Group #27271164 — subgroups

Subgroups — #27271164

Order 13568.

244 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 79 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	#20274765	C₂	1	2	yes	maximal
2	C₅₃×#2113	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133153	—	2	2	no	—
4	#16810607	—	2	2	no	—
5	#16810381	C₄	1	4	yes	—
6	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×M₃₂×C₅₃	C₄	1	1	yes	characteristic
8	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
9	C₅₃×C₂²⋊C₁₆	—	2	2	no	—
10	C₂×#11719917	—	2	4	no	—
11	#13133153	—	4	2	no	—
12	#13133153	—	4	2	no	—
13	#13133136	—	4	2	no	—
14	#13133134	—	4	2	no	—
15	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
16	#131×C₅₃	—	4	1	no	—
17	#131×C₅₃	—	4	1	no	—
18	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
19	C₂×M₁₆×C₅₃	Q₈	1	1	yes	characteristic
20	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₅₃	D₄	1	2	yes	—
22	M₃₂×C₅₃	—	4	1	no	—
23	C₂×C₁₆×C₅₃	—	2	1	no	—
24	C₂×C₁₆×C₅₃	—	2	1	no	—
25	C₂×C₁₆×C₅₃	—	4	1	no	—
26	#41×C₅₃	—	4	1	no	—
27	#41×C₅₃	—	4	1	no	—
28	C₂×Q₈×C₅₃	D₈	1	1	yes	characteristic
29	C₂×D₄×C₅₃	SD₁₆	1	2	yes	—
30	C₂×D₄×C₅₃	Q₁₆	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
32	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
33	M₁₆×C₅₃	—	4	2	no	—
34	M₁₆×C₅₃	—	4	1	no	—
35	C₂×C₈×C₅₃	—	2	1	no	—
36	C₂×C₈×C₅₃	—	2	1	no	—
37	C₂×C₈×C₅₃	—	2	2	no	—
38	C₂×C₈×C₅₃	—	2	2	no	—
39	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
42	#11719917	—	4	4	no	—
43	#11719917	—	4	4	no	—
44	C₂×#88322	—	4	4	no	—
45	C₁₆×C₅₃	—	4	1	no	—
46	C₁₆×C₅₃	—	4	1	no	—
47	C₂³×C₅₃	#103	1	2	yes	—
48	C₂³×C₅₃	#106	1	1	yes	characteristic
49	Q₈×C₅₃	—	4	1	no	—
50	D₄×C₅₃	—	4	2	no	—
51	D₄×C₅₃	—	4	1	no	—
52	C₂×C₄×C₅₃	—	2	2	no	—
53	C₂×C₄×C₅₃	—	2	1	no	—
54	C₂×C₄×C₅₃	—	2	2	no	—
55	C₂×C₄×C₅₃	—	2	1	no	—
56	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
57	C₂×C₄×C₅₃	#102	1	2	yes	—
58	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
59	C₈×C₅₃	—	2	1	no	—
60	C₈×C₅₃	—	2	2	no	—
61	C₈×C₅₃	#99	1	1	yes	characteristic
62	C₈×C₅₃	#99	1	1	yes	characteristic
63	#88322	—	8	4	no	—
64	#7229	—	53	1	no	maximal, Sylow
65	C₂²×C₅₃	—	2	2	no	—
66	C₂²×C₅₃	—	2	1	no	—
67	C₂²×C₅₃	—	4	2	no	—
68	C₂²×C₅₃	—	4	1	no	—
69	C₂²×C₅₃	#340	1	1	yes	characteristic
70	C₄×C₅₃	—	2	2	no	—
71	C₄×C₅₃	—	2	1	no	—
72	C₄×C₅₃	H(3,ℤ/4)	1	1	yes	characteristic
