Group #27217288 — subgroups

Subgroups — C₅₃×#9776 #27217288

Order 13568.

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

normal subgroups only

This group has 390 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×C₅₃×#498	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1837	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1549	C₂	1	2	yes	maximal
4	C₅₃×#1548	C₂	1	2	yes	maximal
5	C₅₃×#1452	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
7	C₅₃×#498	—	2	2	no	—
8	C₅₃×#498	C₄	1	4	yes	—
9	C₅₃×#431	C₂²	1	2	yes	—
10	C₂×#106×C₅₃	—	2	2	no	—
11	C₅₃×#423	C₂²	1	1	yes	characteristic
12	C₅₃×#423	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#341	—	2	2	no	—
15	C₅₃×#333	—	2	4	no	—
16	C₅₃×#332	—	2	4	no	—
17	C₅₃×#322	—	2	2	no	—
18	C₂²×Q₈×C₅₃	—	2	2	no	—
19	C₂×M₁₆×C₅₃	—	2	1	no	—
20	C₂×M₁₆×C₅₃	—	2	1	no	—
21	C₂²×C₈×C₅₃	D₄	1	2	yes	—
22	C₄⋊Q₈×C₅₃	—	2	2	no	—
23	C₄⋊Q₈×C₅₃	—	2	2	no	—
24	C₄⋊Q₈×C₅₃	—	2	2	no	—
25	C₄⋊Q₈×C₅₃	C₂×C₄	1	4	yes	—
26	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
27	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
28	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
29	#106×C₅₃	—	2	8	no	—
30	C₄⋊C₈×C₅₃	—	2	2	no	—
31	C₄⋊C₈×C₅₃	—	2	2	no	—
32	C₄⋊C₈×C₅₃	D₄	1	4	yes	—
33	C₂²⋊C₈×C₅₃	—	2	2	no	—
34	C₄×C₈×C₅₃	—	2	2	no	—
35	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₅₃	D₄	1	2	yes	—
37	C₂×Q₈×C₅₃	—	2	4	no	—
38	C₂×Q₈×C₅₃	—	4	2	no	—
39	C₂×Q₈×C₅₃	—	8	4	no	—
40	C₂²×C₄×C₅₃	—	4	2	no	—
41	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	#41	1	2	yes	—
43	M₁₆×C₅₃	—	4	2	no	—
44	M₁₆×C₅₃	—	4	2	no	—
45	C₂×C₈×C₅₃	—	2	2	no	—
46	C₂×C₈×C₅₃	—	2	2	no	—
47	C₂×C₈×C₅₃	—	2	4	no	—
48	C₂×C₈×C₅₃	—	4	1	no	—
49	C₂×C₈×C₅₃	—	4	1	no	—
50	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
51	C₄⋊C₄×C₅₃	—	2	4	no	—
52	C₄⋊C₄×C₅₃	—	2	4	no	—
53	C₄⋊C₄×C₅₃	—	4	2	no	—
54	C₄⋊C₄×C₅₃	—	4	2	no	—
55	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₅₃	C₂×D₄	1	2	yes	—
58	C₂³×C₅₃	#124	1	1	yes	characteristic
59	Q₈×C₅₃	—	8	4	no	—
60	Q₈×C₅₃	—	8	4	no	—
61	C₂×C₄×C₅₃	—	2	4	no	—
62	C₂×C₄×C₅₃	—	4	4	no	—
63	C₂×C₄×C₅₃	—	8	4	no	—
64	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
66	C₂×C₄×C₅₃	#121	1	2	yes	—
67	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
68	C₂×C₄×C₅₃	#137	1	2	yes	—
69	C₂×C₄×C₅₃	#137	1	2	yes	—
70	C₈×C₅₃	—	2	4	no	—
71	C₈×C₅₃	—	4	4	no	—
72	C₈×C₅₃	—	8	1	no	—
73	C₈×C₅₃	—	8	1	no	—
74	#9776	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂²×C₅₃	—	2	2	no	—
76	C₂²×C₅₃	#390	1	1	yes	characteristic
77	C₂²×C₅₃	#491	1	2	yes	—
78	C₄×C₅₃	—	2	2	no	—
79	C₄×C₅₃	—	8	4	no	—
80	C₄×C₅₃	#419	1	2	yes	—
81	C₄×C₅₃	#471	1	4	yes	—
82	C₄×C₅₃	#497	1	1	yes	characteristic
83	C₄×C₅₃	#497	1	1	yes	characteristic
