Group #27215117 — subgroups

Subgroups — C₅₃×#7605 #27215117

Order 13568.

440 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 154 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	C₄×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1839	C₂	1	2	yes	maximal
3	C₅₃×#1272	C₂	1	4	yes	maximal
4	C₂×C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
5	C₂×C₄×Q₈×C₅₃	C₄	1	2	yes	—
6	C₅₃×#391	C₂²	1	2	yes	—
7	C₅₃×#389	—	2	2	no	—
8	C₅₃×#384	C₄	1	1	yes	characteristic
9	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
10	C₄³×C₅₃	C₂²	1	1	yes	characteristic
11	C₅₃×#336	—	2	2	no	—
12	C₅₃×#336	—	2	2	no	—
13	C₅₃×#336	C₂²	1	4	yes	—
14	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
15	C₂²×C₈×C₅₃	—	2	4	no	—
16	C₄⋊Q₈×C₅₃	—	2	2	no	—
17	C₄⋊Q₈×C₅₃	C₈	1	4	yes	complemented
18	C₄×Q₈×C₅₃	—	4	4	no	—
19	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
20	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
21	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
22	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₅₃	—	2	2	no	—
24	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₅₃	D₄	1	2	yes	—
27	C₂×C₄²×C₅₃	D₄	1	2	yes	—
28	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
29	#95×C₅₃	—	2	4	no	—
30	C₂×Q₈×C₅₃	—	2	2	no	—
31	C₂×Q₈×C₅₃	—	4	2	no	—
32	C₂×Q₈×C₅₃	—	4	2	no	—
33	C₂×Q₈×C₅₃	C₂×C₈	1	4	yes	—
34	C₂²×C₄×C₅₃	—	2	2	no	—
35	C₂²×C₄×C₅₃	—	2	2	no	—
36	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
38	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
41	C₂×C₈×C₅₃	—	4	4	no	—
42	C₂×C₈×C₅₃	—	4	4	no	—
43	C₂×C₈×C₅₃	—	4	4	no	—
44	C₄⋊C₄×C₅₃	—	2	8	no	—
45	C₄⋊C₄×C₅₃	—	4	4	no	—
46	C₄⋊C₄×C₅₃	—	4	4	no	—
47	C₄⋊C₄×C₅₃	—	4	2	no	—
48	C₄²×C₅₃	—	2	2	no	—
49	C₄²×C₅₃	—	2	4	no	—
50	C₄²×C₅₃	—	4	2	no	—
51	C₄²×C₅₃	—	4	1	no	—
52	C₄²×C₅₃	—	4	4	no	—
53	C₄²×C₅₃	C₂×C₈	1	2	yes	—
54	C₄²×C₅₃	M₁₆	1	2	yes	—
55	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	Q₈×C₅₃	—	4	4	no	—
57	Q₈×C₅₃	—	4	4	no	—
58	C₂×C₄×C₅₃	—	2	1	no	—
59	C₂×C₄×C₅₃	—	2	2	no	—
60	C₂×C₄×C₅₃	—	2	1	no	—
61	C₂×C₄×C₅₃	—	2	2	no	—
62	C₂×C₄×C₅₃	—	2	2	no	—
63	C₂×C₄×C₅₃	—	2	2	no	—
64	C₂×C₄×C₅₃	—	2	2	no	—
65	C₂×C₄×C₅₃	—	2	4	no	—
66	C₂×C₄×C₅₃	—	2	1	no	—
67	C₂×C₄×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	—	4	2	no	—
69	C₂×C₄×C₅₃	—	4	2	no	—
70	C₂×C₄×C₅₃	—	4	2	no	—
71	C₂×C₄×C₅₃	—	4	4	no	—
72	C₂×C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
73	C₂×C₄×C₅₃	#100	1	2	yes	—
74	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄×C₅₃	#137	1	4	yes	—
77	C₈×C₅₃	—	8	4	no	—
78	#7605	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
80	C₂²×C₅₃	#410	1	1	yes	characteristic
81	C₂²×C₅₃	C₂×#100	1	1	yes	characteristic
82	C₂²×C₅₃	#419	1	2	yes	—
83	C₂²×C₅₃	#421	1	2	yes	—
84	C₄×C₅₃	—	2	2	no	—
85	C₄×C₅₃	—	2	1	no	—
86	C₄×C₅₃	—	2	1	no	—
87	C₄×C₅₃	—	2	4	no	—
88	C₄×C₅₃	—	2	4	no	—
