C₂×C₄⋊Q₈×C₅₃	—	2	3	no	—
7	C₂×C₄⋊Q₈×C₅₃	C₂²	1	3	yes	—
8	C₅₃×#519	C₂²	1	1	yes	characteristic
9	C₂×C₄×Q₈×C₅₃	—	2	3	no	—
10	C₅₃×#419	—	2	3	no	—
11	C₂×#101×C₅₃	C₂²	1	3	yes	—
12	C₅₃×#356	—	2	6	no	—
13	C₅₃×#339	—	2	3	no	—
14	C₂²×Q₈×C₅₃	—	2	1	no	—
15	C₂²×Q₈×C₅₃	—	2	1	no	—
16	C₂²×Q₈×C₅₃	D₄	1	3	yes	—
17	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
18	C₂×M₁₆×C₅₃	—	2	3	no	—
19	C₄⋊Q₈×C₅₃	—	2	12	no	—
20	C₄⋊Q₈×C₅₃	—	2	12	no	—
21	C₄⋊Q₈×C₅₃	—	2	6	no	—
22	C₄⋊Q₈×C₅₃	C₂×C₄	1	6	yes	—
23	C₄⋊Q₈×C₅₃	D₄	1	6	yes	—
24	C₄×Q₈×C₅₃	—	2	12	no	—
25	C₄×Q₈×C₅₃	—	2	12	no	—
26	C₄×Q₈×C₅₃	—	4	2	no	—
27	#117×C₅₃	—	2	3	no	—
28	C₂×C₄⋊C₄×C₅₃	—	2	3	no	—
29	C₂×C₄⋊C₄×C₅₃	—	2	6	no	—
30	C₂×C₄²×C₅₃	—	2	3	no	—
31	C₂×C₄²×C₅₃	D₄	1	3	yes	—
32	#103×C₅₃	—	4	6	no	—
33	#101×C₅₃	—	2	6	no	—
34	C₂×Q₈×C₅₃	—	2	4	no	—
35	C₂×Q₈×C₅₃	—	2	8	no	—
36	C₂×Q₈×C₅₃	—	2	4	no	—
37	C₂×Q₈×C₅₃	—	2	3	no	—
38	C₂×Q₈×C₅₃	—	4	1	no	—
39	C₂×Q₈×C₅₃	—	4	1	no	—
40	C₂×Q₈×C₅₃	—	4	3	no	—
41	C₂×Q₈×C₅₃	—	4	3	no	—
42	C₂×Q₈×C₅₃	—	8	1	no	—
43	C₂×Q₈×C₅₃	—	8	1	no	—
44	C₂×Q₈×C₅₃	C₂²⋊C₄	1	6	yes	—
45	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×Q₈×C₅₃	C₂×D₄	1	3	yes	—
47	C₂²×C₄×C₅₃	—	2	3	no	—
48	C₂²×C₄×C₅₃	—	2	3	no	—
49	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
50	M₁₆×C₅₃	—	4	3	no	—
51	M₁₆×C₅₃	—	4	3	no	—
52	C₂×C₈×C₅₃	—	4	3	no	—
53	C₄⋊C₄×C₅₃	—	2	12	no	—
54	C₄⋊C₄×C₅₃	—	2	12	no	—
55	C₄⋊C₄×C₅₃	—	2	12	no	—
56	C₄⋊C₄×C₅₃	—	2	6	no	—
57	C₂²⋊C₄×C₅₃	—	4	1	no	—
58	C₄²×C₅₃	—	2	12	no	—
59	C₄²×C₅₃	—	2	3	no	—
60	C₄²×C₅₃	—	4	3	no	—
61	C₄²×C₅₃	C₂²⋊C₄	1	6	yes	—
62	C₂³×C₅₃	#120	1	1	yes	characteristic
63	Q₈×C₅₃	—	2	16	no	—
64	Q₈×C₅₃	—	4	1	no	—
65	Q₈×C₅₃	—	4	3	no	—
