Group #27216301 — subgroups

Subgroups — C₅₃×#8789 #27216301

Order 13568.

342 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 166 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	C₅₃×#1842	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1831	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1744	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1277	C₂	1	4	yes	maximal
5	C₂×C₄⋊C₈×C₅₃	—	2	2	no	—
6	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
7	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
9	C₄³×C₅₃	C₂²	1	1	yes	characteristic
10	C₅₃×#336	—	2	2	no	—
11	C₅₃×#336	C₄	1	2	yes	—
12	C₅₃×#336	C₄	1	2	yes	—
13	C₂²×C₈×C₅₃	—	2	2	no	—
14	C₂²×C₈×C₅₃	—	2	2	no	—
15	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
16	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₅₃	D₄	1	2	yes	—
19	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
20	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
21	C₄⋊C₈×C₅₃	—	2	4	no	—
22	C₄⋊C₈×C₅₃	—	2	4	no	—
23	C₄⋊C₈×C₅₃	—	4	4	no	—
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	C₂×C₈×C₅₃	—	4	2	no	—
35	C₂×C₈×C₅₃	—	4	2	no	—
36	C₂×C₈×C₅₃	—	4	4	no	—
37	C₂×C₈×C₅₃	—	4	2	no	—
38	C₂×C₈×C₅₃	—	4	2	no	—
39	C₄²×C₅₃	—	2	4	no	—
40	C₄²×C₅₃	—	2	2	no	—
41	C₄²×C₅₃	—	2	2	no	—
42	C₄²×C₅₃	—	2	2	no	—
43	C₄²×C₅₃	—	2	2	no	—
44	C₄²×C₅₃	#41	1	2	yes	—
45	C₄²×C₅₃	#41	1	2	yes	—
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	C₈×C₅₃	—	8	2	no	—
64	C₈×C₅₃	—	8	2	no	—
65	#8789	C₅₃	1	1	yes	maximal, characteristic, direct product factor, 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	C₂×C₅₃	1	1	yes	characteristic
79	#1831	C₂×C₅₃	1	1	yes	characteristic
80	#1744	C₂×C₅₃	1	1	yes	characteristic
81	#1277	C₂×C₅₃	1	4	yes	—
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	—	2	2	no	—
88	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
89	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
90	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
91	C₄³	C₂²×C₅₃	1	1	yes	characteristic
92	#336	—	2	2	no	—
93	#336	C₄×C₅₃	1	2	yes	—
94	#336	C₄×C₅₃	1	2	yes	—
95	C₅₃	#8789	1	1	yes	characteristic, direct product factor, Sylow
96	C₂²×C₈	—	2	2	no	—
97	C₂²×C₈	—	2	2	no	—
98	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
99	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
100	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
101	C₂×C₄²	D₄×C₅₃	1	2	yes	—
102	C₂×C₄²	Q₈×C₅₃	1	2	yes	—
103	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
104	C₄⋊C₈	—	2	4	no	—
105	C₄⋊C₈	—	2	4	no	—
106	C₄⋊C₈	—	4	4	no	—
107	C₄×C₈	—	2	2	no	—
108	C₄×C₈	—	2	2	no	—
109	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
110	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
111	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
112	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
113	C₂²×C₄	C₂×Q₈×C₅₃	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	—	4	2	no	—
118	C₂×C₈	—	4	2	no	—
119	C₂×C₈	—	4	4	no	—
120	C₂×C₈	—	4	2	no	—
121	C₂×C₈	—	4	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	#41×C₅₃	1	2	yes	—
128	C₄²	#41×C₅₃	1	2	yes	—
129	C₂³	C₂×C₄⋊C₄×C₅₃	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	#108×C₅₃	1	4	yes	—
137	C₂×C₄	#117×C₅₃	1	2	yes	—
138	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
139	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
140	C₂×C₄	#125×C₅₃	1	2	yes	—
141	C₂×C₄	#131×C₅₃	1	2	yes	—
142	C₂×C₄	#131×C₅₃	1	2	yes	—
143	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
144	C₂×C₄	#137×C₅₃	1	2	yes	—
145	C₈	—	4	4	no	—
146	C₈	—	8	2	no	—
147	C₈	—	8	2	no	—
148	C₂²	C₅₃×#382	1	1	yes	characteristic
149	C₂²	C₅₃×#424	1	1	yes	characteristic
150	C₂²	C₅₃×#424	1	1	yes	characteristic
151	C₂²	C₅₃×#428	1	2	yes	—
152	C₂²	C₂×#108×C₅₃	1	2	yes	—
153	C₄	—	2	4	no	—
154	C₄	—	2	2	no	—
155	C₄	—	4	2	no	—
156	C₄	C₅₃×#433	1	4	yes	—
157	C₄	C₅₃×#489	1	2	yes	—
158	C₄	C₅₃×#490	1	1	yes	characteristic
159	C₄	C₅₃×#491	1	1	yes	characteristic
160	C₂	C₅₃×#1573	1	2	yes	—
161	C₂	C₅₃×#1574	1	2	yes	—
162	C₂	C₅₃×#1832	1	1	yes	characteristic
163	C₂	C₅₃×#1836	1	1	yes	characteristic
164	C₂	C₅₃×#1837	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#1842

