Group #27216950 — subgroups

Subgroups — C₅₃×#9438 #27216950

Order 13568.

452 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 126 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	C₅₃×#3138	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2986	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#1561	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1545	C₂	1	2	yes	maximal
5	C₅₃×#1299	C₂	1	2	yes	maximal, complemented
6	C₅₃×#533	C₂²	1	1	yes	characteristic
7	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
8	C₅₃×#490	C₄	1	4	yes	—
9	C₅₃×#486	—	2	2	no	—
10	C₅₃×#446	—	2	2	no	—
11	C₈×Q₈×C₅₃	—	2	2	no	—
12	C₅₃×#433	—	2	2	no	—
13	C₅₃×#431	C₂²	1	2	yes	—
14	C₅₃×#423	C₂²	1	2	yes	—
15	C₅₃×#418	—	2	2	no	—
16	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#332	—	2	2	no	—
18	C₅₃×#331	—	2	2	no	—
19	C₅₃×#331	—	2	2	no	—
20	C₂²×D₄×C₅₃	—	2	1	no	—
21	C₂²×D₄×C₅₃	—	2	2	no	—
22	C₂×M₁₆×C₅₃	—	2	1	no	—
23	C₂×M₁₆×C₅₃	—	2	2	no	—
24	C₄⋊Q₈×C₅₃	D₄	1	2	yes	—
25	#127×C₅₃	—	2	1	no	—
26	#127×C₅₃	—	2	2	no	—
27	#127×C₅₃	C₂×C₄	1	2	yes	—
28	#125×C₅₃	C₂×C₄	1	2	yes	—
29	C₂²⋊Q₈×C₅₃	—	2	2	no	—
30	C₄×Q₈×C₅₃	—	2	2	no	—
31	#117×C₅₃	D₄	1	2	yes	—
32	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₅₃	—	2	4	no	—
34	C₄⋊C₈×C₅₃	—	2	2	no	—
35	C₄⋊C₈×C₅₃	—	2	2	no	—
36	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
37	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
38	#102×C₅₃	—	2	8	no	—
39	C₂²⋊C₈×C₅₃	—	2	2	no	—
40	#97×C₅₃	—	2	2	no	—
41	#97×C₅₃	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₅₃	—	2	2	no	—
43	C₄×C₈×C₅₃	—	2	1	no	—
44	C₄×C₈×C₅₃	—	2	2	no	—
45	C₂⁴×C₅₃	—	4	1	no	—
46	C₂×Q₈×C₅₃	—	2	2	no	—
47	C₂×D₄×C₅₃	—	2	2	no	—
48	C₂×D₄×C₅₃	—	2	4	no	—
49	C₂×D₄×C₅₃	—	4	4	no	—
50	C₂×D₄×C₅₃	—	4	1	no	—
51	C₂×D₄×C₅₃	—	4	2	no	—
52	C₂×D₄×C₅₃	—	8	2	no	—
53	C₂×D₄×C₅₃	—	8	2	no	—
54	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	#41	1	2	yes	—
56	M₁₆×C₅₃	—	4	2	no	—
57	M₁₆×C₅₃	—	8	2	no	—
58	C₂×C₈×C₅₃	—	2	2	no	—
59	C₂×C₈×C₅₃	—	2	2	no	—
60	C₂×C₈×C₅₃	—	2	2	no	—
61	C₂×C₈×C₅₃	—	4	2	no	—
62	C₄⋊C₄×C₅₃	—	2	4	no	—
63	C₄⋊C₄×C₅₃	—	2	2	no	—
64	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
65	C₂²⋊C₄×C₅₃	—	2	2	no	—
66	C₄²×C₅₃	—	2	2	no	—
67	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₅₃	C₂×D₄	1	2	yes	—
70	C₂³×C₅₃	—	4	2	no	—
71	C₂³×C₅₃	—	8	2	no	—
72	C₂³×C₅₃	—	8	4	no	—
73	C₂³×C₅₃	#123	1	1	yes	characteristic
74	Q₈×C₅₃	—	4	2	no	—
75	Q₈×C₅₃	—	4	2	no	—
76	D₄×C₅₃	—	4	4	no	—
77	D₄×C₅₃	—	8	4	no	—
78	D₄×C₅₃	—	8	2	no	—
79	D₄×C₅₃	—	8	2	no	—
80	D₄×C₅₃	—	8	2	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	2	2	no	—
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	2	4	no	—
85	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
87	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
88	C₂×C₄×C₅₃	#121	1	2	yes	—
89	C₈×C₅₃	—	4	2	no	—
90	C₈×C₅₃	—	4	2	no	—
91	C₈×C₅₃	—	4	2	no	—
92	C₈×C₅₃	—	8	2	no	—
93	#9438	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂²×C₅₃	—	2	1	no	—
95	C₂²×C₅₃	—	2	2	no	—
96	C₂²×C₅₃	—	8	2	no	—
97	C₂²×C₅₃	—	8	4	no	—
98	C₂²×C₅₃	—	16	4	no	—
99	C₂²×C₅₃	#386	1	1	yes	characteristic
100	C₄×C₅₃	—	2	1	no	—
101	C₄×C₅₃	—	2	2	no	—
