Group #27216149 — subgroups

Subgroups — C₅₃×#8637 #27216149

Order 13568.

336 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 138 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×#2960	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1566	C₂	1	2	yes	maximal
3	C₅₃×#1561	C₂	1	2	yes	maximal
4	C₅₃×#1309	C₂	1	2	yes	maximal
5	C₅₃×#533	C₂²	1	1	yes	characteristic
6	C₅₃×#446	—	2	2	no	—
7	C₈×Q₈×C₅₃	—	2	2	no	—
8	C₅₃×#424	C₄	1	2	yes	—
9	C₅₃×#423	C₂²	1	4	yes	—
10	C₅₃×#423	C₂²	1	1	yes	characteristic
11	C₅₃×#405	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#334	—	2	2	no	—
14	C₅₃×#334	—	2	2	no	—
15	C₅₃×#332	—	2	4	no	—
16	C₂×M₁₆×C₅₃	—	2	4	no	—
17	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
18	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₄⋊Q₈×C₅₃	D₄	1	2	yes	—
21	#125×C₅₃	Q₈	1	2	yes	—
22	C₂²⋊Q₈×C₅₃	—	2	2	no	—
23	C₄×Q₈×C₅₃	—	2	2	no	—
24	#117×C₅₃	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
26	C₄⋊C₈×C₅₃	—	2	4	no	—
27	C₄⋊C₈×C₅₃	—	2	2	no	—
28	C₄⋊C₈×C₅₃	—	2	4	no	—
29	C₄⋊C₈×C₅₃	—	2	4	no	—
30	#97×C₅₃	—	2	2	no	—
31	C₄×C₈×C₅₃	—	2	2	no	—
32	C₄×C₈×C₅₃	D₄	1	2	yes	—
33	C₄×C₈×C₅₃	D₄	1	2	yes	—
34	C₂×Q₈×C₅₃	—	2	2	no	—
35	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	#41	1	2	yes	—
37	M₁₆×C₅₃	—	4	2	no	—
38	M₁₆×C₅₃	—	8	4	no	—
39	C₂×C₈×C₅₃	—	2	2	no	—
40	C₂×C₈×C₅₃	—	2	2	no	—
41	C₂×C₈×C₅₃	—	4	1	no	—
42	C₂×C₈×C₅₃	—	4	4	no	—
43	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
45	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₄⋊C₄×C₅₃	—	2	2	no	—
47	C₄⋊C₄×C₅₃	—	2	4	no	—
48	C₄⋊C₄×C₅₃	C₄⋊C₄	1	4	yes	—
49	C₂²⋊C₄×C₅₃	—	2	2	no	—
50	C₄²×C₅₃	—	2	2	no	—
51	C₄²×C₅₃	C₂×D₄	1	2	yes	—
52	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
54	C₂³×C₅₃	#117	1	1	yes	characteristic
55	Q₈×C₅₃	—	4	4	no	—
56	C₂×C₄×C₅₃	—	2	1	no	—
57	C₂×C₄×C₅₃	—	2	1	no	—
58	C₂×C₄×C₅₃	—	2	4	no	—
59	C₂×C₄×C₅₃	—	2	4	no	—
60	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₅₃	#121	1	2	yes	—
63	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
64	C₈×C₅₃	—	2	1	no	—
65	C₈×C₅₃	—	2	2	no	—
66	C₈×C₅₃	—	4	2	no	—
67	C₈×C₅₃	—	8	4	no	—
68	C₈×C₅₃	#101	1	2	yes	—
69	#8637	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂²×C₅₃	—	2	1	no	—
71	C₂²×C₅₃	—	2	2	no	—
72	C₂²×C₅₃	#380	1	1	yes	characteristic
73	C₄×C₅₃	—	2	1	no	—
74	C₄×C₅₃	—	2	2	no	—
75	C₄×C₅₃	—	4	4	no	—
76	C₄×C₅₃	C₂×#101	1	1	yes	characteristic
77	C₄×C₅₃	#413	1	1	yes	characteristic
78	C₄×C₅₃	#471	1	4	yes	—
79	#2960	C₂×C₅₃	1	1	yes	characteristic
