Group #27219945 — subgroups

Subgroups — C₅₃×#12433 #27219945

Order 13568.

262 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 82 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×#2246	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#2169	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#1667	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1372	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1361	C₂	1	2	yes	maximal, complemented
6	C₂×D₁₆×C₅₃	—	2	1	no	—
7	C₂×D₁₆×C₅₃	—	2	1	no	—
8	C₂×M₃₂×C₅₃	—	2	1	no	—
9	C₅₃×#493	C₂²	1	2	yes	—
10	C₅₃×#459	—	2	2	no	—
11	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₅₃×C₂²⋊C₁₆	—	2	1	no	—
14	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#335	C₂²	1	2	yes	—
16	C₅₃×#331	—	2	2	no	—
17	C₂×D₈×C₅₃	—	2	2	no	—
18	C₂×D₈×C₅₃	—	2	2	no	—
19	C₂×D₈×C₅₃	—	4	2	no	—
20	C₂²×C₈×C₅₃	—	2	1	no	—
21	#127×C₅₃	—	2	2	no	—
22	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
23	D₁₆×C₅₃	—	4	2	no	—
24	D₁₆×C₅₃	—	4	2	no	—
25	M₃₂×C₅₃	—	4	2	no	—
26	C₂×C₁₆×C₅₃	—	2	1	no	—
27	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
29	C₄⋊C₈×C₅₃	—	2	2	no	—
30	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
31	#102×C₅₃	—	4	2	no	—
32	C₂²⋊C₈×C₅₃	—	2	1	no	—
33	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	C₂×D₄×C₅₃	—	4	2	no	—
35	C₂×D₄×C₅₃	—	4	2	no	—
36	C₂×D₄×C₅₃	—	4	2	no	—
37	C₂×D₄×C₅₃	D₈	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	—	2	1	no	—
39	D₈×C₅₃	—	4	2	no	—
40	D₈×C₅₃	—	4	2	no	—
41	D₈×C₅₃	—	8	2	no	—
42	D₈×C₅₃	—	8	4	no	—
43	C₂×C₈×C₅₃	—	2	1	no	—
44	C₂×C₈×C₅₃	—	4	1	no	—
45	C₂×C₈×C₅₃	—	4	1	no	—
46	C₂×C₈×C₅₃	—	4	2	no	—
47	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
49	C₄⋊C₄×C₅₃	D₈	1	1	yes	characteristic
50	C₂²⋊C₄×C₅₃	—	2	1	no	—
51	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₁₆×C₅₃	—	2	1	no	—
53	C₁₆×C₅₃	—	4	1	no	—
54	C₁₆×C₅₃	D₈	1	2	yes	complemented
55	C₂³×C₅₃	—	2	1	no	—
56	C₂³×C₅₃	—	8	2	no	—
57	D₄×C₅₃	—	2	2	no	—
58	D₄×C₅₃	—	8	2	no	—
59	D₄×C₅₃	—	8	2	no	—
60	D₄×C₅₃	—	8	2	no	—
61	D₄×C₅₃	—	8	2	no	—
62	D₄×C₅₃	—	16	2	no	—
63	C₂×C₄×C₅₃	—	2	1	no	—
64	C₂×C₄×C₅₃	—	4	1	no	—
65	C₂×C₄×C₅₃	—	4	1	no	—
66	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
69	C₈×C₅₃	—	2	1	no	—
70	C₈×C₅₃	—	4	1	no	—
71	C₈×C₅₃	—	8	2	no	—
72	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
73	C₈×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
74	#12433	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂²×C₅₃	—	4	1	no	—
76	C₂²×C₅₃	—	4	1	no	—
77	C₂²×C₅₃	—	4	1	no	—
78	C₂²×C₅₃	—	16	2	no	—
79	C₂²×C₅₃	—	16	2	no	—
80	C₂²×C₅₃	—	16	2	no	—
81	C₂²×C₅₃	#466	1	1	yes	characteristic
82	C₄×C₅₃	—	2	1	no	—
83	C₄×C₅₃	—	4	1	no	—
84	C₄×C₅₃	#459	1	1	yes	characteristic
85	C₄×C₅₃	#472	1	1	yes	characteristic
86	C₄×C₅₃	#509	1	2	yes	—
87	#2246	C₂×C₅₃	1	1	yes	characteristic, complemented
