Group #27271876 — subgroups

Subgroups — #27271876

Order 13568.

286 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 59 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	#20276400	C₂	1	2	yes	maximal
2	C₅₃×#1962	C₂	1	1	yes	maximal, characteristic
3	#16810607	—	2	2	no	—
4	#16810422	—	2	2	no	—
5	#16810420	C₄	1	4	yes	complemented
6	C₅₃×#532	C₂²	1	1	yes	characteristic
7	C₅₃×#423	C₄	1	1	yes	characteristic
8	C₂×#104×C₅₃	—	2	2	no	—
9	C₄³×C₅₃	C₄	1	1	yes	characteristic
10	C₂×#11719917	—	4	2	no	—
11	#13133136	—	2	4	no	—
12	#13133134	—	4	2	no	—
13	C₂×#41×C₅₃	D₄	1	2	yes	—
14	C₂×M₁₆×C₅₃	—	2	1	no	—
15	C₄⋊Q₈×C₅₃	D₄	1	1	yes	characteristic
16	#127×C₅₃	Q₈	1	1	yes	characteristic
17	#124×C₅₃	C₂×C₄	1	2	yes	—
18	#121×C₅₃	—	4	1	no	—
19	C₄×D₄×C₅₃	—	4	1	no	—
20	C₂×C₄²×C₅₃	—	2	1	no	—
21	C₂×C₄²×C₅₃	—	4	1	no	—
22	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₄⋊C₈×C₅₃	—	4	1	no	—
24	#104×C₅₃	—	4	4	no	—
25	#41×C₅₃	—	4	2	no	—
26	#41×C₅₃	—	4	2	no	—
27	C₂×Q₈×C₅₃	C₂²⋊C₄	1	2	yes	—
28	C₂×D₄×C₅₃	—	2	2	no	—
29	C₂×D₄×C₅₃	—	4	1	no	—
30	C₂×D₄×C₅₃	C₄⋊C₄	1	2	yes	—
31	C₂²×C₄×C₅₃	—	2	1	no	—
32	C₂²×C₄×C₅₃	—	2	2	no	—
33	C₂²×C₄×C₅₃	—	4	1	no	—
34	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
35	M₁₆×C₅₃	—	4	2	no	—
36	C₂×C₈×C₅₃	—	4	1	no	—
37	C₄⋊C₄×C₅₃	—	4	1	no	—
38	C₂²⋊C₄×C₅₃	—	4	2	no	—
39	C₄²×C₅₃	—	2	1	no	—
40	C₄²×C₅₃	—	2	2	no	—
41	C₄²×C₅₃	—	4	1	no	—
42	C₄²×C₅₃	—	4	4	no	—
43	C₄²×C₅₃	C₄²	1	1	yes	characteristic
44	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
45	#11719917	—	4	4	no	—
46	#11719917	—	8	2	no	—
47	C₂×#88322	—	8	2	no	—
48	C₂³×C₅₃	—	2	2	no	—
49	C₂³×C₅₃	#99	1	1	yes	characteristic
50	Q₈×C₅₃	—	4	2	no	—
51	D₄×C₅₃	—	4	2	no	—
52	D₄×C₅₃	—	8	2	no	—
53	D₄×C₅₃	—	8	2	no	—
54	C₂×C₄×C₅₃	—	2	1	no	—
55	C₂×C₄×C₅₃	—	2	1	no	—
56	C₂×C₄×C₅₃	—	2	1	no	—
57	C₂×C₄×C₅₃	—	2	2	no	—
58	C₂×C₄×C₅₃	—	2	1	no	—
59	C₂×C₄×C₅₃	—	4	2	no	—
60	C₂×C₄×C₅₃	—	4	2	no	—
61	C₂×C₄×C₅₃	—	4	4	no	—
62	C₂×C₄×C₅₃	—	4	2	no	—
63	C₂×C₄×C₅₃	—	4	2	no	—
64	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
65	C₂×C₄×C₅₃	#99	1	1	yes	characteristic
66	C₈×C₅₃	—	8	1	no	—
67	#88322	—	8	4	no	—
68	#7501	—	53	1	no	maximal, Sylow
69	C₂²×C₅₃	—	2	1	no	—
70	C₂²×C₅₃	—	4	1	no	—
71	C₂²×C₅₃	—	4	2	no	—
72	C₂²×C₅₃	—	8	2	no	—
73	C₂²×C₅₃	#342	1	1	yes	characteristic
74	C₄×C₅₃	—	2	2	no	—
75	C₄×C₅₃	—	2	1	no	—
76	C₄×C₅₃	—	4	1	no	—
77	C₄×C₅₃	—	4	4	no	—