80	#1566	C₂×C₅₃	1	2	yes	—
81	#1561	C₂×C₅₃	1	2	yes	—
82	#1309	C₂×C₅₃	1	2	yes	—
83	C₂×C₅₃	—	4	1	no	—
84	C₂×C₅₃	#1801	1	1	yes	characteristic
85	C₂×C₅₃	#1815	1	2	yes	—
86	#533	C₂²×C₅₃	1	1	yes	characteristic
87	#446	—	2	2	no	—
88	C₈×Q₈	—	2	2	no	—
89	#424	C₄×C₅₃	1	2	yes	—
90	#423	C₂²×C₅₃	1	4	yes	—
91	#423	C₂²×C₅₃	1	1	yes	characteristic
92	#405	C₄×C₅₃	1	2	yes	—
93	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
94	#334	—	2	2	no	—
95	#334	—	2	2	no	—
96	#332	—	2	4	no	—
97	C₅₃	#8637	1	1	yes	characteristic, direct product factor, Sylow
98	C₂×M₁₆	—	2	4	no	—
99	C₂×M₁₆	C₂×C₄×C₅₃	1	2	yes	—
100	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
101	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
102	C₄⋊Q₈	D₄×C₅₃	1	2	yes	—
103	#125	Q₈×C₅₃	1	2	yes	—
104	C₂²⋊Q₈	—	2	2	no	—
105	C₄×Q₈	—	2	2	no	—
106	#117	C₂×C₄×C₅₃	1	2	yes	—
107	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
108	C₄⋊C₈	—	2	4	no	—
109	C₄⋊C₈	—	2	2	no	—
110	C₄⋊C₈	—	2	4	no	—
111	C₄⋊C₈	—	2	4	no	—
112	#97	—	2	2	no	—
113	C₄×C₈	—	2	2	no	—
114	C₄×C₈	D₄×C₅₃	1	2	yes	—
115	C₄×C₈	D₄×C₅₃	1	2	yes	—
116	C₂×Q₈	—	2	2	no	—
117	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
118	C₂²×C₄	#41×C₅₃	1	2	yes	—
119	M₁₆	—	4	2	no	—
120	M₁₆	—	8	4	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	4	1	no	—
124	C₂×C₈	—	4	4	no	—
125	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
126	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
127	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	—	2	4	no	—
130	C₄⋊C₄	C₄⋊C₄×C₅₃	1	4	yes	—
131	C₂²⋊C₄	—	2	2	no	—
132	C₄²	—	2	2	no	—
133	C₄²	C₂×D₄×C₅₃	1	2	yes	—
134	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
135	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
136	C₂³	#117×C₅₃	1	1	yes	characteristic
137	Q₈	—	4	4	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
143	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
144	C₂×C₄	#121×C₅₃	1	2	yes	—
145	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
146	C₈	—	2	1	no	—
147	C₈	—	2	2	no	—
148	C₈	—	4	2	no	—
149	C₈	—	8	4	no	—
150	C₈	#101×C₅₃	1	2	yes	—
151	C₂²	—	2	1	no	—
152	C₂²	—	2	2	no	—
153	C₂²	C₅₃×#380	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	2	2	no	—
156	C₄	—	4	4	no	—
157	C₄	C₂×#101×C₅₃	1	1	yes	characteristic
158	C₄	C₅₃×#413	1	1	yes	characteristic
159	C₄	C₅₃×#471	1	4	yes	—
160	C₂	—	4	1	no	—
161	C₂	C₅₃×#1801	1	1	yes	characteristic
162	C₂	C₅₃×#1815	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#2960

