Group #27302689 — subgroups

Subgroups — #27302689

Order 13568.

366 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 163 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	C₂×#16811025	C₂	1	1	yes	maximal, characteristic
2	#20274879	C₂	1	1	yes	maximal, characteristic
3	#20272807	C₂	1	4	yes	maximal
4	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133209	—	2	1	no	—
6	#16811025	—	2	2	no	—
7	#16811025	C₄	1	4	yes	complemented
8	C₂×#13133151	C₂²	1	2	yes	—
9	C₂×#13133151	C₂²	1	2	yes	—
10	C₂×#13133149	—	2	1	no	—
11	C₂×#13133149	—	2	1	no	—
12	#16810489	—	2	1	no	—
13	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₄×C₁₆×C₅₃	—	2	2	no	—
16	#13133209	—	2	4	no	—
17	#13133209	—	4	2	no	—
18	C₂×C₄⋊Dic₅₃	—	2	2	no	—
19	C₂×C₄⋊Dic₅₃	—	2	2	no	—
20	C₂×#11719919	—	4	2	no	—
21	#13133151	—	2	2	no	—
22	#13133151	—	2	4	no	—
23	#13133151	C₂×C₄	1	4	yes	—
24	#13133149	—	2	4	no	—
25	#13133149	—	4	2	no	—
26	#13133124	—	4	2	no	—
27	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
29	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
30	C₂×C₁₆×C₅₃	—	2	2	no	—
31	C₂×C₁₆×C₅₃	—	2	2	no	—
32	C₂×C₁₆×C₅₃	Q₈	1	4	yes	—
33	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₅₃	D₄	1	2	yes	—
35	C₂²×C₄×C₅₃	D₈	1	2	yes	—
36	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₅₃	—	4	2	no	—
38	C₂²×Dic₅₃	—	4	2	no	—
39	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
40	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
41	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
42	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
45	C₂×C₈×C₅₃	#41	1	2	yes	—
46	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
47	C₄²×C₅₃	SD₁₆	1	2	yes	—
48	C₄⋊Dic₅₃	—	2	4	no	—
49	C₄⋊Dic₅₃	—	4	2	no	—
50	C₄⋊Dic₅₃	—	4	2	no	—
51	C₄⋊Dic₅₃	—	4	2	no	—
52	#11719919	—	4	8	no	—
53	C₁₆×C₅₃	—	2	4	no	—
54	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₅₃	—	2	4	no	—
56	C₂×C₄×C₅₃	#102	1	4	yes	—
57	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
63	C₂×Dic₅₃	—	4	4	no	—
64	C₂×Dic₅₃	—	8	4	no	—
65	C₂×Dic₅₃	—	8	2	no	—
66	C₂×Dic₅₃	—	8	4	no	—
67	C₈×C₅₃	#103	1	4	yes	—
68	C₈×C₅₃	C₄×Q₈	1	2	yes	—
69	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
70	#13281	—	53	1	no	maximal, Sylow
71	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
72	C₂²×C₅₃	#486	1	1	yes	characteristic
73	C₂²×C₅₃	#500	1	1	yes	characteristic
74	C₂²×C₅₃	#508	1	4	yes	—
75	C₄×C₅₃	—	2	4	no	—
76	C₄×C₅₃	#391	1	1	yes	characteristic
77	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
78	C₄×C₅₃	#487	1	1	yes	characteristic
79	C₄×C₅₃	#498	1	1	yes	characteristic
80	Dic₅₃	—	8	4	no	—
81	Dic₅₃	—	16	2	no	—
82	C₂×#499	—	53	1	no	—
83	C₂×C₄×C₁₆	D₅₃	1	1	yes	characteristic
84	#1846	—	53	1	no	—
85	#1380	—	53	4	no	—
86	C₂×C₅₃	#1990	1	1	yes	characteristic
87	C₂×C₅₃	#2138	1	2	yes	—
