Group #27289080 — subgroups

Subgroups — #27289080

Order 13568.

320 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 99 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	#20274890	C₂	1	1	yes	maximal, characteristic
2	#20273489	C₂	1	2	yes	maximal
3	#20273144	C₂	1	2	yes	maximal
4	#20272546	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1985	C₂	1	1	yes	maximal, characteristic
6	#16810676	C₂²	1	2	yes	—
7	C₂×#13133153	—	2	2	no	—
8	#16810647	C₂²	1	2	yes	—
9	#16810375	—	2	4	no	—
10	Q₈⋊#88322	—	2	4	no	—
11	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#336	—	2	2	no	—
15	C₂×#11719917	—	2	2	no	—
16	#13133153	—	4	4	no	—
17	C₄⋊#88322	—	2	2	no	—
18	C₄⋊#88322	—	2	2	no	—
19	C₄⋊#88322	—	2	2	no	—
20	C₂²⋊#88322	—	2	2	no	—
21	C₄×#88322	D₄	1	4	yes	—
22	C₂²×Q₈×C₅₃	Q₈	1	1	yes	characteristic
23	C₂²×C₈×C₅₃	—	2	1	no	—
24	C₂²×C₈×C₅₃	D₄	1	2	yes	—
25	C₄×Q₈×C₅₃	—	2	2	no	—
26	C₄×Q₈×C₅₃	—	2	2	no	—
27	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
28	C₂×C₄⋊C₄×C₅₃	Q₈	1	1	yes	characteristic
29	C₂×C₄²×C₅₃	—	2	1	no	—
30	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₅₃	—	2	2	no	—
32	#97×C₅₃	—	2	2	no	—
33	C₂×Q₈×C₅₃	—	2	2	no	—
34	C₂×Q₈×C₅₃	—	4	2	no	—
35	C₂²×C₄×C₅₃	—	2	1	no	—
36	C₂²×C₄×C₅₃	—	2	1	no	—
37	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
40	C₂×C₈×C₅₃	—	2	2	no	—
41	C₂×C₈×C₅₃	—	2	2	no	—
42	C₂×C₈×C₅₃	—	4	1	no	—
43	C₂×C₈×C₅₃	—	4	2	no	—
44	C₂×C₈×C₅₃	Q₁₆	1	4	yes	—
45	C₄⋊C₄×C₅₃	—	2	2	no	—
46	C₄⋊C₄×C₅₃	—	4	2	no	—
47	C₄²×C₅₃	—	4	1	no	—
48	C₄²×C₅₃	—	4	1	no	—
49	C₄²×C₅₃	C₂×D₄	1	2	yes	—
50	C₄²×C₅₃	#41	1	2	yes	—
51	#11719917	—	4	4	no	—
52	C₂×#88322	—	2	4	no	—
53	C₂×#88322	—	4	2	no	—
54	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
55	Q₈×C₅₃	—	2	4	no	—
56	Q₈×C₅₃	—	4	2	no	—
57	C₂×C₄×C₅₃	—	2	2	no	—
58	C₂×C₄×C₅₃	—	4	1	no	—
59	C₂×C₄×C₅₃	—	4	1	no	—
60	C₂×C₄×C₅₃	—	4	1	no	—
61	C₂×C₄×C₅₃	—	4	1	no	—
62	C₂×C₄×C₅₃	—	4	2	no	—
63	C₂×C₄×C₅₃	—	4	1	no	—
64	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
65	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
66	C₂×C₄×C₅₃	#123	1	2	yes	—
67	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
69	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
70	C₈×C₅₃	—	2	4	no	—
71	C₈×C₅₃	—	8	1	no	—
72	#88322	—	4	4	no	—
73	#88322	—	8	2	no	—
74	#10662	—	53	1	no	maximal, Sylow
75	C₂²×C₅₃	—	2	1	no	—
76	C₂²×C₅₃	—	2	1	no	—
77	C₂²×C₅₃	#399	1	1	yes	characteristic
78	C₂²×C₅₃	#477	1	2	yes	—
79	C₄×C₅₃	—	2	2	no	—
80	C₄×C₅₃	—	2	2	no	—
81	C₄×C₅₃	—	4	2	no	—
82	C₄×C₅₃	—	8	1	no	—
83	C₄×C₅₃	#484	1	2	yes	—
84	C₄×C₅₃	#494	1	1	yes	characteristic
85	C₄×C₅₃	#496	1	1	yes	characteristic
