Group #27221029 — subgroups

Subgroups — C₅₃×#13517 #27221029

Order 13568.

250 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 110 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	C₂×C₅₃×#366	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2151	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1952	C₂	1	2	yes	maximal
4	C₅₃×#1387	C₂	1	2	yes	maximal
5	C₅₃×#1381	C₂	1	1	yes	maximal, characteristic
6	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×#108×C₅₃	C₂²	1	2	yes	—
9	C₅₃×#428	—	2	2	no	—
10	C₂×#107×C₅₃	C₂²	1	2	yes	—
11	C₅₃×#405	C₂²	1	1	yes	characteristic
12	C₅₃×#366	—	2	2	no	—
13	C₅₃×C₄⋊C₁₆	—	2	2	no	—
14	C₅₃×#343	—	2	2	no	—
15	C₂×M₁₆×C₅₃	—	2	2	no	—
16	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
18	#117×C₅₃	D₄	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
20	M₃₂×C₅₃	—	2	2	no	—
21	C₂×C₁₆×C₅₃	—	2	2	no	—
22	C₂×C₁₆×C₅₃	—	2	1	no	—
23	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
24	#108×C₅₃	—	2	2	no	—
25	#108×C₅₃	Q₈	1	4	yes	—
26	#107×C₅₃	—	2	2	no	—
27	#107×C₅₃	—	2	2	no	—
28	#107×C₅₃	—	4	2	no	—
29	#106×C₅₃	—	4	2	no	—
30	#97×C₅₃	—	2	1	no	—
31	C₄×C₈×C₅₃	—	2	1	no	—
32	C₂²×C₄×C₅₃	—	4	2	no	—
33	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
34	M₁₆×C₅₃	—	2	4	no	—
35	M₁₆×C₅₃	—	2	2	no	—
36	M₁₆×C₅₃	—	4	2	no	—
37	C₂×C₈×C₅₃	—	2	1	no	—
38	C₂×C₈×C₅₃	—	4	2	no	—
39	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
41	C₂×C₈×C₅₃	#41	1	2	yes	—
42	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
43	C₄⋊C₄×C₅₃	—	4	2	no	—
44	C₄⋊C₄×C₅₃	—	4	2	no	—
45	C₄⋊C₄×C₅₃	D₈	1	1	yes	characteristic
46	C₄⋊C₄×C₅₃	D₈	1	1	yes	characteristic
47	C₂²⋊C₄×C₅₃	—	2	1	no	—
48	C₄²×C₅₃	—	2	1	no	—
49	C₁₆×C₅₃	—	2	2	no	—
50	C₁₆×C₅₃	—	4	1	no	—
51	C₂³×C₅₃	#135	1	1	yes	characteristic
52	C₂×C₄×C₅₃	—	2	1	no	—
53	C₂×C₄×C₅₃	—	2	1	no	—
54	C₂×C₄×C₅₃	—	8	2	no	—
55	C₂×C₄×C₅₃	—	16	2	no	—
56	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
57	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
58	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₅₃	#137	1	2	yes	—
60	C₈×C₅₃	—	4	1	no	—
61	C₈×C₅₃	—	4	4	no	—
62	C₈×C₅₃	#123	1	1	yes	characteristic
63	C₈×C₅₃	#125	1	2	yes	—
64	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
65	#13517	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂²×C₅₃	—	2	1	no	—
67	C₂²×C₅₃	—	2	1	no	—
68	C₂²×C₅₃	#466	1	1	yes	characteristic
69	C₂²×C₅₃	#479	1	2	yes	—
70	C₄×C₅₃	—	4	1	no	—
71	C₄×C₅₃	—	4	1	no	—
72	C₄×C₅₃	—	16	2	no	—
73	C₄×C₅₃	#398	1	1	yes	characteristic
74	C₄×C₅₃	#470	1	1	yes	characteristic
75	C₄×C₅₃	#474	1	2	yes	—
76	C₂×#366	C₂×C₅₃	1	1	yes	characteristic
