Group #27272410 — subgroups

Subgroups — #27272410

Order 13568.

384 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 119 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	#20274858	C₂	1	2	yes	maximal
2	#20272538	C₂	1	4	yes	maximal
3	C₄×#125×C₅₃	C₂	1	1	yes	maximal, characteristic
4	#16810378	—	2	2	no	—
5	#16810378	—	2	2	no	—
6	#16810378	C₂²	1	4	yes	—
7	C₂×#125×C₅₃	C₄	1	1	yes	characteristic
8	C₅₃×#389	C₂²	1	2	yes	—
9	C₅₃×#387	—	2	2	no	—
10	C₅₃×#382	C₄	1	1	yes	characteristic
11	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
12	C₄×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
13	C₄³×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×#88322	—	2	4	no	—
15	#125×C₅₃	—	2	4	no	—
16	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
17	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
18	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
19	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₅₃	—	2	2	no	—
21	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₅₃	D₄	1	2	yes	—
24	C₂×C₄²×C₅₃	D₄	1	2	yes	—
25	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
26	#95×C₅₃	—	2	2	no	—
27	#95×C₅₃	—	2	2	no	—
28	#95×C₅₃	—	2	2	no	—
29	C₂²×C₄×C₅₃	—	2	2	no	—
30	C₂²×C₄×C₅₃	—	2	2	no	—
31	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
33	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
36	C₄⋊C₄×C₅₃	—	2	8	no	—
37	C₄⋊C₄×C₅₃	—	2	2	no	—
38	C₄⋊C₄×C₅₃	—	4	2	no	—
39	C₄⋊C₄×C₅₃	—	4	4	no	—
40	C₄⋊C₄×C₅₃	M₁₆	1	4	yes	—
41	C₄²×C₅₃	—	2	2	no	—
42	C₄²×C₅₃	—	2	2	no	—
43	C₄²×C₅₃	—	2	4	no	—
44	C₄²×C₅₃	—	4	2	no	—
45	C₄²×C₅₃	—	4	1	no	—
46	C₄²×C₅₃	—	4	4	no	—
47	C₂×#88322	—	4	4	no	—
48	C₂×#88322	—	4	4	no	—
49	C₂×#88322	—	4	4	no	—
50	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₅₃	—	2	1	no	—
52	C₂×C₄×C₅₃	—	2	2	no	—
53	C₂×C₄×C₅₃	—	2	1	no	—
54	C₂×C₄×C₅₃	—	2	2	no	—
55	C₂×C₄×C₅₃	—	2	2	no	—
56	C₂×C₄×C₅₃	—	2	4	no	—
57	C₂×C₄×C₅₃	—	2	2	no	—
58	C₂×C₄×C₅₃	—	2	1	no	—
59	C₂×C₄×C₅₃	—	2	2	no	—
60	C₂×C₄×C₅₃	—	2	2	no	—
61	C₂×C₄×C₅₃	—	2	2	no	—
62	C₂×C₄×C₅₃	—	4	2	no	—
63	C₂×C₄×C₅₃	—	4	2	no	—
64	C₂×C₄×C₅₃	—	4	2	no	—
65	C₂×C₄×C₅₃	—	4	2	no	—
66	C₂×C₄×C₅₃	—	4	2	no	—
67	C₂×C₄×C₅₃	#101	1	2	yes	—
68	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
69	C₂×C₄×C₅₃	#135	1	4	yes	—
70	#88322	—	8	4	no	—
71	#7650	—	53	1	no	maximal, Sylow
72	C₂²×C₅₃	#407	1	1	yes	characteristic
73	C₂²×C₅₃	#410	1	1	yes	characteristic
74	C₂²×C₅₃	C₂×#101	1	1	yes	characteristic
75	C₂²×C₅₃	#416	1	2	yes	—
76	C₂²×C₅₃	#421	1	2	yes	—
77	C₄×C₅₃	—	2	2	no	—
78	C₄×C₅₃	—	2	4	no	—
79	C₄×C₅₃	—	2	2	no	—
80	C₄×C₅₃	—	4	1	no	—
