Group #27271025 — subgroups

Subgroups — #27271025

Order 13568.

238 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 115 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	C₄×#13133153	C₂	1	1	yes	maximal, characteristic
2	#20273463	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic
4	#16810676	—	2	1	no	—
5	C₂×#13133153	C₄	1	2	yes	—
6	C₄×#11719917	—	2	1	no	—
7	#16810388	—	2	1	no	—
8	#16810376	—	2	1	no	—
9	#16810374	C₄	1	2	yes	—
10	C₂²×C₁₆×C₅₃	—	2	1	no	—
11	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#346	—	2	1	no	—
13	C₅₃×#346	C₄	1	2	yes	—
14	C₂×#11719917	—	2	2	no	—
15	#13133153	—	2	2	no	—
16	#13133153	C₈	1	4	yes	complemented
17	C₄⋊#88322	—	2	2	no	—
18	C₂²⋊#88322	—	4	1	no	—
19	#13133128	—	4	1	no	—
20	C₄×#88322	—	4	1	no	—
21	C₄×#88322	—	4	1	no	—
22	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
25	C₂×C₁₆×C₅₃	—	2	4	no	—
26	C₂×C₁₆×C₅₃	—	2	2	no	—
27	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
30	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
34	C₂×C₈×C₅₃	—	2	2	no	—
35	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
40	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
41	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
43	C₄²×C₅₃	D₈	1	1	yes	characteristic
44	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
45	#11719917	—	2	4	no	—
46	#11719917	—	4	2	no	—
47	C₂×#88322	—	4	2	no	—
48	C₂×#88322	—	4	2	no	—
49	C₁₆×C₅₃	—	2	4	no	—
50	C₂³×C₅₃	#103	1	1	yes	characteristic
51	C₂×C₄×C₅₃	—	4	1	no	—
52	C₂×C₄×C₅₃	—	4	1	no	—
53	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
55	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
56	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
57	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
58	C₂×C₄×C₅₃	#104	1	2	yes	—
59	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
60	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
61	C₈×C₅₃	—	2	1	no	—
62	C₈×C₅₃	—	2	1	no	—
63	C₈×C₅₃	C₄×C₈	1	1	yes	characteristic
64	C₈×C₅₃	#97	1	1	yes	characteristic
65	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
66	#88322	—	4	4	no	—
67	#88322	—	8	2	no	—
68	#7177	—	53	1	no	maximal, Sylow
69	C₂²×C₅₃	—	2	1	no	—
70	C₂²×C₅₃	—	2	1	no	—
71	C₂²×C₅₃	Q₈⋊C₈	1	2	yes	—
72	C₂²×C₅₃	#340	1	1	yes	characteristic
73	C₄×C₅₃	—	2	1	no	—
74	C₄×C₅₃	—	2	1	no	—
75	C₄×C₅₃	—	2	1	no	—
76	C₄×C₅₃	—	2	1	no	—
77	C₄×C₅₃	D₄⋊C₈	1	2	yes	—
78	C₄×C₅₃	#336	1	1	yes	characteristic
79	C₄×C₅₃	#339	1	1	yes	characteristic
80	C₂×#346	D₅₃	1	1	yes	characteristic
81	C₄×#108	—	53	1	no	—
82	#1564	—	53	1	no	—
83	C₂×C₅₃	—	2	2	no	—
84	C₂×C₅₃	#1279	1	1	yes	characteristic
85	C₂×C₅₃	#1388	1	1	yes	characteristic
86	C₂×C₅₃	#1389	1	1	yes	characteristic
87	C₂²×C₁₆	—	2	1	no	—