86	#1985	D₅₃	1	1	yes	characteristic
87	#1849	—	53	1	no	—
88	#1571	—	53	2	no	—
89	#1487	—	53	2	no	—
90	#1275	—	53	1	no	—
91	C₂×C₅₃	—	2	2	no	—
92	C₂×C₅₃	#2091	1	1	yes	characteristic
93	C₂×C₅₃	#2095	1	1	yes	characteristic
94	C₂×C₅₃	#2098	1	1	yes	characteristic
95	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
96	#431	—	53	2	no	—
97	C₂×#108	—	106	2	no	—
98	#423	—	53	2	no	—
99	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
100	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
101	#336	—	2	2	no	—
102	#335	—	106	4	no	—
103	Q₈⋊C₈	—	106	4	no	—
104	C₅₃	#10662	1	1	yes	characteristic, complemented, Sylow
105	C₂²×Q₈	Dic₁₀₆	1	1	yes	characteristic
106	C₂×M₁₆	—	106	2	no	—
107	C₂²×C₈	—	2	1	no	—
108	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
109	C₄×Q₈	—	2	2	no	—
110	C₄×Q₈	—	2	2	no	—
111	C₂×C₄⋊C₄	—	2	1	no	—
112	C₂×C₄⋊C₄	Dic₁₀₆	1	1	yes	characteristic
113	C₂×C₄²	—	2	1	no	—
114	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
115	#108	—	212	4	no	—
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	106	2	no	—
118	C₄⋊C₈	—	106	2	no	—
119	C₄⋊C₈	—	106	2	no	—
120	C₂²⋊C₈	—	106	2	no	—
121	#97	—	2	2	no	—
122	C₄×C₈	—	53	4	no	—
123	C₂×Q₈	—	2	2	no	—
124	C₂×Q₈	—	4	2	no	—
125	C₂²×C₄	—	2	1	no	—
126	C₂²×C₄	—	2	1	no	—
127	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
128	C₂²×C₄	#11719945	1	1	yes	characteristic
129	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
130	M₁₆	—	212	4	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	4	1	no	—
134	C₂×C₈	—	4	2	no	—
135	C₂×C₈	—	106	4	no	—
136	C₂×C₈	—	212	2	no	—
137	C₂×C₈	#11719925	1	4	yes	—
138	C₄⋊C₄	—	2	2	no	—
139	C₄⋊C₄	—	4	2	no	—
140	C₄²	—	4	1	no	—
141	C₄²	—	4	1	no	—
142	C₄²	D₄×D₅₃	1	2	yes	—
143	C₄²	#11719947	1	2	yes	—
144	C₂³	#13133196	1	1	yes	characteristic
145	Q₈	—	2	4	no	—
146	Q₈	—	4	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	#13133194	1	2	yes	—
155	C₂×C₄	#13133198	1	1	yes	characteristic
156	C₂×C₄	#13133199	1	2	yes	—
157	C₂×C₄	#13133213	1	1	yes	characteristic
158	C₂×C₄	C₂×#11719925	1	2	yes	—
159	C₂×C₄	#13133242	1	2	yes	—
160	C₈	—	2	4	no	—
161	C₈	—	8	1	no	—
162	C₈	—	424	2	no	—
163	C₈	—	212	4	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	#16810559	1	1	yes	characteristic
167	C₂²	#16810918	1	2	yes	—
168	C₄	—	2	2	no	—
169	C₄	—	2	2	no	—
170	C₄	—	4	2	no	—
171	C₄	—	8	1	no	—
172	C₄	#16810956	1	2	yes	—
173	C₄	#16811002	1	1	yes	characteristic
174	C₄	#16811013	1	1	yes	characteristic
175	C₂	—	2	2	no	—
176	C₂	#20276231	1	1	yes	characteristic
177	C₂	#20276253	1	1	yes	characteristic
178	C₂	#20276268	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20274890

