Group #27281615 — subgroups

Subgroups — #27281615

Order 13568.

384 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 103 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	#20283821	C₂	1	1	yes	maximal, characteristic
2	#20273432	C₂	1	2	yes	maximal
3	#20273353	C₂	1	1	yes	maximal, characteristic
4	#20272587	C₂	1	2	yes	maximal
5	C₅₃×#3139	C₂	1	1	yes	maximal, characteristic
6	#16810754	—	2	4	no	—
7	#16810684	—	2	2	no	—
8	#16810680	C₂²	1	2	yes	—
9	#16810680	C₂²	1	2	yes	—
10	#16810369	—	2	4	no	—
11	#16810365	—	2	2	no	—
12	C₅₃×#533	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#491	—	2	2	no	—
15	C₅₃×#487	C₄	1	4	yes	—
16	C₅₃×#419	—	2	2	no	—
17	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
18	C₄⋊#88322	—	2	4	no	—
19	C₄⋊#88322	—	2	2	no	—
20	C₄⋊#88322	D₄	1	4	yes	—
21	C₂²⋊#88322	—	2	2	no	—
22	C₂²⋊#88322	—	2	2	no	—
23	#13133128	—	2	2	no	—
24	#13133128	—	2	2	no	—
25	#13133128	—	2	2	no	—
26	C₄×#88322	—	2	2	no	—
27	C₂²×Q₈×C₅₃	—	2	1	no	—
28	C₂×M₁₆×C₅₃	—	2	1	no	—
29	C₄⋊Q₈×C₅₃	—	2	1	no	—
30	C₄⋊Q₈×C₅₃	—	2	2	no	—
31	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
32	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
33	#125×C₅₃	D₄	1	2	yes	—
34	C₂²⋊Q₈×C₅₃	—	2	2	no	—
35	C₄×Q₈×C₅₃	—	2	2	no	—
36	#117×C₅₃	D₄	1	2	yes	—
37	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
38	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
39	#103×C₅₃	—	2	8	no	—
40	#97×C₅₃	—	2	1	no	—
41	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
42	C₂×Q₈×C₅₃	—	2	2	no	—
43	C₂×Q₈×C₅₃	—	2	2	no	—
44	C₂×Q₈×C₅₃	—	4	4	no	—
45	C₂×Q₈×C₅₃	—	4	1	no	—
46	C₂²×C₄×C₅₃	—	4	1	no	—
47	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	#41	1	2	yes	—
49	M₁₆×C₅₃	—	4	2	no	—
50	C₂×C₈×C₅₃	—	2	2	no	—
51	C₄⋊C₄×C₅₃	—	2	2	no	—
52	C₄⋊C₄×C₅₃	—	2	4	no	—
53	C₄⋊C₄×C₅₃	—	2	4	no	—
54	C₄⋊C₄×C₅₃	—	4	2	no	—
55	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
56	C₂²⋊C₄×C₅₃	—	2	2	no	—
57	C₄²×C₅₃	—	2	2	no	—
58	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₅₃	C₂×D₄	1	2	yes	—
61	C₂×#88322	—	2	4	no	—
62	C₂×#88322	—	4	2	no	—
63	C₂³×C₅₃	#123	1	1	yes	characteristic
64	Q₈×C₅₃	—	4	4	no	—
65	Q₈×C₅₃	—	8	2	no	—
66	Q₈×C₅₃	—	8	2	no	—
67	C₂×C₄×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	—	2	4	no	—
69	C₂×C₄×C₅₃	—	4	2	no	—
70	C₂×C₄×C₅₃	—	4	2	no	—
71	C₂×C₄×C₅₃	—	8	2	no	—
72	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
74	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
75	C₂×C₄×C₅₃	#121	1	2	yes	—
76	C₈×C₅₃	—	4	2	no	—
77	#88322	—	4	4	no	—
78	#88322	—	8	2	no	—
79	#9422	—	53	1	no	maximal, Sylow
80	C₂²×C₅₃	—	2	1	no	—
81	C₂²×C₅₃	—	2	2	no	—
82	C₂²×C₅₃	#386	1	1	yes	characteristic
83	C₄×C₅₃	—	2	1	no	—
84	C₄×C₅₃	—	2	2	no	—
