Group #27279047 — subgroups

Subgroups — #27279047

Order 13568.

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

normal subgroups only

This group has 376 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	#20273380	C₂	1	1	yes	maximal, characteristic
3	#20273188	C₂	1	2	yes	maximal
4	#20272614	C₂	1	2	yes	maximal
5	C₅₃×#3162	C₂	1	1	yes	maximal, characteristic
6	#16810754	—	2	4	no	—
7	#16810684	—	2	2	no	—
8	#16810680	C₂²	1	2	yes	—
9	#16810676	C₂²	1	2	yes	—
10	#16810369	—	2	2	no	—
11	Q₈⋊#88322	—	2	4	no	—
12	C₅₃×#533	C₂²	1	1	yes	characteristic
13	C₅₃×#533	C₂²	1	1	yes	characteristic
14	C₅₃×#501	—	2	2	no	—
15	C₅₃×#501	C₄	1	4	yes	—
16	C₅₃×#427	—	2	2	no	—
17	C₂×#97×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	C₄×#88322	—	2	2	no	—
26	C₄×#88322	—	2	2	no	—
27	C₂²×C₈×C₅₃	—	2	1	no	—
28	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
29	C₄⋊Q₈×C₅₃	D₄	1	2	yes	—
30	#125×C₅₃	—	2	1	no	—
31	#125×C₅₃	C₂×C₄	1	2	yes	—
32	C₂²⋊Q₈×C₅₃	—	2	2	no	—
33	C₂²⋊Q₈×C₅₃	—	2	2	no	—
34	C₄×Q₈×C₅₃	—	2	2	no	—
35	C₄×Q₈×C₅₃	—	2	2	no	—
36	#117×C₅₃	—	2	1	no	—
37	#117×C₅₃	D₄	1	2	yes	—
38	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
39	#107×C₅₃	—	2	4	no	—
40	#106×C₅₃	—	2	4	no	—
41	#97×C₅₃	—	2	1	no	—
42	#97×C₅₃	C₂×C₄	1	2	yes	—
43	C₂×Q₈×C₅₃	—	2	2	no	—
44	C₂×Q₈×C₅₃	—	2	2	no	—
45	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	#41	1	2	yes	—
47	C₂×C₈×C₅₃	—	2	4	no	—
48	C₂×C₈×C₅₃	—	2	2	no	—
49	C₄⋊C₄×C₅₃	—	2	4	no	—
50	C₄⋊C₄×C₅₃	—	2	2	no	—
51	C₄⋊C₄×C₅₃	—	2	2	no	—
52	C₄⋊C₄×C₅₃	—	2	2	no	—
53	C₄⋊C₄×C₅₃	—	4	2	no	—
54	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
55	C₂²⋊C₄×C₅₃	—	2	2	no	—
56	C₂²⋊C₄×C₅₃	—	4	1	no	—
57	C₄²×C₅₃	—	2	2	no	—
58	C₄²×C₅₃	—	4	1	no	—
59	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₅₃	C₂×D₄	1	2	yes	—
62	C₂×#88322	—	2	4	no	—
63	C₂×#88322	—	4	2	no	—
64	C₂³×C₅₃	#123	1	1	yes	characteristic
65	Q₈×C₅₃	—	4	4	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	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
73	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
74	C₂×C₄×C₅₃	#121	1	2	yes	—
75	C₈×C₅₃	—	2	4	no	—
76	#88322	—	4	4	no	—
77	#88322	—	8	2	no	—
78	#9006	—	53	1	no	maximal, Sylow
79	C₂²×C₅₃	—	2	2	no	—
80	C₂²×C₅₃	—	2	1	no	—
81	C₂²×C₅₃	#386	1	1	yes	characteristic
82	C₄×C₅₃	—	2	2	no	—
