Group #27277879 — subgroups

Subgroups — #27277879

Order 13568.

438 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 203 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	#20274875	C₂	1	1	yes	maximal, characteristic
2	#20274874	C₂	1	2	yes	maximal
3	#20274842	C₂	1	2	yes	maximal
4	C₂×#16810344	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1844	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133151	—	2	2	no	—
7	C₂×#13133150	—	2	4	no	—
8	C₂×C₄×#88322	C₂²	1	2	yes	—
9	#16810493	C₂²	1	2	yes	—
10	#16810385	—	2	4	no	—
11	#16810344	C₄	1	4	yes	complemented
12	C₂×#106×C₅₃	—	2	2	no	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#384	C₂²	1	1	yes	characteristic
15	C₅₃×#384	C₂²	1	1	yes	characteristic
16	C₂²×#88322	D₄	1	4	yes	—
17	C₂×C₄⋊Dic₅₃	—	2	2	no	—
18	C₂×C₄⋊Dic₅₃	—	2	2	no	—
19	C₂×C₄⋊Dic₅₃	—	2	2	no	—
20	#13133151	—	4	4	no	—
21	#13133150	—	4	8	no	—
22	C₄×#88322	C₂×C₄	1	4	yes	—
23	C₄×#88322	Q₈	1	4	yes	—
24	C₂²×C₈×C₅₃	D₄	1	2	yes	—
25	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
26	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
27	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
28	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
29	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
30	#106×C₅₃	—	4	4	no	—
31	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₅₃	Q₈	1	2	yes	—
33	C₂²×C₄×C₅₃	—	4	1	no	—
34	C₂²×C₄×C₅₃	—	4	1	no	—
35	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
37	C₂²×Dic₅₃	—	4	2	no	—
38	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
39	C₂×C₈×C₅₃	D₈	1	4	yes	—
40	C₂×C₈×C₅₃	Q₁₆	1	4	yes	—
41	C₄⋊C₄×C₅₃	—	4	2	no	—
42	C₄⋊C₄×C₅₃	—	4	2	no	—
43	C₄⋊C₄×C₅₃	—	4	4	no	—
44	C₄⋊C₄×C₅₃	—	4	4	no	—
45	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
47	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
48	C₄⋊Dic₅₃	—	4	4	no	—
49	C₄⋊Dic₅₃	—	4	4	no	—
50	C₄⋊Dic₅₃	—	4	4	no	—
51	C₂×#88322	—	2	4	no	—
52	C₂×#88322	—	2	4	no	—
53	C₂×#88322	C₄⋊C₄	1	8	yes	—
54	C₂³×C₅₃	#127	1	1	yes	characteristic
55	C₂×C₄×C₅₃	—	8	1	no	—
56	C₂×C₄×C₅₃	—	8	1	no	—
57	C₂×C₄×C₅₃	—	8	2	no	—
58	C₂×C₄×C₅₃	—	8	2	no	—
59	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
60	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
63	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
64	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
65	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
66	C₂×C₄×C₅₃	#137	1	4	yes	—
67	C₂×Dic₅₃	—	8	2	no	—
68	C₂×Dic₅₃	—	8	2	no	—
69	C₂×Dic₅₃	—	8	2	no	—
70	C₈×C₅₃	#107	1	8	yes	complemented
71	#88322	—	2	8	no	—
72	#8820	—	53	1	no	maximal, Sylow
73	C₂²×C₅₃	#384	1	1	yes	characteristic
74	C₂²×C₅₃	#493	1	1	yes	characteristic
75	C₂²×C₅₃	#494	1	1	yes	characteristic
76	C₂²×C₅₃	#496	1	2	yes	—
77	C₂²×C₅₃	#497	1	2	yes	—
78	C₄×C₅₃	—	16	1	no	—
79	C₄×C₅₃	—	16	1	no	—
80	C₄×C₅₃	C₂×#107	1	2	yes	—
81	C₄×C₅₃	#428	1	4	yes	—
82	C₄×C₅₃	#500	1	2	yes	—
83	C₄×C₅₃	#501	1	4	yes	—
84	Dic₅₃	—	16	2	no	—