85	C₄×C₅₃	—	2	2	no	—
86	C₄×C₅₃	—	4	4	no	—
87	C₄×C₅₃	—	8	2	no	—
88	C₄×C₅₃	#454	1	2	yes	—
89	#3139	D₅₃	1	1	yes	characteristic
90	#2987	—	53	1	no	—
91	#1556	—	53	2	no	—
92	#1537	—	53	1	no	—
93	#1291	—	53	2	no	—
94	C₂×C₅₃	—	4	1	no	—
95	C₂×C₅₃	#1888	1	1	yes	characteristic
96	C₂×C₅₃	#1902	1	2	yes	—
97	#533	C₂×D₅₃	1	1	yes	characteristic
98	C₂×C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
99	#491	—	2	2	no	—
100	#487	Dic₅₃	1	4	yes	—
101	#446	—	106	4	no	—
102	#433	—	106	2	no	—
103	#432	—	53	2	no	—
104	#432	—	53	2	no	—
105	#419	—	2	2	no	—
106	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
107	#332	—	106	4	no	—
108	#330	—	106	2	no	—
109	C₅₃	#9422	1	1	yes	characteristic, complemented, Sylow
110	C₂²×Q₈	—	2	1	no	—
111	C₂×M₁₆	—	2	1	no	—
112	C₄⋊Q₈	—	2	2	no	—
113	C₄⋊Q₈	—	2	1	no	—
114	C₄⋊Q₈	C₂×Dic₅₃	1	2	yes	—
115	C₄⋊Q₈	C₂×Dic₅₃	1	2	yes	—
116	#125	C₂²⋊D₅₃	1	2	yes	—
117	C₂²⋊Q₈	—	2	2	no	—
118	C₄×Q₈	—	2	2	no	—
119	#117	C₂²⋊D₅₃	1	2	yes	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
122	C₄⋊C₈	—	53	4	no	—
123	C₄⋊C₈	—	106	4	no	—
124	C₄⋊C₈	—	106	2	no	—
125	#103	—	2	8	no	—
126	C₂²⋊C₈	—	106	2	no	—
127	C₂²⋊C₈	—	106	2	no	—
128	#97	—	2	1	no	—
129	#97	—	106	2	no	—
130	#97	—	106	2	no	—
131	#97	—	106	2	no	—
132	C₄×C₈	—	106	2	no	—
133	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
134	C₂×Q₈	—	2	2	no	—
135	C₂×Q₈	—	2	2	no	—
136	C₂×Q₈	—	4	4	no	—
137	C₂×Q₈	—	4	1	no	—
138	C₂²×C₄	—	4	1	no	—
139	C₂²×C₄	#11719947	1	2	yes	—
140	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
141	M₁₆	—	4	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	106	4	no	—
144	C₂×C₈	—	212	2	no	—
145	C₄⋊C₄	—	2	2	no	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	—	2	4	no	—
148	C₄⋊C₄	—	4	2	no	—
149	C₄⋊C₄	C₂²⋊Dic₅₃	1	4	yes	—
150	C₂²⋊C₄	—	2	2	no	—
151	C₄²	—	2	2	no	—
152	C₄²	D₄×D₅₃	1	2	yes	—
153	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
154	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
155	C₂³	#13133198	1	1	yes	characteristic
156	Q₈	—	4	4	no	—
157	Q₈	—	8	2	no	—
158	Q₈	—	8	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	8	2	no	—
164	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
165	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
166	C₂×C₄	#13133185	1	1	yes	characteristic
167	C₂×C₄	#13133189	1	2	yes	—
168	C₈	—	4	2	no	—
169	C₈	—	424	2	no	—
170	C₈	—	212	4	no	—
171	C₂²	—	2	1	no	—
172	C₂²	—	2	2	no	—
173	C₂²	#16810503	1	1	yes	characteristic
174	C₄	—	2	1	no	—
175	C₄	—	2	2	no	—
176	C₄	—	2	2	no	—
177	C₄	—	4	4	no	—
178	C₄	—	8	2	no	—
179	C₄	#16810795	1	2	yes	—
180	C₂	—	4	1	no	—
181	C₂	#20275119	1	1	yes	characteristic
182	C₂	#20275190	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20283821