88	#431	—	106	1	no	—
89	C₂×#108	—	53	2	no	—
90	C₄×M₁₆	—	106	1	no	—
91	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
92	#346	—	2	1	no	—
93	#346	Dic₅₃	1	2	yes	—
94	#341	—	106	1	no	—
95	#335	—	106	1	no	—
96	#334	—	53	2	no	—
97	C₅₃	#7177	1	1	yes	characteristic, complemented, Sylow
98	C₂×M₁₆	—	106	2	no	—
99	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
100	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
101	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
102	C₂×C₁₆	—	2	4	no	—
103	C₂×C₁₆	—	2	2	no	—
104	#108	—	53	4	no	—
105	#108	—	106	2	no	—
106	C₄⋊C₈	—	106	2	no	—
107	C₂²⋊C₈	—	212	1	no	—
108	#97	—	212	1	no	—
109	C₄×C₈	—	212	1	no	—
110	C₄×C₈	—	212	1	no	—
111	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
112	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
113	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
114	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
115	C₂²×C₄	#11719913	1	1	yes	characteristic
116	C₂²×C₄	Dic₂₁₂	1	1	yes	characteristic
117	C₂²×C₄	#11719921	1	1	yes	characteristic
118	M₁₆	—	106	4	no	—
119	M₁₆	—	212	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	212	2	no	—
122	C₂×C₈	—	212	2	no	—
123	C₂×C₈	C₈×D₅₃	1	2	yes	—
124	C₂×C₈	#11719912	1	2	yes	—
125	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
126	C₂×C₈	#11719919	1	1	yes	characteristic
127	C₂×C₈	#11719919	1	1	yes	characteristic
128	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
129	C₄²	#11719913	1	1	yes	characteristic
130	C₄²	D₄₂₄	1	1	yes	characteristic
131	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
132	C₄²	#11719921	1	1	yes	characteristic
133	C₁₆	—	2	4	no	—
134	C₂³	#13133142	1	1	yes	characteristic
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	#13133124	1	1	yes	characteristic
138	C₂×C₄	#13133129	1	2	yes	—
139	C₂×C₄	#13133139	1	1	yes	characteristic
140	C₂×C₄	#13133139	1	1	yes	characteristic
141	C₂×C₄	#13133142	1	1	yes	characteristic
142	C₂×C₄	#13133145	1	2	yes	—
143	C₂×C₄	#13133149	1	1	yes	characteristic
144	C₂×C₄	#13133151	1	1	yes	characteristic
145	C₈	—	2	1	no	—
146	C₈	—	2	1	no	—
147	C₈	—	424	2	no	—
148	C₈	—	212	4	no	—
149	C₈	C₈×Dic₅₃	1	1	yes	characteristic
150	C₈	#13133127	1	1	yes	characteristic
151	C₈	#13133146	1	2	yes	—
152	C₂²	—	2	1	no	—
153	C₂²	—	2	1	no	—
154	C₂²	#16810355	1	2	yes	—
155	C₂²	#16810386	1	1	yes	characteristic
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	#16810352	1	2	yes	—
161	C₄	#16810377	1	1	yes	characteristic
162	C₄	#16810383	1	1	yes	characteristic
163	C₂	—	2	2	no	—
164	C₂	#20272555	1	1	yes	characteristic
165	C₂	#20272827	1	1	yes	characteristic
166	C₂	#20272830	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13133153

