Group #27288904 — subgroups

Subgroups — #27288904

Order 13568.

294 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 85 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	#20275535	C₂	1	1	yes	maximal, characteristic
2	#20274856	C₂	1	1	yes	maximal, characteristic
3	#20273386	C₂	1	2	yes	maximal
4	#20272574	C₂	1	2	yes	maximal
5	C₅₃×#1281	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133217	C₂²	1	1	yes	characteristic
7	#16810647	C₂²	1	2	yes	—
8	C₂×#13133142	—	2	2	no	—
9	#16810489	C₂²	1	1	yes	characteristic
10	#16810388	—	2	2	no	—
11	#16810368	—	2	4	no	—
12	C₅₃×#431	C₂²	1	2	yes	—
13	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₀₆	—	2	1	no	—
15	C₂×#11719917	—	2	2	no	—
16	#13133217	—	2	1	no	—
17	#13133217	—	2	2	no	—
18	#13133217	D₄	1	2	yes	—
19	C₂×C₄⋊Dic₅₃	—	2	1	no	—
20	C₂×C₄⋊Dic₅₃	—	2	2	no	—
21	C₄⋊#88322	—	2	2	no	—
22	C₄⋊#88322	—	2	2	no	—
23	#13133142	—	4	4	no	—
24	#13133124	—	2	2	no	—
25	C₂²×C₈×C₅₃	D₄	1	2	yes	—
26	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
27	C₄⋊C₈×C₅₃	—	2	2	no	—
28	C₂²⋊C₈×C₅₃	—	2	2	no	—
29	#97×C₅₃	—	2	2	no	—
30	C₄×C₈×C₅₃	—	2	2	no	—
31	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	#41	1	2	yes	—
33	C₂²×Dic₅₃	—	4	1	no	—
34	C₂²×Dic₅₃	—	4	1	no	—
35	C₂×Dic₁₀₆	—	2	2	no	—
36	C₂×Dic₁₀₆	—	4	2	no	—
37	C₂×Dic₁₀₆	—	4	2	no	—
38	C₂×Dic₁₀₆	—	4	1	no	—
39	C₂×C₈×C₅₃	—	2	2	no	—
40	C₂×C₈×C₅₃	—	2	2	no	—
41	C₂×C₈×C₅₃	—	2	2	no	—
42	C₂×C₈×C₅₃	—	4	2	no	—
43	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₄²×C₅₃	#41	1	1	yes	characteristic
45	C₄²×C₅₃	#41	1	2	yes	—
46	C₄⋊Dic₅₃	—	2	4	no	—
47	C₄⋊Dic₅₃	—	4	2	no	—
48	C₄⋊Dic₅₃	—	4	2	no	—
49	#11719917	—	4	2	no	—
50	#11719917	—	4	2	no	—
51	C₂×#88322	—	4	2	no	—
52	C₂³×C₅₃	#124	1	1	yes	characteristic
53	C₂×C₄×C₅₃	—	4	2	no	—
54	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
55	C₂×C₄×C₅₃	#123	1	2	yes	—
56	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
57	C₂×C₄×C₅₃	#126	1	2	yes	—
58	C₂×C₄×C₅₃	#135	1	2	yes	—
59	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
60	C₂×Dic₅₃	—	4	2	no	—
61	C₂×Dic₅₃	—	8	1	no	—
62	C₂×Dic₅₃	—	8	1	no	—
63	C₂×Dic₅₃	—	8	2	no	—
64	C₂×Dic₅₃	—	8	1	no	—
65	Dic₁₀₆	—	4	4	no	—
66	Dic₁₀₆	—	8	2	no	—
67	C₈×C₅₃	—	4	2	no	—
68	C₈×C₅₃	—	8	2	no	—
69	#88322	—	8	2	no	—
70	#10635	—	53	1	no	maximal, Sylow
71	C₂²×C₅₃	—	2	2	no	—
72	C₂²×C₅₃	#397	1	1	yes	characteristic
73	C₂²×C₅₃	#488	1	1	yes	characteristic
74	C₂²×C₅₃	#490	1	1	yes	characteristic
75	C₄×C₅₃	—	2	2	no	—
76	C₄×C₅₃	—	2	2	no	—
77	C₄×C₅₃	#463	1	2	yes	—