85	#1845	—	53	1	no	—
86	#1845	—	53	2	no	—
87	#1844	D₅₃	1	1	yes	characteristic
88	#1834	—	53	2	no	—
89	C₂×#322	—	53	1	no	—
90	C₂×C₅₃	#1721	1	1	yes	characteristic
91	C₂×C₅₃	#1722	1	1	yes	characteristic
92	C₂×C₅₃	#1723	1	2	yes	—
93	C₂×C₅₃	#1845	1	1	yes	characteristic
94	C₂×C₅₃	#1847	1	2	yes	—
95	C₂×#107	—	106	4	no	—
96	C₂×#107	—	106	2	no	—
97	C₂×#106	—	2	2	no	—
98	C₂×C₄×C₈	—	53	2	no	—
99	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
100	#384	—	53	2	no	—
101	#384	C₂×D₅₃	1	1	yes	characteristic
102	#384	C₂×D₅₃	1	1	yes	characteristic
103	#340	—	106	4	no	—
104	#322	—	53	4	no	—
105	C₅₃	#8820	1	1	yes	characteristic, complemented, Sylow
106	C₂²×C₈	—	53	4	no	—
107	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
108	C₂×C₄⋊C₄	—	2	2	no	—
109	C₂×C₄⋊C₄	—	2	2	no	—
110	C₂×C₄⋊C₄	—	2	1	no	—
111	C₂×C₄⋊C₄	—	2	1	no	—
112	C₂×C₄⋊C₄	—	106	2	no	—
113	C₂×C₄⋊C₄	—	106	2	no	—
114	C₂×C₄⋊C₄	—	106	2	no	—
115	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
116	#107	—	212	8	no	—
117	#107	—	212	4	no	—
118	#106	—	4	4	no	—
119	C₄×C₈	—	53	4	no	—
120	C₄×C₈	—	53	4	no	—
121	C₄×C₈	Dic₁₀₆	1	2	yes	—
122	C₄×C₈	C₄×D₅₃	1	2	yes	—
123	C₂²×C₄	—	4	1	no	—
124	C₂²×C₄	—	4	1	no	—
125	C₂²×C₄	—	212	2	no	—
126	C₂²×C₄	D₄×D₅₃	1	2	yes	—
127	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
128	C₂×C₈	—	53	8	no	—
129	C₂×C₈	—	106	4	no	—
130	C₂×C₈	—	106	4	no	—
131	C₂×C₈	#11719919	1	4	yes	—
132	C₂×C₈	#11719922	1	4	yes	—
133	C₂×C₈	#11719925	1	4	yes	—
134	C₄⋊C₄	—	4	2	no	—
135	C₄⋊C₄	—	4	2	no	—
136	C₄⋊C₄	—	4	4	no	—
137	C₄⋊C₄	—	4	4	no	—
138	C₄⋊C₄	—	212	4	no	—
139	C₄⋊C₄	—	212	4	no	—
140	C₄⋊C₄	—	212	4	no	—
141	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
142	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
143	C₄²	Q₈×D₅₃	1	2	yes	—
144	C₂³	#13133213	1	1	yes	characteristic
145	C₂×C₄	—	8	2	no	—
146	C₂×C₄	—	8	2	no	—
147	C₂×C₄	—	8	1	no	—
148	C₂×C₄	—	8	1	no	—
149	C₂×C₄	—	424	2	no	—
150	C₂×C₄	—	424	2	no	—
151	C₂×C₄	—	424	2	no	—
152	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
153	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
154	C₂×C₄	#13133215	1	1	yes	characteristic
155	C₂×C₄	#13133216	1	2	yes	—
156	C₂×C₄	C₂×#11719922	1	2	yes	—
157	C₂×C₄	C₂×#11719925	1	2	yes	—
158	C₂×C₄	#13133244	1	4	yes	—
159	C₂×C₄	#13133246	1	4	yes	—
160	C₈	—	106	8	no	—
161	C₈	#13133150	1	8	yes	complemented
162	C₂²	#16810492	1	1	yes	characteristic
163	C₂²	#16810999	1	1	yes	characteristic
164	C₂²	#16811002	1	1	yes	characteristic
165	C₂²	#16811009	1	2	yes	—
166	C₂²	#16811014	1	2	yes	—
167	C₄	—	16	1	no	—
168	C₄	—	16	1	no	—
169	C₄	—	848	2	no	—
170	C₄	C₂×#13133150	1	2	yes	—
171	C₄	#16810662	1	4	yes	—
172	C₄	#16811026	1	2	yes	—
173	C₄	#16811030	1	4	yes	—
174	C₂	#20274379	1	1	yes	characteristic
175	C₂	#20274384	1	1	yes	characteristic
176	C₂	#20274389	1	2	yes	—
177	C₂	#20274874	1	1	yes	characteristic
178	C₂	#20274880	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20274875

