Group #27304241 — subgroups

Subgroups — #27304241

Order 13568.

268 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 93 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	#20275463	C₂	1	1	yes	maximal, characteristic
2	#20272822	C₂	1	2	yes	maximal
3	#20272813	C₂	1	2	yes	maximal
4	#20272797	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1844	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	C₂²	1	2	yes	—
7	C₂×#13133153	C₂²	1	1	yes	characteristic
8	#16810661	C₂²	1	1	yes	characteristic
9	#16810649	—	2	1	no	—
10	#16810383	—	2	1	no	—
11	C₂×#106×C₅₃	—	2	1	no	—
12	C₂×#106×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#384	—	2	1	no	—
15	C₂×#11719917	—	2	1	no	—
16	#13133176	—	2	1	no	—
17	#13133157	—	2	2	no	—
18	#13133157	—	2	2	no	—
19	C₂×#11719908	—	2	2	no	—
20	#13133153	—	2	2	no	—
21	#13133151	Q₈	1	2	yes	—
22	#13133149	—	2	1	no	—
23	#13133146	—	4	2	no	—
24	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
26	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
27	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
28	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
29	#106×C₅₃	—	2	4	no	—
30	#106×C₅₃	—	4	2	no	—
31	C₄×C₈×C₅₃	—	2	2	no	—
32	C₂²×C₄×C₅₃	—	2	1	no	—
33	C₂²×C₄×C₅₃	—	4	2	no	—
34	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₅₃	—	2	1	no	—
36	C₂×C₈×C₅₃	—	2	1	no	—
37	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
38	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈×C₅₃	#41	1	2	yes	—
40	C₂×C₈×C₅₃	#41	1	2	yes	—
41	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
42	C₄⋊C₄×C₅₃	—	4	2	no	—
43	C₄⋊C₄×C₅₃	—	4	2	no	—
44	C₄⋊C₄×C₅₃	—	8	4	no	—
45	C₄²×C₅₃	—	2	1	no	—
46	C₄²×C₅₃	Q₁₆	1	2	yes	—
47	C₂²⋊Dic₅₃	—	4	1	no	—
48	C₄⋊Dic₅₃	—	2	2	no	—
49	C₄×Dic₅₃	—	4	1	no	—
50	#11719917	—	4	1	no	—
51	#11719917	—	4	1	no	—
52	C₂×#88322	—	4	1	no	—
53	#11719908	—	4	2	no	—
54	C₂³×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
55	C₂×C₄×C₅₃	—	2	4	no	—
56	C₂×C₄×C₅₃	—	2	2	no	—
57	C₂×C₄×C₅₃	—	8	4	no	—
58	C₂×C₄×C₅₃	—	8	2	no	—
59	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
60	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
61	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
64	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₂×Dic₅₃	—	4	2	no	—
66	C₈×C₅₃	—	2	2	no	—
67	C₈×C₅₃	#125	1	2	yes	—
68	C₈×C₅₃	#126	1	2	yes	—
69	#88322	—	8	1	no	—
70	#13543	—	53	1	no	maximal, Sylow
71	C₂²×C₅₃	—	2	2	no	—
72	C₂²×C₅₃	#476	1	1	yes	characteristic
73	C₂²×C₅₃	#484	1	1	yes	characteristic
74	C₂²×C₅₃	#490	1	1	yes	characteristic
75	C₄×C₅₃	—	2	4	no	—
76	C₄×C₅₃	—	16	2	no	—
77	C₄×C₅₃	#400	1	1	yes	characteristic
78	C₄×C₅₃	#477	1	1	yes	characteristic
79	C₄×C₅₃	#481	1	1	yes	characteristic
80	C₄×C₅₃	#487	1	1	yes	characteristic