78	C₄×C₅₃	—	4	2	no	—
79	C₄×C₅₃	#353	1	1	yes	characteristic
80	C₄×C₅₃	#354	1	1	yes	characteristic
81	#2132	—	53	2	no	—
82	#1962	D₅₃	1	1	yes	characteristic
83	C₂×C₅₃	—	2	1	no	—
84	C₂×C₅₃	—	4	1	no	—
85	C₂×C₅₃	—	8	2	no	—
86	C₂×C₅₃	#1393	1	1	yes	characteristic
87	#532	C₂×D₅₃	1	1	yes	characteristic
88	#423	Dic₅₃	1	1	yes	characteristic
89	C₂×#104	—	2	2	no	—
90	#413	—	106	2	no	—
91	C₄³	Dic₅₃	1	1	yes	characteristic
92	#356	—	106	2	no	—
93	#355	—	53	4	no	—
94	C₅₃	#7501	1	1	yes	characteristic, complemented, Sylow
95	C₂×#41	C₂²⋊D₅₃	1	2	yes	—
96	C₂×M₁₆	—	2	1	no	—
97	C₂×M₁₆	—	212	2	no	—
98	C₄⋊Q₈	D₂₁₂	1	1	yes	characteristic
99	#127	Dic₁₀₆	1	1	yes	characteristic
100	#124	C₄×D₅₃	1	2	yes	—
101	#121	—	4	1	no	—
102	C₄×D₄	—	4	1	no	—
103	C₂×C₄²	—	2	1	no	—
104	C₂×C₄²	—	4	1	no	—
105	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
106	C₄⋊C₈	—	4	1	no	—
107	#104	—	4	4	no	—
108	#101	—	106	4	no	—
109	#100	—	212	2	no	—
110	#41	—	4	2	no	—
111	#41	—	4	2	no	—
112	C₂×Q₈	#11719921	1	2	yes	—
113	C₂×D₄	—	2	2	no	—
114	C₂×D₄	—	4	1	no	—
115	C₂×D₄	#11719919	1	2	yes	—
116	C₂²×C₄	—	2	1	no	—
117	C₂²×C₄	—	2	2	no	—
118	C₂²×C₄	—	4	1	no	—
119	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
120	M₁₆	—	4	2	no	—
121	M₁₆	—	424	2	no	—
122	M₁₆	—	212	4	no	—
123	C₂×C₈	—	4	1	no	—
124	C₂×C₈	—	424	2	no	—
125	C₄⋊C₄	—	4	1	no	—
126	C₂²⋊C₄	—	4	2	no	—
127	C₄²	—	2	1	no	—
128	C₄²	—	2	2	no	—
129	C₄²	—	4	1	no	—
130	C₄²	—	4	4	no	—
131	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
132	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
133	C₂³	—	2	2	no	—
134	C₂³	#13133132	1	1	yes	characteristic
135	Q₈	—	4	2	no	—
136	D₄	—	4	2	no	—
137	D₄	—	8	2	no	—
138	D₄	—	8	2	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	4	4	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	#13133124	1	1	yes	characteristic
150	C₂×C₄	#13133132	1	1	yes	characteristic
151	C₈	—	8	1	no	—
152	C₈	—	424	4	no	—
153	C₂²	—	2	1	no	—
154	C₂²	—	4	1	no	—
155	C₂²	—	4	2	no	—
156	C₂²	—	8	2	no	—
157	C₂²	#16810390	1	1	yes	characteristic
158	C₄	—	2	2	no	—
159	C₄	—	2	1	no	—
160	C₄	—	4	1	no	—
161	C₄	—	4	4	no	—
162	C₄	—	4	2	no	—
163	C₄	#16810416	1	1	yes	characteristic
164	C₄	#16810418	1	1	yes	characteristic
165	C₂	—	2	1	no	—
166	C₂	—	4	1	no	—
167	C₂	—	8	2	no	—
168	C₂	#20272840	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276400