77	#2151	C₂×C₅₃	1	1	yes	characteristic
78	#1952	C₂×C₅₃	1	2	yes	—
79	#1387	C₂×C₅₃	1	2	yes	—
80	#1381	C₂×C₅₃	1	1	yes	characteristic
81	C₂×C₅₃	—	2	2	no	—
82	C₂×C₅₃	#2086	1	1	yes	characteristic
83	C₂×C₅₃	#2221	1	1	yes	characteristic
84	C₂×C₅₃	#2222	1	1	yes	characteristic
85	C₂×M₃₂	C₂²×C₅₃	1	1	yes	characteristic
86	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
87	C₂×#108	C₂²×C₅₃	1	2	yes	—
88	#428	—	2	2	no	—
89	C₂×#107	C₂²×C₅₃	1	2	yes	—
90	#405	C₂²×C₅₃	1	1	yes	characteristic
91	#366	—	2	2	no	—
92	C₄⋊C₁₆	—	2	2	no	—
93	#343	—	2	2	no	—
94	C₅₃	#13517	1	1	yes	characteristic, direct product factor, Sylow
95	C₂×M₁₆	—	2	2	no	—
96	C₂×M₁₆	D₄×C₅₃	1	1	yes	characteristic
97	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
98	#117	D₄×C₅₃	1	1	yes	characteristic
99	C₂×C₄⋊C₄	—	2	2	no	—
100	M₃₂	—	2	2	no	—
101	C₂×C₁₆	—	2	2	no	—
102	C₂×C₁₆	—	2	1	no	—
103	C₂×C₁₆	D₄×C₅₃	1	2	yes	—
104	#108	—	2	2	no	—
105	#108	Q₈×C₅₃	1	4	yes	—
106	#107	—	2	2	no	—
107	#107	—	2	2	no	—
108	#107	—	4	2	no	—
109	#106	—	4	2	no	—
110	#97	—	2	1	no	—
111	C₄×C₈	—	2	1	no	—
112	C₂²×C₄	—	4	2	no	—
113	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
114	M₁₆	—	2	4	no	—
115	M₁₆	—	2	2	no	—
116	M₁₆	—	4	2	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	4	2	no	—
119	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
120	C₂×C₈	C₂×Q₈×C₅₃	1	2	yes	—
121	C₂×C₈	#41×C₅₃	1	2	yes	—
122	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
123	C₄⋊C₄	—	4	2	no	—
124	C₄⋊C₄	—	4	2	no	—
125	C₄⋊C₄	D₈×C₅₃	1	1	yes	characteristic
126	C₄⋊C₄	D₈×C₅₃	1	1	yes	characteristic
127	C₂²⋊C₄	—	2	1	no	—
128	C₄²	—	2	1	no	—
129	C₁₆	—	2	2	no	—
130	C₁₆	—	4	1	no	—
131	C₂³	#135×C₅₃	1	1	yes	characteristic
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	8	2	no	—
135	C₂×C₄	—	16	2	no	—
136	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
137	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
138	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
139	C₂×C₄	#137×C₅₃	1	2	yes	—
140	C₈	—	4	1	no	—
141	C₈	—	4	4	no	—
142	C₈	#123×C₅₃	1	1	yes	characteristic
143	C₈	#125×C₅₃	1	2	yes	—
144	C₈	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	C₅₃×#466	1	1	yes	characteristic
148	C₂²	C₅₃×#479	1	2	yes	—
149	C₄	—	4	1	no	—
150	C₄	—	4	1	no	—
151	C₄	—	16	2	no	—
152	C₄	C₅₃×#398	1	1	yes	characteristic
153	C₄	C₅₃×#470	1	1	yes	characteristic
154	C₄	C₅₃×#474	1	2	yes	—
155	C₂	—	2	2	no	—
156	C₂	C₅₃×#2086	1	1	yes	characteristic
157	C₂	C₅₃×#2221	1	1	yes	characteristic
158	C₂	C₅₃×#2222	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅₃×#366

