Group #27277769 — subgroups

Subgroups — #27277769

Order 13568.

342 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 155 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	#20274867	C₂	1	1	yes	maximal, characteristic
2	#20274832	C₂	1	1	yes	maximal, characteristic
3	#20272550	C₂	1	4	yes	maximal
4	C₅₃×#1744	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#88322	—	2	2	no	—
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	C₂×C₄⋊#88322	C₂²	1	2	yes	—
8	#16810378	—	2	2	no	—
9	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
10	C₄³×C₅₃	C₂²	1	1	yes	characteristic
11	C₅₃×#336	C₄	1	2	yes	—
12	C₅₃×#336	C₄	1	2	yes	—
13	C₂²×#88322	—	2	2	no	—
14	C₂²×#88322	—	2	2	no	—
15	C₄⋊#88322	—	2	4	no	—
16	C₄⋊#88322	—	2	4	no	—
17	C₄⋊#88322	—	4	4	no	—
18	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
19	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₅₃	D₄	1	2	yes	—
22	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
23	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
24	C₄×C₈×C₅₃	—	2	2	no	—
25	C₄×C₈×C₅₃	—	2	2	no	—
26	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
27	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
28	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
31	C₂×C₈×C₅₃	—	2	4	no	—
32	C₂×C₈×C₅₃	—	2	4	no	—
33	C₂×C₈×C₅₃	—	2	4	no	—
34	C₄²×C₅₃	—	2	4	no	—
35	C₄²×C₅₃	—	2	2	no	—
36	C₄²×C₅₃	—	2	2	no	—
37	C₄²×C₅₃	—	2	2	no	—
38	C₄²×C₅₃	—	2	2	no	—
39	C₄²×C₅₃	#41	1	2	yes	—
40	C₄²×C₅₃	#41	1	2	yes	—
41	C₂×#88322	—	4	2	no	—
42	C₂×#88322	—	4	2	no	—
43	C₂×#88322	—	4	4	no	—
44	C₂×#88322	—	4	2	no	—
45	C₂×#88322	—	4	2	no	—
46	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₅₃	—	2	2	no	—
48	C₂×C₄×C₅₃	—	2	2	no	—
49	C₂×C₄×C₅₃	—	2	2	no	—
50	C₂×C₄×C₅₃	—	2	2	no	—
51	C₂×C₄×C₅₃	—	2	2	no	—
52	C₂×C₄×C₅₃	—	2	2	no	—
53	C₂×C₄×C₅₃	#108	1	4	yes	—
54	C₂×C₄×C₅₃	#117	1	2	yes	—
55	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
56	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
57	C₂×C₄×C₅₃	#125	1	2	yes	—
58	C₂×C₄×C₅₃	#131	1	2	yes	—
59	C₂×C₄×C₅₃	#131	1	2	yes	—
60	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
61	C₂×C₄×C₅₃	#137	1	2	yes	—
62	C₈×C₅₃	—	4	4	no	—
63	#88322	—	8	2	no	—
64	#88322	—	8	2	no	—
65	#8789	—	53	1	no	maximal, Sylow
66	C₂²×C₅₃	#382	1	1	yes	characteristic
67	C₂²×C₅₃	#424	1	1	yes	characteristic
68	C₂²×C₅₃	#424	1	1	yes	characteristic
69	C₂²×C₅₃	#428	1	2	yes	—
70	C₂²×C₅₃	C₂×#108	1	2	yes	—
71	C₄×C₅₃	—	2	4	no	—
72	C₄×C₅₃	—	2	2	no	—
73	C₄×C₅₃	—	4	2	no	—
74	C₄×C₅₃	#433	1	4	yes	—
75	C₄×C₅₃	#489	1	2	yes	—
76	C₄×C₅₃	#490	1	1	yes	characteristic
77	C₄×C₅₃	#491	1	1	yes	characteristic
78	#1842	—	53	1	no	—