C₂

yes

maximal, characteristic

#20273432

C₂

yes

maximal

#20273353

C₂

yes

maximal, characteristic

#20272587

C₂

yes

maximal

C₅₃×#3139

C₂

yes

maximal, characteristic

#16810754

—

#16810684

—

#16810680

C₂²

yes

—

#16810680

C₂²

yes

—

#16810369

—

#16810365

—

C₅₃×#533

C₂²

yes

characteristic

C₂×C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

C₅₃×#491

—

C₅₃×#487

C₄

yes

—

C₅₃×#419

—

C₄×M₁₆×C₅₃

C₂²

yes

—

—

D₄

yes

—

—

—

—

—

—

—

C₂²×Q₈×C₅₃

—

C₂×M₁₆×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

C₂×C₄

yes

—

C₄⋊Q₈×C₅₃

C₂×C₄

yes

—

#125×C₅₃

D₄

yes

—

C₂²⋊Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

#117×C₅₃

D₄

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#103×C₅₃

—

#97×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

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

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

#123

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×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

—

C₂×C₄×C₅₃

#120

yes

characteristic

C₂×C₄×C₅₃

#121

yes

—

C₈×C₅₃

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#386

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#454

yes

—

#3139

D₅₃

yes

—

—

—

—

C₂×C₅₃

—

C₂×C₅₃

#1888

yes

characteristic

C₂×C₅₃

#1902

yes

—

#533

C₂×D₅₃

yes

characteristic

C₂×C₄⋊Q₈

C₂×D₅₃

yes

characteristic

#491

—

100

#487

Dic₅₃

yes

—

101

#446

—

106

—

102

#433

—

106

—

103

#432

—

104

#432

—

105

#419

—

106

C₄×M₁₆

C₂×D₅₃

yes

characteristic

107

#332

—

106

—

108

#330

—

106

—

109

C₅₃

#9422

yes

characteristic, complemented, Sylow

110

C₂²×Q₈

—

111

C₂×M₁₆

—

112

C₄⋊Q₈

—

113

C₄⋊Q₈

—

114

C₄⋊Q₈

C₂×Dic₅₃

yes

—

115

C₄⋊Q₈

C₂×Dic₅₃

yes

—

116

#125

C₂²⋊D₅₃

yes

—

117

C₂²⋊Q₈

—

118

C₄×Q₈

—

119

#117

C₂²⋊D₅₃

yes

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄²

C₂²×D₅₃

yes

characteristic

122

C₄⋊C₈

—

123

C₄⋊C₈

—

106

—

124

C₄⋊C₈

—

106

—

125

#103

—

126

C₂²⋊C₈

—

106

—

127

C₂²⋊C₈

—

106

—

128

#97

—

129

#97

—

106

—

130

#97

—

106

—

131

#97

—

106

—

132

C₄×C₈

—

106

—

133

C₄×C₈

C₂×Dic₅₃

yes

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂×Q₈

—

137

C₂×Q₈

—

138

C₂²×C₄

—

139

C₂²×C₄

#11719947

yes

—

140

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

141

M₁₆

—

142

C₂×C₈

—

143

C₂×C₈

—

106

—

144

C₂×C₈

—

212

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

C₂²⋊Dic₅₃

yes

—

150

C₂²⋊C₄

—

151

C₄²

—

152

C₄²

D₄×D₅₃

yes

—

153

C₄²

C₂²×Dic₅₃

yes

characteristic

154

C₄²

C₂×C₂²⋊D₅₃

yes

characteristic

155

C₂³

#13133198

yes

characteristic

156

Q₈

—

157

Q₈

—

158

Q₈

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

C₂×C₂²⋊Dic₅₃

yes

characteristic

165

C₂×C₄

D₄×Dic₅₃

yes

—

166

C₂×C₄

#13133185

yes

characteristic

167

C₂×C₄

#13133189

yes

—

168

C₈

—

169

C₈

—

424

—

170

C₈

—

212

—

171

C₂²

—

172

C₂²

—

173

C₂²

#16810503

yes

characteristic

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

#16810795

yes

—

180

C₂

—

181

C₂

#20275119

yes

characteristic

182

C₂

#20275190

yes

—

	subgroup	G/H
center	C₂²	#16810503
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#3139	C₂
socle	C₂²×C₅₃	#386
prime radical	C₂³×C₅₃	#123
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#3139	D₅₃
hypocenter	C₅₃	#9422
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3139

subgroup

G/H

center

C₂²

#16810503

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#3139

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3139

N	H ≅ G/N	type
109. C₅₃	#9422	semidirect

H ≅ G/N

type

109. C₅₃

#9422

semidirect

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

subgroup	index
#20283821	2
#20273432	2
#20273353	2
#20272587	2
C₅₃×#3139	2
#9422	53

subgroup

index

C₅₃×#3139

Subgroups — #27281615

Conjugacy classes of subgroups

Complements

Maximal subgroups