81	Dic₅₃	—	8	2	no	—
82	#1945	—	53	1	no	—
83	#1844	D₅₃	1	1	yes	characteristic
84	#1386	—	53	2	no	—
85	#1383	—	53	2	no	—
86	#1376	—	53	1	no	—
87	C₂×C₅₃	—	2	1	no	—
88	C₂×C₅₃	—	2	1	no	—
89	C₂×C₅₃	#2100	1	1	yes	characteristic
90	C₂×C₅₃	#2230	1	2	yes	—
91	C₂×M₃₂	—	53	2	no	—
92	C₂×#108	—	53	1	no	—
93	#427	—	53	1	no	—
94	C₂×#106	—	2	1	no	—
95	C₂×#106	C₂×D₅₃	1	2	yes	—
96	#424	—	106	1	no	—
97	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
98	#384	—	2	1	no	—
99	#339	—	106	1	no	—
100	C₅₃	#13543	1	1	yes	characteristic, complemented, Sylow
101	C₂×M₁₆	—	106	1	no	—
102	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
103	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
104	#117	—	106	1	no	—
105	C₂×C₄⋊C₄	—	2	2	no	—
106	C₂×C₄⋊C₄	—	4	2	no	—
107	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
108	M₃₂	—	106	2	no	—
109	M₃₂	—	106	2	no	—
110	C₂×C₁₆	—	106	2	no	—
111	#108	—	106	2	no	—
112	#107	—	53	2	no	—
113	#106	—	2	4	no	—
114	#106	—	4	2	no	—
115	#106	—	106	1	no	—
116	C₄⋊C₈	—	212	2	no	—
117	C₄×C₈	—	2	2	no	—
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	—	4	2	no	—
120	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
121	M₁₆	—	212	1	no	—
122	M₁₆	—	212	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	212	1	no	—
126	C₂×C₈	#11719924	1	2	yes	—
127	C₂×C₈	#11719945	1	2	yes	—
128	C₂×C₈	#11719947	1	2	yes	—
129	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
130	C₂×C₈	#11719949	1	1	yes	characteristic
131	C₄⋊C₄	—	4	2	no	—
132	C₄⋊C₄	—	4	2	no	—
133	C₄⋊C₄	—	8	4	no	—
134	C₄⋊C₄	—	106	2	no	—
135	C₂²⋊C₄	—	212	1	no	—
136	C₄²	—	2	1	no	—
137	C₄²	—	212	1	no	—
138	C₄²	#11719925	1	2	yes	—
139	C₁₆	—	212	2	no	—
140	C₂³	#13133243	1	1	yes	characteristic
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	8	4	no	—
144	C₂×C₄	—	8	2	no	—
145	C₂×C₄	—	212	2	no	—
146	C₂×C₄	#13133193	1	1	yes	characteristic
147	C₂×C₄	#13133200	1	1	yes	characteristic
148	C₂×C₄	#13133204	1	1	yes	characteristic
149	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
150	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
151	C₂×C₄	#13133243	1	1	yes	characteristic
152	C₈	—	2	2	no	—
153	C₈	—	424	1	no	—
154	C₈	#13133207	1	2	yes	—
155	C₈	#13133212	1	2	yes	—
156	C₂²	—	2	2	no	—
157	C₂²	#16810910	1	1	yes	characteristic
158	C₂²	#16810958	1	1	yes	characteristic
159	C₂²	#16810987	1	1	yes	characteristic
160	C₄	—	2	4	no	—
161	C₄	—	16	2	no	—
162	C₄	—	424	2	no	—
163	C₄	#16810561	1	1	yes	characteristic
164	C₄	#16810915	1	1	yes	characteristic
165	C₄	#16810941	1	1	yes	characteristic
166	C₄	#16810973	1	1	yes	characteristic
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	#20276277	1	1	yes	characteristic
170	C₂	#20276883	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20275463

