Group #27283589 — subgroups

Subgroups — #27283589

Order 13568.

390 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 127 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	#20274853	C₂	1	1	yes	maximal, characteristic
2	#20273400	C₂	1	2	yes	maximal
3	#20273396	C₂	1	2	yes	maximal
4	#20273007	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₃×#500	C₂	1	1	yes	maximal, characteristic
6	#16810676	C₂²	1	1	yes	characteristic
7	#16810676	C₂²	1	1	yes	characteristic
8	#16810647	C₂²	1	2	yes	—
9	#16810388	—	2	2	no	—
10	#16810372	—	2	4	no	—
11	#16810369	—	2	4	no	—
12	#16810345	—	2	2	no	—
13	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
14	C₅₃×#500	—	2	2	no	—
15	C₅₃×#500	C₄	1	4	yes	—
16	C₂×#107×C₅₃	—	2	2	no	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×#11719917	—	2	2	no	—
19	C₄⋊#88322	—	2	2	no	—
20	C₄⋊#88322	—	2	2	no	—
21	C₄⋊#88322	D₄	1	2	yes	—
22	C₄⋊#88322	D₄	1	2	yes	—
23	C₂²⋊#88322	—	2	1	no	—
24	C₂²⋊#88322	—	2	1	no	—
25	C₄×#88322	—	2	1	no	—
26	C₄×#88322	—	2	1	no	—
27	C₂²×Q₈×C₅₃	—	2	2	no	—
28	C₂²×C₈×C₅₃	D₄	1	2	yes	—
29	C₄⋊Q₈×C₅₃	—	2	2	no	—
30	C₄⋊Q₈×C₅₃	—	2	2	no	—
31	C₄⋊Q₈×C₅₃	—	2	2	no	—
32	C₄⋊Q₈×C₅₃	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
35	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
36	#107×C₅₃	—	2	8	no	—
37	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₅₃	D₄	1	2	yes	—
39	C₂×Q₈×C₅₃	—	2	4	no	—
40	C₂×Q₈×C₅₃	—	4	2	no	—
41	C₂×Q₈×C₅₃	—	8	4	no	—
42	C₂²×C₄×C₅₃	—	4	2	no	—
43	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
46	C₂×C₈×C₅₃	—	2	4	no	—
47	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
48	C₄⋊C₄×C₅₃	—	2	4	no	—
49	C₄⋊C₄×C₅₃	—	2	4	no	—
50	C₄⋊C₄×C₅₃	—	4	2	no	—
51	C₄⋊C₄×C₅₃	—	4	2	no	—
52	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
56	#11719917	—	4	4	no	—
57	C₂×#88322	—	2	2	no	—
58	C₂×#88322	—	2	2	no	—
59	C₂×#88322	—	4	2	no	—
60	C₂³×C₅₃	#124	1	1	yes	characteristic
61	Q₈×C₅₃	—	8	4	no	—
62	Q₈×C₅₃	—	8	4	no	—
63	C₂×C₄×C₅₃	—	2	2	no	—
64	C₂×C₄×C₅₃	—	2	2	no	—
65	C₂×C₄×C₅₃	—	4	4	no	—
66	C₂×C₄×C₅₃	—	8	4	no	—
67	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
71	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
72	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
73	C₂×C₄×C₅₃	#137	1	4	yes	—
74	C₈×C₅₃	—	2	4	no	—
75	#88322	—	4	2	no	—
76	#88322	—	4	2	no	—
77	#88322	—	8	2	no	—
78	#9778	—	53	1	no	maximal, Sylow
79	C₂²×C₅₃	—	2	1	no	—
80	C₂²×C₅₃	—	2	1	no	—
81	C₂²×C₅₃	#390	1	1	yes	characteristic
82	C₂²×C₅₃	#491	1	2	yes	—
83	C₄×C₅₃	—	2	1	no	—
84	C₄×C₅₃	—	2	1	no	—
85	C₄×C₅₃	—	8	4	no	—
86	C₄×C₅₃	#419	1	2	yes	—
87	C₄×C₅₃	#472	1	2	yes	—
88	C₄×C₅₃	#473	1	2	yes	—
89	C₄×C₅₃	#497	1	2	yes	—
90	C₂×#500	D₅₃	1	1	yes	characteristic
91	#1837	—	53	1	no	—
92	#1549	—	53	2	no	—
93	#1548	—	53	2	no	—
94	#1452	—	53	1	no	—
95	C₂×C₅₃	—	2	2	no	—
96	C₂×C₅₃	#1960	1	1	yes	characteristic