73	C₄×C₅₃	#342	1	1	yes	characteristic
74	#2113	D₅₃	1	1	yes	characteristic
75	#1813	—	53	2	no	—
76	C₂×C₅₃	—	2	1	no	—
77	C₂×C₅₃	—	4	2	no	—
78	C₂×C₅₃	—	4	1	no	—
79	C₂×C₅₃	#1289	1	1	yes	characteristic
80	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
81	C₂×M₃₂	Dic₅₃	1	1	yes	characteristic
82	C₂²×C₁₆	Dic₅₃	1	1	yes	characteristic
83	C₂×#108	—	106	2	no	—
84	#413	—	106	2	no	—
85	C₂²⋊C₁₆	—	2	2	no	—
86	#338	—	53	4	no	—
87	C₅₃	#7229	1	1	yes	characteristic, complemented, Sylow
88	C₂×#41	D₂₁₂	1	1	yes	characteristic
89	#131	—	4	1	no	—
90	#131	—	4	1	no	—
91	C₂×M₁₆	—	106	4	no	—
92	C₂×M₁₆	Dic₁₀₆	1	1	yes	characteristic
93	C₂×M₁₆	C₄×D₅₃	1	2	yes	—
94	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
95	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
96	M₃₂	—	4	1	no	—
97	C₂×C₁₆	—	2	1	no	—
98	C₂×C₁₆	—	2	1	no	—
99	C₂×C₁₆	—	4	1	no	—
100	#108	—	212	2	no	—
101	#108	—	212	2	no	—
102	#101	—	212	2	no	—
103	#100	—	212	2	no	—
104	#41	—	4	1	no	—
105	#41	—	4	1	no	—
106	C₂×Q₈	D₄₂₄	1	1	yes	characteristic
107	C₂×D₄	#11719913	1	2	yes	—
108	C₂×D₄	Dic₂₁₂	1	1	yes	characteristic
109	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
110	C₂²×C₄	#11719921	1	2	yes	—
111	M₁₆	—	4	1	no	—
112	M₁₆	—	4	2	no	—
113	M₁₆	—	212	4	no	—
114	M₁₆	—	212	4	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	212	4	no	—
120	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
121	C₂×C₈	#11719919	1	2	yes	—
122	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
123	C₁₆	—	4	1	no	—
124	C₁₆	—	4	1	no	—
125	C₂³	#13133142	1	2	yes	—
126	C₂³	#13133149	1	1	yes	characteristic
127	Q₈	—	4	1	no	—
128	D₄	—	4	2	no	—
129	D₄	—	4	1	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	#13133124	1	1	yes	characteristic
135	C₂×C₄	#13133139	1	2	yes	—
136	C₂×C₄	#13133151	1	1	yes	characteristic
137	C₈	—	2	1	no	—
138	C₈	—	2	2	no	—
139	C₈	—	424	4	no	—
140	C₈	#13133132	1	1	yes	characteristic
141	C₈	#13133132	1	1	yes	characteristic
142	C₂²	—	2	2	no	—
143	C₂²	—	2	1	no	—
144	C₂²	—	4	2	no	—
145	C₂²	—	4	1	no	—
146	C₂²	#16810386	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	1	no	—
149	C₄	#16810380	1	1	yes	characteristic
150	C₄	#16810390	1	1	yes	characteristic
151	C₂	—	2	1	no	—
152	C₂	—	4	2	no	—
153	C₂	—	4	1	no	—
154	C₂	#20272583	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20274765