79	#1831	—	53	1	no	—
80	#1744	D₅₃	1	1	yes	characteristic
81	#1277	—	53	4	no	—
82	C₂×C₅₃	#1573	1	2	yes	—
83	C₂×C₅₃	#1574	1	2	yes	—
84	C₂×C₅₃	#1832	1	1	yes	characteristic
85	C₂×C₅₃	#1836	1	1	yes	characteristic
86	C₂×C₅₃	#1837	1	1	yes	characteristic
87	C₂×C₄⋊C₈	—	53	2	no	—
88	C₂×C₄⋊C₈	—	53	2	no	—
89	C₂×C₄⋊C₈	—	106	2	no	—
90	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
91	C₄³	C₂×D₅₃	1	1	yes	characteristic
92	#336	—	106	2	no	—
93	#336	Dic₅₃	1	2	yes	—
94	#336	Dic₅₃	1	2	yes	—
95	C₅₃	#8789	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₈	—	106	2	no	—
97	C₂²×C₈	—	106	2	no	—
98	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
99	C₂×C₄²	Dic₁₀₆	1	2	yes	—
100	C₂×C₄²	D₂₁₂	1	2	yes	—
101	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
102	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
103	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
104	C₄⋊C₈	—	106	4	no	—
105	C₄⋊C₈	—	106	4	no	—
106	C₄⋊C₈	—	212	4	no	—
107	C₄×C₈	—	2	2	no	—
108	C₄×C₈	—	2	2	no	—
109	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
110	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
111	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
112	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
113	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
114	C₂×C₈	—	2	4	no	—
115	C₂×C₈	—	2	4	no	—
116	C₂×C₈	—	2	4	no	—
117	C₂×C₈	—	212	2	no	—
118	C₂×C₈	—	212	2	no	—
119	C₂×C₈	—	212	4	no	—
120	C₂×C₈	—	212	2	no	—
121	C₂×C₈	—	212	2	no	—
122	C₄²	—	2	4	no	—
123	C₄²	—	2	2	no	—
124	C₄²	—	2	2	no	—
125	C₄²	—	2	2	no	—
126	C₄²	—	2	2	no	—
127	C₄²	#11719945	1	2	yes	—
128	C₄²	#11719945	1	2	yes	—
129	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	#13133153	1	4	yes	—
137	C₂×C₄	#13133176	1	2	yes	—
138	C₂×C₄	#13133205	1	1	yes	characteristic
139	C₂×C₄	#13133205	1	1	yes	characteristic
140	C₂×C₄	#13133209	1	2	yes	—
141	C₂×C₄	#13133224	1	2	yes	—
142	C₂×C₄	#13133224	1	2	yes	—
143	C₂×C₄	#13133245	1	2	yes	—
144	C₂×C₄	#13133250	1	2	yes	—
145	C₈	—	4	4	no	—
146	C₈	—	424	2	no	—
147	C₈	—	424	2	no	—
148	C₂²	#16810489	1	1	yes	characteristic
149	C₂²	#16810650	1	1	yes	characteristic
150	C₂²	#16810650	1	1	yes	characteristic
151	C₂²	#16810665	1	2	yes	—
152	C₂²	C₂×#13133153	1	2	yes	—
153	C₄	—	2	4	no	—
154	C₄	—	2	2	no	—
155	C₄	—	4	2	no	—
156	C₄	#16810684	1	4	yes	—
157	C₄	#16810983	1	2	yes	—
158	C₄	#16810988	1	1	yes	characteristic
159	C₄	#16810992	1	1	yes	characteristic
160	C₂	#20273500	1	2	yes	—
161	C₂	#20273504	1	2	yes	—
162	C₂	#20274835	1	1	yes	characteristic
163	C₂	#20274849	1	1	yes	characteristic
164	C₂	#20274853	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20274867