C₂

yes

maximal, characteristic

#20272822

C₂

yes

maximal

#20272813

C₂

yes

maximal

#20272797

C₂

yes

maximal, characteristic

C₅₃×#1844

C₂

yes

maximal, characteristic

C₂×#13133157

C₂²

yes

—

C₂×#13133153

C₂²

yes

characteristic

#16810661

C₂²

yes

characteristic

#16810649

—

#16810383

—

C₂×#106×C₅₃

—

C₂×#106×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#384

—

—

—

—

—

—

—

Q₈

yes

—

#13133149

—

#13133146

—

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

D₄

yes

characteristic

#106×C₅₃

—

#106×C₅₃

—

C₄×C₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₂×C₈×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₃

#41

yes

—

C₂×C₈×C₅₃

#41

yes

—

C₂×C₈×C₅₃

#41

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

Q₁₆

yes

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

—

—

—

—

C₂³×C₅₃

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#125

yes

—

C₈×C₅₃

#126

yes

—

#88322

—

#13543

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

#1945

—

#1844

D₅₃

yes

—

—

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#2100

yes

characteristic

C₂×C₅₃

#2230

yes

—

C₂×M₃₂

—

—

—

—

C₂×D₅₃

yes

—

#424

—

106

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

#384

—

#339

—

106

—

100

C₅₃

#13543

yes

characteristic, complemented, Sylow

101

C₂×M₁₆

—

106

—

102

C₂²×C₈

C₂²⋊D₅₃

yes

characteristic

103

C₂²×C₈

C₂²×D₅₃

yes

characteristic

104

#117

—

106

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄²

C₂²⋊D₅₃

yes

characteristic

108

M₃₂

—

106

—

109

M₃₂

—

106

—

110

C₂×C₁₆

—

106

—

111

#108

—

106

—

112

#107

—

113

#106

—

114

#106

—

115

#106

—

106

—

116

C₄⋊C₈

—

212

—

117

C₄×C₈

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

121

M₁₆

—

212

—

122

M₁₆

—

212

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

212

—

126

C₂×C₈

#11719924

yes

—

127

C₂×C₈

#11719945

yes

—

128

C₂×C₈

#11719947

yes

—

129

C₂×C₈

Q₈×D₅₃

yes

characteristic

130

C₂×C₈

#11719949

yes

characteristic

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

106

—

135

C₂²⋊C₄

—

212

—

136

C₄²

—

137

C₄²

—

212

—

138

C₄²

#11719925

yes

—

139

C₁₆

—

212

—

140

C₂³

#13133243

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

212

—

146

C₂×C₄

#13133193

yes

characteristic

147

C₂×C₄

#13133200

yes

characteristic

148

C₂×C₄

#13133204

yes

characteristic

149

C₂×C₄

C₂×#11719924

yes

characteristic

150

C₂×C₄

C₂×#11719925

yes

characteristic

151

C₂×C₄

#13133243

yes

characteristic

152

C₈

—

153

C₈

—

424

—

154

C₈

#13133207

yes

—

155

C₈

#13133212

yes

—

156

C₂²

—

157

C₂²

#16810910

yes

characteristic

158

C₂²

#16810958

yes

characteristic

159

C₂²

#16810987

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

—

424

—

163

C₄

#16810561

yes

characteristic

164

C₄

#16810915

yes

characteristic

165

C₄

#16810941

yes

characteristic

166

C₄

#16810973

yes

characteristic

167

C₂

—

168

C₂

—

169

C₂

#20276277

yes

characteristic

170

C₂

#20276883

yes

—

	subgroup	G/H
center	C₂²	#16810910
commutator, cosocle	C₂²×C₈×C₅₃	C₂³
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1844	C₂
socle	C₂²×C₅₃	#476
prime radical	C₂³×C₅₃	AGL(1,ℤ/8)
hypercenter	#1844	D₅₃
hypocenter	C₅₃	#13543
LCS	G ▷ C₂²×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1844

subgroup

G/H

center

C₂²

#16810910

commutator, cosocle

C₂²×C₈×C₅₃

C₂³

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₅₃×#1844

C₂

C₂²×C₅₃

C₂³×C₅₃

D₅₃

C₅₃

G ▷ C₂²×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1844

N	H ≅ G/N	type
100. C₅₃	#13543	semidirect

H ≅ G/N

type

100. C₅₃

#13543

semidirect

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

subgroup	index
#20275463	2
#20272822	2
#20272813	2
#20272797	2
C₅₃×#1844	2
#13543	53

subgroup

index

C₅₃×#1844

Subgroups — #27304241

Conjugacy classes of subgroups

Complements

Maximal subgroups