84	C₂×#498	C₂×C₅₃	1	1	yes	characteristic
85	#1837	C₂×C₅₃	1	1	yes	characteristic
86	#1549	C₂×C₅₃	1	2	yes	—
87	#1548	C₂×C₅₃	1	2	yes	—
88	#1452	C₂×C₅₃	1	1	yes	characteristic
89	C₂×C₅₃	—	2	2	no	—
90	C₂×C₅₃	#1960	1	1	yes	characteristic
91	C₂×C₅₃	#1977	1	2	yes	—
92	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
93	#498	—	2	2	no	—
94	#498	C₄×C₅₃	1	4	yes	—
95	#431	C₂²×C₅₃	1	2	yes	—
96	C₂×#106	—	2	2	no	—
97	#423	C₂²×C₅₃	1	1	yes	characteristic
98	#423	C₂²×C₅₃	1	1	yes	characteristic
99	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
100	#341	—	2	2	no	—
101	#333	—	2	4	no	—
102	#332	—	2	4	no	—
103	#322	—	2	2	no	—
104	C₅₃	#9776	1	1	yes	characteristic, direct product factor, Sylow
105	C₂²×Q₈	—	2	2	no	—
106	C₂×M₁₆	—	2	1	no	—
107	C₂×M₁₆	—	2	1	no	—
108	C₂²×C₈	D₄×C₅₃	1	2	yes	—
109	C₄⋊Q₈	—	2	2	no	—
110	C₄⋊Q₈	—	2	2	no	—
111	C₄⋊Q₈	—	2	2	no	—
112	C₄⋊Q₈	C₂×C₄×C₅₃	1	4	yes	—
113	C₂×C₄⋊C₄	—	2	2	no	—
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
116	#106	—	2	8	no	—
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	2	2	no	—
119	C₄⋊C₈	D₄×C₅₃	1	4	yes	—
120	C₂²⋊C₈	—	2	2	no	—
121	C₄×C₈	—	2	2	no	—
122	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
123	C₄×C₈	D₄×C₅₃	1	2	yes	—
124	C₂×Q₈	—	2	4	no	—
125	C₂×Q₈	—	4	2	no	—
126	C₂×Q₈	—	8	4	no	—
127	C₂²×C₄	—	4	2	no	—
128	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
129	C₂²×C₄	#41×C₅₃	1	2	yes	—
130	M₁₆	—	4	2	no	—
131	M₁₆	—	4	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	4	no	—
135	C₂×C₈	—	4	1	no	—
136	C₂×C₈	—	4	1	no	—
137	C₂×C₈	C₂²⋊C₄×C₅₃	1	4	yes	—
138	C₄⋊C₄	—	2	4	no	—
139	C₄⋊C₄	—	2	4	no	—
140	C₄⋊C₄	—	4	2	no	—
141	C₄⋊C₄	—	4	2	no	—
142	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
143	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
144	C₄²	C₂×D₄×C₅₃	1	2	yes	—
145	C₂³	#124×C₅₃	1	1	yes	characteristic
146	Q₈	—	8	4	no	—
147	Q₈	—	8	4	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	4	4	no	—
150	C₂×C₄	—	8	4	no	—
151	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
152	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
153	C₂×C₄	#121×C₅₃	1	2	yes	—
154	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
155	C₂×C₄	#137×C₅₃	1	2	yes	—
156	C₂×C₄	#137×C₅₃	1	2	yes	—
157	C₈	—	2	4	no	—
158	C₈	—	4	4	no	—
159	C₈	—	8	1	no	—
160	C₈	—	8	1	no	—
161	C₂²	—	2	2	no	—
162	C₂²	C₅₃×#390	1	1	yes	characteristic
163	C₂²	C₅₃×#491	1	2	yes	—
164	C₄	—	2	2	no	—
165	C₄	—	8	4	no	—
166	C₄	C₅₃×#419	1	2	yes	—
167	C₄	C₅₃×#471	1	4	yes	—
168	C₄	C₅₃×#497	1	1	yes	characteristic
169	C₄	C₅₃×#497	1	1	yes	characteristic
170	C₂	—	2	2	no	—
171	C₂	C₅₃×#1960	1	1	yes	characteristic
172	C₂	C₅₃×#1977	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅₃×#498