97	C₂×C₅₃	#1978	1	1	yes	characteristic
98	C₂×C₅₃	#1979	1	1	yes	characteristic
99	C₂×C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
100	#500	—	2	2	no	—
101	#500	Dic₅₃	1	4	yes	—
102	#431	—	53	1	no	—
103	#431	—	53	1	no	—
104	C₂×#107	—	2	2	no	—
105	#423	—	53	2	no	—
106	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
107	#341	—	106	2	no	—
108	#333	—	106	4	no	—
109	#332	—	106	4	no	—
110	#322	—	106	2	no	—
111	C₅₃	#9778	1	1	yes	characteristic, complemented, Sylow
112	C₂²×Q₈	—	2	2	no	—
113	C₂×M₁₆	—	106	2	no	—
114	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
115	C₄⋊Q₈	—	2	2	no	—
116	C₄⋊Q₈	—	2	2	no	—
117	C₄⋊Q₈	—	2	2	no	—
118	C₄⋊Q₈	C₂×Dic₅₃	1	4	yes	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
122	#107	—	2	8	no	—
123	C₄⋊C₈	—	53	2	no	—
124	C₄⋊C₈	—	53	2	no	—
125	C₄⋊C₈	—	106	2	no	—
126	C₄⋊C₈	—	106	2	no	—
127	C₂²⋊C₈	—	106	1	no	—
128	C₂²⋊C₈	—	106	1	no	—
129	C₄×C₈	—	106	1	no	—
130	C₄×C₈	—	106	1	no	—
131	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
132	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
133	C₂×Q₈	—	2	4	no	—
134	C₂×Q₈	—	4	2	no	—
135	C₂×Q₈	—	8	4	no	—
136	C₂²×C₄	—	4	2	no	—
137	C₂²×C₄	#11719947	1	1	yes	characteristic
138	C₂²×C₄	#11719947	1	1	yes	characteristic
139	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
140	M₁₆	—	212	4	no	—
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	—	106	2	no	—
143	C₂×C₈	—	106	2	no	—
144	C₂×C₈	—	212	2	no	—
145	C₂×C₈	C₂²⋊Dic₅₃	1	4	yes	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	—	2	4	no	—
148	C₄⋊C₄	—	4	2	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄²	D₄×D₅₃	1	1	yes	characteristic
151	C₄²	D₄×D₅₃	1	1	yes	characteristic
152	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
153	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
154	C₂³	#13133204	1	1	yes	characteristic
155	Q₈	—	8	4	no	—
156	Q₈	—	8	4	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	4	4	no	—
160	C₂×C₄	—	8	4	no	—
161	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
162	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
163	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
164	C₂×C₄	#13133188	1	1	yes	characteristic
165	C₂×C₄	#13133188	1	1	yes	characteristic
166	C₂×C₄	#13133213	1	1	yes	characteristic
167	C₂×C₄	#13133248	1	4	yes	—
168	C₈	—	2	4	no	—
169	C₈	—	424	2	no	—
170	C₈	—	212	2	no	—
171	C₈	—	212	2	no	—
172	C₂²	—	2	1	no	—
173	C₂²	—	2	1	no	—
174	C₂²	#16810526	1	1	yes	characteristic
175	C₂²	#16810991	1	2	yes	—
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	—	8	4	no	—
179	C₄	#16810632	1	2	yes	—
180	C₄	#16810890	1	2	yes	—
181	C₄	#16810895	1	2	yes	—
182	C₄	#16811015	1	2	yes	—
183	C₂	—	2	2	no	—
184	C₂	#20275536	1	1	yes	characteristic
185	C₂	#20275626	1	1	yes	characteristic
186	C₂	#20275631	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20274853