102	C₄×C₅₃	—	4	4	no	—
103	C₄×C₅₃	#454	1	2	yes	—
104	C₄×C₅₃	#471	1	2	yes	—
105	C₄×C₅₃	#472	1	2	yes	—
106	#3138	C₂×C₅₃	1	1	yes	characteristic
107	#2986	C₂×C₅₃	1	1	yes	characteristic, complemented
108	#1561	C₂×C₅₃	1	1	yes	characteristic, complemented
109	#1545	C₂×C₅₃	1	2	yes	—
110	#1299	C₂×C₅₃	1	2	yes	complemented
111	C₂×C₅₃	—	4	1	no	—
112	C₂×C₅₃	—	16	2	no	—
113	C₂×C₅₃	#1888	1	1	yes	characteristic
114	C₂×C₅₃	#1907	1	2	yes	—
115	#533	C₂²×C₅₃	1	1	yes	characteristic
116	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
117	#490	C₄×C₅₃	1	4	yes	—
118	#486	—	2	2	no	—
119	#446	—	2	2	no	—
120	C₈×Q₈	—	2	2	no	—
121	#433	—	2	2	no	—
122	#431	C₂²×C₅₃	1	2	yes	—
123	#423	C₂²×C₅₃	1	2	yes	—
124	#418	—	2	2	no	—
125	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
126	#332	—	2	2	no	—
127	#331	—	2	2	no	—
128	#331	—	2	2	no	—
129	C₅₃	#9438	1	1	yes	characteristic, direct product factor, Sylow
130	C₂²×D₄	—	2	1	no	—
131	C₂²×D₄	—	2	2	no	—
132	C₂×M₁₆	—	2	1	no	—
133	C₂×M₁₆	—	2	2	no	—
134	C₄⋊Q₈	D₄×C₅₃	1	2	yes	—
135	#127	—	2	2	no	—
136	#127	—	2	1	no	—
137	#127	C₂×C₄×C₅₃	1	2	yes	—
138	#125	C₂×C₄×C₅₃	1	2	yes	—
139	C₂²⋊Q₈	—	2	2	no	—
140	C₄×Q₈	—	2	2	no	—
141	#117	D₄×C₅₃	1	2	yes	—
142	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
143	C₄⋊C₈	—	2	4	no	—
144	C₄⋊C₈	—	2	2	no	—
145	C₄⋊C₈	—	2	2	no	—
146	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
147	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
148	#102	—	2	8	no	—
149	C₂²⋊C₈	—	2	2	no	—
150	#97	—	2	2	no	—
151	#97	C₂×C₄×C₅₃	1	2	yes	—
152	C₄×C₈	—	2	2	no	—
153	C₄×C₈	—	2	1	no	—
154	C₄×C₈	—	2	2	no	—
155	C₂⁴	—	4	1	no	—
156	C₂×Q₈	—	2	2	no	—
157	C₂×D₄	—	2	2	no	—
158	C₂×D₄	—	2	4	no	—
159	C₂×D₄	—	4	4	no	—
160	C₂×D₄	—	4	1	no	—
161	C₂×D₄	—	4	2	no	—
162	C₂×D₄	—	8	2	no	—
163	C₂×D₄	—	8	2	no	—
164	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
165	C₂²×C₄	#41×C₅₃	1	2	yes	—
166	M₁₆	—	4	2	no	—
167	M₁₆	—	8	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	4	2	no	—
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
175	C₂²⋊C₄	—	2	2	no	—
176	C₄²	—	2	2	no	—
177	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
178	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
179	C₄²	C₂×D₄×C₅₃	1	2	yes	—
180	C₂³	—	4	2	no	—
181	C₂³	—	8	2	no	—
182	C₂³	—	8	4	no	—
183	C₂³	#123×C₅₃	1	1	yes	characteristic
184	Q₈	—	4	2	no	—
185	Q₈	—	4	2	no	—
186	D₄	—	4	4	no	—
187	D₄	—	8	4	no	—
188	D₄	—	8	2	no	—
189	D₄	—	8	2	no	—
190	D₄	—	8	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
196	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
197	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
198	C₂×C₄	#121×C₅₃	1	2	yes	—
199	C₈	—	4	2	no	—
200	C₈	—	4	2	no	—
201	C₈	—	4	2	no	—
202	C₈	—	8	2	no	—
203	C₂²	—	2	1	no	—
204	C₂²	—	2	2	no	—
205	C₂²	—	8	2	no	—
206	C₂²	—	8	4	no	—
207	C₂²	—	16	4	no	—
208	C₂²	C₅₃×#386	1	1	yes	characteristic
209	C₄	—	2	1	no	—
210	C₄	—	2	2	no	—
211	C₄	—	4	4	no	—
212	C₄	C₅₃×#454	1	2	yes	—
213	C₄	C₅₃×#471	1	2	yes	—
214	C₄	C₅₃×#472	1	2	yes	—
215	C₂	—	4	1	no	—
216	C₂	—	16	2	no	—
217	C₂	C₅₃×#1888	1	1	yes	characteristic
218	C₂	C₅₃×#1907	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#3138