78	C₄×C₅₃	#482	1	2	yes	—
79	Dic₅₃	—	8	2	no	—
80	Dic₅₃	—	16	1	no	—
81	#1960	—	53	1	no	—
82	#1838	—	53	1	no	—
83	#1546	—	53	2	no	—
84	#1286	—	53	2	no	—
85	#1281	D₅₃	1	1	yes	characteristic
86	C₂×C₅₃	—	2	1	no	—
87	C₂×C₅₃	—	2	1	no	—
88	C₂×C₅₃	#2073	1	1	yes	characteristic
89	C₂×C₅₃	#2082	1	2	yes	—
90	C₂×C₄⋊Q₈	—	53	1	no	—
91	#431	C₂×D₅₃	1	2	yes	—
92	#423	—	53	2	no	—
93	C₂×#103	—	106	2	no	—
94	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
95	#382	—	53	1	no	—
96	#341	—	106	2	no	—
97	#332	—	106	4	no	—
98	C₅₃	#10635	1	1	yes	characteristic, complemented, Sylow
99	C₂²×Q₈	—	106	1	no	—
100	C₂×M₁₆	—	106	2	no	—
101	C₂²×C₈	D₂₁₂	1	2	yes	—
102	C₄⋊Q₈	—	53	2	no	—
103	C₄⋊Q₈	—	106	2	no	—
104	C₄⋊Q₈	—	106	1	no	—
105	C₂×C₄⋊C₄	—	106	2	no	—
106	C₂×C₄⋊C₄	—	106	1	no	—
107	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
108	C₄⋊C₈	—	2	2	no	—
109	C₄⋊C₈	—	106	2	no	—
110	C₄⋊C₈	—	106	2	no	—
111	#103	—	212	4	no	—
112	C₂²⋊C₈	—	2	2	no	—
113	#97	—	2	2	no	—
114	C₄×C₈	—	2	2	no	—
115	#95	—	106	2	no	—
116	C₂×Q₈	—	106	2	no	—
117	C₂×Q₈	—	212	2	no	—
118	C₂×Q₈	—	212	2	no	—
119	C₂×Q₈	—	212	1	no	—
120	C₂²×C₄	—	212	1	no	—
121	C₂²×C₄	—	212	1	no	—
122	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
123	C₂²×C₄	#11719945	1	2	yes	—
124	M₁₆	—	212	2	no	—
125	M₁₆	—	212	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	4	2	no	—
130	C₂×C₈	—	212	2	no	—
131	C₄⋊C₄	—	106	4	no	—
132	C₄⋊C₄	—	212	2	no	—
133	C₄⋊C₄	—	212	2	no	—
134	C₄²	D₄×D₅₃	1	1	yes	characteristic
135	C₄²	#11719947	1	2	yes	—
136	C₄²	#11719949	1	1	yes	characteristic
137	C₂³	#13133205	1	1	yes	characteristic
138	Q₈	—	424	2	no	—
139	Q₈	—	212	4	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	424	2	no	—
142	C₂×C₄	—	424	1	no	—
143	C₂×C₄	—	424	1	no	—
144	C₂×C₄	—	424	1	no	—
145	C₂×C₄	—	212	2	no	—
146	C₂×C₄	#13133192	1	1	yes	characteristic
147	C₂×C₄	#13133199	1	2	yes	—
148	C₂×C₄	#13133201	1	1	yes	characteristic
149	C₂×C₄	#13133212	1	2	yes	—
150	C₂×C₄	#13133239	1	2	yes	—
151	C₂×C₄	#13133245	1	2	yes	—
152	C₈	—	4	2	no	—
153	C₈	—	8	2	no	—
154	C₈	—	424	2	no	—
155	C₂²	—	2	2	no	—
156	C₂²	#16810550	1	1	yes	characteristic
157	C₂²	#16810978	1	1	yes	characteristic
158	C₂²	#16810988	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	2	2	no	—
161	C₄	—	848	1	no	—
162	C₄	—	424	2	no	—
163	C₄	#16810839	1	2	yes	—
164	C₄	#16810948	1	2	yes	—
165	C₂	—	2	1	no	—
166	C₂	—	2	1	no	—
167	C₂	#20276130	1	1	yes	characteristic
168	C₂	#20276181	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20275535