C₂

yes

maximal, characteristic

C₅₃×#1837

C₂

yes

maximal, characteristic

C₅₃×#1549

C₂

yes

maximal

C₅₃×#1548

C₂

yes

maximal

C₅₃×#1452

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₅₃

C₂²

yes

—

C₅₃×#498

—

C₅₃×#498

C₄

yes

—

C₅₃×#431

C₂²

yes

—

C₂×#106×C₅₃

—

C₅₃×#423

C₂²

yes

characteristic

C₅₃×#423

C₂²

yes

characteristic

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#341

—

C₅₃×#333

—

C₅₃×#332

—

C₅₃×#322

—

C₂²×Q₈×C₅₃

—

C₂×M₁₆×C₅₃

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#106×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

D₄

yes

—

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×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

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

—

C₂³×C₅₃

#124

yes

characteristic

Q₈×C₅₃

—

Q₈×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

#121

yes

—

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

#137

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#9776

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

#390

yes

characteristic

C₂²×C₅₃

#491

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#419

yes

—

C₄×C₅₃

#471

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₂×C₅₃

yes

characteristic

#1837

C₂×C₅₃

yes

characteristic

#1549

C₂×C₅₃

yes

—

#1548

C₂×C₅₃

yes

—

#1452

C₂×C₅₃

yes

characteristic

C₂×C₅₃

—

C₂×C₅₃

#1960

yes

characteristic

C₂×C₅₃

#1977

yes

—

C₂×C₄⋊Q₈

C₂²×C₅₃

yes

—

#498

—

#498

C₄×C₅₃

yes

—

#431

C₂²×C₅₃

yes

—

C₂×#106

—

#423

C₂²×C₅₃

yes

characteristic

#423

C₂²×C₅₃

yes

characteristic

C₂×C₄×C₈

C₂²×C₅₃

yes

characteristic

100

#341

—

101

#333

—

102

#332

—

103

#322

—

104

C₅₃

#9776

yes

characteristic, direct product factor, Sylow

105

C₂²×Q₈

—

106

C₂×M₁₆

—

107

C₂×M₁₆

—

108

C₂²×C₈

D₄×C₅₃

yes

—

109

C₄⋊Q₈

—

110

C₄⋊Q₈

—

111

C₄⋊Q₈

—

112

C₄⋊Q₈

C₂×C₄×C₅₃

yes

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄²

C₂³×C₅₃

yes

characteristic

116

#106

—

117

C₄⋊C₈

—

118

C₄⋊C₈

—

119

C₄⋊C₈

D₄×C₅₃

yes

—

120

C₂²⋊C₈

—

121

C₄×C₈

—

122

C₄×C₈

C₂×C₄×C₅₃

yes

—

123

C₄×C₈

D₄×C₅₃

yes

—

124

C₂×Q₈

—

125

C₂×Q₈

—

126

C₂×Q₈

—

127

C₂²×C₄

—

128

C₂²×C₄

C₂×D₄×C₅₃

yes

characteristic

129

C₂²×C₄

#41×C₅₃

yes

—

130

M₁₆

—

131

M₁₆

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

C₂²⋊C₄×C₅₃

yes

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄²

C₂²×C₄×C₅₃

yes

characteristic

143

C₄²

C₂×D₄×C₅₃

yes

characteristic

144

C₄²

C₂×D₄×C₅₃

yes

—

145

C₂³

#124×C₅₃

yes

characteristic

146

Q₈

—

147

Q₈

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

C₂×C₂²⋊C₄×C₅₃

yes

characteristic

152

C₂×C₄

C₄×D₄×C₅₃

yes

—

153

C₂×C₄

#121×C₅₃

yes

—

154

C₂×C₄

#127×C₅₃

yes

characteristic

155

C₂×C₄

#137×C₅₃

yes

—

156

C₂×C₄

#137×C₅₃

yes

—

157

C₈

—

158

C₈

—

159

C₈

—

160

C₈

—

161

C₂²

—

162

C₂²

C₅₃×#390

yes

characteristic

163

C₂²

C₅₃×#491

yes

—

164

C₄

—

165

C₄

—

166

C₄

C₅₃×#419

yes

—

167

C₄

C₅₃×#471

yes

—

168

C₄

C₅₃×#497

yes

characteristic

169

C₄

C₅₃×#497

yes

characteristic

170

C₂

—

171

C₂

C₅₃×#1960

yes

characteristic

172

C₂

C₅₃×#1977

yes

—

	subgroup	G/H
center, socle	C₂²×C₅₃	#390
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂³×C₅₃	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₅₃

#390

commutator

C₄²

C₂²×C₄×C₅₃

Frattini, cosocle

C₂×C₄²

C₂³×C₅₃

C₂³×C₅₃

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

N	H ≅ G/N	type
74. #9776	C₅₃	direct (N × H)
104. C₅₃	#9776	direct (N × H)

H ≅ G/N

type

74. #9776

C₅₃

direct (N × H)

104. C₅₃

#9776

direct (N × H)

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

subgroup	index
C₂×C₅₃×#498	2
C₅₃×#1837	2
C₅₃×#1549	2
C₅₃×#1548	2
C₅₃×#1452	2
#9776	53

subgroup

index

C₂×C₅₃×#498

C₅₃×#1837

C₅₃×#1549

C₅₃×#1548

C₅₃×#1452

#9776

Subgroups — C₅₃×#9776 #27217288

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#9776 #27217288

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9776 #27217288