66	Q₈×C₅₃	—	4	6	no	—
67	Q₈×C₅₃	—	8	3	no	—
68	Q₈×C₅₃	—	8	4	no	—
69	C₂×C₄×C₅₃	—	2	12	no	—
70	C₂×C₄×C₅₃	—	2	6	no	—
71	C₂×C₄×C₅₃	—	2	3	no	—
72	C₂×C₄×C₅₃	—	2	3	no	—
73	C₂×C₄×C₅₃	—	2	6	no	—
74	C₂×C₄×C₅₃	—	4	3	no	—
75	C₂×C₄×C₅₃	—	4	3	no	—
76	C₂×C₄×C₅₃	—	4	2	no	—
77	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	3	yes	—
78	C₂×C₄×C₅₃	#120	1	3	yes	—
79	C₈×C₅₃	—	8	3	no	—
80	#12777	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂²×C₅₃	—	2	2	no	—
82	C₂²×C₅₃	#379	1	1	yes	characteristic
83	C₂²×C₅₃	B(4,𝔽₂)	1	1	yes	characteristic
84	C₂²×C₅₃	#458	1	1	yes	characteristic
85	C₄×C₅₃	—	2	12	no	—
86	C₄×C₅₃	—	2	3	no	—
87	C₄×C₅₃	—	2	3	no	—
88	C₄×C₅₃	—	4	6	no	—
89	C₄×C₅₃	—	8	2	no	—
90	C₂×Q₈²	C₂×C₅₃	1	1	yes	characteristic
91	C₂×#356	C₂×C₅₃	1	3	yes	—
92	#1886	C₂×C₅₃	1	3	yes	—
93	C₂×C₅₃	—	2	1	no	—
94	C₂×C₅₃	—	2	1	no	—
95	C₂×C₅₃	#1790	1	1	yes	characteristic
96	C₂×C₅₃	#2205	1	2	yes	—
97	Q₈²	—	2	6	no	—
98	Q₈²	C₄×C₅₃	1	4	yes	complemented
99	C₂×C₄⋊Q₈	—	2	3	no	—
100	C₂×C₄⋊Q₈	C₂²×C₅₃	1	3	yes	—
101	#519	C₂²×C₅₃	1	1	yes	characteristic
102	C₂×C₄×Q₈	—	2	3	no	—
103	#419	—	2	3	no	—
104	C₂×#101	C₂²×C₅₃	1	3	yes	—
105	#356	—	2	6	no	—
106	#339	—	2	3	no	—
107	C₅₃	#12777	1	1	yes	characteristic, direct product factor, Sylow
108	C₂²×Q₈	—	2	1	no	—
109	C₂²×Q₈	—	2	1	no	—
110	C₂²×Q₈	D₄×C₅₃	1	3	yes	—
111	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
112	C₂×M₁₆	—	2	3	no	—
113	C₄⋊Q₈	—	2	12	no	—
114	C₄⋊Q₈	—	2	6	no	—
115	C₄⋊Q₈	—	2	12	no	—
116	C₄⋊Q₈	C₂×C₄×C₅₃	1	6	yes	—
117	C₄⋊Q₈	D₄×C₅₃	1	6	yes	—
118	C₄×Q₈	—	2	12	no	—
119	C₄×Q₈	—	2	12	no	—
120	C₄×Q₈	—	4	2	no	—
121	#117	—	2	3	no	—
122	C₂×C₄⋊C₄	—	2	3	no	—