81	C₄×C₅₃	—	4	2	no	—
82	C₄×C₅₃	—	4	4	no	—
83	C₄×C₅₃	—	8	2	no	—
84	C₄×#125	D₅₃	1	1	yes	characteristic
85	#1839	—	53	2	no	—
86	#1271	—	53	4	no	—
87	C₂×C₅₃	#1467	1	1	yes	characteristic
88	C₂×C₅₃	#1488	1	2	yes	—
89	C₂×C₅₃	#1521	1	2	yes	—
90	C₂×C₅₃	#1534	1	1	yes	characteristic
91	C₂×C₅₃	#1544	1	1	yes	characteristic
92	C₂×#125	Dic₅₃	1	1	yes	characteristic
93	#389	C₂×D₅₃	1	2	yes	—
94	#387	—	2	2	no	—
95	#382	Dic₅₃	1	1	yes	characteristic
96	C₄×C₄⋊C₄	—	2	2	no	—
97	C₄×C₄⋊C₄	Dic₅₃	1	2	yes	—
98	C₄³	C₂×D₅₃	1	1	yes	characteristic
99	#336	—	53	4	no	—
100	#336	—	106	2	no	—
101	#336	—	106	2	no	—
102	C₅₃	#7650	1	1	yes	characteristic, complemented, Sylow
103	C₂²×C₈	—	106	4	no	—
104	#125	—	2	4	no	—
105	C₂×C₄⋊C₄	—	2	2	no	—
106	C₂×C₄⋊C₄	—	2	2	no	—
107	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
108	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
109	C₂×C₄²	—	2	2	no	—
110	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
111	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
112	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
113	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
114	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
115	#95	—	2	2	no	—
116	#95	—	2	2	no	—
117	#95	—	2	2	no	—
118	C₂²×C₄	—	2	2	no	—
119	C₂²×C₄	—	2	2	no	—
120	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
121	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
122	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
123	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
124	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
125	C₂×C₈	—	212	4	no	—
126	C₂×C₈	—	212	4	no	—
127	C₂×C₈	—	212	4	no	—
128	C₄⋊C₄	—	2	8	no	—
129	C₄⋊C₄	—	2	2	no	—
130	C₄⋊C₄	—	4	2	no	—
131	C₄⋊C₄	—	4	4	no	—
132	C₄⋊C₄	#11719917	1	4	yes	—
133	C₄²	—	2	2	no	—
134	C₄²	—	2	2	no	—
135	C₄²	—	2	4	no	—
136	C₄²	—	4	2	no	—
137	C₄²	—	4	1	no	—
138	C₄²	—	4	4	no	—
139	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	#13133136	1	2	yes	—
157	C₂×C₄	C₂×#11719917	1	2	yes	—
158	C₂×C₄	#13133237	1	4	yes	—
159	C₈	—	424	4	no	—
160	C₂²	#16810587	1	1	yes	characteristic
161	C₂²	#16810597	1	1	yes	characteristic
162	C₂²	C₂×#13133136	1	1	yes	characteristic
163	C₂²	#16810619	1	2	yes	—
164	C₂²	#16810640	1	2	yes	—
165	C₄	—	2	4	no	—
166	C₄	—	2	2	no	—
167	C₄	—	2	2	no	—
168	C₄	—	4	1	no	—
169	C₄	—	4	2	no	—
170	C₄	—	4	4	no	—
171	C₄	—	8	2	no	—
172	C₂	#20273059	1	1	yes	characteristic
173	C₂	#20273149	1	2	yes	—
174	C₂	#20273292	1	2	yes	—
175	C₂	#20273342	1	1	yes	characteristic
176	C₂	#20273380	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20274858