88	C₂×C₅₃	#2242	1	2	yes	—
89	C₂×C₅₃	#2253	1	2	yes	—
90	C₂×#125	—	106	1	no	—
91	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
92	#499	—	53	4	no	—
93	#499	—	106	2	no	—
94	C₂×#107	—	53	2	no	—
95	C₂×#107	—	53	2	no	—
96	C₂×#106	—	106	1	no	—
97	C₂×#106	—	106	1	no	—
98	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
99	#382	—	106	1	no	—
100	C₄×C₁₆	—	2	2	no	—
101	C₅₃	#13281	1	1	yes	characteristic, complemented, Sylow
102	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
103	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
104	#125	—	106	4	no	—
105	#125	—	212	2	no	—
106	C₂×C₄⋊C₄	—	106	2	no	—
107	C₂×C₄⋊C₄	—	106	2	no	—
108	C₂×C₄⋊C₄	—	212	2	no	—
109	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
110	C₂×C₁₆	—	2	2	no	—
111	C₂×C₁₆	—	2	2	no	—
112	C₂×C₁₆	Dic₁₀₆	1	4	yes	—
113	#107	—	53	4	no	—
114	#107	—	106	2	no	—
115	#107	—	106	4	no	—
116	#106	—	106	4	no	—
117	#106	—	212	2	no	—
118	C₄×C₈	C₄×D₅₃	1	2	yes	—
119	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
120	#95	—	212	2	no	—
121	C₂²×C₄	—	212	2	no	—
122	C₂²×C₄	—	212	2	no	—
123	C₂²×C₄	D₄₂₄	1	2	yes	—
124	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
125	C₂×C₈	#11719913	1	2	yes	—
126	C₂×C₈	Dic₂₁₂	1	2	yes	—
127	C₂×C₈	#11719921	1	2	yes	—
128	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
129	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
130	C₂×C₈	#11719945	1	2	yes	—
131	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
132	C₄⋊C₄	—	106	4	no	—
133	C₄⋊C₄	—	212	2	no	—
134	C₄⋊C₄	—	212	2	no	—
135	C₄⋊C₄	—	212	8	no	—
136	C₄⋊C₄	—	212	2	no	—
137	C₄²	#11719913	1	2	yes	—
138	C₄²	#11719921	1	2	yes	—
139	C₁₆	—	2	4	no	—
140	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	424	4	no	—
143	C₂×C₄	—	424	2	no	—
144	C₂×C₄	—	424	4	no	—
145	C₂×C₄	—	212	4	no	—
146	C₂×C₄	#13133139	1	4	yes	—
147	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
148	C₂×C₄	#13133205	1	1	yes	characteristic
149	C₂×C₄	#13133217	1	1	yes	characteristic
150	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
151	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
152	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
153	C₈	#13133142	1	4	yes	—
154	C₈	C₄×Dic₁₀₆	1	2	yes	—
155	C₈	#13133196	1	2	yes	—
156	C₂²	C₂×#13133139	1	1	yes	characteristic
157	C₂²	#16810970	1	1	yes	characteristic
158	C₂²	#16811028	1	1	yes	characteristic
159	C₂²	#16811055	1	4	yes	—
160	C₄	—	2	4	no	—
161	C₄	—	848	2	no	—
162	C₄	—	424	4	no	—
163	C₄	#16810529	1	1	yes	characteristic
164	C₄	C₂×#13133142	1	1	yes	characteristic
165	C₄	#16810974	1	1	yes	characteristic
166	C₄	#16811021	1	1	yes	characteristic
167	C₂	#20275676	1	1	yes	characteristic
168	C₂	#20276423	1	2	yes	—
169	C₂	#20276947	1	2	yes	—
170	C₂	#20276994	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16811025