C₂

yes

maximal, characteristic

C₅₃×#1831

C₂

yes

maximal, characteristic

C₅₃×#1744

C₂

yes

maximal, characteristic

C₅₃×#1277

C₂

yes

maximal

C₂×C₄⋊C₈×C₅₃

—

C₂×C₄⋊C₈×C₅₃

C₂²

yes

—

C₂×C₄⋊C₈×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

—

C₄³×C₅₃

C₂²

yes

characteristic

C₅₃×#336

—

C₅₃×#336

C₄

yes

—

C₅₃×#336

C₄

yes

—

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

—

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

—

C₄⋊C₈×C₅₃

—

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

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

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

—

C₈×C₅₃

—

C₈×C₅₃

—

#8789

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

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

C₂×C₅₃

yes

characteristic

#1831

C₂×C₅₃

yes

characteristic

#1744

C₂×C₅₃

yes

characteristic

#1277

C₂×C₅₃

yes

—

C₂×C₅₃

#1573

yes

—

C₂×C₅₃

#1574

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂²×C₅₃

yes

—

C₂×C₄⋊C₈

C₂²×C₅₃

yes

—

C₂×C₄×C₈

C₂²×C₅₃

yes

—

C₄³

C₂²×C₅₃

yes

characteristic

#336

—

#336

C₄×C₅₃

yes

—

#336

C₄×C₅₃

yes

—

C₅₃

#8789

yes

characteristic, direct product factor, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄×C₅₃

yes

—

C₂×C₄²

C₂×C₄×C₅₃

yes

characteristic

100

C₂×C₄²

C₂×C₄×C₅₃

yes

characteristic

101

C₂×C₄²

D₄×C₅₃

yes

—

102

C₂×C₄²

Q₈×C₅₃

yes

—

103

C₂×C₄²

C₂³×C₅₃

yes

characteristic

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

C₄⋊C₈

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₂²×C₄

C₄⋊C₄×C₅₃

yes

—

110

C₂²×C₄

C₄⋊C₄×C₅₃

yes

—

111

C₂²×C₄

C₂²×C₄×C₅₃

yes

characteristic

112

C₂²×C₄

C₂×D₄×C₅₃

yes

characteristic

113

C₂²×C₄

C₂×Q₈×C₅₃

yes

characteristic

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

#41×C₅₃

yes

—

128

C₄²

#41×C₅₃

yes

—

129

C₂³

C₂×C₄⋊C₄×C₅₃

yes

characteristic

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

#108×C₅₃

yes

—

137

C₂×C₄

#117×C₅₃

yes

—

138

C₂×C₄

#124×C₅₃

yes

characteristic

139

C₂×C₄

#124×C₅₃

yes

characteristic

140

C₂×C₄

#125×C₅₃

yes

—

141

C₂×C₄

#131×C₅₃

yes

—

142

C₂×C₄

#131×C₅₃

yes

—

143

C₂×C₄

AGL(1,ℤ/8)×C₅₃

yes

—

144

C₂×C₄

#137×C₅₃

yes

—

145

C₈

—

146

C₈

—

147

C₈

—

148

C₂²

C₅₃×#382

yes

characteristic

149

C₂²

C₅₃×#424

yes

characteristic

150

C₂²

C₅₃×#424

yes

characteristic

151

C₂²

C₅₃×#428

yes

—

152

C₂²

C₂×#108×C₅₃

yes

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

C₅₃×#433

yes

—

157

C₄

C₅₃×#489

yes

—

158

C₄

C₅₃×#490

yes

characteristic

159

C₄

C₅₃×#491

yes

characteristic

160

C₂

C₅₃×#1573

yes

—

161

C₂

C₅₃×#1574

yes

—

162

C₂

C₅₃×#1832

yes

characteristic

163

C₂

C₅₃×#1836

yes

characteristic

164

C₂

C₅₃×#1837

yes

characteristic

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₄³×C₅₃ ◁ G

subgroup

G/H

center

C₂²×C₄×C₅₃

C₂×D₄

commutator

C₂²×C₄

C₂²×C₄×C₅₃

Frattini, cosocle

C₂×C₄²

C₂³×C₅₃

socle, prime radical

C₂³×C₅₃

C₂×C₄⋊C₄

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₄×C₅₃ ◁ C₄³×C₅₃ ◁ G

N	H ≅ G/N	type
65. #8789	C₅₃	direct (N × H)
95. C₅₃	#8789	direct (N × H)

H ≅ G/N

type

65. #8789

C₅₃

direct (N × H)

95. C₅₃

#8789

direct (N × H)

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

subgroup	index
C₅₃×#1842	2
C₅₃×#1831	2
C₅₃×#1744	2
C₅₃×#1277	2
#8789	53

subgroup

index

C₅₃×#1842

C₅₃×#1831

C₅₃×#1744

C₅₃×#1277

#8789

Subgroups — C₅₃×#8789 #27216301

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#8789 #27216301

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8789 #27216301