C₂

yes

maximal

#20272538

C₂

yes

maximal

C₄×#125×C₅₃

C₂

yes

maximal, characteristic

#16810378

—

#16810378

—

#16810378

C₂²

yes

—

C₂×#125×C₅₃

C₄

yes

characteristic

C₅₃×#389

C₂²

yes

—

C₅₃×#387

—

C₅₃×#382

C₄

yes

characteristic

C₄×C₄⋊C₄×C₅₃

—

C₄×C₄⋊C₄×C₅₃

C₄

yes

—

C₄³×C₅₃

C₂²

yes

characteristic

C₂²×#88322

—

#125×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂×C₄

yes

—

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

—

#95×C₅₃

—

#95×C₅₃

—

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

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

M₁₆

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

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

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#101

yes

—

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₂×C₄×C₅₃

#135

yes

—

—

—

C₂²×C₅₃

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×#125

D₅₃

yes

characteristic

#1839

—

#1271

—

C₂×C₅₃

#1467

yes

characteristic

C₂×C₅₃

#1488

yes

—

C₂×C₅₃

#1521

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

Dic₅₃

yes

characteristic

#389

C₂×D₅₃

yes

—

#387

—

#382

Dic₅₃

yes

characteristic

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

Dic₅₃

yes

—

C₄³

C₂×D₅₃

yes

characteristic

#336

—

100

#336

—

106

—

101

#336

—

106

—

102

C₅₃

#7650

yes

characteristic, complemented, Sylow

103

C₂²×C₈

—

106

—

104

#125

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

C₂×Dic₅₃

yes

—

108

C₂×C₄⋊C₄

C₂×Dic₅₃

yes

—

109

C₂×C₄²

—

110

C₂×C₄²

C₂×Dic₅₃

yes

characteristic

111

C₂×C₄²

C₂×Dic₅₃

yes

characteristic

112

C₂×C₄²

C₂²⋊D₅₃

yes

—

113

C₂×C₄²

C₂²⋊D₅₃

yes

—

114

C₂×C₄²

C₂²×D₅₃

yes

characteristic

115

#95

—

116

#95

—

117

#95

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

C₂²⋊Dic₅₃

yes

characteristic

121

C₂²×C₄

C₂²⋊Dic₅₃

yes

—

122

C₂²×C₄

C₂²⋊Dic₅₃

yes

characteristic

123

C₂²×C₄

C₂²×Dic₅₃

yes

characteristic

124

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

—

125

C₂×C₈

—

212

—

126

C₂×C₈

—

212

—

127

C₂×C₈

—

212

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

#11719917

yes

—

133

C₄²

—

134

C₄²

—

135

C₄²

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

139

C₂³

C₂×C₂²⋊Dic₅₃

yes

characteristic

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

#13133136

yes

—

157

C₂×C₄

C₂×#11719917

yes

—

158

C₂×C₄

#13133237

yes

—

159

C₈

—

424

—

160

C₂²

#16810587

yes

characteristic

161

C₂²

#16810597

yes

characteristic

162

C₂²

C₂×#13133136

yes

characteristic

163

C₂²

#16810619

yes

—

164

C₂²

#16810640

yes

—

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

172

C₂

#20273059

yes

characteristic

173

C₂

#20273149

yes

—

174

C₂

#20273292

yes

—

175

C₂

#20273342

yes

characteristic

176

C₂

#20273380

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×#125×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×#125	D₅₃
hypocenter	C₅₃	#7650
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₅₃

commutator

C₂²×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₄×#125×C₅₃

C₂

socle, prime radical

C₂³×C₅₃

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

N	H ≅ G/N	type
102. C₅₃	#7650	semidirect

H ≅ G/N

type

102. C₅₃

#7650

semidirect

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

subgroup	index
#20274858	2
#20272538	2
C₄×#125×C₅₃	2
#7650	53

subgroup

index

#20274858

#20272538

C₄×#125×C₅₃

#7650

Subgroups — #27272410

Conjugacy classes of subgroups

Complements

Maximal subgroups