Group #27302628 — subgroups

Subgroups — #27302628

Order 13568.

334 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 151 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	#20276471	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810399	C₂	1	1	yes	maximal, characteristic
3	#20272804	C₂	1	4	yes	maximal
4	C₅₃×#2960	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	C₂²	1	2	yes	—
6	C₂²×#11719908	C₂²	1	2	yes	—
7	C₄⋊#11719908	—	2	2	no	—
8	#16810399	—	2	2	no	—
9	C₅₃×#533	C₄	1	1	yes	characteristic
10	C₅₃×#446	—	2	2	no	—
11	C₈×Q₈×C₅₃	—	2	2	no	—
12	C₅₃×#424	C₄	1	2	yes	—
13	C₅₃×#423	C₄	1	1	yes	characteristic
14	C₅₃×#405	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	#13133157	—	2	4	no	—
17	C₂×#11719908	—	2	2	no	—
18	C₂×#11719908	—	2	2	no	—
19	C₂×#11719908	—	2	2	no	—
20	C₂×#11719908	Q₈	1	4	yes	complemented
21	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
24	C₄⋊Q₈×C₅₃	C₈	1	2	yes	—
25	#125×C₅₃	C₈	1	2	yes	—
26	C₂²⋊Q₈×C₅₃	—	2	2	no	—
27	C₄×Q₈×C₅₃	—	2	2	no	—
28	#117×C₅₃	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄⋊C₈×C₅₃	—	2	4	no	—
31	C₄⋊C₈×C₅₃	—	2	2	no	—
32	#97×C₅₃	—	2	2	no	—
33	C₄×C₈×C₅₃	—	2	2	no	—
34	C₄×C₈×C₅₃	D₄	1	2	yes	—
35	C₄×C₈×C₅₃	D₄	1	2	yes	—
36	C₂×Q₈×C₅₃	—	2	2	no	—
37	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
38	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	M₁₆×C₅₃	—	2	4	no	—
40	C₂×C₈×C₅₃	—	2	2	no	—
41	C₂×C₈×C₅₃	—	2	2	no	—
42	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
43	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
46	C₂×C₈×C₅₃	#41	1	2	yes	—
47	C₄⋊C₄×C₅₃	—	2	4	no	—
48	C₄⋊C₄×C₅₃	—	2	2	no	—
49	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
50	C₂²⋊C₄×C₅₃	—	2	2	no	—
51	C₄²×C₅₃	—	2	2	no	—
52	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
53	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
54	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
55	#11719908	—	2	4	no	—
56	#11719908	—	4	2	no	—
57	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
58	Q₈×C₅₃	—	4	4	no	—
59	C₂×C₄×C₅₃	—	2	4	no	—
60	C₂×C₄×C₅₃	—	4	2	no	—
61	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
62	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
63	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
65	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₅₃	#131	1	2	yes	—
67	C₈×C₅₃	—	2	2	no	—
68	C₈×C₅₃	—	4	2	no	—
69	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
70	C₈×C₅₃	#124	1	1	yes	characteristic
71	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
72	#13267	—	53	1	no	maximal, Sylow
73	C₂²×C₅₃	—	2	2	no	—
74	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
75	C₂²×C₅₃	#446	1	2	yes	—
76	C₄×C₅₃	—	2	2	no	—
77	C₄×C₅₃	—	2	2	no	—
78	C₄×C₅₃	—	4	4	no	—
79	C₄×C₅₃	#391	1	1	yes	characteristic
80	C₄×C₅₃	#408	1	1	yes	characteristic
81	C₄×C₅₃	C₈×Q₈	1	2	yes	—
