Group #27216249 — subgroups

Subgroups — C₅₃×#8737 #27216249

Order 13568.

418 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 126 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×#2313	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#324	C₂	1	2	yes	maximal
3	C₅₃×#1282	C₂	1	4	yes	maximal
4	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
5	C₂×C₂²⋊C₈×C₅₃	C₂²	1	4	yes	—
6	C₅₃×#387	—	2	4	no	—
7	C₅₃×#381	C₄	1	2	yes	—
8	C₅₃×#380	C₄	1	2	yes	—
9	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
10	C₅₃×#324	—	2	4	no	—
11	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
14	C₂²×C₈×C₅₃	—	2	4	no	—
15	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
16	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
17	C₂×C₄⋊C₄×C₅₃	D₄	1	4	yes	—
18	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
19	C₂×C₄²×C₅₃	—	2	2	no	—
20	C₂²⋊C₈×C₅₃	—	2	4	no	—
21	C₂²⋊C₈×C₅₃	—	2	4	no	—
22	#95×C₅₃	—	2	2	no	—
23	#95×C₅₃	—	2	4	no	—
24	#95×C₅₃	—	2	2	no	—
25	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	—	2	2	no	—
27	C₂²×C₄×C₅₃	—	2	4	no	—
28	C₂²×C₄×C₅₃	—	2	2	no	—
29	C₂²×C₄×C₅₃	—	2	4	no	—
30	C₂²×C₄×C₅₃	—	8	1	no	—
31	C₂²×C₄×C₅₃	—	8	1	no	—
32	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
33	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
34	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
35	C₂²×C₄×C₅₃	#41	1	4	yes	—
36	C₂×C₈×C₅₃	—	4	4	no	—
37	C₂×C₈×C₅₃	—	8	4	no	—
38	C₄⋊C₄×C₅₃	—	2	8	no	—
39	C₄⋊C₄×C₅₃	—	8	2	no	—
40	C₂²⋊C₄×C₅₃	—	2	8	no	—
41	C₄²×C₅₃	—	8	2	no	—
42	C₂³×C₅₃	—	4	1	no	—
43	C₂³×C₅₃	—	4	1	no	—
44	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂³×C₅₃	#117	1	2	yes	—
46	C₂³×C₅₃	#131	1	4	yes	—
47	C₂×C₄×C₅₃	—	2	2	no	—
48	C₂×C₄×C₅₃	—	2	2	no	—
49	C₂×C₄×C₅₃	—	2	4	no	—
50	C₂×C₄×C₅₃	—	4	2	no	—
51	C₂×C₄×C₅₃	—	4	4	no	—
52	C₂×C₄×C₅₃	—	4	4	no	—
53	C₂×C₄×C₅₃	—	4	2	no	—
54	C₂×C₄×C₅₃	—	4	4	no	—
55	C₂×C₄×C₅₃	—	4	4	no	—
56	C₂×C₄×C₅₃	—	4	4	no	—
57	C₂×C₄×C₅₃	—	8	2	no	—
58	C₂×C₄×C₅₃	#123	1	4	yes	—
59	C₈×C₅₃	—	8	4	no	—
60	#8737	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂²×C₅₃	—	2	1	no	—
62	C₂²×C₅₃	—	2	2	no	—
63	C₂²×C₅₃	—	2	1	no	—
64	C₂²×C₅₃	—	2	2	no	—
65	C₂²×C₅₃	—	4	4	no	—
66	C₂²×C₅₃	#381	1	1	yes	characteristic
67	C₂²×C₅₃	#408	1	2	yes	—
68	C₂²×C₅₃	#433	1	4	yes	—
69	C₄×C₅₃	—	2	4	no	—
70	C₄×C₅₃	—	4	2	no	—
71	C₄×C₅₃	—	4	2	no	—
72	C₄×C₅₃	—	8	4	no	—
73	#2313	C₂×C₅₃	1	1	yes	characteristic