88	#2169	C₂×C₅₃	1	1	yes	characteristic, complemented
89	#1667	C₂×C₅₃	1	2	yes	complemented
90	#1372	C₂×C₅₃	1	1	yes	characteristic, complemented
91	#1361	C₂×C₅₃	1	2	yes	complemented
92	C₂×C₅₃	—	8	1	no	—
93	C₂×C₅₃	—	32	2	no	—
94	C₂×C₅₃	#1667	1	1	yes	characteristic
95	C₂×C₅₃	#2218	1	1	yes	characteristic
96	C₂×C₅₃	#2221	1	1	yes	characteristic
97	C₂×D₁₆	—	2	1	no	—
98	C₂×D₁₆	—	2	1	no	—
99	C₂×M₃₂	—	2	1	no	—
100	#493	C₂²×C₅₃	1	2	yes	—
101	#459	—	2	2	no	—
102	C₈×D₄	C₂²×C₅₃	1	1	yes	characteristic
103	C₄⋊C₁₆	C₂²×C₅₃	1	1	yes	characteristic
104	C₂²⋊C₁₆	—	2	1	no	—
105	C₄×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
106	#335	C₂²×C₅₃	1	2	yes	—
107	#331	—	2	2	no	—
108	C₅₃	#12433	1	1	yes	characteristic, direct product factor, Sylow
109	C₂×D₈	—	2	2	no	—
110	C₂×D₈	—	2	2	no	—
111	C₂×D₈	—	4	2	no	—
112	C₂²×C₈	—	2	1	no	—
113	#127	—	2	2	no	—
114	C₄×D₄	D₄×C₅₃	1	1	yes	characteristic
115	D₁₆	—	4	2	no	—
116	D₁₆	—	4	2	no	—
117	M₃₂	—	4	2	no	—
118	C₂×C₁₆	—	2	1	no	—
119	C₂×C₁₆	D₄×C₅₃	1	1	yes	characteristic
120	C₂×C₁₆	D₄×C₅₃	1	1	yes	characteristic
121	C₄⋊C₈	—	2	2	no	—
122	C₄⋊C₈	D₄×C₅₃	1	1	yes	characteristic
123	#102	—	4	2	no	—
124	C₂²⋊C₈	—	2	1	no	—
125	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
126	C₂×D₄	—	4	2	no	—
127	C₂×D₄	—	4	2	no	—
128	C₂×D₄	—	4	2	no	—
129	C₂×D₄	D₈×C₅₃	1	1	yes	characteristic
130	C₂²×C₄	—	2	1	no	—
131	D₈	—	4	2	no	—
132	D₈	—	4	2	no	—
133	D₈	—	8	2	no	—
134	D₈	—	8	4	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	4	1	no	—
137	C₂×C₈	—	4	1	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
140	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
141	C₄⋊C₄	D₈×C₅₃	1	1	yes	characteristic
142	C₂²⋊C₄	—	2	1	no	—
143	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
144	C₁₆	—	2	1	no	—
145	C₁₆	—	4	1	no	—
146	C₁₆	D₈×C₅₃	1	2	yes	complemented
147	C₂³	—	2	1	no	—
148	C₂³	—	8	2	no	—
149	D₄	—	2	2	no	—
150	D₄	—	8	2	no	—
151	D₄	—	8	2	no	—
152	D₄	—	8	2	no	—
153	D₄	—	8	2	no	—
154	D₄	—	16	2	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	4	1	no	—
158	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
159	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
160	C₂×C₄	#135×C₅₃	1	1	yes	characteristic
161	C₈	—	2	1	no	—
162	C₈	—	4	1	no	—
163	C₈	—	8	2	no	—
164	C₈	C₂×D₈×C₅₃	1	1	yes	characteristic
165	C₈	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
166	C₂²	—	4	1	no	—
167	C₂²	—	4	1	no	—
168	C₂²	—	4	1	no	—
169	C₂²	—	16	2	no	—
170	C₂²	—	16	2	no	—
171	C₂²	—	16	2	no	—
172	C₂²	C₅₃×#466	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	4	1	no	—
175	C₄	C₅₃×#459	1	1	yes	characteristic
176	C₄	C₅₃×#472	1	1	yes	characteristic
177	C₄	C₅₃×#509	1	2	yes	—
178	C₂	—	8	1	no	—
179	C₂	—	32	2	no	—
180	C₂	C₅₃×#1667	1	1	yes	characteristic
181	C₂	C₅₃×#2218	1	1	yes	characteristic
182	C₂	C₅₃×#2221	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#2246