C₂

yes

maximal, characteristic

#20274856

C₂

yes

maximal, characteristic

#20273386

C₂

yes

maximal

#20272574

C₂

yes

maximal

C₅₃×#1281

C₂

yes

maximal, characteristic

C₂×#13133217

C₂²

yes

characteristic

#16810647

C₂²

yes

—

C₂×#13133142

—

#16810489

C₂²

yes

characteristic

#16810388

—

#16810368

—

C₅₃×#431

C₂²

yes

—

C₂×#97×C₅₃

C₂²

yes

characteristic

C₂²×Dic₁₀₆

—

—

—

—

D₄

yes

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

—

—

—

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

#97×C₅₃

—

C₄×C₈×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

C₄²×C₅₃

yes

C₄²×C₅₃

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719917

—

#11719917

—

C₂×#88322

—

C₂³×C₅₃

#124

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#121

yes

characteristic

C₂×C₄×C₅₃

#123

yes

—

C₂×C₄×C₅₃

#123

yes

characteristic

C₂×C₄×C₅₃

#126

yes

—

C₂×C₄×C₅₃

#135

yes

—

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#10635

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#463

yes

—

C₄×C₅₃

#482

yes

—

Dic₅₃

—

Dic₅₃

—

—

—

—

—

D₅₃

yes

characteristic

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#2073

yes

characteristic

C₂×C₅₃

#2082

yes

—

C₂×C₄⋊Q₈

—

#431

C₂×D₅₃

yes

—

#423

—

C₂×#103

—

106

—

C₂×#97

C₂×D₅₃

yes

characteristic

#382

—

#341

—

106

—

#332

—

106

—

C₅₃

#10635

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

106

—

100

C₂×M₁₆

—

106

—

101

C₂²×C₈

D₂₁₂

yes

—

102

C₄⋊Q₈

—

103

C₄⋊Q₈

—

106

—

104

C₄⋊Q₈

—

106

—

105

C₂×C₄⋊C₄

—

106

—

106

C₂×C₄⋊C₄

—

106

—

107

C₂×C₄²

C₂²×D₅₃

yes

characteristic

108

C₄⋊C₈

—

109

C₄⋊C₈

—

106

—

110

C₄⋊C₈

—

106

—

111

#103

—

212

—

112

C₂²⋊C₈

—

113

#97

—

114

C₄×C₈

—

115

#95

—

106

—

116

C₂×Q₈

—

106

—

117

C₂×Q₈

—

212

—

118

C₂×Q₈

—

212

—

119

C₂×Q₈

—

212

—

120

C₂²×C₄

—

212

—

121

C₂²×C₄

—

212

—

122

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

123

C₂²×C₄

#11719945

yes

—

124

M₁₆

—

212

—

125

M₁₆

—

212

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

212

—

131

C₄⋊C₄

—

106

—

132

C₄⋊C₄

—

212

—

133

C₄⋊C₄

—

212

—

134

C₄²

D₄×D₅₃

yes

characteristic

135

C₄²

#11719947

yes

—

136

C₄²

#11719949

yes

characteristic

137

C₂³

#13133205

yes

characteristic

138

Q₈

—

424

—

139

Q₈

—

212

—

140

C₂×C₄

—

141

C₂×C₄

—

424

—

142

C₂×C₄

—

424

—

143

C₂×C₄

—

424

—

144

C₂×C₄

—

424

—

145

C₂×C₄

—

212

—

146

C₂×C₄

#13133192

yes

characteristic

147

C₂×C₄

#13133199

yes

—

148

C₂×C₄

#13133201

yes

characteristic

149

C₂×C₄

#13133212

yes

—

150

C₂×C₄

#13133239

yes

—

151

C₂×C₄

#13133245

yes

—

152

C₈

—

153

C₈

—

154

C₈

—

424

—

155

C₂²

—

156

C₂²

#16810550

yes

characteristic

157

C₂²

#16810978

yes

characteristic

158

C₂²

#16810988

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

848

—

162

C₄

—

424

—

163

C₄

#16810839

yes

—

164

C₄

#16810948

yes

—

165

C₂

—

166

C₂

—

167

C₂

#20276130

yes

characteristic

168

C₂

#20276181

yes

—

	subgroup	G/H
center	C₂²	#16810550
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1281	C₂
socle	C₂²×C₅₃	#397
prime radical	C₂³×C₅₃	#124
hypercenter	#1281	D₅₃
hypocenter	C₅₃	#10635
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1281

subgroup

G/H

center

C₂²

#16810550

commutator, cosocle

C₂×C₄²×C₅₃

C₂³

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1281

C₂

C₂²×C₅₃

C₂³×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1281

N	H ≅ G/N	type
98. C₅₃	#10635	semidirect

H ≅ G/N

type

98. C₅₃

#10635

semidirect

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

subgroup	index
#20275535	2
#20274856	2
#20273386	2
#20272574	2
C₅₃×#1281	2
#10635	53

subgroup

index

C₅₃×#1281

Subgroups — #27288904

Conjugacy classes of subgroups

Complements

Maximal subgroups