C₂

yes

maximal, characteristic

#20274879

C₂

yes

maximal, characteristic

#20272807

C₂

yes

maximal

C₂×C₄×C₁₆×C₅₃

C₂

yes

maximal, characteristic

C₂×#13133209

—

#16811025

—

#16811025

C₄

yes

complemented

C₂×#13133151

C₂²

yes

—

C₂×#13133151

C₂²

yes

—

C₂×#13133149

—

C₂×#13133149

—

#16810489

—

C₂²×C₁₆×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₄×C₁₆×C₅₃

—

#13133209

—

#13133209

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

—

—

—

C₂×C₄

yes

—

#13133149

—

#13133149

—

#13133124

—

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

characteristic

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

Q₈

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

D₈

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₂×C₈×C₅₃

Q₁₆

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×Q₈

yes

—

C₂×C₈×C₅₃

#41

yes

—

C₄²×C₅₃

C₂²⋊C₄

yes

—

C₄²×C₅₃

SD₁₆

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

C₁₆×C₅₃

—

C₂³×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#102

yes

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#124

yes

characteristic

C₂×C₄×C₅₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₁₆

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

#103

yes

—

C₈×C₅₃

C₄×Q₈

yes

—

C₈×C₅₃

C₂²⋊Q₈

yes

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

C₂×#499

—

C₂×C₄×C₁₆

D₅₃

yes

characteristic

#1846

—

#1380

—

C₂×C₅₃

#1990

yes

characteristic

C₂×C₅₃

#2138

yes

—

C₂×C₅₃

#2242

yes

—

C₂×C₅₃

#2253

yes

—

C₂×#125

—

106

—

C₂²×C₁₆

C₂×D₅₃

yes

—

#499

—

#499

—

106

—

C₂×#107

—

C₂×#107

—

C₂×#106

—

106

—

C₂×#106

—

106

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

#382

—

106

—

100

C₄×C₁₆

—

101

C₅₃

#13281

yes

characteristic, complemented, Sylow

102

C₂²×C₈

D₂₁₂

yes

characteristic

103

C₂²×C₈

C₂²×D₅₃

yes

characteristic

104

#125

—

106

—

105

#125

—

212

—

106

C₂×C₄⋊C₄

—

106

—

107

C₂×C₄⋊C₄

—

106

—

108

C₂×C₄⋊C₄

—

212

—

109

C₂×C₄²

D₂₁₂

yes

characteristic

110

C₂×C₁₆

—

111

C₂×C₁₆

—

112

C₂×C₁₆

Dic₁₀₆

yes

—

113

#107

—

114

#107

—

106

—

115

#107

—

106

—

116

#106

—

106

—

117

#106

—

212

—

118

C₄×C₈

C₄×D₅₃

yes

—

119

C₄×C₈

C₂²⋊D₅₃

yes

—

120

#95

—

212

—

121

C₂²×C₄

—

212

—

122

C₂²×C₄

—

212

—

123

C₂²×C₄

D₄₂₄

yes

—

124

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

125

C₂×C₈

#11719913

yes

—

126

C₂×C₈

Dic₂₁₂

yes

—

127

C₂×C₈

#11719921

yes

—

128

C₂×C₈

C₂×Dic₁₀₆

yes

—

129

C₂×C₈

C₂×C₄×D₅₃

yes

characteristic

130

C₂×C₈

#11719945

yes

—

131

C₂×C₈

C₂×C₂²⋊D₅₃

yes

characteristic

132

C₄⋊C₄

—

106

—

133

C₄⋊C₄

—

212

—

134

C₄⋊C₄

—

212

—

135

C₄⋊C₄

—

212

—

136

C₄⋊C₄

—

212

—

137

C₄²

#11719913

yes

—

138

C₄²

#11719921

yes

—

139

C₁₆

—

140

C₂³

C₂×D₄₂₄

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

424

—

143

C₂×C₄

—

424

—

144

C₂×C₄

—

424

—

145

C₂×C₄

—

212

—

146

C₂×C₄

#13133139

yes

—

147

C₂×C₄

C₂×#11719921

yes

characteristic

148

C₂×C₄

#13133205

yes

characteristic

149

C₂×C₄

#13133217

yes

characteristic

150

C₂×C₄

C₂×#11719913

yes

characteristic

151

C₂×C₄

C₂×#11719913

yes

characteristic

152

C₂×C₄

C₂×Dic₂₁₂

yes

characteristic

153

C₈

#13133142

yes

—

154

C₈

C₄×Dic₁₀₆

yes

—

155

C₈

#13133196

yes

—

156

C₂²

C₂×#13133139

yes

characteristic

157

C₂²

#16810970

yes

characteristic

158

C₂²

#16811028

yes

characteristic

159

C₂²

#16811055

yes

—

160

C₄

—

161

C₄

—

848

—

162

C₄

—

424

—

163

C₄

#16810529

yes

characteristic

164

C₄

C₂×#13133142

yes

characteristic

165

C₄

#16810974

yes

characteristic

166

C₄

#16811021

yes

characteristic

167

C₂

#20275676

yes

characteristic

168

C₂

#20276423

yes

—

169

C₂

#20276947

yes

—

170

C₂

#20276994

yes

—

	subgroup	G/H
center	C₂³	C₂×D₄₂₄
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₄×C₁₆×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₅₃
hypocenter	C₅₃	#13281
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

subgroup

G/H

center

C₂³

C₂×D₄₂₄

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₂×C₄×C₁₆×C₅₃

C₂

socle, prime radical

C₂³×C₅₃

C₂×D₈

cosocle

C₂²×C₈×C₅₃

C₂³

hypercenter

C₂×C₄×C₁₆

D₅₃

hypocenter

C₅₃

#13281

LCS

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

N	H ≅ G/N	type
7. #16811025	C₄	semidirect
101. C₅₃	#13281	semidirect

H ≅ G/N

type

7. #16811025

C₄

semidirect

101. C₅₃

#13281

semidirect

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

subgroup	index
C₂×#16811025	2
#20274879	2
#20272807	2
C₂×C₄×C₁₆×C₅₃	2
#13281	53

subgroup

index

C₂×C₄×C₁₆×C₅₃

Subgroups — #27302689

Conjugacy classes of subgroups

Complements

Maximal subgroups