83	C₄×C₅₃	—	2	2	no	—
84	C₄×C₅₃	—	2	1	no	—
85	C₄×C₅₃	—	4	4	no	—
86	C₄×C₅₃	—	8	2	no	—
87	C₄×C₅₃	#456	1	2	yes	—
88	#3162	D₅₃	1	1	yes	characteristic
89	#2987	—	53	1	no	—
90	#1544	—	53	1	no	—
91	#1497	—	53	2	no	—
92	#1302	—	53	2	no	—
93	C₂×C₅₃	—	4	1	no	—
94	C₂×C₅₃	#1881	1	2	yes	—
95	C₂×C₅₃	#1886	1	1	yes	characteristic
96	#533	C₂×D₅₃	1	1	yes	characteristic
97	#533	C₂×D₅₃	1	1	yes	characteristic
98	#501	—	2	2	no	—
99	#501	Dic₅₃	1	4	yes	—
100	#446	—	106	4	no	—
101	#433	—	106	2	no	—
102	#432	—	53	2	no	—
103	#431	—	53	2	no	—
104	#427	—	2	2	no	—
105	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
106	#332	—	106	2	no	—
107	Q₈⋊C₈	—	106	4	no	—
108	C₅₃	#9006	1	1	yes	characteristic, complemented, Sylow
109	C₂²×C₈	—	2	1	no	—
110	C₄⋊Q₈	C₂×Dic₅₃	1	2	yes	—
111	C₄⋊Q₈	C₂²⋊D₅₃	1	2	yes	—
112	#125	—	2	1	no	—
113	#125	C₂×Dic₅₃	1	2	yes	—
114	C₂²⋊Q₈	—	2	2	no	—
115	C₂²⋊Q₈	—	2	2	no	—
116	C₄×Q₈	—	2	2	no	—
117	C₄×Q₈	—	2	2	no	—
118	#117	—	2	1	no	—
119	#117	C₂²⋊D₅₃	1	2	yes	—
120	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
121	#107	—	2	4	no	—
122	#106	—	2	4	no	—
123	C₄⋊C₈	—	53	4	no	—
124	C₄⋊C₈	—	106	4	no	—
125	C₄⋊C₈	—	106	2	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	C₂×Dic₅₃	1	2	yes	—
132	C₄×C₈	—	106	2	no	—
133	C₄×C₈	—	106	2	no	—
134	C₂×Q₈	—	2	2	no	—
135	C₂×Q₈	—	2	2	no	—
136	C₂²×C₄	#11719947	1	2	yes	—
137	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	106	4	no	—
141	C₂×C₈	—	212	2	no	—
142	C₄⋊C₄	—	2	4	no	—
143	C₄⋊C₄	—	2	2	no	—
144	C₄⋊C₄	—	2	2	no	—
145	C₄⋊C₄	—	2	2	no	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	C₂²⋊Dic₅₃	1	4	yes	—
148	C₂²⋊C₄	—	2	2	no	—
149	C₂²⋊C₄	—	4	1	no	—
150	C₄²	—	2	2	no	—
151	C₄²	—	4	1	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	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
163	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
164	C₂×C₄	#13133185	1	1	yes	characteristic
165	C₂×C₄	#13133189	1	2	yes	—
166	C₈	—	2	4	no	—
167	C₈	—	424	2	no	—
168	C₈	—	212	4	no	—
169	C₂²	—	2	2	no	—
170	C₂²	—	2	1	no	—
171	C₂²	#16810503	1	1	yes	characteristic
172	C₄	—	2	2	no	—
173	C₄	—	2	2	no	—
174	C₄	—	2	1	no	—
175	C₄	—	4	4	no	—
176	C₄	—	8	2	no	—
177	C₄	#16810805	1	2	yes	—
178	C₂	—	4	1	no	—
179	C₂	#20275079	1	2	yes	—
180	C₂	#20275109	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20283821

