Group #27217284 — subgroups

Subgroups — C₅₃×#9772 #27217284

Order 13568.

478 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 174 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×#494	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1848	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1546	C₂	1	4	yes	maximal
4	C₅₃×#1450	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₁₆×C₅₃	—	2	2	no	—
6	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
7	C₅₃×#494	—	2	1	no	—
8	C₅₃×#494	—	2	1	no	—
9	C₅₃×#494	C₄	1	4	yes	—
10	C₅₃×#431	C₂²	1	2	yes	—
11	C₂×#108×C₅₃	—	2	2	no	—
12	C₅₃×#423	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#332	—	2	8	no	—
15	C₅₃×#322	—	2	1	no	—
16	C₈²×C₅₃	—	2	1	no	—
17	C₂²×Q₈×C₅₃	—	2	2	no	—
18	C₂×Q₁₆×C₅₃	—	2	4	no	—
19	C₂×Q₁₆×C₅₃	—	2	4	no	—
20	C₂×Q₁₆×C₅₃	—	4	4	no	—
21	C₂×Q₁₆×C₅₃	—	4	4	no	—
22	C₂×M₁₆×C₅₃	—	2	2	no	—
23	C₂²×C₈×C₅₃	D₄	1	2	yes	—
24	C₄⋊Q₈×C₅₃	—	2	2	no	—
25	C₄⋊Q₈×C₅₃	—	2	4	no	—
26	C₄⋊Q₈×C₅₃	C₂×C₄	1	4	yes	—
27	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
28	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
29	#108×C₅₃	—	4	4	no	—
30	C₄⋊C₈×C₅₃	—	2	4	no	—
31	C₄⋊C₈×C₅₃	D₄	1	4	yes	—
32	C₂²⋊C₈×C₅₃	—	2	2	no	—
33	C₄×C₈×C₅₃	—	2	2	no	—
34	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
36	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
37	C₂×Q₈×C₅₃	—	2	4	no	—
38	C₂×Q₈×C₅₃	—	4	2	no	—
39	C₂×Q₈×C₅₃	—	4	4	no	—
40	C₂×Q₈×C₅₃	—	4	4	no	—
41	C₂²×C₄×C₅₃	—	4	2	no	—
42	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	#41	1	2	yes	—
44	Q₁₆×C₅₃	—	4	8	no	—
45	Q₁₆×C₅₃	—	4	8	no	—
46	M₁₆×C₅₃	—	4	4	no	—
47	C₂×C₈×C₅₃	—	2	4	no	—
48	C₂×C₈×C₅₃	—	4	2	no	—
49	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
50	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
51	C₂×C₈×C₅₃	D₈	1	4	yes	—
52	C₂×C₈×C₅₃	SD₁₆	1	4	yes	—
53	C₄⋊C₄×C₅₃	—	2	8	no	—
54	C₄⋊C₄×C₅₃	—	4	4	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	—	4	8	no	—
60	Q₈×C₅₃	—	8	2	no	—
61	Q₈×C₅₃	—	8	2	no	—
62	C₂×C₄×C₅₃	—	2	4	no	—
63	C₂×C₄×C₅₃	—	4	4	no	—
64	C₂×C₄×C₅₃	—	8	4	no	—
65	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
67	C₂×C₄×C₅₃	#121	1	2	yes	—
68	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
70	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
71	C₈×C₅₃	—	4	4	no	—
72	C₈×C₅₃	—	8	2	no	—
73	C₈×C₅₃	#102	1	4	yes	—
74	C₈×C₅₃	#102	1	4	yes	—
75	#9772	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂²×C₅₃	—	2	2	no	—
77	C₂²×C₅₃	#390	1	1	yes	characteristic
78	C₂²×C₅₃	#486	1	2	yes	—
79	C₄×C₅₃	—	2	2	no	—
80	C₄×C₅₃	—	8	4	no	—
81	C₄×C₅₃	C₂×#102	1	2	yes	—
82	C₄×C₅₃	#473	1	4	yes	—
83	C₄×C₅₃	#492	1	1	yes	characteristic
84	C₄×C₅₃	#493	1	1	yes	characteristic
85	C₂×#494	C₂×C₅₃	1	1	yes	characteristic