89	C₄×C₅₃	—	4	1	no	—
90	C₄×C₅₃	—	4	4	no	—
91	C₄×C₅₃	—	8	2	no	—
92	C₄×C₄⋊Q₈	C₂×C₅₃	1	1	yes	characteristic
93	#1839	C₂×C₅₃	1	2	yes	—
94	#1272	C₂×C₅₃	1	4	yes	—
95	C₂×C₅₃	#1460	1	1	yes	characteristic
96	C₂×C₅₃	#1479	1	2	yes	—
97	C₂×C₅₃	#1515	1	2	yes	—
98	C₂×C₅₃	#1536	1	1	yes	characteristic
99	C₂×C₅₃	#1558	1	1	yes	characteristic
100	C₂×C₄⋊Q₈	C₄×C₅₃	1	1	yes	characteristic
101	C₂×C₄×Q₈	C₄×C₅₃	1	2	yes	—
102	#391	C₂²×C₅₃	1	2	yes	—
103	#389	—	2	2	no	—
104	#384	C₄×C₅₃	1	1	yes	characteristic
105	C₄×C₄⋊C₄	—	2	2	no	—
106	C₄³	C₂²×C₅₃	1	1	yes	characteristic
107	#336	—	2	2	no	—
108	#336	—	2	2	no	—
109	#336	C₂²×C₅₃	1	4	yes	—
110	C₅₃	#7605	1	1	yes	characteristic, direct product factor, Sylow
111	C₂²×Q₈	C₂×C₄×C₅₃	1	2	yes	—
112	C₂²×C₈	—	2	4	no	—
113	C₄⋊Q₈	—	2	2	no	—
114	C₄⋊Q₈	C₈×C₅₃	1	4	yes	complemented
115	C₄×Q₈	—	4	4	no	—
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
120	C₂×C₄²	—	2	2	no	—
121	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
122	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
123	C₂×C₄²	D₄×C₅₃	1	2	yes	—
124	C₂×C₄²	D₄×C₅₃	1	2	yes	—
125	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
126	#95	—	2	4	no	—
127	C₂×Q₈	—	2	2	no	—
128	C₂×Q₈	—	4	2	no	—
129	C₂×Q₈	—	4	2	no	—
130	C₂×Q₈	C₂×C₈×C₅₃	1	4	yes	—
131	C₂²×C₄	—	2	2	no	—
132	C₂²×C₄	—	2	2	no	—
133	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
134	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
135	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
136	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
137	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
138	C₂×C₈	—	4	4	no	—
139	C₂×C₈	—	4	4	no	—
140	C₂×C₈	—	4	4	no	—
141	C₄⋊C₄	—	2	8	no	—
142	C₄⋊C₄	—	4	2	no	—
143	C₄⋊C₄	—	4	4	no	—
144	C₄⋊C₄	—	4	4	no	—
145	C₄²	—	2	2	no	—
146	C₄²	—	2	4	no	—
147	C₄²	—	4	2	no	—
148	C₄²	—	4	1	no	—
149	C₄²	—	4	4	no	—
150	C₄²	C₂×C₈×C₅₃	1	2	yes	—
151	C₄²	M₁₆×C₅₃	1	2	yes	—
152	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
153	Q₈	—	4	4	no	—
154	Q₈	—	4	4	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	C₂²⋊C₈×C₅₃	1	4	yes	—
170	C₂×C₄	#100×C₅₃	1	2	yes	—
171	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
172	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
173	C₂×C₄	#137×C₅₃	1	4	yes	—
174	C₈	—	8	4	no	—
175	C₂²	C₂×C₂²⋊C₈×C₅₃	1	1	yes	characteristic
176	C₂²	C₅₃×#410	1	1	yes	characteristic
177	C₂²	C₂×#100×C₅₃	1	1	yes	characteristic
178	C₂²	C₅₃×#419	1	2	yes	—
179	C₂²	C₅₃×#421	1	2	yes	—
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	2	4	no	—
183	C₄	—	2	4	no	—
184	C₄	—	2	2	no	—
185	C₄	—	4	4	no	—
186	C₄	—	4	1	no	—
187	C₄	—	8	2	no	—
188	C₂	C₅₃×#1460	1	1	yes	characteristic
189	C₂	C₅₃×#1479	1	2	yes	—
190	C₂	C₅₃×#1515	1	2	yes	—
191	C₂	C₅₃×#1536	1	1	yes	characteristic
192	C₂	C₅₃×#1558	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₄⋊Q₈×C₅₃