C₂

yes

maximal, characteristic

#20273380

C₂

yes

maximal, characteristic

#20273188

C₂

yes

maximal

#20272614

C₂

yes

maximal

C₅₃×#3162

C₂

yes

maximal, characteristic

#16810754

—

#16810684

—

#16810680

C₂²

yes

—

#16810676

C₂²

yes

—

#16810369

—

Q₈⋊#88322

—

C₅₃×#533

C₂²

yes

characteristic

C₅₃×#533

C₂²

yes

characteristic

C₅₃×#501

—

C₅₃×#501

C₄

yes

—

C₅₃×#427

—

C₂×#97×C₅₃

C₂²

yes

—

—

D₄

yes

—

—

—

—

—

—

—

C₂²×C₈×C₅₃

—

C₄⋊Q₈×C₅₃

C₂×C₄

yes

—

C₄⋊Q₈×C₅₃

D₄

yes

—

#125×C₅₃

—

#125×C₅₃

C₂×C₄

yes

—

C₂²⋊Q₈×C₅₃

—

C₂²⋊Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

#117×C₅₃

—

#117×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#107×C₅₃

—

#106×C₅₃

—

#97×C₅₃

—

#97×C₅₃

C₂×C₄

yes

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×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

—

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

#456

yes

—

#3162

D₅₃

yes

—

—

—

—

C₂×C₅₃

—

C₂×C₅₃

#1881

yes

—

C₂×C₅₃

#1886

yes

characteristic

#533

C₂×D₅₃

yes

characteristic

#533

C₂×D₅₃

yes

characteristic

#501

—

#501

Dic₅₃

yes

—

100

#446

—

106

—

101

#433

—

106

—

102

#432

—

103

#431

—

104

#427

—

105

C₂×#97

C₂×D₅₃

yes

characteristic

106

#332

—

106

—

107

Q₈⋊C₈

—

106

—

108

C₅₃

#9006

yes

characteristic, complemented, Sylow

109

C₂²×C₈

—

110

C₄⋊Q₈

C₂×Dic₅₃

yes

—

111

C₄⋊Q₈

C₂²⋊D₅₃

yes

—

112

#125

—

113

#125

C₂×Dic₅₃

yes

—

114

C₂²⋊Q₈

—

115

C₂²⋊Q₈

—

116

C₄×Q₈

—

117

C₄×Q₈

—

118

#117

—

119

#117

C₂²⋊D₅₃

yes

—

120

C₂×C₄²

C₂²×D₅₃

yes

characteristic

121

#107

—

122

#106

—

123

C₄⋊C₈

—

124

C₄⋊C₈

—

106

—

125

C₄⋊C₈

—

106

—

126

C₂²⋊C₈

—

106

—

127

C₂²⋊C₈

—

106

—

128

#97

—

129

#97

—

106

—

130

#97

—

106

—

131

#97

C₂×Dic₅₃

yes

—

132

C₄×C₈

—

106

—

133

C₄×C₈

—

106

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂²×C₄

#11719947

yes

—

137

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

106

—

141

C₂×C₈

—

212

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

C₂²⋊Dic₅₃

yes

—

148

C₂²⋊C₄

—

149

C₂²⋊C₄

—

150

C₄²

—

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

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

C₂×C₂²⋊Dic₅₃

yes

characteristic

163

C₂×C₄

D₄×Dic₅₃

yes

—

164

C₂×C₄

#13133185

yes

characteristic

165

C₂×C₄

#13133189

yes

—

166

C₈

—

167

C₈

—

424

—

168

C₈

—

212

—

169

C₂²

—

170

C₂²

—

171

C₂²

#16810503

yes

characteristic

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

#16810805

yes

—

178

C₂

—

179

C₂

#20275079

yes

—

180

C₂

#20275109

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13133185
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#3162	C₂
socle	C₂²×C₅₃	#386
prime radical	C₂³×C₅₃	#123
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#3162	D₅₃
hypocenter	C₅₃	#9006
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3162

subgroup

G/H

center

C₂×C₄

#13133185

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#3162

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3162

N	H ≅ G/N	type
108. C₅₃	#9006	semidirect

H ≅ G/N

type

108. C₅₃

#9006

semidirect

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

subgroup	index
#20283821	2
#20273380	2
#20273188	2
#20272614	2
C₅₃×#3162	2
#9006	53

subgroup

index

C₅₃×#3162

Subgroups — #27279047

Conjugacy classes of subgroups

Complements

Maximal subgroups