C₂

yes

maximal, characteristic

#20273463

C₂

yes

maximal, characteristic

C₂×C₅₃×#346

C₂

yes

maximal, characteristic

#16810676

—

C₂×#13133153

C₄

yes

—

—

—

—

C₄

yes

—

C₂²×C₁₆×C₅₃

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#346

—

C₅₃×#346

C₄

yes

—

C₂×#11719917

—

#13133153

—

#13133153

C₈

yes

—

—

—

—

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

characteristic

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

Q₈

yes

characteristic

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₅₃

SD₁₆

yes

characteristic

C₂²×C₄×C₅₃

Q₁₆

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄²

yes

characteristic

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

M₁₆

yes

—

C₄²×C₅₃

C₂²⋊C₄

yes

characteristic

C₄²×C₅₃

C₄⋊C₄

yes

characteristic

C₄²×C₅₃

D₈

yes

characteristic

C₄²×C₅₃

SD₁₆

yes

—

—

—

—

C₁₆×C₅₃

—

C₂³×C₅₃

#103

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#95

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₈

yes

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₄×C₈

yes

characteristic

C₈×C₅₃

#97

yes

characteristic

C₈×C₅₃

C₄⋊C₈

yes

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

Q₈⋊C₈

yes

—

C₂²×C₅₃

#340

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

D₄⋊C₈

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

D₅₃

yes

characteristic

C₄×#108

—

#1564

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂²×C₁₆

—

#431

—

106

—

C₂×#108

—

C₄×M₁₆

—

106

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

#346

—

#346

Dic₅₃

yes

—

#341

—

106

—

#335

—

106

—

#334

—

C₅₃

#7177

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

106

—

C₂²×C₈

C₄×D₅₃

yes

characteristic

100

C₂²×C₈

C₂²⋊D₅₃

yes

characteristic

101

C₂×C₄²

D₂₁₂

yes

characteristic

102

C₂×C₁₆

—

103

C₂×C₁₆

—

104

#108

—

105

#108

—

106

—

106

C₄⋊C₈

—

106

—

107

C₂²⋊C₈

—

212

—

108

#97

—

212

—

109

C₄×C₈

—

212

—

110

C₄×C₈

—

212

—

111

C₄×C₈

Dic₁₀₆

yes

characteristic

112

C₄×C₈

C₄×D₅₃

yes

characteristic

113

C₄×C₈

C₂×Dic₅₃

yes

characteristic

114

C₄×C₈

C₂²⋊D₅₃

yes

characteristic

115

C₂²×C₄

#11719913

yes

characteristic

116

C₂²×C₄

Dic₂₁₂

yes

characteristic

117

C₂²×C₄

#11719921

yes

characteristic

118

M₁₆

—

106

—

119

M₁₆

—

212

—

120

C₂×C₈

—

121

C₂×C₈

—

212

—

122

C₂×C₈

—

212

—

123

C₂×C₈

C₈×D₅₃

yes

—

124

C₂×C₈

#11719912

yes

—

125

C₂×C₈

C₄×Dic₅₃

yes

characteristic

126

C₂×C₈

#11719919

yes

characteristic

127

C₂×C₈

#11719919

yes

characteristic

128

C₂×C₈

C₂²⋊Dic₅₃

yes

characteristic

129

C₄²

#11719913

yes

characteristic

130

C₄²

D₄₂₄

yes

characteristic

131

C₄²

C₄⋊Dic₅₃

yes

characteristic

132

C₄²

#11719921

yes

characteristic

133

C₁₆

—

134

C₂³

#13133142

yes

characteristic

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

#13133124

yes

characteristic

138

C₂×C₄

#13133129

yes

—

139

C₂×C₄

#13133139

yes

characteristic

140

C₂×C₄

#13133139

yes

characteristic

141

C₂×C₄

#13133142

yes

characteristic

142

C₂×C₄

#13133145

yes

—

143

C₂×C₄

#13133149

yes

characteristic

144

C₂×C₄

#13133151

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

424

—

148

C₈

—

212

—

149

C₈

C₈×Dic₅₃

yes

characteristic

150

C₈

#13133127

yes

characteristic

151

C₈

#13133146

yes

—

152

C₂²

—

153

C₂²

—

154

C₂²

#16810355

yes

—

155

C₂²

#16810386

yes

characteristic

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

#16810352

yes

—

161

C₄

#16810377

yes

characteristic

162

C₄

#16810383

yes

characteristic

163

C₂

—

164

C₂

#20272555

yes

characteristic

165

C₂

#20272827

yes

characteristic

166

C₂

#20272830

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13133139
commutator	C₈×C₅₃	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₅₃
Fitting	C₂×C₅₃×#346	C₂
socle	C₂²×C₅₃	#340
prime radical	C₂³×C₅₃	#103
cosocle	C₂×C₄×C₈×C₅₃	C₂²
hypercenter	C₂×#346	D₅₃
hypocenter	C₅₃	#7177
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₄

#13133139

commutator

C₈×C₅₃

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₅₃

Fitting

C₂×C₅₃×#346

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂×C₄×C₈×C₅₃

C₂²

D₅₃

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
16. #13133153	C₈	semidirect
97. C₅₃	#7177	semidirect

H ≅ G/N

type

16. #13133153

C₈

semidirect

97. C₅₃

#7177

semidirect

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

subgroup	index
C₄×#13133153	2
#20273463	2
C₂×C₅₃×#346	2
#7177	53

subgroup

index

C₄×#13133153

#20273463

C₂×C₅₃×#346

#7177

Subgroups — #27271025

Conjugacy classes of subgroups

Complements

Maximal subgroups