C₂

yes

maximal, characteristic

C₅₃×#1839

C₂

yes

maximal

C₅₃×#1272

C₂

yes

maximal

C₂×C₄⋊Q₈×C₅₃

C₄

yes

characteristic

C₂×C₄×Q₈×C₅₃

C₄

yes

—

C₅₃×#391

C₂²

yes

—

C₅₃×#389

—

C₅₃×#384

C₄

yes

characteristic

C₄×C₄⋊C₄×C₅₃

—

C₄³×C₅₃

C₂²

yes

characteristic

C₅₃×#336

—

C₅₃×#336

—

C₅₃×#336

C₂²

yes

—

C₂²×Q₈×C₅₃

C₂×C₄

yes

—

C₂²×C₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

C₈

yes

complemented

C₄×Q₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#95×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

C₂×C₈

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×C₈

yes

—

C₄²×C₅₃

M₁₆

yes

—

C₂³×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

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×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₈

yes

—

C₂×C₄×C₅₃

#100

yes

—

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅₃

#137

yes

—

C₈×C₅₃

—

#7605

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

C₂×C₂²⋊C₈

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#421

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₄⋊Q₈

C₂×C₅₃

yes

characteristic

#1839

C₂×C₅₃

yes

—

#1272

C₂×C₅₃

yes

—

C₂×C₅₃

#1460

yes

characteristic

C₂×C₅₃

#1479

yes

—

C₂×C₅₃

#1515

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

100

C₂×C₄⋊Q₈

C₄×C₅₃

yes

characteristic

101

C₂×C₄×Q₈

C₄×C₅₃

yes

—

102

#391

C₂²×C₅₃

yes

—

103

#389

—

104

#384

C₄×C₅₃

yes

characteristic

105

C₄×C₄⋊C₄

—

106

C₄³

C₂²×C₅₃

yes

characteristic

107

#336

—

108

#336

—

109

#336

C₂²×C₅₃

yes

—

110

C₅₃

#7605

yes

characteristic, direct product factor, Sylow

111

C₂²×Q₈

C₂×C₄×C₅₃

yes

—

112

C₂²×C₈

—

113

C₄⋊Q₈

—

114

C₄⋊Q₈

C₈×C₅₃

yes

complemented

115

C₄×Q₈

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

C₂×C₄×C₅₃

yes

—

120

C₂×C₄²

—

121

C₂×C₄²

C₂×C₄×C₅₃

yes

characteristic

122

C₂×C₄²

C₂×C₄×C₅₃

yes

characteristic

123

C₂×C₄²

D₄×C₅₃

yes

—

124

C₂×C₄²

D₄×C₅₃

yes

—

125

C₂×C₄²

C₂³×C₅₃

yes

characteristic

126

#95

—

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂×Q₈

C₂×C₈×C₅₃

yes

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

C₂²⋊C₄×C₅₃

yes

characteristic

134

C₂²×C₄

C₂²⋊C₄×C₅₃

yes

—

135

C₂²×C₄

C₂²⋊C₄×C₅₃

yes

characteristic

136

C₂²×C₄

C₂²×C₄×C₅₃

yes

characteristic

137

C₂²×C₄

C₂×D₄×C₅₃

yes

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄²

—

146

C₄²

—

147

C₄²

—

148

C₄²

—

149

C₄²

—

150

C₄²

C₂×C₈×C₅₃

yes

—

151

C₄²

M₁₆×C₅₃

yes

—

152

C₂³

C₂×C₂²⋊C₄×C₅₃

yes

characteristic

153

Q₈

—

154

Q₈

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

C₂²⋊C₈×C₅₃

yes

—

170

C₂×C₄

#100×C₅₃

yes

—

171

C₂×C₄

C₂²×C₈×C₅₃

yes

characteristic

172

C₂×C₄

C₂×M₁₆×C₅₃

yes

characteristic

173

C₂×C₄

#137×C₅₃

yes

—

174

C₈

—

175

C₂²

C₂×C₂²⋊C₈×C₅₃

yes

characteristic

176

C₂²

C₅₃×#410

yes

characteristic

177

C₂²

C₂×#100×C₅₃

yes

characteristic

178

C₂²

C₅₃×#419

yes

—

179

C₂²

C₅₃×#421

yes

—

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

—

188

C₂

C₅₃×#1460

yes

characteristic

189

C₂

C₅₃×#1479

yes

—

190

C₂

C₅₃×#1515

yes

—

191

C₂

C₅₃×#1536

yes

characteristic

192

C₂

C₅₃×#1558

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₄³×C₅₃ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₅₃

C₂×C₂²⋊C₄

commutator

C₂×C₄

C₂²×C₈×C₅₃

Frattini, cosocle

C₂×C₄²

C₂³×C₅₃

LCS

G ▷ C₂×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₅₃ ◁ C₄³×C₅₃ ◁ G

N	H ≅ G/N	type
17. C₄⋊Q₈×C₅₃	C₈	semidirect
78. #7605	C₅₃	direct (N × H)
110. C₅₃	#7605	direct (N × H)
114. C₄⋊Q₈	C₈×C₅₃	semidirect

H ≅ G/N

type

17. C₄⋊Q₈×C₅₃

C₈

semidirect

78. #7605

C₅₃

direct (N × H)

110. C₅₃

#7605

direct (N × H)

114. C₄⋊Q₈

C₈×C₅₃

semidirect

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

subgroup	index
C₄×C₄⋊Q₈×C₅₃	2
C₅₃×#1839	2
C₅₃×#1272	2
#7605	53

subgroup

index

C₄×C₄⋊Q₈×C₅₃

C₅₃×#1839

C₅₃×#1272

#7605

Subgroups — C₅₃×#7605 #27215117

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#7605 #27215117

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7605 #27215117