123	C₂×C₄⋊C₄	—	2	6	no	—
124	C₂×C₄²	—	2	3	no	—
125	C₂×C₄²	D₄×C₅₃	1	3	yes	—
126	#103	—	4	6	no	—
127	#101	—	2	6	no	—
128	C₂×Q₈	—	2	4	no	—
129	C₂×Q₈	—	2	8	no	—
130	C₂×Q₈	—	2	4	no	—
131	C₂×Q₈	—	2	3	no	—
132	C₂×Q₈	—	4	1	no	—
133	C₂×Q₈	—	4	1	no	—
134	C₂×Q₈	—	4	3	no	—
135	C₂×Q₈	—	4	3	no	—
136	C₂×Q₈	—	8	1	no	—
137	C₂×Q₈	—	8	1	no	—
138	C₂×Q₈	C₂²⋊C₄×C₅₃	1	6	yes	—
139	C₂×Q₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
140	C₂×Q₈	C₂×D₄×C₅₃	1	3	yes	—
141	C₂²×C₄	—	2	3	no	—
142	C₂²×C₄	—	2	3	no	—
143	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
144	M₁₆	—	4	3	no	—
145	M₁₆	—	4	3	no	—
146	C₂×C₈	—	4	3	no	—
147	C₄⋊C₄	—	2	12	no	—
148	C₄⋊C₄	—	2	12	no	—
149	C₄⋊C₄	—	2	12	no	—
150	C₄⋊C₄	—	2	6	no	—
151	C₂²⋊C₄	—	4	1	no	—
152	C₄²	—	2	12	no	—
153	C₄²	—	2	3	no	—
154	C₄²	—	4	3	no	—
155	C₄²	C₂²⋊C₄×C₅₃	1	6	yes	—
156	C₂³	#120×C₅₃	1	1	yes	characteristic
157	Q₈	—	2	16	no	—
158	Q₈	—	4	1	no	—
159	Q₈	—	4	3	no	—
160	Q₈	—	4	6	no	—
161	Q₈	—	8	3	no	—
162	Q₈	—	8	4	no	—
163	C₂×C₄	—	2	12	no	—
164	C₂×C₄	—	2	6	no	—
165	C₂×C₄	—	2	3	no	—
166	C₂×C₄	—	2	3	no	—
167	C₂×C₄	—	2	6	no	—
168	C₂×C₄	—	4	3	no	—
169	C₂×C₄	—	4	3	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	3	yes	—
172	C₂×C₄	#120×C₅₃	1	3	yes	—
173	C₈	—	8	3	no	—
174	C₂²	—	2	2	no	—
175	C₂²	C₅₃×#379	1	1	yes	characteristic
176	C₂²	C₅₃×B(4,𝔽₂)	1	1	yes	characteristic
177	C₂²	C₅₃×#458	1	1	yes	characteristic
178	C₄	—	2	12	no	—
179	C₄	—	2	3	no	—
180	C₄	—	2	3	no	—
181	C₄	—	4	6	no	—
182	C₄	—	8	2	no	—
183	C₂	—	2	1	no	—
184	C₂	—	2	1	no	—
185	C₂	C₅₃×#1790	1	1	yes	characteristic
186	C₂	C₅₃×#2205	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈²×C₅₃