C₂

yes

maximal, characteristic

C₅₃×#2151

C₂

yes

maximal, characteristic

C₅₃×#1952

C₂

yes

maximal

C₅₃×#1387

C₂

yes

maximal

C₅₃×#1381

C₂

yes

maximal, characteristic

C₂×M₃₂×C₅₃

C₂²

yes

characteristic

C₂²×C₁₆×C₅₃

C₂²

yes

characteristic

C₂×#108×C₅₃

C₂²

yes

—

C₅₃×#428

—

C₂×#107×C₅₃

C₂²

yes

—

C₅₃×#405

C₂²

yes

characteristic

C₅₃×#366

—

C₅₃×C₄⋊C₁₆

—

C₅₃×#343

—

C₂×M₁₆×C₅₃

—

C₂×M₁₆×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

#117×C₅₃

D₄

yes

characteristic

C₂×C₄⋊C₄×C₅₃

—

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

D₄

yes

—

#108×C₅₃

—

#108×C₅₃

Q₈

yes

—

#107×C₅₃

—

#107×C₅₃

—

#107×C₅₃

—

#106×C₅₃

—

#97×C₅₃

—

C₄×C₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×Q₈

yes

—

C₂×C₈×C₅₃

#41

yes

—

C₂×C₈×C₅₃

#41

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₈

yes

characteristic

C₄⋊C₄×C₅₃

D₈

yes

characteristic

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₁₆×C₅₃

—

C₁₆×C₅₃

—

C₂³×C₅₃

#135

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#121

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

#137

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#123

yes

characteristic

C₈×C₅₃

#125

yes

—

C₈×C₅₃

C₄⋊Q₈

yes

characteristic

#13517

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#466

yes

characteristic

C₂²×C₅₃

#479

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₂×#366

C₂×C₅₃

yes

characteristic

#2151

C₂×C₅₃

yes

characteristic

#1952

C₂×C₅₃

yes

—

#1387

C₂×C₅₃

yes

—

#1381

C₂×C₅₃

yes

characteristic

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×M₃₂

C₂²×C₅₃

yes

characteristic

C₂²×C₁₆

C₂²×C₅₃

yes

characteristic

C₂×#108

C₂²×C₅₃

yes

—

#428

—

C₂×#107

C₂²×C₅₃

yes

—

#405

C₂²×C₅₃

yes

characteristic

#366

—

C₄⋊C₁₆

—

#343

—

C₅₃

#13517

yes

characteristic, direct product factor, Sylow

C₂×M₁₆

—

C₂×M₁₆

D₄×C₅₃

yes

characteristic

C₂²×C₈

C₂³×C₅₃

yes

characteristic

#117

D₄×C₅₃

yes

characteristic

C₂×C₄⋊C₄

—

100

M₃₂

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

103

C₂×C₁₆

D₄×C₅₃

yes

—

104

#108

—

105

#108

Q₈×C₅₃

yes

—

106

#107

—

107

#107

—

108

#107

—

109

#106

—

110

#97

—

111

C₄×C₈

—

112

C₂²×C₄

—

113

C₂²×C₄

C₂×D₄×C₅₃

yes

characteristic

114

M₁₆

—

115

M₁₆

—

116

M₁₆

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

C₂×D₄×C₅₃

yes

characteristic

120

C₂×C₈

C₂×Q₈×C₅₃

yes

—

121

C₂×C₈

#41×C₅₃

yes

—

122

C₂×C₈

#41×C₅₃

yes

characteristic

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

D₈×C₅₃

yes

characteristic

126

C₄⋊C₄

D₈×C₅₃

yes

characteristic

127

C₂²⋊C₄

—

128

C₄²

—

129

C₁₆

—

130

C₁₆

—

131

C₂³

#135×C₅₃

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

#121×C₅₃

yes

characteristic

137

C₂×C₄

C₂²⋊Q₈×C₅₃

yes

—

138

C₂×C₄

C₂×D₈×C₅₃

yes

characteristic

139

C₂×C₄

#137×C₅₃

yes

—

140

C₈

—

141

C₈

—

142

C₈

#123×C₅₃

yes

characteristic

143

C₈

#125×C₅₃

yes

—

144

C₈

C₄⋊Q₈×C₅₃

yes

characteristic

145

C₂²

—

146

C₂²

—

147

C₂²

C₅₃×#466

yes

characteristic

148

C₂²

C₅₃×#479

yes

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

C₅₃×#398

yes

characteristic

153

C₄

C₅₃×#470

yes

characteristic

154

C₄

C₅₃×#474

yes

—

155

C₂

—

156

C₂

C₅₃×#2086

yes

characteristic

157

C₂

C₅₃×#2221

yes

characteristic

158

C₂

C₅₃×#2222

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	#466
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
prime radical	C₂³×C₅₃	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₅₃×#405 ◁ G

subgroup

G/H

center, socle

C₂²×C₅₃

#466

commutator, Frattini, cosocle

C₂²×C₈

C₂³×C₅₃

prime radical

C₂³×C₅₃

#135

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₅₃×#405 ◁ G

N	H ≅ G/N	type
65. #13517	C₅₃	direct (N × H)
94. C₅₃	#13517	direct (N × H)

H ≅ G/N

type

65. #13517

C₅₃

direct (N × H)

94. C₅₃

#13517

direct (N × H)

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

subgroup	index
C₂×C₅₃×#366	2
C₅₃×#2151	2
C₅₃×#1952	2
C₅₃×#1387	2
C₅₃×#1381	2
#13517	53

subgroup

index

C₂×C₅₃×#366

C₅₃×#2151

C₅₃×#1952

C₅₃×#1387

C₅₃×#1381

#13517

Subgroups — C₅₃×#13517 #27221029

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#13517 #27221029

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#13517 #27221029