C₂

yes

maximal, characteristic, complemented

C₅₃×#2169

C₂

yes

maximal, characteristic, complemented

C₅₃×#1667

C₂

yes

maximal, complemented

C₅₃×#1372

C₂

yes

maximal, characteristic, complemented

C₅₃×#1361

C₂

yes

maximal, complemented

C₂×D₁₆×C₅₃

—

C₂×D₁₆×C₅₃

—

C₂×M₃₂×C₅₃

—

C₅₃×#493

C₂²

yes

—

C₅₃×#459

—

C₈×D₄×C₅₃

C₂²

yes

characteristic

C₅₃×C₄⋊C₁₆

C₂²

yes

characteristic

C₅₃×C₂²⋊C₁₆

—

C₄×C₁₆×C₅₃

C₂²

yes

characteristic

C₅₃×#335

C₂²

yes

—

C₅₃×#331

—

C₂×D₈×C₅₃

—

C₂×D₈×C₅₃

—

C₂×D₈×C₅₃

—

C₂²×C₈×C₅₃

—

#127×C₅₃

—

C₄×D₄×C₅₃

D₄

yes

characteristic

D₁₆×C₅₃

—

D₁₆×C₅₃

—

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

D₄

yes

characteristic

C₂×C₁₆×C₅₃

D₄

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

D₄

yes

characteristic

#102×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

C₂³

yes

characteristic

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

D₈

yes

characteristic

C₂²×C₄×C₅₃

—

D₈×C₅₃

—

D₈×C₅₃

—

D₈×C₅₃

—

D₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

#41

yes

characteristic

C₄⋊C₄×C₅₃

D₈

yes

characteristic

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₁₆×C₅₃

—

C₁₆×C₅₃

—

C₁₆×C₅₃

D₈

yes

complemented

C₂³×C₅₃

—

C₂³×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#121

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

#135

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₂×D₈

yes

C₈×C₅₃

yes

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#466

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

#2246

C₂×C₅₃

yes

characteristic, complemented

#2169

C₂×C₅₃

yes

characteristic, complemented

#1667

C₂×C₅₃

yes

complemented

#1372

C₂×C₅₃

yes

characteristic, complemented

#1361

C₂×C₅₃

yes

complemented

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×D₁₆

—

C₂×D₁₆

—

C₂×M₃₂

—

100

#493

C₂²×C₅₃

yes

—

101

#459

—

102

C₈×D₄

C₂²×C₅₃

yes

characteristic

103

C₄⋊C₁₆

C₂²×C₅₃

yes

characteristic

104

C₂²⋊C₁₆

—

105

C₄×C₁₆

C₂²×C₅₃

yes

characteristic

106

#335

C₂²×C₅₃

yes

—

107

#331

—

108

C₅₃

#12433

yes

characteristic, direct product factor, Sylow

109

C₂×D₈

—

110

C₂×D₈

—

111

C₂×D₈

—

112

C₂²×C₈

—

113

#127

—

114

C₄×D₄

D₄×C₅₃

yes

characteristic

115

D₁₆

—

116

D₁₆

—

117

M₃₂

—

118

C₂×C₁₆

—

119

C₂×C₁₆

D₄×C₅₃

yes

characteristic

120

C₂×C₁₆

D₄×C₅₃

yes

characteristic

121

C₄⋊C₈

—

122

C₄⋊C₈

D₄×C₅₃

yes

characteristic

123

#102

—

124

C₂²⋊C₈

—

125

C₄×C₈

C₂³×C₅₃

yes

characteristic

126

C₂×D₄

—

127

C₂×D₄

—

128

C₂×D₄

—

129

C₂×D₄

D₈×C₅₃

yes

characteristic

130

C₂²×C₄

—

131

D₈

—

132

D₈

—

133

D₈