C₂

yes

maximal, characteristic

#20274832

C₂

yes

maximal, characteristic

#20272550

C₂

yes

maximal

C₅₃×#1744

C₂

yes

maximal, characteristic

C₂×C₄⋊#88322

—

C₂×C₄⋊#88322

C₂²

yes

—

C₂×C₄⋊#88322

C₂²

yes

—

#16810378

—

C₂×C₄×C₈×C₅₃

C₂²

yes

—

C₄³×C₅₃

C₂²

yes

characteristic

C₅₃×#336

C₄

yes

—

C₅₃×#336

C₄

yes

—

C₂²×#88322

—

C₂²×#88322

—

C₄⋊#88322

—

C₄⋊#88322

—

C₄⋊#88322

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

Q₈

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

#41

yes

—

C₄²×C₅₃

#41

yes

—

—

—

—

—

—

C₂³×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#108

yes

—

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₈×C₅₃

—

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

C₂×#108

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#433

yes

—

C₄×C₅₃

#489

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

—

—

D₅₃

yes

characteristic

#1277

—

C₂×C₅₃

#1573

yes

—

C₂×C₅₃

#1574

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

106

—

C₂×C₄×C₈

C₂×D₅₃

yes

—

C₄³

C₂×D₅₃

yes

characteristic

#336

—

106

—

#336

Dic₅₃

yes

—

#336

Dic₅₃

yes

—

C₅₃

#8789

yes

characteristic, complemented, Sylow

C₂²×C₈

—

106

—

C₂²×C₈

—

106

—

C₂²×C₈

C₂×Dic₅₃

yes

—

C₂×C₄²

Dic₁₀₆

yes

—

100

C₂×C₄²

D₂₁₂

yes

—

101

C₂×C₄²

C₂×Dic₅₃

yes

characteristic

102

C₂×C₄²

C₂×Dic₅₃

yes

characteristic

103

C₂×C₄²

C₂²×D₅₃

yes

characteristic

104

C₄⋊C₈

—

106

—

105

C₄⋊C₈

—

106

—

106

C₄⋊C₈

—

212

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₂²×C₄

C₄⋊Dic₅₃

yes

—

110

C₂²×C₄

C₄⋊Dic₅₃

yes

—

111

C₂²×C₄

C₂×Dic₁₀₆

yes

characteristic

112

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

113

C₂²×C₄

C₂²×Dic₅₃

yes

characteristic

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

212

—

118

C₂×C₈

—

212

—

119

C₂×C₈

—

212

—

120

C₂×C₈

—

212

—

121

C₂×C₈

—

212

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

#11719945

yes

—

128

C₄²

#11719945

yes

—

129

C₂³

C₂×C₄⋊Dic₅₃

yes

characteristic

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

#13133153

yes

—

137

C₂×C₄

#13133176

yes

—

138

C₂×C₄

#13133205

yes

characteristic

139

C₂×C₄

#13133205

yes

characteristic

140

C₂×C₄

#13133209

yes

—

141

C₂×C₄

#13133224

yes

—

142

C₂×C₄

#13133224

yes

—

143

C₂×C₄

#13133245

yes

—

144

C₂×C₄

#13133250

yes

—

145

C₈

—

146

C₈

—

424

—

147

C₈

—

424

—

148

C₂²

#16810489

yes

characteristic

149

C₂²

#16810650

yes

characteristic

150

C₂²

#16810650

yes

characteristic

151

C₂²

#16810665

yes

—

152

C₂²

C₂×#13133153

yes

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

#16810684

yes

—

157

C₄

#16810983

yes

—

158

C₄

#16810988

yes

characteristic

159

C₄

#16810992

yes

characteristic

160

C₂

#20273500

yes

—

161

C₂

#20273504

yes

—

162

C₂

#20274835

yes

characteristic

163

C₂

#20274849

yes

characteristic

164

C₂

#20274853

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1744	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1744	D₅₃
hypocenter	C₅₃	#8789
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

subgroup

G/H

center

C₂²×C₄

C₂×D₂₁₂

commutator

C₂²×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1744

C₂

socle, prime radical

C₂³×C₅₃

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

N	H ≅ G/N	type
95. C₅₃	#8789	semidirect

H ≅ G/N

type

95. C₅₃

#8789

semidirect

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

subgroup	index
#20274867	2
#20274832	2
#20272550	2
C₅₃×#1744	2
#8789	53

subgroup

index

C₅₃×#1744

Subgroups — #27277769

Conjugacy classes of subgroups

Complements

Maximal subgroups