C₂

yes

maximal, characteristic

C₅₃×#2986

C₂

yes

maximal, characteristic, complemented

C₅₃×#1561

C₂

yes

maximal, characteristic, complemented

C₅₃×#1545

C₂

yes

maximal

C₅₃×#1299

C₂

yes

maximal, complemented

C₅₃×#533

C₂²

yes

characteristic

C₂×#127×C₅₃

C₂²

yes

characteristic

C₅₃×#490

C₄

yes

—

C₅₃×#486

—

C₅₃×#446

—

C₈×Q₈×C₅₃

—

C₅₃×#433

—

C₅₃×#431

C₂²

yes

—

C₅₃×#423

C₂²

yes

—

C₅₃×#418

—

C₄×M₁₆×C₅₃

C₂²

yes

characteristic

C₅₃×#332

—

C₅₃×#331

—

C₅₃×#331

—

C₂²×D₄×C₅₃

—

C₂²×D₄×C₅₃

—

C₂×M₁₆×C₅₃

—

C₂×M₁₆×C₅₃

—

C₄⋊Q₈×C₅₃

D₄

yes

—

#127×C₅₃

—

#127×C₅₃

—

#127×C₅₃

C₂×C₄

yes

—

#125×C₅₃

C₂×C₄

yes

—

C₂²⋊Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

#117×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

D₄

yes

—

C₄⋊C₈×C₅₃

D₄

yes

—

#102×C₅₃

—

C₂²⋊C₈×C₅₃

—

#97×C₅₃

—

#97×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₂⁴×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

