Group #27218142 — subgroups

Subgroups — C₅₃×#10630 #27218142

Order 13568.

272 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 98 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×#1995	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1832	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1542	C₂	1	2	yes	maximal
4	C₅₃×#1301	C₂	1	2	yes	maximal
5	C₅₃×#1274	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#533	C₂²	1	1	yes	characteristic
7	C₅₃×#431	C₂²	1	2	yes	—
8	C₅₃×#423	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
10	C₅₃×#376	C₂²	1	1	yes	characteristic
11	C₅₃×#339	—	2	2	no	—
12	C₅₃×#336	—	2	2	no	—
13	C₅₃×#333	—	2	2	no	—
14	C₅₃×#332	—	2	2	no	—
15	C₂×M₁₆×C₅₃	—	2	1	no	—
16	C₂²×C₈×C₅₃	—	2	2	no	—
17	C₄⋊Q₈×C₅₃	D₄	1	2	yes	—
18	#125×C₅₃	—	2	1	no	—
19	C₂²⋊Q₈×C₅₃	—	2	2	no	—
20	C₄×Q₈×C₅₃	—	2	2	no	—
21	#117×C₅₃	—	2	1	no	—
22	C₂×C₄²×C₅₃	D₄	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	C₄⋊C₈×C₅₃	—	2	2	no	—
27	C₄⋊C₈×C₅₃	—	2	2	no	—
28	C₂²⋊C₈×C₅₃	—	2	2	no	—
29	C₄×C₈×C₅₃	—	2	2	no	—
30	#95×C₅₃	—	2	2	no	—
31	C₂×Q₈×C₅₃	—	2	2	no	—
32	C₂²×C₄×C₅₃	—	2	2	no	—
33	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	#41	1	2	yes	—
35	M₁₆×C₅₃	—	4	2	no	—
36	C₂×C₈×C₅₃	—	4	2	no	—
37	C₂×C₈×C₅₃	—	4	2	no	—
38	C₂×C₈×C₅₃	—	4	1	no	—
39	C₂×C₈×C₅₃	—	4	2	no	—
40	C₂×C₈×C₅₃	—	4	2	no	—
41	C₄⋊C₄×C₅₃	—	2	2	no	—
42	C₄⋊C₄×C₅₃	—	2	2	no	—
43	C₄⋊C₄×C₅₃	—	4	2	no	—
44	C₂²⋊C₄×C₅₃	—	4	1	no	—
45	C₄²×C₅₃	—	2	2	no	—
46	C₄²×C₅₃	—	2	2	no	—
47	C₄²×C₅₃	—	4	1	no	—
48	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₅₃	#41	1	1	yes	characteristic
50	C₄²×C₅₃	#41	1	2	yes	—
51	C₂³×C₅₃	#123	1	1	yes	characteristic
52	Q₈×C₅₃	—	8	2	no	—
53	C₂×C₄×C₅₃	—	4	2	no	—
54	C₂×C₄×C₅₃	—	4	4	no	—
55	C₂×C₄×C₅₃	—	4	2	no	—
56	C₂×C₄×C₅₃	—	4	2	no	—
57	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
58	C₂×C₄×C₅₃	#123	1	2	yes	—
59	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
60	C₂×C₄×C₅₃	#126	1	2	yes	—
61	C₂×C₄×C₅₃	#135	1	2	yes	—
62	C₂×C₄×C₅₃	#137	1	2	yes	—
63	C₈×C₅₃	—	8	2	no	—
64	C₈×C₅₃	—	8	2	no	—
65	C₈×C₅₃	—	8	1	no	—
66	#10630	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂²×C₅₃	—	2	2	no	—
68	C₂²×C₅₃	#397	1	1	yes	characteristic
69	C₂²×C₅₃	#485	1	2	yes	—
70	C₄×C₅₃	—	2	2	no	—
71	C₄×C₅₃	—	2	2	no	—
72	C₄×C₅₃	—	8	2	no	—
73	C₄×C₅₃	—	8	2	no	—
74	C₄×C₅₃	#464	1	2	yes	—
75	C₄×C₅₃	#488	1	1	yes	characteristic
76	C₄×C₅₃	#491	1	1	yes	characteristic