86	#1848	C₂×C₅₃	1	1	yes	characteristic
87	#1546	C₂×C₅₃	1	4	yes	—
88	#1450	C₂×C₅₃	1	1	yes	characteristic
89	C₂×C₅₃	—	2	2	no	—
90	C₂×C₅₃	#1957	1	1	yes	characteristic
91	C₂×C₅₃	#1979	1	2	yes	—
92	C₂²×Q₁₆	—	2	2	no	—
93	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
94	#494	—	2	1	no	—
95	#494	—	2	1	no	—
96	#494	C₄×C₅₃	1	4	yes	—
97	#431	C₂²×C₅₃	1	2	yes	—
98	C₂×#108	—	2	2	no	—
99	#423	C₂²×C₅₃	1	2	yes	—
100	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
101	#332	—	2	8	no	—
102	#322	—	2	1	no	—
103	C₈²	—	2	1	no	—
104	C₅₃	#9772	1	1	yes	characteristic, direct product factor, Sylow
105	C₂²×Q₈	—	2	2	no	—
106	C₂×Q₁₆	—	2	4	no	—
107	C₂×Q₁₆	—	2	4	no	—
108	C₂×Q₁₆	—	4	4	no	—
109	C₂×Q₁₆	—	4	4	no	—
110	C₂×M₁₆	—	2	2	no	—
111	C₂²×C₈	D₄×C₅₃	1	2	yes	—
112	C₄⋊Q₈	—	2	4	no	—
113	C₄⋊Q₈	—	2	2	no	—
114	C₄⋊Q₈	C₂×C₄×C₅₃	1	4	yes	—
115	C₂×C₄⋊C₄	—	2	4	no	—
116	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
117	#108	—	4	4	no	—
118	C₄⋊C₈	—	2	4	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	1	yes	characteristic
124	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
125	C₂×Q₈	—	2	4	no	—
126	C₂×Q₈	—	4	2	no	—
127	C₂×Q₈	—	4	4	no	—
128	C₂×Q₈	—	4	4	no	—
129	C₂²×C₄	—	4	2	no	—
130	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
131	C₂²×C₄	#41×C₅₃	1	2	yes	—
132	Q₁₆	—	4	8	no	—
133	Q₁₆	—	4	8	no	—
134	M₁₆	—	4	4	no	—
135	C₂×C₈	—	2	4	no	—
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
138	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
139	C₂×C₈	D₈×C₅₃	1	4	yes	—
140	C₂×C₈	SD₁₆×C₅₃	1	4	yes	—
141	C₄⋊C₄	—	2	8	no	—
142	C₄⋊C₄	—	4	4	no	—
143	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
144	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
145	C₄²	C₂×D₄×C₅₃	1	2	yes	—
146	C₂³	#124×C₅₃	1	1	yes	characteristic
147	Q₈	—	4	8	no	—
148	Q₈	—	8	2	no	—
149	Q₈	—	8	2	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	4	4	no	—
152	C₂×C₄	—	8	4	no	—
153	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
154	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
155	C₂×C₄	#121×C₅₃	1	2	yes	—
156	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
157	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
158	C₂×C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
159	C₈	—	4	4	no	—
160	C₈	—	8	2	no	—
161	C₈	#102×C₅₃	1	4	yes	—
162	C₈	#102×C₅₃	1	4	yes	—
163	C₂²	—	2	2	no	—
164	C₂²	C₅₃×#390	1	1	yes	characteristic
165	C₂²	C₅₃×#486	1	2	yes	—
166	C₄	—	2	2	no	—
167	C₄	—	8	4	no	—
168	C₄	C₂×#102×C₅₃	1	2	yes	—
169	C₄	C₅₃×#473	1	4	yes	—
170	C₄	C₅₃×#492	1	1	yes	characteristic
171	C₄	C₅₃×#493	1	1	yes	characteristic
172	C₂	—	2	2	no	—
173	C₂	C₅₃×#1957	1	1	yes	characteristic
174	C₂	C₅₃×#1979	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅₃×#494