C₂

yes

maximal, characteristic

C₂×C₅₃×#356

C₂

yes

maximal

C₅₃×#1886

C₂

yes

maximal

Q₈²×C₅₃

—

Q₈²×C₅₃

C₄

yes

complemented

C₂×C₄⋊Q₈×C₅₃

—

C₂×C₄⋊Q₈×C₅₃

C₂²

yes

—

C₅₃×#519

C₂²

yes

characteristic

C₂×C₄×Q₈×C₅₃

—

C₅₃×#419

—

C₂×#101×C₅₃

C₂²

yes

—

C₅₃×#356

—

C₅₃×#339

—

C₂²×Q₈×C₅₃

—

C₂²×Q₈×C₅₃

—

C₂²×Q₈×C₅₃

D₄

yes

—

C₂²×Q₈×C₅₃

C₂³

yes

characteristic

C₂×M₁₆×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

C₂×C₄

yes

—

C₄⋊Q₈×C₅₃

D₄

yes

—

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

#117×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

D₄

yes

—

#103×C₅₃

—

#101×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

C₂²⋊C₄

yes

—

C₂×Q₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×Q₈×C₅₃

C₂×D₄

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂²⋊C₄

yes

—

C₂³×C₅₃

#120

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

#120

yes

—

C₈×C₅₃

—

#12777

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₂×Q₈²

C₂×C₅₃

yes

characteristic

C₂×#356

C₂×C₅₃

yes

—

#1886

C₂×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#1790

yes

characteristic

C₂×C₅₃

#2205

yes

—

Q₈²

—

Q₈²

C₄×C₅₃

yes

complemented

C₂×C₄⋊Q₈

—

100

C₂×C₄⋊Q₈

C₂²×C₅₃

yes

—

101

#519

C₂²×C₅₃

yes

characteristic

102

C₂×C₄×Q₈

—

103

#419

—

104

C₂×#101

C₂²×C₅₃

yes

—

105

#356

—

106

#339

—

107

C₅₃

#12777

yes

characteristic, direct product factor, Sylow

108

C₂²×Q₈

—

109

C₂²×Q₈

—

110

C₂²×Q₈

D₄×C₅₃

yes

—

111

C₂²×Q₈

C₂³×C₅₃

yes

characteristic

112

C₂×M₁₆

—

113

C₄⋊Q₈

—

114

C₄⋊Q₈

—

115

C₄⋊Q₈

—

116

C₄⋊Q₈

C₂×C₄×C₅₃

yes

—

117

C₄⋊Q₈

D₄×C₅₃

yes

—

118

C₄×Q₈

—

119

C₄×Q₈

—

120

C₄×Q₈

—

121

#117

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄²

—

125

C₂×C₄²

D₄×C₅₃

yes

—

126

#103

—

127

#101

—

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂×Q₈

—

137

C₂×Q₈

—

138

C₂×Q₈

C₂²⋊C₄×C₅₃

yes

—

139

C₂×Q₈

C₂²×C₄×C₅₃

yes

characteristic

140

C₂×Q₈

C₂×D₄×C₅₃

yes

—

141

C₂²×C₄

—

142

C₂²×C₄

—

143

C₂²×C₄

C₂×D₄×C₅₃

yes

—

144

M₁₆

—

145

M₁₆

—

146

C₂×C₈

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₂²⋊C₄

—

152

C₄²

—

153

C₄²

—

154

C₄²

—

155

C₄²

C₂²⋊C₄×C₅₃

yes

—

156

C₂³

#120×C₅₃

yes

characteristic

157

Q₈

—

158

Q₈

—

159

Q₈

—

160

Q₈

—

161

Q₈

—

162

Q₈

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

C₂×C₂²⋊C₄×C₅₃

yes

—

172

C₂×C₄

#120×C₅₃

yes

—

173

C₈

—

174

C₂²

—

175

C₂²

C₅₃×#379

yes

characteristic

176

C₂²

C₅₃×B(4,𝔽₂)

yes

characteristic

177

C₂²

C₅₃×#458

yes

characteristic

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

—

183

C₂

—

184

C₂

—

185

C₂

C₅₃×#1790

yes

characteristic

186

C₂

C₅₃×#2205

yes

—

	subgroup	G/H
center, socle	C₂²×C₅₃	B(4,𝔽₂)
commutator	C₂×Q₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₂²×Q₈	C₂³×C₅₃
prime radical	C₂³×C₅₃	#120
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂³×C₅₃ ◁ C₂²×Q₈×C₅₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₅₃

B(4,𝔽₂)

commutator

C₂×Q₈

C₂²×C₄×C₅₃

Frattini, cosocle

C₂²×Q₈

C₂³×C₅₃

prime radical

C₂³×C₅₃

#120

LCS

G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₅₃ ◁ C₂³×C₅₃ ◁ C₂²×Q₈×C₅₃ ◁ G

N	H ≅ G/N	type
5. Q₈²×C₅₃	C₄	semidirect
80. #12777	C₅₃	direct (N × H)
98. Q₈²	C₄×C₅₃	semidirect
107. C₅₃	#12777	direct (N × H)

H ≅ G/N

type

5. Q₈²×C₅₃

C₄

semidirect

80. #12777

C₅₃

direct (N × H)

98. Q₈²

C₄×C₅₃

semidirect

107. C₅₃

#12777

direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×Q₈²×C₅₃	2
C₂×C₅₃×#356	2
C₅₃×#1886	2
#12777	53

subgroup

index

C₂×Q₈²×C₅₃

C₂×C₅₃×#356

C₅₃×#1886

#12777

Subgroups — C₅₃×#12777 #27220289

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#12777 #27220289

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12777 #27220289