74	C₂×#324	C₂×C₅₃	1	2	yes	—
75	#1282	C₂×C₅₃	1	4	yes	—
76	C₂×C₅₃	—	2	2	no	—
77	C₂×C₅₃	—	2	4	no	—
78	C₂×C₅₃	#1818	1	1	yes	characteristic
79	C₂×C₅₃	#1828	1	1	yes	characteristic
80	C₂×C₅₃	#1830	1	1	yes	characteristic
81	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
82	C₂×C₂²⋊C₈	C₂²×C₅₃	1	4	yes	—
83	#387	—	2	4	no	—
84	#381	C₄×C₅₃	1	2	yes	—
85	#380	C₄×C₅₃	1	2	yes	—
86	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
87	#324	—	2	4	no	—
88	C₅₃	#8737	1	1	yes	characteristic, direct product factor, Sylow
89	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
90	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
91	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
92	C₂²×C₈	—	2	4	no	—
93	C₂×C₄⋊C₄	—	2	4	no	—
94	C₂×C₄⋊C₄	—	2	2	no	—
95	C₂×C₄⋊C₄	D₄×C₅₃	1	4	yes	—
96	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
97	C₂×C₄²	—	2	2	no	—
98	C₂²⋊C₈	—	2	4	no	—
99	C₂²⋊C₈	—	2	4	no	—
100	#95	—	2	2	no	—
101	#95	—	2	2	no	—
102	#95	—	2	4	no	—
103	C₂⁴	C₂²×C₄×C₅₃	1	1	yes	characteristic
104	C₂²×C₄	—	2	2	no	—
105	C₂²×C₄	—	2	4	no	—
106	C₂²×C₄	—	2	2	no	—
107	C₂²×C₄	—	2	4	no	—
108	C₂²×C₄	—	8	1	no	—
109	C₂²×C₄	—	8	1	no	—
110	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
111	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
112	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
113	C₂²×C₄	#41×C₅₃	1	4	yes	—
114	C₂×C₈	—	4	4	no	—
115	C₂×C₈	—	8	4	no	—
116	C₄⋊C₄	—	2	8	no	—
117	C₄⋊C₄	—	8	2	no	—
118	C₂²⋊C₄	—	2	8	no	—
119	C₄²	—	8	2	no	—
120	C₂³	—	4	1	no	—
121	C₂³	—	4	1	no	—
122	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
123	C₂³	#117×C₅₃	1	2	yes	—
124	C₂³	#131×C₅₃	1	4	yes	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	4	4	no	—
130	C₂×C₄	—	4	4	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	4	4	no	—
133	C₂×C₄	—	4	4	no	—
134	C₂×C₄	—	4	4	no	—
135	C₂×C₄	—	8	2	no	—
136	C₂×C₄	#123×C₅₃	1	4	yes	—
137	C₈	—	8	4	no	—
138	C₂²	—	2	1	no	—
139	C₂²	—	2	2	no	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	4	4	no	—
143	C₂²	C₅₃×#381	1	1	yes	characteristic
144	C₂²	C₅₃×#408	1	2	yes	—
145	C₂²	C₅₃×#433	1	4	yes	—
146	C₄	—	2	4	no	—
147	C₄	—	4	2	no	—
148	C₄	—	4	2	no	—
149	C₄	—	8	4	no	—
150	C₂	—	2	2	no	—
151	C₂	—	2	4	no	—
152	C₂	C₅₃×#1818	1	1	yes	characteristic
153	C₂	C₅₃×#1828	1	1	yes	characteristic
154	C₂	C₅₃×#1830	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#2313