#123

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×C₄×C₅₃

#120

yes

characteristic

C₂×C₄×C₅₃

#121

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#9438

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#386

yes

characteristic

100

C₄×C₅₃

—

101

C₄×C₅₃

—

102

C₄×C₅₃

—

103

C₄×C₅₃

#454

yes

—

104

C₄×C₅₃

#471

yes

—

105

C₄×C₅₃

#472

yes

—

106

#3138

C₂×C₅₃

yes

characteristic

107

#2986

C₂×C₅₃

yes

characteristic, complemented

108

#1561

C₂×C₅₃

yes

characteristic, complemented

109

#1545

C₂×C₅₃

yes

—

110

#1299

C₂×C₅₃

yes

complemented

111

C₂×C₅₃

—

112

C₂×C₅₃

—

113

C₂×C₅₃

#1888

yes

characteristic

114

C₂×C₅₃

#1907

yes

—

115

#533

C₂²×C₅₃

yes

characteristic

116

C₂×#127

C₂²×C₅₃

yes

characteristic

117

#490

C₄×C₅₃

yes

—

118

#486

—

119

#446

—

120

C₈×Q₈

—

121

#433

—

122

#431

C₂²×C₅₃

yes

—

123

#423

C₂²×C₅₃

yes

—

124

#418

—

125

C₄×M₁₆

C₂²×C₅₃

yes

characteristic

126

#332

—

127

#331

—

128

#331

—

129

C₅₃

#9438

yes

characteristic, direct product factor, Sylow

130

C₂²×D₄

—

131

C₂²×D₄

—

132

C₂×M₁₆

—

133

C₂×M₁₆

—

134

C₄⋊Q₈

D₄×C₅₃

yes

—

135

#127

—

136

#127

—

137

#127

C₂×C₄×C₅₃

yes

—

138

#125

C₂×C₄×C₅₃

yes

—

139

C₂²⋊Q₈

—

140

C₄×Q₈

—

141

#117

D₄×C₅₃

yes

—

142

C₂×C₄²

C₂³×C₅₃

yes

characteristic

143

C₄⋊C₈

—

144

C₄⋊C₈

—

145

C₄⋊C₈

—

146

C₄⋊C₈

D₄×C₅₃

yes

—

147

C₄⋊C₈

D₄×C₅₃

yes

—

148

#102

—

149

C₂²⋊C₈

—

150

#97

—

151

#97

C₂×C₄×C₅₃

yes

—

152

C₄×C₈

—

153

C₄×C₈

—

154

C₄×C₈

—

155

C₂⁴

—

156

C₂×Q₈

—

157

C₂×D₄

—

158

C₂×D₄

—

159

C₂×D₄

—

160

C₂×D₄

—

161

C₂×D₄

—

162

C₂×D₄

—

163

C₂×D₄

—

164

C₂²×C₄

C₂×D₄×C₅₃

yes

characteristic

165

C₂²×C₄

#41×C₅₃

yes

—

166

M₁₆

—

167

M₁₆

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

—

171

C₂×C₈

—

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₄⋊C₄

C₂²⋊C₄×C₅₃

yes

—

175

C₂²⋊C₄

—

176

C₄²

—

177

C₄²

C₂²×C₄×C₅₃

yes

characteristic

178

C₄²

C₂×D₄×C₅₃

yes

characteristic

179

C₄²

C₂×D₄×C₅₃

yes

—

180

C₂³

—

181

C₂³

—

182

C₂³

—

183

C₂³

#123×C₅₃

yes

characteristic

184

Q₈

—

185

Q₈

—

186

D₄

—

187

D₄

—

188

D₄

—

189

D₄

—

190

D₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

C₂×C₂²⋊C₄×C₅₃

yes

characteristic

196

C₂×C₄

C₄×D₄×C₅₃

yes

—

197

C₂×C₄

#120×C₅₃

yes

characteristic

198

C₂×C₄

#121×C₅₃

yes

—

199

C₈

—

200

C₈

—

201

C₈

—

202

C₈

—

203

C₂²

—

204

C₂²

—

205

C₂²

—

206

C₂²

—

207

C₂²

—

208

C₂²

C₅₃×#386

yes

characteristic

209

C₄

—

210

C₄

—

211

C₄

—

212

C₄

C₅₃×#454

yes

—

213

C₄

C₅₃×#471

yes

—

214

C₄

C₅₃×#472

yes

—

215

C₂

—

216

C₂

—

217

C₂

C₅₃×#1888

yes

characteristic

218

C₂

C₅₃×#1907

yes

—

	subgroup	G/H
center, socle	C₂²×C₅₃	#386
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂×#127×C₅₃	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₅₃

#386

commutator

C₄²

C₂²×C₄×C₅₃

Frattini, cosocle

C₂×C₄²

C₂³×C₅₃

prime radical

C₂×#127×C₅₃

C₂²

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

N	H ≅ G/N	type
2. C₅₃×#2986	C₂	semidirect
3. C₅₃×#1561	C₂	semidirect
5. C₅₃×#1299	C₂	semidirect
93. #9438	C₅₃	direct (N × H)
107. #2986	C₂×C₅₃	semidirect
108. #1561	C₂×C₅₃	semidirect
110. #1299	C₂×C₅₃	semidirect
129. C₅₃	#9438	direct (N × H)

H ≅ G/N

type

2. C₅₃×#2986

C₂

semidirect

3. C₅₃×#1561

C₂

semidirect

5. C₅₃×#1299

C₂

semidirect

93. #9438

C₅₃

direct (N × H)

107. #2986

C₂×C₅₃

semidirect

108. #1561

C₂×C₅₃

semidirect

110. #1299

C₂×C₅₃

semidirect

129. C₅₃

#9438

direct (N × H)

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

subgroup	index
C₅₃×#3138	2
C₅₃×#2986	2
C₅₃×#1561	2
C₅₃×#1545	2
C₅₃×#1299	2
#9438	53

subgroup

index

C₅₃×#3138

C₅₃×#2986

C₅₃×#1561

C₅₃×#1545

C₅₃×#1299

#9438

Subgroups — C₅₃×#9438 #27216950

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#9438 #27216950

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9438 #27216950