Group #20274548 — subgroups

Subgroups — #20274548

Order 6784.

324 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 129 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×C₄×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133140	C₂	1	2	yes	maximal
3	C₂×#13133128	C₂	1	1	yes	maximal, characteristic
4	#16810385	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₀₆	C₂²	1	1	yes	characteristic
7	C₂²×#88322	C₂²	1	2	yes	—
8	C₄×Dic₁₀₆	—	2	2	no	—
9	C₄×Dic₁₀₆	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂×#11719919	—	2	1	no	—
12	C₂×C₄×Dic₅₃	—	2	1	no	—
13	#13133140	C₄	1	8	yes	complemented
14	#13133128	—	2	2	no	—
15	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄²×C₅₃	—	2	1	no	—
17	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
18	#95×C₅₃	—	2	1	no	—
19	C₂²×C₄×C₅₃	—	2	2	no	—
20	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
23	C₂²×Dic₅₃	—	2	1	no	—
24	C₂²×Dic₅₃	—	2	1	no	—
25	C₂×Dic₁₀₆	—	2	1	no	—
26	C₂×Dic₁₀₆	—	2	2	no	—
27	C₂×Dic₁₀₆	C₂×C₄	1	4	yes	—
28	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
29	C₄⋊C₄×C₅₃	—	2	2	no	—
30	C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
31	C₄²×C₅₃	—	4	2	no	—
32	C₄²×C₅₃	C₂×C₄	1	2	yes	—
33	C₄²×C₅₃	D₄	1	2	yes	—
34	C₄⋊Dic₅₃	—	2	1	no	—
35	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
36	#11719919	—	2	4	no	—
37	C₄×Dic₅₃	—	2	4	no	—
38	C₂×#88322	—	2	2	no	—
39	C₂×#88322	—	2	2	no	—
40	C₂×#88322	C₂×C₄	1	4	yes	—
41	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₅₃	—	2	1	no	—
43	C₂×C₄×C₅₃	—	2	2	no	—
44	C₂×C₄×C₅₃	—	2	1	no	—
45	C₂×C₄×C₅₃	—	2	4	no	—
46	C₂×C₄×C₅₃	—	4	2	no	—
47	C₂×C₄×C₅₃	—	4	2	no	—
48	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
49	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
50	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
52	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₅₃	#41	1	2	yes	—
54	C₂×Dic₅₃	—	2	2	no	—
55	C₂×Dic₅₃	—	2	4	no	—
56	C₂×Dic₅₃	—	2	2	no	—
57	C₂×Dic₅₃	—	4	1	no	—
58	C₂×Dic₅₃	—	4	1	no	—
59	Dic₁₀₆	—	2	2	no	—
60	Dic₁₀₆	—	2	4	no	—
61	Dic₁₀₆	C₄²	1	4	yes	complemented
62	#88322	—	2	4	no	—
63	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₅₃	C₄×D₄	1	2	yes	—
65	C₂²×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₂²×C₅₃	#137	1	2	yes	—
67	C₂²×C₅₃	#137	1	1	yes	characteristic
68	C₄×C₅₃	—	2	2	no	—
69	C₄×C₅₃	—	2	2	no	—
70	C₄×C₅₃	—	4	4	no	—
71	C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
72	C₄×C₅₃	#117	1	1	yes	characteristic
73	C₄×C₅₃	C₄×D₄	1	2	yes	—
74	Dic₅₃	—	2	4	no	—
75	Dic₅₃	—	4	2	no	—
76	#1763	—	53	1	no	maximal, Sylow
77	C₂×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₅₃	#417	1	1	yes	characteristic
79	C₂×C₅₃	#419	1	1	yes	characteristic
80	C₂×C₅₃	#440	1	2	yes	—
81	C₂×C₅₃	#441	1	2	yes	—
82	C₂×C₄×Q₈	—	53	1	no	—
83	C₂×#103	—	53	2	no	—
84	C₂×#97	—	53	1	no	—
85	C₄×C₄⋊C₄	D₅₃	1	1	yes	characteristic
86	#340	—	53	2	no	—
87	C₅₃	#1763	1	1	yes	characteristic, complemented, Sylow
88	C₂²×Q₈	—	53	1	no	—
89	C₂²×C₈	—	53	2	no	—
90	C₄×Q₈	—	53	4	no	—
91	C₄×Q₈	—	106	2	no	—
92	C₂×C₄⋊C₄	—	53	1	no	—
93	C₂×C₄⋊C₄	—	106	1	no	—
94	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
95	C₂×C₄²	—	2	1	no	—
96	C₂×C₄²	—	106	1	no	—
97	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
98	#103	—	53	8	no	—
99	#97	—	106	2	no	—
100	#95	—	2	1	no	—
101	C₂×Q₈	—	53	4	no	—
102	C₂×Q₈	—	53	2	no	—
103	C₂×Q₈	—	106	1	no	—
104	C₂×Q₈	—	106	2	no	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	—	106	1	no	—
107	C₂²×C₄	—	106	1	no	—
108	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
109	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
110	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
111	C₂×C₈	—	53	4	no	—
112	C₂×C₈	—	106	2	no	—
113	C₂×C₈	—	106	2	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	53	2	no	—
116	C₄⋊C₄	—	106	1	no	—
117	C₄⋊C₄	—	106	4	no	—
118	C₄⋊C₄	C₄×D₅₃	1	4	yes	—
119	C₄²	—	4	2	no	—
120	C₄²	—	106	4	no	—
121	C₄²	C₄×D₅₃	1	2	yes	—
122	C₄²	D₂₁₂	1	2	yes	—
123	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
124	Q₈	—	53	4	no	—
125	Q₈	—	106	2	no	—
126	Q₈	—	106	4	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	4	2	no	—
133	C₂×C₄	—	106	2	no	—
134	C₂×C₄	—	106	4	no	—
135	C₂×C₄	—	106	2	no	—
136	C₂×C₄	—	212	1	no	—
137	C₂×C₄	—	212	1	no	—
138	C₂×C₄	#11719921	1	2	yes	—
139	C₂×C₄	#11719921	1	2	yes	—
140	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
142	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
143	C₂×C₄	#11719945	1	2	yes	—
144	C₈	—	106	4	no	—
145	C₂²	C₂×#11719921	1	1	yes	characteristic
146	C₂²	C₄×C₂²⋊D₅₃	1	2	yes	—
147	C₂²	#13133243	1	1	yes	characteristic
148	C₂²	#13133247	1	2	yes	—
149	C₂²	#13133248	1	1	yes	characteristic
150	C₄	—	2	2	no	—
151	C₄	—	2	2	no	—
152	C₄	—	4	4	no	—
153	C₄	—	106	4	no	—
154	C₄	—	212	2	no	—
155	C₄	C₄²×D₅₃	1	1	yes	characteristic
156	C₄	#13133174	1	1	yes	characteristic
157	C₄	C₄×D₂₁₂	1	2	yes	—
158	C₂	C₄×#11719921	1	1	yes	characteristic
159	C₂	#16810622	1	1	yes	characteristic
160	C₂	#16810631	1	1	yes	characteristic
161	C₂	#16810723	1	2	yes	—
162	C₂	#16810728	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₀₆