77	#1995	C₂×C₅₃	1	1	yes	characteristic
78	#1832	C₂×C₅₃	1	1	yes	characteristic
79	#1542	C₂×C₅₃	1	2	yes	—
80	#1301	C₂×C₅₃	1	2	yes	—
81	#1274	C₂×C₅₃	1	1	yes	characteristic
82	C₂×C₅₃	—	2	2	no	—
83	C₂×C₅₃	#2074	1	1	yes	characteristic
84	C₂×C₅₃	#2080	1	2	yes	—
85	#533	C₂²×C₅₃	1	1	yes	characteristic
86	#431	C₂²×C₅₃	1	2	yes	—
87	#423	C₂²×C₅₃	1	1	yes	characteristic
88	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
89	#376	C₂²×C₅₃	1	1	yes	characteristic
90	#339	—	2	2	no	—
91	#336	—	2	2	no	—
92	#333	—	2	2	no	—
93	#332	—	2	2	no	—
94	C₅₃	#10630	1	1	yes	characteristic, direct product factor, Sylow
95	C₂×M₁₆	—	2	1	no	—
96	C₂²×C₈	—	2	2	no	—
97	C₄⋊Q₈	D₄×C₅₃	1	2	yes	—
98	#125	—	2	1	no	—
99	C₂²⋊Q₈	—	2	2	no	—
100	C₄×Q₈	—	2	2	no	—
101	#117	—	2	1	no	—
102	C₂×C₄²	D₄×C₅₃	1	2	yes	—
103	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	2	2	no	—
106	C₄⋊C₈	—	2	2	no	—
107	C₄⋊C₈	—	2	2	no	—
108	C₂²⋊C₈	—	2	2	no	—
109	C₄×C₈	—	2	2	no	—
110	#95	—	2	2	no	—
111	C₂×Q₈	—	2	2	no	—
112	C₂²×C₄	—	2	2	no	—
113	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
114	C₂²×C₄	#41×C₅₃	1	2	yes	—
115	M₁₆	—	4	2	no	—
116	C₂×C₈	—	4	2	no	—
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	—	4	1	no	—
119	C₂×C₈	—	4	2	no	—
120	C₂×C₈	—	4	2	no	—
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	2	2	no	—
123	C₄⋊C₄	—	4	2	no	—
124	C₂²⋊C₄	—	4	1	no	—
125	C₄²	—	2	2	no	—
126	C₄²	—	2	2	no	—
127	C₄²	—	4	1	no	—
128	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
129	C₄²	#41×C₅₃	1	1	yes	characteristic
130	C₄²	#41×C₅₃	1	2	yes	—
131	C₂³	#123×C₅₃	1	1	yes	characteristic
132	Q₈	—	8	2	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	4	no	—
135	C₂×C₄	—	4	2	no	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
138	C₂×C₄	#123×C₅₃	1	2	yes	—
139	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
140	C₂×C₄	#126×C₅₃	1	2	yes	—
141	C₂×C₄	#135×C₅₃	1	2	yes	—
142	C₂×C₄	#137×C₅₃	1	2	yes	—
143	C₈	—	8	2	no	—
144	C₈	—	8	1	no	—
145	C₈	—	8	2	no	—
146	C₂²	—	2	2	no	—
147	C₂²	C₅₃×#397	1	1	yes	characteristic
148	C₂²	C₅₃×#485	1	2	yes	—
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	8	2	no	—
152	C₄	—	8	2	no	—
153	C₄	C₅₃×#464	1	2	yes	—
154	C₄	C₅₃×#488	1	1	yes	characteristic
155	C₄	C₅₃×#491	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	C₅₃×#2074	1	1	yes	characteristic
158	C₂	C₅₃×#2080	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#1995