C₂

yes

maximal, characteristic

#20273400

C₂

yes

maximal

#20273396

C₂

yes

maximal

#20273007

C₂

yes

maximal, characteristic

C₂×C₅₃×#500

C₂

yes

maximal, characteristic

#16810676

C₂²

yes

characteristic

#16810676

C₂²

yes

characteristic

#16810647

C₂²

yes

—

—

—

—

—

C₂×C₄⋊Q₈×C₅₃

C₂²

yes

—

C₅₃×#500

—

C₅₃×#500

C₄

yes

—

C₂×#107×C₅₃

—

C₂×C₄×C₈×C₅₃

C₂²

yes

—

—

—

D₄

yes

—

C₄⋊#88322

D₄

yes

—

—

—

—

—

C₂²×Q₈×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

#107×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

C₂²×C₄×C₅₃

yes

C₂²×C₄×C₅₃

yes

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

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×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₈×C₅₃

—

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#390

yes

characteristic

C₂²×C₅₃

#491

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#419

yes

—

C₄×C₅₃

#472

yes

—

C₄×C₅₃

#473

yes

—

C₄×C₅₃

#497

yes

—

C₂×#500

D₅₃

yes

—

—

—

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄⋊Q₈

C₂×D₅₃

yes

—

100

#500

—

101

#500

Dic₅₃

yes

—

102

#431

—

103

#431

—

104

C₂×#107

—

105

#423

—

106

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

107

#341

—

106

—

108

#333

—

106

—

109

#332

—

106

—

110

#322

—

106

—

111

C₅₃

#9778

yes

characteristic, complemented, Sylow

112

C₂²×Q₈

—

113

C₂×M₁₆

—

106

—

114

C₂²×C₈

C₂²⋊D₅₃

yes

—

115

C₄⋊Q₈

—

116

C₄⋊Q₈

—

117

C₄⋊Q₈

—

118

C₄⋊Q₈

C₂×Dic₅₃

yes

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄²

C₂²×D₅₃

yes

characteristic

122

#107

—

123

C₄⋊C₈

—

124

C₄⋊C₈

—

125

C₄⋊C₈

—

106

—

126

C₄⋊C₈

—

106

—

127

C₂²⋊C₈

—

106

—

128

C₂²⋊C₈

—

106

—

129

C₄×C₈

—

106

—

130

C₄×C₈

—

106

—

131

C₄×C₈

C₂×Dic₅₃

yes

—

132

C₄×C₈

C₂²⋊D₅₃

yes

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂²×C₄

—

137

C₂²×C₄

#11719947

yes

characteristic

138

C₂²×C₄

#11719947

yes

characteristic

139

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

140

M₁₆

—

212

—

141

C₂×C₈

—

142

C₂×C₈

—

106

—

143

C₂×C₈

—

106

—

144

C₂×C₈

—

212

—

145

C₂×C₈

C₂²⋊Dic₅₃

yes

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄²

D₄×D₅₃

yes

characteristic

151

C₄²

D₄×D₅₃

yes

characteristic

152

C₄²

C₂²×Dic₅₃

yes

characteristic

153

C₄²

C₂×C₂²⋊D₅₃

yes

characteristic

154

C₂³

#13133204

yes

characteristic

155

Q₈

—

156

Q₈

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

C₂×C₂²⋊Dic₅₃

yes

characteristic

162

C₂×C₄

D₄×Dic₅₃

yes

characteristic

163

C₂×C₄

D₄×Dic₅₃

yes

characteristic

164

C₂×C₄

#13133188

yes

characteristic

165

C₂×C₄

#13133188

yes

characteristic

166

C₂×C₄

#13133213

yes

characteristic

167

C₂×C₄

#13133248

yes

—

168

C₈

—

169

C₈

—

424

—

170

C₈

—

212

—

171

C₈

—

212

—

172

C₂²

—

173

C₂²

—

174

C₂²

#16810526

yes

characteristic

175

C₂²

#16810991

yes

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

#16810632

yes

—

180

C₄

#16810890

yes

—

181

C₄

#16810895

yes

—

182

C₄

#16811015

yes

—

183

C₂

—

184

C₂

#20275536

yes

characteristic

185

C₂

#20275626

yes

characteristic

186

C₂

#20275631

yes

characteristic

	subgroup	G/H
center	C₂²	#16810526
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#500	C₂
socle	C₂²×C₅₃	#390
prime radical	C₂³×C₅₃	#124
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#500	D₅₃
hypocenter	C₅₃	#9778
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

subgroup

G/H

center

C₂²

#16810526

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₂×C₅₃×#500

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

N	H ≅ G/N	type
111. C₅₃	#9778	semidirect

H ≅ G/N

type

111. C₅₃

#9778

semidirect

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

subgroup	index
#20274853	2
#20273400	2
#20273396	2
#20273007	2
C₂×C₅₃×#500	2
#9778	53

subgroup

index

C₂×C₅₃×#500

Subgroups — #27283589

Conjugacy classes of subgroups

Complements

Maximal subgroups