C₂

yes

maximal

C₅₃×#1962

C₂

yes

maximal, characteristic

#16810607

—

#16810422

—

#16810420

C₄

yes

complemented

C₅₃×#532

C₂²

yes

characteristic

C₅₃×#423

C₄

yes

characteristic

C₂×#104×C₅₃

—

C₄³×C₅₃

C₄

yes

—

—

—

C₂×#41×C₅₃

D₄

yes

—

C₂×M₁₆×C₅₃

—

C₄⋊Q₈×C₅₃

D₄

yes

characteristic

#127×C₅₃

Q₈

yes

characteristic

#124×C₅₃

C₂×C₄

yes

—

#121×C₅₃

—

C₄×D₄×C₅₃

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄⋊C₈×C₅₃

—

#104×C₅₃

—

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

C₂²⋊C₄

yes

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₄⋊C₄

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₄²

yes

characteristic

C₄²×C₅₃

C₄⋊C₄

yes

—

—

—

C₂³×C₅₃

—

C₂³×C₅₃

#99

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×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

yes

C₂×C₄×C₅₃

yes

C₈×C₅₃

—

#88322

—

#7501

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#342

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

—

D₅₃

yes

characteristic

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#1393

yes

characteristic

#532

C₂×D₅₃

yes

characteristic

#423

Dic₅₃

yes

characteristic

C₂×#104

—

#413

—

106

—

C₄³

Dic₅₃

yes

characteristic

#356

—

106

—

#355

—

C₅₃

#7501

yes

characteristic, complemented, Sylow

C₂×#41

C₂²⋊D₅₃

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

212

—

C₄⋊Q₈

D₂₁₂

yes

characteristic

#127

Dic₁₀₆

yes

characteristic

100

#124

C₄×D₅₃

yes

—

101

#121

—

102

C₄×D₄

—

103

C₂×C₄²

—

104

C₂×C₄²

—

105

C₂×C₄²

C₂×Dic₅₃

yes

characteristic

106

C₄⋊C₈

—

107

#104

—

108

#101

—

106

—

109

#100

—

212

—

110

#41

—

111

#41

—

112

C₂×Q₈

#11719921

yes

—

113

C₂×D₄

—

114

C₂×D₄

—

115

C₂×D₄

#11719919

yes

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

C₂²⋊Dic₅₃

yes

characteristic

120

M₁₆

—

121

M₁₆

—

424

—

122

M₁₆

—

212

—

123

C₂×C₈

—

124

C₂×C₈

—

424

—

125

C₄⋊C₄

—

126

C₂²⋊C₄

—

127

C₄²

—

128

C₄²

—

129

C₄²

—

130

C₄²

—

131

C₄²

C₄×Dic₅₃

yes

characteristic

132

C₄²

C₄⋊Dic₅₃

yes

characteristic

133

C₂³

—

134

C₂³

#13133132

yes

characteristic

135

Q₈

—

136

D₄

—

137

D₄

—

138

D₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

#13133124

yes

characteristic

150

C₂×C₄

#13133132

yes

characteristic

151

C₈

—

152

C₈

—

424

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

#16810390

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

#16810416

yes

characteristic

164

C₄

#16810418

yes

characteristic

165

C₂

—

166

C₂

—

167

C₂

—

168

C₂

#20272840

yes

characteristic

	subgroup	G/H
center	C₄	#16810418
commutator	C₄²×C₅₃	C₄²
Frattini	#532	C₂×D₅₃
Fitting	C₅₃×#1962	C₂
socle	C₂×C₅₃	#1393
prime radical, cosocle	C₅₃×#532	C₂²
hypercenter	#1962	D₅₃
hypocenter	C₅₃	#7501
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #532 ◁ #1962

subgroup

G/H

center

C₄

#16810418

commutator

C₄²×C₅₃

C₄²

Frattini

#532

C₂×D₅₃

Fitting

C₅₃×#1962

C₂

socle

C₂×C₅₃

#1393

prime radical, cosocle

C₅₃×#532

C₂²

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #532 ◁ #1962

N	H ≅ G/N	type
5. #16810420	C₄	semidirect
94. C₅₃	#7501	semidirect

H ≅ G/N

type

5. #16810420

C₄

semidirect

94. C₅₃

#7501

semidirect

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

subgroup	index
#20276400	2
C₅₃×#1962	2
#7501	53

subgroup

index

#20276400

C₅₃×#1962

#7501

Subgroups — #27271876

Conjugacy classes of subgroups

Complements

Maximal subgroups