C₂

yes

maximal, characteristic

C₅₃×#1566

C₂

yes

maximal

C₅₃×#1561

C₂

yes

maximal

C₅₃×#1309

C₂

yes

maximal

C₅₃×#533

C₂²

yes

characteristic

C₅₃×#446

—

C₈×Q₈×C₅₃

—

C₅₃×#424

C₄

yes

—

C₅₃×#423

C₂²

yes

—

C₅₃×#423

C₂²

yes

characteristic

C₅₃×#405

C₄

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#334

—

C₅₃×#334

—

C₅₃×#332

—

C₂×M₁₆×C₅₃

—

C₂×M₁₆×C₅₃

C₂×C₄

yes

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄⋊Q₈×C₅₃

D₄

yes

—

#125×C₅₃

Q₈

yes

—

C₂²⋊Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

#117×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

#97×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₄⋊C₄

yes

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×Q₈

yes

characteristic

C₂³×C₅₃

#117

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#121

yes

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#101

yes

—

#8637

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#380

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

#2960

C₂×C₅₃

yes

characteristic

#1566

C₂×C₅₃

yes

—

#1561

C₂×C₅₃

yes

—

#1309

C₂×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

#1801

yes

characteristic

C₂×C₅₃

#1815

yes

—

#533

C₂²×C₅₃

yes

characteristic

#446

—

C₈×Q₈

—

#424

C₄×C₅₃

yes

—

#423

C₂²×C₅₃

yes

—

#423

C₂²×C₅₃

yes

characteristic

#405

C₄×C₅₃

yes

—

C₂×C₄×C₈

C₂²×C₅₃

yes

—

—

—

C₅₃

#8637

yes

characteristic, direct product factor, Sylow

C₂×M₁₆

—

C₂×M₁₆

C₂×C₄×C₅₃

yes

—

100

C₂²×C₈

C₂×C₄×C₅₃

yes

characteristic

101

C₂²×C₈

C₂×C₄×C₅₃

yes

characteristic

102

C₄⋊Q₈

D₄×C₅₃

yes

—

103

#125

Q₈×C₅₃

yes

—

104

C₂²⋊Q₈

—

105

C₄×Q₈

—

106

#117

C₂×C₄×C₅₃

yes

—

107

C₂×C₄²

C₂³×C₅₃

yes

characteristic

108

C₄⋊C₈

—

109

C₄⋊C₈

—

110

C₄⋊C₈

—

111

C₄⋊C₈

—

112

#97

—

113

C₄×C₈

—

114

C₄×C₈

D₄×C₅₃

yes

—

115

C₄×C₈

D₄×C₅₃

yes

—

116

C₂×Q₈

—

117

C₂²×C₄

C₂²×C₄×C₅₃

yes

characteristic

118

C₂²×C₄

#41×C₅₃

yes

—

119

M₁₆

—

120

M₁₆

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

C₂²⋊C₄×C₅₃

yes

characteristic

126

C₂×C₈

C₂²⋊C₄×C₅₃

yes

—

127

C₂×C₈

C₂²⋊C₄×C₅₃

yes

characteristic

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

C₄⋊C₄×C₅₃

yes

—

131

C₂²⋊C₄

—

132

C₄²

—

133

C₄²

C₂×D₄×C₅₃

yes

—

134

C₄²

C₂×D₄×C₅₃

yes

characteristic

135

C₄²

C₂×Q₈×C₅₃

yes

characteristic

136

C₂³

#117×C₅₃

yes

characteristic

137

Q₈

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

C₂×C₂²⋊C₄×C₅₃

yes

characteristic

143

C₂×C₄

C₂×C₄⋊C₄×C₅₃

yes

characteristic

144

C₂×C₄

#121×C₅₃

yes

—

145

C₂×C₄

C₂²⋊Q₈×C₅₃

yes

—

146

C₈

—

147

C₈

—

148

C₈

—

149

C₈

—

150

C₈

#101×C₅₃

yes

—

151

C₂²

—

152

C₂²

—

153

C₂²

C₅₃×#380

yes

characteristic

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

C₂×#101×C₅₃

yes

characteristic

158

C₄

C₅₃×#413

yes

characteristic

159

C₄

C₅₃×#471

yes

—

160

C₂

—

161

C₂

C₅₃×#1801

yes

characteristic

162

C₂

C₅₃×#1815

yes

—

	subgroup	G/H
center, socle	C₂²×C₅₃	#380
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂³×C₅₃	#117
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₅₃

#380

commutator

C₂²×C₄

C₂²×C₄×C₅₃

Frattini, cosocle

C₂×C₄²

C₂³×C₅₃

prime radical

C₂³×C₅₃

#117

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

N	H ≅ G/N	type
69. #8637	C₅₃	direct (N × H)
97. C₅₃	#8637	direct (N × H)

H ≅ G/N

type

69. #8637

C₅₃

direct (N × H)

97. C₅₃

#8637

direct (N × H)

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

subgroup	index
C₅₃×#2960	2
C₅₃×#1566	2
C₅₃×#1561	2
C₅₃×#1309	2
#8637	53

subgroup

index

C₅₃×#2960

C₅₃×#1566

C₅₃×#1561

C₅₃×#1309

#8637

Subgroups — C₅₃×#8637 #27216149

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#8637 #27216149

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8637 #27216149