C₂

yes

maximal

C₅₃×#2113

C₂

yes

maximal, characteristic

C₂×#13133153

—

#16810607

—

#16810381

C₄

yes

—

C₂×#131×C₅₃

C₂²

yes

characteristic

C₂×M₃₂×C₅₃

C₄

yes

characteristic

C₂²×C₁₆×C₅₃

C₄

yes

characteristic

C₅₃×C₂²⋊C₁₆

—

—

—

—

—

—

C₂×#41×C₅₃

D₄

yes

characteristic

#131×C₅₃

—

#131×C₅₃

—

C₂×M₁₆×C₅₃

C₂×C₄

yes

—

C₂×M₁₆×C₅₃

Q₈

yes

characteristic

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

D₄

yes

—

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

—

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

D₈

yes

characteristic

C₂×D₄×C₅₃

SD₁₆

yes

—

C₂×D₄×C₅₃

Q₁₆

yes

characteristic

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄²

yes

characteristic

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

#11719917

—

#11719917

—

C₂×#88322

—

C₁₆×C₅₃

—

C₁₆×C₅₃

—

C₂³×C₅₃

#103

yes

—

C₂³×C₅₃

#106

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#95

yes

characteristic

C₂×C₄×C₅₃

#102

yes

—

C₂×C₄×C₅₃

#107

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

yes

C₈×C₅₃

yes

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#340

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

D₅₃

yes

characteristic

#1813

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#1289

yes

characteristic

C₂×#131

C₂×D₅₃

yes

characteristic

C₂×M₃₂

Dic₅₃

yes

characteristic

C₂²×C₁₆

Dic₅₃

yes

characteristic

C₂×#108

—

106

—

#413

—

106

—

C₂²⋊C₁₆

—

#338

—

C₅₃

#7229

yes

characteristic, complemented, Sylow

C₂×#41

D₂₁₂

yes

characteristic

#131

—

#131

—

C₂×M₁₆

—

106

—

C₂×M₁₆

Dic₁₀₆

yes

characteristic

C₂×M₁₆

C₄×D₅₃

yes

—

C₂²×C₈

C₂×Dic₅₃

yes

characteristic

C₂²×C₈

C₂²⋊D₅₃

yes

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

100

#108

—

212

—

101

#108

—

212

—

102

#101

—

212

—

103

#100

—

212

—

104

#41

—

105

#41

—

106

C₂×Q₈

D₄₂₄

yes

characteristic

107

C₂×D₄

#11719913

yes

—

108

C₂×D₄

Dic₂₁₂

yes

characteristic

109

C₂²×C₄

C₄⋊Dic₅₃

yes

characteristic

110

C₂²×C₄

#11719921

yes

—

111

M₁₆

—

112

M₁₆

—

113

M₁₆

—

212

—

114

M₁₆

—

212

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

212

—

120

C₂×C₈

C₄×Dic₅₃

yes

characteristic

121

C₂×C₈

#11719919

yes

—

122

C₂×C₈

C₂²⋊Dic₅₃

yes

characteristic

123

C₁₆

—

124

C₁₆

—

125

C₂³

#13133142

yes

—

126

C₂³

#13133149

yes

characteristic

127

Q₈

—

128

D₄

—

129

D₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

#13133124

yes

characteristic

135

C₂×C₄

#13133139

yes

—

136

C₂×C₄

#13133151

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

—

424

—

140

C₈

#13133132

yes

characteristic

141

C₈

#13133132

yes

characteristic

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

#16810386

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

#16810380

yes

characteristic

150

C₄

#16810390

yes

characteristic

151

C₂

—

152

C₂

—

153

C₂

—

154

C₂

#20272583

yes

characteristic

	subgroup	G/H
center	C₄	#16810380
commutator	C₂×C₈×C₅₃	C₄²
Frattini	C₂×#131	C₂×D₅₃
Fitting	C₅₃×#2113	C₂
socle	C₂×C₅₃	#1289
prime radical	C₂×#41×C₅₃	D₄
cosocle	C₂×#131×C₅₃	C₂²
hypercenter	#2113	D₅₃
hypocenter	C₅₃	#7229
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #2113

subgroup

G/H

center

C₄

#16810380

commutator

C₂×C₈×C₅₃

C₄²

Frattini

C₂×#131

C₂×D₅₃

Fitting

C₅₃×#2113

C₂

socle

C₂×C₅₃

#1289

prime radical

C₂×#41×C₅₃

D₄

cosocle

C₂×#131×C₅₃

C₂²

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ #2113

N	H ≅ G/N	type
87. C₅₃	#7229	semidirect

H ≅ G/N

type

87. C₅₃

#7229

semidirect

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

subgroup	index
#20274765	2
C₅₃×#2113	2
#7229	53

subgroup

index

#20274765

C₅₃×#2113

#7229

Subgroups — #27271164

Conjugacy classes of subgroups

Complements

Maximal subgroups