—

134

D₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

C₂×D₄×C₅₃

yes

characteristic

140

C₂×C₈

#41×C₅₃

yes

characteristic

141

C₄⋊C₄

D₈×C₅₃

yes

characteristic

142

C₂²⋊C₄

—

143

C₄²

C₂×D₄×C₅₃

yes

characteristic

144

C₁₆

—

145

C₁₆

—

146

C₁₆

D₈×C₅₃

yes

complemented

147

C₂³

—

148

C₂³

—

149

D₄

—

150

D₄

—

151

D₄

—

152

D₄

—

153

D₄

—

154

D₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

#121×C₅₃

yes

characteristic

159

C₂×C₄

C₂×D₈×C₅₃

yes

characteristic

160

C₂×C₄

#135×C₅₃

yes

characteristic

161

C₈

—

162

C₈

—

163

C₈

—

164

C₈

C₂×D₈×C₅₃

yes

characteristic

165

C₈

AGL(1,ℤ/8)×C₅₃

yes

characteristic

166

C₂²

—

167

C₂²

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

C₅₃×#466

yes

characteristic

173

C₄

—

174

C₄

—

175

C₄

C₅₃×#459

yes

characteristic

176

C₄

C₅₃×#472

yes

characteristic

177

C₄

C₅₃×#509

yes

—

178

C₂

—

179

C₂

—

180

C₂

C₅₃×#1667

yes

characteristic

181

C₂

C₅₃×#2218

yes

characteristic

182

C₂

C₅₃×#2221

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	#466
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₅₃
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄²×C₅₃ ◁ C₈×D₄×C₅₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₅₃

#466

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₅₃

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₅₃ ◁ C₄²×C₅₃ ◁ C₈×D₄×C₅₃ ◁ G

N	H ≅ G/N	type
1. C₅₃×#2246	C₂	semidirect
2. C₅₃×#2169	C₂	semidirect
3. C₅₃×#1667	C₂	semidirect
4. C₅₃×#1372	C₂	semidirect
5. C₅₃×#1361	C₂	semidirect
54. C₁₆×C₅₃	D₈	semidirect
74. #12433	C₅₃	direct (N × H)
87. #2246	C₂×C₅₃	semidirect
88. #2169	C₂×C₅₃	semidirect
89. #1667	C₂×C₅₃	semidirect
90. #1372	C₂×C₅₃	semidirect
91. #1361	C₂×C₅₃	semidirect
108. C₅₃	#12433	direct (N × H)
146. C₁₆	D₈×C₅₃	semidirect

H ≅ G/N

type

1. C₅₃×#2246

C₂

semidirect

2. C₅₃×#2169

C₂

semidirect

3. C₅₃×#1667

C₂

semidirect

4. C₅₃×#1372

C₂

semidirect

5. C₅₃×#1361

C₂

semidirect

54. C₁₆×C₅₃

D₈

semidirect

74. #12433

C₅₃

direct (N × H)

87. #2246

C₂×C₅₃

semidirect

88. #2169

C₂×C₅₃

semidirect

89. #1667

C₂×C₅₃

semidirect

90. #1372

C₂×C₅₃

semidirect

91. #1361

C₂×C₅₃

semidirect

108. C₅₃

#12433

direct (N × H)

146. C₁₆

D₈×C₅₃

semidirect

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

subgroup	index
C₅₃×#2246	2
C₅₃×#2169	2
C₅₃×#1667	2
C₅₃×#1372	2
C₅₃×#1361	2
#12433	53

subgroup

index

C₅₃×#2246

C₅₃×#2169

C₅₃×#1667

C₅₃×#1372

C₅₃×#1361

#12433

Subgroups — C₅₃×#12433 #27219945

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#12433 #27219945

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12433 #27219945