C₂

yes

maximal, characteristic

#20274874

C₂

yes

maximal

#20274842

C₂

yes

maximal

C₂×#16810344

C₂

yes

maximal, characteristic

C₅₃×#1844

C₂

yes

maximal, characteristic

C₂×#13133151

—

C₂×#13133150

—

C₂×C₄×#88322

C₂²

yes

—

#16810493

C₂²

yes

—

#16810385

—

#16810344

C₄

yes

complemented

C₂×#106×C₅₃

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#384

C₂²

yes

characteristic

C₅₃×#384

C₂²

yes

characteristic

C₂²×#88322

D₄

yes

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

#13133151

—

#13133150

—

C₄×#88322

C₂×C₄

yes

—

C₄×#88322

Q₈

yes

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#106×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×Dic₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

Q₁₆

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×Q₈

yes

characteristic

C₄²×C₅₃

C₂×Q₈

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

C₄⋊C₄

yes

—

C₂³×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

C₄⋊Q₈

yes

—

C₂×C₄×C₅₃

C₂×D₈

yes

—

C₂×C₄×C₅₃

C₂×Q₁₆

yes

—

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

yes

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#497

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×#107

yes

—

C₄×C₅₃

#428

yes

—

C₄×C₅₃

#500

yes

—

C₄×C₅₃

#501

yes

—

Dic₅₃

—

#1845

—

#1845

—

#1844

D₅₃

yes

characteristic

#1834

—

C₂×#322

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#1845

yes

characteristic

C₂×C₅₃

#1847

yes

—

C₂×#107

—

106

—

C₂×#107

—

106

—

C₂×#106

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

100

#384

—

101

#384

C₂×D₅₃

yes

characteristic

102

#384

C₂×D₅₃

yes

characteristic

103

#340

—

106

—

104

#322

—

105

C₅₃

#8820

yes

characteristic, complemented, Sylow

106

C₂²×C₈

—

107

C₂²×C₈

C₂²⋊D₅₃

yes

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

106

—

113

C₂×C₄⋊C₄

—

106

—

114

C₂×C₄⋊C₄

—

106

—

115

C₂×C₄²

C₂²×D₅₃

yes

characteristic

116

#107

—

212

—

117

#107

—

212

—

118

#106

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₄×C₈

Dic₁₀₆

yes

—

122

C₄×C₈

C₄×D₅₃

yes

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

212

—

126

C₂²×C₄

D₄×D₅₃

yes

—

127

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

128

C₂×C₈

—

129

C₂×C₈

—

106

—

130

C₂×C₈

—

106

—

131

C₂×C₈

#11719919

yes

—

132

C₂×C₈

#11719922

yes

—

133

C₂×C₈

#11719925

yes

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

212

—

139

C₄⋊C₄

—

212

—

140

C₄⋊C₄

—

212

—

141

C₄²

C₂×Dic₁₀₆

yes

characteristic

142

C₄²

C₂×C₄×D₅₃

yes

characteristic

143

C₄²

Q₈×D₅₃

yes

—

144

C₂³

#13133213

yes

characteristic

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

424

—

150

C₂×C₄

—

424

—

151

C₂×C₄

—

424

—

152

C₂×C₄

C₄⋊C₄×D₅₃

yes

—

153

C₂×C₄

C₂×#11719919

yes

characteristic

154

C₂×C₄

#13133215

yes

characteristic

155

C₂×C₄

#13133216

yes

—

156

C₂×C₄

C₂×#11719922

yes

—

157

C₂×C₄

C₂×#11719925

yes

—

158

C₂×C₄

#13133244

yes

—

159

C₂×C₄

#13133246

yes

—

160

C₈

—

106

—

161

C₈

#13133150

yes

complemented

162

C₂²

#16810492

yes

characteristic

163

C₂²

#16810999

yes

characteristic

164

C₂²

#16811002

yes

characteristic

165

C₂²

#16811009

yes

—

166

C₂²

#16811014

yes

—

167

C₄

—

168

C₄

—

169

C₄

—

848

—

170

C₄

C₂×#13133150

yes

—

171

C₄

#16810662

yes

—

172

C₄

#16811026

yes

—

173

C₄

#16811030

yes

—

174

C₂

#20274379

yes

characteristic

175

C₂

#20274384

yes

characteristic

176

C₂

#20274389

yes

—

177

C₂

#20274874

yes

characteristic

178

C₂

#20274880

yes

—

	subgroup	G/H
center	C₂³	#13133213
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1844	C₂
socle, prime radical	C₂³×C₅₃	#127
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1844	D₅₃
hypocenter	C₅₃	#8820
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

subgroup

G/H

center

C₂³

#13133213

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1844

C₂

socle, prime radical

C₂³×C₅₃

#127

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

N	H ≅ G/N	type
11. #16810344	C₄	semidirect
70. C₈×C₅₃	#107	semidirect
105. C₅₃	#8820	semidirect
161. C₈	#13133150	semidirect

H ≅ G/N

type

11. #16810344

C₄

semidirect

70. C₈×C₅₃

#107

semidirect

105. C₅₃

#8820

semidirect

161. C₈

#13133150

semidirect

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

subgroup	index
#20274875	2
#20274874	2
#20274842	2
C₂×#16810344	2
C₅₃×#1844	2
#8820	53

subgroup

index

C₅₃×#1844

Subgroups — #27277879

Conjugacy classes of subgroups

Complements

Maximal subgroups