82	#2960	D₅₃	1	1	yes	characteristic
83	#2150	—	53	1	no	—
84	C₂×#346	—	53	1	no	—
85	#1379	—	53	4	no	—
86	C₂×C₅₃	—	2	2	no	—
87	C₂×C₅₃	#1984	1	1	yes	characteristic
88	C₂×C₅₃	#2117	1	2	yes	—
89	#533	Dic₅₃	1	1	yes	characteristic
90	C₂×M₃₂	—	53	2	no	—
91	C₂²×C₁₆	—	53	2	no	—
92	#446	—	2	2	no	—
93	C₈×Q₈	—	2	2	no	—
94	#424	Dic₅₃	1	2	yes	—
95	#423	Dic₅₃	1	1	yes	characteristic
96	#405	C₂×D₅₃	1	2	yes	—
97	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
98	C₄⋊C₁₆	—	106	2	no	—
99	#346	—	106	2	no	—
100	C₅₃	#13267	1	1	yes	characteristic, complemented, Sylow
101	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
102	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
103	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
104	C₄⋊Q₈	#88322	1	2	yes	—
105	#125	#88322	1	2	yes	—
106	C₂²⋊Q₈	—	2	2	no	—
107	C₄×Q₈	—	2	2	no	—
108	#117	C₂×Dic₅₃	1	2	yes	—
109	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
110	M₃₂	—	106	4	no	—
111	C₂×C₁₆	—	53	4	no	—
112	C₂×C₁₆	—	106	2	no	—
113	C₂×C₁₆	—	106	2	no	—
114	C₂×C₁₆	—	106	2	no	—
115	C₄⋊C₈	—	2	4	no	—
116	C₄⋊C₈	—	2	2	no	—
117	#97	—	2	2	no	—
118	C₄×C₈	—	2	2	no	—
119	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
120	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
121	C₂×Q₈	—	2	2	no	—
122	C₂²×C₄	#11719917	1	2	yes	—
123	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
124	M₁₆	—	2	4	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
128	C₂×C₈	Q₈×D₅₃	1	2	yes	—
129	C₂×C₈	#11719949	1	2	yes	—
130	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
131	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
132	C₄⋊C₄	—	2	4	no	—
133	C₄⋊C₄	—	2	2	no	—
134	C₄⋊C₄	C₂×#88322	1	4	yes	—
135	C₂²⋊C₄	—	2	2	no	—
136	C₄²	—	2	2	no	—
137	C₄²	C₂×#88322	1	1	yes	characteristic
138	C₄²	C₂×#88322	1	1	yes	characteristic
139	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
140	C₁₆	—	106	4	no	—
141	C₁₆	—	212	2	no	—
142	C₂³	C₂×#11719917	1	1	yes	characteristic
143	Q₈	—	4	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	C₂²⋊#88322	1	2	yes	—
147	C₂×C₄	C₂²⋊#88322	1	2	yes	—
148	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
149	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
150	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
151	C₂×C₄	#13133224	1	2	yes	—
152	C₈	—	2	2	no	—
153	C₈	—	4	2	no	—
154	C₈	#13133193	1	2	yes	—
155	C₈	#13133206	1	1	yes	characteristic
156	C₈	#13133215	1	1	yes	characteristic
157	C₂²	—	2	2	no	—
158	C₂²	C₂×C₂²⋊#88322	1	1	yes	characteristic
159	C₂²	#16810754	1	2	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	4	4	no	—
163	C₄	#16810530	1	1	yes	characteristic
164	C₄	#16810590	1	1	yes	characteristic
165	C₄	Q₈×#88322	1	2	yes	—
166	C₂	—	2	2	no	—
167	C₂	#20275652	1	1	yes	characteristic
168	C₂	#20276343	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20276471