C₂

yes

maximal, characteristic

C₅₃×#1848

C₂

yes

maximal, characteristic

C₅₃×#1546

C₂

yes

maximal

C₅₃×#1450

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₅₃

—

C₂×C₄⋊Q₈×C₅₃

C₂²

yes

—

C₅₃×#494

—

C₅₃×#494

—

C₅₃×#494

C₄

yes

—

C₅₃×#431

C₂²

yes

—

C₂×#108×C₅₃

—

C₅₃×#423

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#332

—

C₅₃×#322

—

C₈²×C₅₃

—

C₂²×Q₈×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#108×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

characteristic

C₄×C₈×C₅₃

D₄

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

SD₁₆

yes

—

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

—

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

C₂×D₈

yes

—

C₂×C₄×C₅₃

C₂×SD₁₆

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#102

yes

—

C₈×C₅₃

#102

yes

—

#9772

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

#390

yes

characteristic

C₂²×C₅₃

#486

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×#102

yes

—

C₄×C₅₃

#473

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₂×C₅₃

yes

characteristic

#1848

C₂×C₅₃

yes

characteristic

#1546

C₂×C₅₃

yes

—

#1450

C₂×C₅₃

yes

characteristic

C₂×C₅₃

—

C₂×C₅₃

#1957

yes

characteristic

C₂×C₅₃

#1979

yes

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

C₂²×C₅₃

yes

—

#494

—

#494

—

#494

C₄×C₅₃

yes

—

#431

C₂²×C₅₃

yes

—

C₂×#108

—

#423

C₂²×C₅₃

yes

—

100

C₂×C₄×C₈

C₂²×C₅₃

yes

characteristic

101

#332

—

102

#322

—

103

C₈²

—

104

C₅₃

#9772

yes

characteristic, direct product factor, Sylow

105

C₂²×Q₈

—

106

C₂×Q₁₆

—

107

C₂×Q₁₆

—

108

C₂×Q₁₆

—

109

C₂×Q₁₆

—

110

C₂×M₁₆

—

111

C₂²×C₈

D₄×C₅₃

yes

—

112

C₄⋊Q₈

—

113

C₄⋊Q₈

—

114

C₄⋊Q₈

C₂×C₄×C₅₃

yes

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄²

C₂³×C₅₃

yes

characteristic

117

#108

—

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

characteristic

124

C₄×C₈

D₄×C₅₃

yes

characteristic

125

C₂×Q₈

—

126

C₂×Q₈

—

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂²×C₄

—

130

C₂²×C₄

C₂×D₄×C₅₃

yes

characteristic

131

C₂²×C₄

#41×C₅₃

yes

—

132

Q₁₆

—

133

Q₁₆

—

134

M₁₆

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

C₂²⋊C₄×C₅₃

yes

—

138

C₂×C₈

C₂²⋊C₄×C₅₃

yes

—

139

C₂×C₈

D₈×C₅₃

yes

—

140

C₂×C₈

SD₁₆×C₅₃

yes

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄²

C₂²×C₄×C₅₃

yes

characteristic

144

C₄²

C₂×D₄×C₅₃

yes

characteristic

145

C₄²

C₂×D₄×C₅₃

yes

—

146

C₂³

#124×C₅₃

yes

characteristic

147

Q₈

—

148

Q₈

—

149

Q₈

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

C₂×C₂²⋊C₄×C₅₃

yes

characteristic

154

C₂×C₄

C₄×D₄×C₅₃

yes

—

155

C₂×C₄

#121×C₅₃

yes

—

156

C₂×C₄

#127×C₅₃

yes

characteristic

157

C₂×C₄

C₂×D₈×C₅₃

yes

—

158

C₂×C₄

C₂×SD₁₆×C₅₃

yes

—

159

C₈

—

160

C₈

—

161

C₈

#102×C₅₃

yes

—

162

C₈

#102×C₅₃

yes

—

163

C₂²

—

164

C₂²

C₅₃×#390

yes

characteristic

165

C₂²

C₅₃×#486

yes

—

166

C₄

—

167

C₄

—

168

C₄

C₂×#102×C₅₃

yes

—

169

C₄

C₅₃×#473

yes

—

170

C₄

C₅₃×#492

yes

characteristic

171

C₄

C₅₃×#493

yes

characteristic

172

C₂

—

173

C₂

C₅₃×#1957

yes

characteristic

174

C₂

C₅₃×#1979

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
75. #9772	C₅₃	direct (N × H)
104. C₅₃	#9772	direct (N × H)

H ≅ G/N

type

75. #9772

C₅₃

direct (N × H)

104. C₅₃

#9772

direct (N × H)

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

subgroup	index
C₂×C₅₃×#494	2
C₅₃×#1848	2
C₅₃×#1546	2
C₅₃×#1450	2
#9772	53

subgroup

index

C₂×C₅₃×#494

C₅₃×#1848

C₅₃×#1546

C₅₃×#1450

#9772

Subgroups — C₅₃×#9772 #27217284

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#9772 #27217284

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9772 #27217284