C₂

yes

maximal, characteristic

#20273489

C₂

yes

maximal

#20273144

C₂

yes

maximal

#20272546

C₂

yes

maximal, characteristic

C₅₃×#1985

C₂

yes

maximal, characteristic

#16810676

C₂²

yes

—

C₂×#13133153

—

#16810647

C₂²

yes

—

#16810375

—

Q₈⋊#88322

—

C₂×C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₂×#97×C₅₃

C₂²

yes

characteristic

C₅₃×#336

—

—

—

—

—

—

—

D₄

yes

—

C₂²×Q₈×C₅₃

Q₈

yes

characteristic

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

—

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

Q₈

yes

characteristic

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

#97×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

Q₁₆

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

#41

yes

—

#11719917

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂²⋊Q₈

yes

characteristic

Q₈×C₅₃

—

Q₈×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

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₃

#123

yes

characteristic

C₂×C₄×C₅₃

#123

yes

—

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₁₆

yes

—

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#399

yes

characteristic

C₂²×C₅₃

#477

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#484

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

D₅₃

yes

—

—

—

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄×Q₈

C₂×D₅₃

yes

characteristic

#431

—

C₂×#108

—

106

—

#423

—

C₂×C₄⋊C₈

C₂×D₅₃

yes

characteristic

100

C₂×#97

C₂×D₅₃

yes

characteristic

101

#336

—

102

#335

—

106

—

103

Q₈⋊C₈

—

106

—

104

C₅₃

#10662

yes

characteristic, complemented, Sylow

105

C₂²×Q₈

Dic₁₀₆

yes

characteristic

106

C₂×M₁₆

—

106

—

107

C₂²×C₈

—

108

C₂²×C₈

C₂²⋊D₅₃

yes

—

109

C₄×Q₈

—

110

C₄×Q₈

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

Dic₁₀₆

yes

characteristic

113

C₂×C₄²

—

114

C₂×C₄²

C₂²×D₅₃

yes

characteristic

115

#108

—

212

—

116

C₄⋊C₈

—

117

C₄⋊C₈

—

106

—

118

C₄⋊C₈

—

106

—

119

C₄⋊C₈

—

106

—

120

C₂²⋊C₈

—

106

—

121

#97

—

122

C₄×C₈

—

123

C₂×Q₈

—

124

C₂×Q₈

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

C₂×Dic₁₀₆

yes

characteristic

128

C₂²×C₄

#11719945

yes

characteristic

129

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

130

M₁₆

—

212

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

106

—

136

C₂×C₈

—

212

—

137

C₂×C₈

#11719925

yes

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄²

—

141

C₄²

—

142

C₄²

D₄×D₅₃

yes

—

143

C₄²

#11719947

yes

—

144

C₂³

#13133196

yes

characteristic

145

Q₈

—

146

Q₈

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

#13133194

yes

—

155

C₂×C₄

#13133198

yes

characteristic

156

C₂×C₄

#13133199

yes

—

157

C₂×C₄

#13133213

yes

characteristic

158

C₂×C₄

C₂×#11719925

yes

—

159

C₂×C₄

#13133242

yes

—

160

C₈

—

161

C₈

—

162

C₈

—

424

—

163

C₈

—

212

—

164

C₂²

—

165

C₂²

—

166

C₂²

#16810559

yes

characteristic

167

C₂²

#16810918

yes

—

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

#16810956

yes

—

173

C₄

#16811002

yes

characteristic

174

C₄

#16811013

yes

characteristic

175

C₂

—

176

C₂

#20276231

yes

characteristic

177

C₂

#20276253

yes

characteristic

178

C₂

#20276268

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13133213
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1985	C₂
socle	C₂²×C₅₃	#399
prime radical	C₂³×C₅₃	C₂²⋊Q₈
hypercenter	#1985	D₅₃
hypocenter	C₅₃	#10662
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1985

subgroup

G/H

center

C₂×C₄

#13133213

commutator, cosocle

C₂×C₄²×C₅₃

C₂³

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1985

C₂

socle

C₂²×C₅₃

#399

prime radical

C₂³×C₅₃

C₂²⋊Q₈

D₅₃

C₅₃

G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1985

N	H ≅ G/N	type
104. C₅₃	#10662	semidirect

H ≅ G/N

type

104. C₅₃

#10662

semidirect

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

subgroup	index
#20274890	2
#20273489	2
#20273144	2
#20272546	2
C₅₃×#1985	2
#10662	53

subgroup

index

C₅₃×#1985

Subgroups — #27289080

Conjugacy classes of subgroups

Complements

Maximal subgroups