C₂

yes

maximal, characteristic

C₂×C₅₃×#324

C₂

yes

maximal

C₅₃×#1282

C₂

yes

maximal

C₂²×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₂×C₂²⋊C₈×C₅₃

C₂²

yes

—

C₅₃×#387

—

C₅₃×#381

C₄

yes

—

C₅₃×#380

C₄

yes

—

C₄×C₂²⋊C₄×C₅₃

C₂²

yes

—

C₅₃×#324

—

C₂³×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂³×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂³×C₄×C₅₃

C₂³

yes

characteristic

C₂²×C₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

D₄

yes

—

C₂×C₂²⋊C₄×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

#95×C₅₃

—

#95×C₅₃

—

#95×C₅₃

—

C₂⁴×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×C₄×C₅₃

#41

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂³×C₅₃

#117

yes

—

C₂³×C₅₃

#131

yes

—

C₂×C₄×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

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#123

yes

—

C₈×C₅₃

—

#8737

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#381

yes

characteristic

C₂²×C₅₃

#408

yes

—

C₂²×C₅₃

#433

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

#2313

C₂×C₅₃

yes

characteristic

C₂×#324

C₂×C₅₃

yes

—

#1282

C₂×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂²×C₄⋊C₄

C₂²×C₅₃

yes

characteristic

C₂×C₂²⋊C₈

C₂²×C₅₃

yes

—

#387

—

#381

C₄×C₅₃

yes

—

#380

C₄×C₅₃

yes

—

C₄×C₂²⋊C₄

C₂²×C₅₃

yes

—

#324

—

C₅₃

#8737

yes

characteristic, direct product factor, Sylow

C₂³×C₄

C₂×C₄×C₅₃

yes

characteristic

C₂³×C₄

C₂×C₄×C₅₃

yes

characteristic

C₂³×C₄

C₂³×C₅₃

yes

characteristic

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₄×C₅₃

yes

—

C₂×C₂²⋊C₄

C₂×C₄×C₅₃

yes

—

C₂×C₄²

—

C₂²⋊C₈

—

C₂²⋊C₈

—

100

#95

—

101

#95

—

102

#95

—

103

C₂⁴

C₂²×C₄×C₅₃

yes

characteristic

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₂²⋊C₄×C₅₃

yes

—

111

C₂²×C₄

C₂²⋊C₄×C₅₃

yes

—

112

C₂²×C₄

C₂×D₄×C₅₃

yes

—

113

C₂²×C₄

#41×C₅₃

yes

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₂²⋊C₄

—

119

C₄²

—

120

C₂³

—

121

C₂³

—

122

C₂³

C₂×C₂²⋊C₄×C₅₃

yes

characteristic

123

C₂³

#117×C₅₃

yes

—

124

C₂³

#131×C₅₃

yes

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

#123×C₅₃

yes

—

137

C₈

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

C₅₃×#381

yes

characteristic

144

C₂²

C₅₃×#408

yes

—

145

C₂²

C₅₃×#433

yes

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₂

—

151

C₂

—

152

C₂

C₅₃×#1818

yes

characteristic

153

C₂

C₅₃×#1828

yes

characteristic

154

C₂

C₅₃×#1830

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	#381
commutator	C₂⁴	C₂²×C₄×C₅₃
Frattini, cosocle	C₂³×C₄	C₂³×C₅₃
prime radical	C₂⁴×C₅₃	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂³×C₄×C₅₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₅₃

#381

commutator

C₂⁴

C₂²×C₄×C₅₃

Frattini, cosocle

C₂³×C₄

C₂³×C₅₃

prime radical

C₂⁴×C₅₃

C₂²×C₄

LCS

G ▷ C₂⁴ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₅₃ ◁ C₂³×C₄×C₅₃ ◁ G

N	H ≅ G/N	type
60. #8737	C₅₃	direct (N × H)
88. C₅₃	#8737	direct (N × H)

H ≅ G/N

type

60. #8737

C₅₃

direct (N × H)

88. C₅₃

#8737

direct (N × H)

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

subgroup	index
C₅₃×#2313	2
C₂×C₅₃×#324	2
C₅₃×#1282	2
#8737	53

subgroup

index

C₅₃×#2313

C₂×C₅₃×#324

C₅₃×#1282

#8737

Subgroups — C₅₃×#8737 #27216249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#8737 #27216249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8737 #27216249