C₂

yes

maximal, characteristic

C₂×#13133140

C₂

yes

maximal

C₂×#13133128

C₂

yes

maximal, characteristic

#16810385

C₂

yes

maximal

C₄×C₄⋊C₄×C₅₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₂²

yes

characteristic

C₂²×#88322

C₂²

yes

—

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

C₄

yes

complemented

C₂×C₄⋊Dic₅₃

C₂²

yes

characteristic

C₂×#11719919

—

C₂×C₄×Dic₅₃

—

#13133140

C₄

yes

complemented

#13133128

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#95×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂×C₄

yes

—

C₂×Dic₁₀₆

C₂×C₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

D₄

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

#11719919

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

C₂×C₄

yes

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

C₄²

yes

complemented

#88322

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

C₄×D₄

yes

—

C₂²×C₅₃

AGL(1,ℤ/8)

yes

characteristic

C₂²×C₅₃

#137

yes

—

C₂²×C₅₃

#137

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄²

yes

characteristic

C₄×C₅₃

#117

yes

characteristic

C₄×C₅₃

C₄×D₄

yes

—

Dic₅₃

—

Dic₅₃

—

#1763

—

maximal, Sylow

C₂×C₅₃

C₄×C₂²⋊C₄

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#441

yes

—

C₂×C₄×Q₈

—

C₂×#103

—

C₂×#97

—

C₄×C₄⋊C₄

D₅₃

yes

characteristic

#340

—

C₅₃

#1763

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

—

106

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

106

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#103

—

#97

—

106

—

100

#95

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

106

—

104

C₂×Q₈

—

106

—

105

C₂²×C₄

—

106

C₂²×C₄

—

106

—

107

C₂²×C₄

—

106

—

108

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

109

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

110

C₂²×C₄

C₂²×D₅₃

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

106

—

113

C₂×C₈

—

106

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

106

—

117

C₄⋊C₄

—

106

—

118

C₄⋊C₄

C₄×D₅₃

yes

—

119

C₄²

—

120

C₄²

—

106

—

121

C₄²

C₄×D₅₃

yes

—

122

C₄²

D₂₁₂

yes

—

123

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

124

Q₈

—

125

Q₈

—

106

—

126

Q₈

—

106

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

106

—

134

C₂×C₄

—

106

—

135

C₂×C₄

—

106

—

136

C₂×C₄

—

212

—

137

C₂×C₄

—

212

—

138

C₂×C₄

#11719921

yes

—

139

C₂×C₄

#11719921

yes

—

140

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

141

C₂×C₄

C₂×C₄×D₅₃

yes

—

142

C₂×C₄

C₂×D₂₁₂

yes

characteristic

143

C₂×C₄

#11719945

yes

—

144

C₈

—

106

—

145

C₂²

C₂×#11719921

yes

characteristic

146

C₂²

C₄×C₂²⋊D₅₃

yes

—

147

C₂²

#13133243

yes

characteristic

148

C₂²

#13133247

yes

—

149

C₂²

#13133248

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

106

—

154

C₄

—

212

—

155

C₄

C₄²×D₅₃

yes

characteristic

156

C₄

#13133174

yes

characteristic

157

C₄

C₄×D₂₁₂

yes

—

158

C₂

C₄×#11719921

yes

characteristic

159

C₂

#16810622

yes

characteristic

160

C₂

#16810631

yes

characteristic

161

C₂

#16810723

yes

—

162

C₂

#16810728

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₄×C₄⋊C₄×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₄×C₄⋊C₄	D₅₃
hypocenter	C₅₃	#1763
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₃

commutator

C₄×C₅₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₄×C₄⋊C₄×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

hypercenter

C₄×C₄⋊C₄

D₅₃

hypocenter

C₅₃

#1763

LCS

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
9. C₄×Dic₁₀₆	C₄	semidirect
13. #13133140	C₄	semidirect
61. Dic₁₀₆	C₄²	semidirect
87. C₅₃	#1763	semidirect

H ≅ G/N

type

9. C₄×Dic₁₀₆

C₄

semidirect

13. #13133140

C₄

semidirect

61. Dic₁₀₆

C₄²

semidirect

87. C₅₃

#1763

semidirect

Subgroups — #20274548

Conjugacy classes of subgroups

Complements