C₂

yes

maximal, characteristic

C₅₃×#1832

C₂

yes

maximal, characteristic

C₅₃×#1542

C₂

yes

maximal

C₅₃×#1301

C₂

yes

maximal

C₅₃×#1274

C₂

yes

maximal, characteristic

C₅₃×#533

C₂²

yes

characteristic

C₅₃×#431

C₂²

yes

—

C₅₃×#423

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₅₃

C₂²

yes

—

C₅₃×#376

C₂²

yes

characteristic

C₅₃×#339

—

C₅₃×#336

—

C₅₃×#333

—

C₅₃×#332

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

—

C₄⋊Q₈×C₅₃

D₄

yes

—

#125×C₅₃

—

C₂²⋊Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

#117×C₅₃

—

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

—

#95×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

C₄²×C₅₃

yes

C₄²×C₅₃

yes

—

C₂³×C₅₃

#123

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#121

yes

characteristic

C₂×C₄×C₅₃

#123

yes

—

C₂×C₄×C₅₃

#124

yes

characteristic

C₂×C₄×C₅₃

#126

yes

—

C₂×C₄×C₅₃

#135

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#10630

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

#397

yes

characteristic

C₂²×C₅₃

#485

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#464

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₂×C₅₃

yes

characteristic

#1832

C₂×C₅₃

yes

characteristic

#1542

C₂×C₅₃

yes

—

#1301

C₂×C₅₃

yes

—

#1274

C₂×C₅₃

yes

characteristic

C₂×C₅₃

—

C₂×C₅₃

#2074

yes

characteristic

C₂×C₅₃

#2080

yes

—

#533

C₂²×C₅₃

yes

characteristic

#431

C₂²×C₅₃

yes

—

#423

C₂²×C₅₃

yes

characteristic

C₂×C₄⋊C₈

C₂²×C₅₃

yes

—

#376

C₂²×C₅₃

yes

—

—

—

—

C₅₃

#10630

yes

characteristic, direct product factor, Sylow

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

D₄×C₅₃

yes

—

#125

—

C₂²⋊Q₈

—

100

C₄×Q₈

—

101

#117

—

102

C₂×C₄²

D₄×C₅₃

yes

—

103

C₂×C₄²

C₂³×C₅₃

yes

characteristic

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₂²⋊C₈

—

109

C₄×C₈

—

110

#95

—

111

C₂×Q₈

—

112

C₂²×C₄

—

113

C₂²×C₄

C₂×D₄×C₅₃

yes

characteristic

114

C₂²×C₄

#41×C₅₃

yes

—

115

M₁₆

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₂²⋊C₄

—

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₄²

C₂×D₄×C₅₃

yes

characteristic

129

C₄²

#41×C₅₃

yes

characteristic

130

C₄²

#41×C₅₃

yes

—

131

C₂³

#123×C₅₃

yes

characteristic

132

Q₈

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

#121×C₅₃

yes

characteristic

138

C₂×C₄

#123×C₅₃

yes

—

139

C₂×C₄

#124×C₅₃

yes

characteristic

140

C₂×C₄

#126×C₅₃

yes

—

141

C₂×C₄

#135×C₅₃

yes

—

142

C₂×C₄

#137×C₅₃

yes

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₂²

—

147

C₂²

C₅₃×#397

yes

characteristic

148

C₂²

C₅₃×#485

yes

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

C₅₃×#464

yes

—

154

C₄

C₅₃×#488

yes

characteristic

155

C₄

C₅₃×#491

yes

characteristic

156

C₂

—

157

C₂

C₅₃×#2074

yes

characteristic

158

C₂

C₅₃×#2080

yes

—

	subgroup	G/H
center	C₂×C₄×C₅₃	#124
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	#397
prime radical	C₂³×C₅₃	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

subgroup

G/H

center

C₂×C₄×C₅₃

#124

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₅₃

C₂²×C₅₃

C₂³×C₅₃

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

N	H ≅ G/N	type
66. #10630	C₅₃	direct (N × H)
94. C₅₃	#10630	direct (N × H)

H ≅ G/N

type

66. #10630

C₅₃

direct (N × H)

94. C₅₃

#10630

direct (N × H)

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

subgroup	index
C₅₃×#1995	2
C₅₃×#1832	2
C₅₃×#1542	2
C₅₃×#1301	2
C₅₃×#1274	2
#10630	53

subgroup

index

C₅₃×#1995

C₅₃×#1832

C₅₃×#1542

C₅₃×#1301

C₅₃×#1274

#10630

Subgroups — C₅₃×#10630 #27218142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#10630 #27218142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10630 #27218142