C₂

yes

maximal, characteristic

C₂×#16810399

C₂

yes

maximal, characteristic

#20272804

C₂

yes

maximal

C₅₃×#2960

C₂

yes

maximal, characteristic

C₂×#13133157

C₂²

yes

—

C₂²×#11719908

C₂²

yes

—

C₄⋊#11719908

—

#16810399

—

C₅₃×#533

C₄

yes

characteristic

C₅₃×#446

—

C₈×Q₈×C₅₃

—

C₅₃×#424

C₄

yes

—

C₅₃×#423

C₄

yes

characteristic

C₅₃×#405

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

—

—

—

—

Q₈

yes

complemented

C₂×M₁₆×C₅₃

C₂×C₄

yes

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊Q₈×C₅₃

C₈

yes

—

#125×C₅₃

C₈

yes

—

C₂²⋊Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

#117×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

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

M₁₆

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×Q₈

yes

—

C₂×C₈×C₅₃

#41

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₈

yes

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂²⋊C₄

yes

—

C₄²×C₅₃

C₂×C₈

yes

characteristic

C₄²×C₅₃

C₂×C₈

yes

characteristic

#11719908

—

#11719908

—

C₂³×C₅₃

C₂×M₁₆

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₈

yes

—

C₂×C₄×C₅₃

C₂²⋊C₈

yes

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×Q₈

yes

—

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₃

#131

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₂²⋊Q₈

yes

—

C₈×C₅₃

#124

yes

characteristic

C₈×C₅₃

C₄⋊Q₈

yes

characteristic

#13267

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₅₃

#446

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

C₈×Q₈

yes

—

#2960

D₅₃

yes

—

—

—

C₂×C₅₃

—

C₂×C₅₃

#1984

yes

characteristic

C₂×C₅₃

#2117

yes

—

#533

Dic₅₃

yes

characteristic

C₂×M₃₂

—

C₂²×C₁₆

—

#446

—

C₈×Q₈

—

#424

Dic₅₃

yes

—

#423

Dic₅₃

yes

characteristic

#405

C₂×D₅₃

yes

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

C₄⋊C₁₆

—

106

—

#346

—

106

—

100

C₅₃

#13267

yes

characteristic, complemented, Sylow

101

C₂×M₁₆

C₂×Dic₅₃

yes

—

102

C₂²×C₈

C₂×Dic₅₃

yes

characteristic

103

C₂²×C₈

C₂²×D₅₃

yes

characteristic

104

C₄⋊Q₈

#88322

yes

—

105

#125

#88322

yes

—

106

C₂²⋊Q₈

—

107

C₄×Q₈

—

108

#117

C₂×Dic₅₃

yes

—

109

C₂×C₄²

C₂×Dic₅₃

yes

characteristic

110

M₃₂

—

106

—

111

C₂×C₁₆

—

112

C₂×C₁₆

—

106

—

113

C₂×C₁₆

—

106

—

114

C₂×C₁₆

—

106

—

115

C₄⋊C₈

—

116

C₄⋊C₈

—

117

#97

—

118

C₄×C₈

—

119

C₄×C₈

C₂²⋊D₅₃

yes

—

120

C₄×C₈

C₂²⋊D₅₃

yes

—

121

C₂×Q₈

—

122

C₂²×C₄

#11719917

yes

—

123

C₂²×C₄

C₂²×Dic₅₃

yes

characteristic

124

M₁₆

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

C₂²⋊Dic₅₃

yes

—

128

C₂×C₈

Q₈×D₅₃

yes

—

129

C₂×C₈

#11719949

yes

—

130

C₂×C₈

C₂×C₂²⋊D₅₃

yes

characteristic

131

C₂×C₈

C₂×C₂²⋊D₅₃

yes

characteristic

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

C₂×#88322

yes

—

135

C₂²⋊C₄

—

136

C₄²

—

137

C₄²

C₂×#88322

yes

characteristic

138

C₄²

C₂×#88322

yes

characteristic

139

C₄²

C₂²⋊Dic₅₃

yes

—

140

C₁₆

—

106

—

141

C₁₆

—

212

—

142

C₂³

C₂×#11719917

yes

characteristic

143

Q₈

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

C₂²⋊#88322

yes

—

147

C₂×C₄

C₂²⋊#88322

yes

—

148

C₂×C₄

C₂×C₂²⋊Dic₅₃

yes

characteristic

149

C₂×C₄

Q₈×Dic₅₃

yes

—

150

C₂×C₄

C₂²×#88322

yes

characteristic

151

C₂×C₄

#13133224

yes

—

152

C₈

—

153

C₈

—

154

C₈

#13133193

yes

—

155

C₈

#13133206

yes

characteristic

156

C₈

#13133215

yes

characteristic

157

C₂²

—

158

C₂²

C₂×C₂²⋊#88322

yes

characteristic

159

C₂²

#16810754

yes

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

#16810530

yes

characteristic

164

C₄

#16810590

yes

characteristic

165

C₄

Q₈×#88322

yes

—

166

C₂

—

167

C₂

#20275652

yes

characteristic

168

C₂

#20276343

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#2960	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂³×C₅₃	C₂×M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#2960	D₅₃
hypocenter	C₅₃	#13267
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2960

subgroup

G/H

center

C₂×C₈

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₅₃×#2960

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₈

prime radical

C₂³×C₅₃

C₂×M₁₆

cosocle

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2960

N	H ≅ G/N	type
20. C₂×#11719908	Q₈	semidirect
100. C₅₃	#13267	semidirect

H ≅ G/N

type

20. C₂×#11719908

Q₈

semidirect

100. C₅₃

#13267

semidirect

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

subgroup	index
#20276471	2
C₂×#16810399	2
#20272804	2
C₅₃×#2960	2
#13267	53

subgroup

index

C₅₃×#2960

Subgroups — #27302628

Conjugacy classes of subgroups

Complements

Maximal subgroups