Group #20274727 — subgroups

Subgroups — #20274727

Order 6784.

292 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 109 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×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133140	C₂	1	2	yes	maximal
4	#16810385	C₂	1	2	yes	maximal
5	C₅₃×#382	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	C₄⋊#88322	—	2	2	no	—
14	#13133140	—	2	4	no	—
15	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	#95×C₅₃	—	2	1	no	—
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	1	no	—
27	C₂×Dic₁₀₆	—	2	1	no	—
28	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
29	C₂×Dic₁₀₆	D₄	1	2	yes	—
30	C₂×Dic₁₀₆	Q₈	1	2	yes	—
31	C₄⋊C₄×C₅₃	—	2	2	no	—
32	C₄⋊C₄×C₅₃	—	2	2	no	—
33	C₄²×C₅₃	C₂×C₄	1	2	yes	—
34	C₄²×C₅₃	D₄	1	2	yes	—
35	C₄⋊Dic₅₃	—	2	1	no	—
36	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
37	#11719919	—	2	4	no	—
38	C₄×Dic₅₃	—	2	4	no	—
39	C₂×#88322	—	2	2	no	—
40	C₂×#88322	—	2	2	no	—
41	C₂×#88322	—	2	2	no	—
42	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₅₃	—	2	2	no	—
44	C₂×C₄×C₅₃	—	2	2	no	—
45	C₂×C₄×C₅₃	—	4	2	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	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
55	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
56	C₂×Dic₅₃	—	2	2	no	—
57	C₂×Dic₅₃	—	2	2	no	—
58	C₂×Dic₅₃	—	2	2	no	—
59	C₂×Dic₅₃	—	2	2	no	—
60	C₂×Dic₅₃	—	4	2	no	—
61	Dic₁₀₆	—	2	2	no	—
62	Dic₁₀₆	—	2	2	no	—
63	Dic₁₀₆	—	2	2	no	—
64	Dic₁₀₆	C₄⋊C₄	1	4	yes	—
65	#88322	—	4	2	no	—
66	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₅₃	#121	1	1	yes	characteristic
68	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
69	C₂²×C₅₃	#135	1	2	yes	—
70	C₂²×C₅₃	#137	1	2	yes	—
71	C₄×C₅₃	—	2	2	no	—
72	C₄×C₅₃	—	2	2	no	—
73	C₄×C₅₃	—	8	2	no	—
74	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₄×C₅₃	#117	1	1	yes	characteristic
76	C₄×C₅₃	#121	1	1	yes	characteristic
77	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
78	Dic₅₃	—	2	4	no	—
79	Dic₅₃	—	4	2	no	—
80	#1805	—	53	1	no	maximal, Sylow
81	C₂×C₅₃	#380	1	1	yes	characteristic
82	C₂×C₅₃	#416	1	1	yes	characteristic
83	C₂×C₅₃	#419	1	1	yes	characteristic
84	C₂×C₅₃	#464	1	2	yes	—
85	C₂×C₅₃	#479	1	2	yes	—
86	C₂×C₄×Q₈	—	53	1	no	—
87	C₂×C₄⋊C₈	—	53	1	no	—
88	C₂×#103	—	53	2	no	—
89	#382	D₅₃	1	1	yes	characteristic
90	#340	—	53	2	no	—
91	C₅₃	#1805	1	1	yes	characteristic, complemented, Sylow
92	C₂²×Q₈	—	53	1	no	—
93	C₂²×C₈	—	53	2	no	—
94	C₄×Q₈	—	53	4	no	—
95	C₄×Q₈	—	106	2	no	—
96	C₂×C₄⋊C₄	—	53	1	no	—
97	C₂×C₄⋊C₄	—	106	1	no	—
98	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
99	C₂×C₄²	—	106	1	no	—
100	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
101	C₄⋊C₈	—	106	2	no	—
102	#103	—	106	4	no	—
103	#95	—	2	1	no	—
104	#95	—	2	1	no	—
105	C₂×Q₈	—	53	2	no	—
106	C₂×Q₈	—	53	2	no	—
107	C₂×Q₈	—	53	2	no	—
108	C₂×Q₈	—	106	1	no	—
109	C₂×Q₈	—	106	1	no	—
110	C₂×Q₈	—	106	1	no	—
111	C₂²×C₄	—	2	2	no	—
112	C₂²×C₄	—	106	1	no	—
113	C₂²×C₄	—	106	1	no	—
114	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
115	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
116	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
117	C₂×C₈	—	106	2	no	—
118	C₂×C₈	—	106	2	no	—
119	C₂×C₈	—	106	2	no	—
120	C₄⋊C₄	—	2	2	no	—
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	53	2	no	—
123	C₄⋊C₄	—	106	1	no	—
124	C₄⋊C₄	—	106	4	no	—
125	C₄²	—	106	4	no	—
126	C₄²	C₄×D₅₃	1	2	yes	—
127	C₄²	D₂₁₂	1	2	yes	—
128	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
129	Q₈	—	53	4	no	—
130	Q₈	—	106	2	no	—
131	Q₈	—	106	2	no	—
132	Q₈	—	106	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	4	2	no	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	—	106	2	no	—
139	C₂×C₄	—	106	2	no	—
140	C₂×C₄	—	106	2	no	—
141	C₂×C₄	—	106	2	no	—
142	C₂×C₄	—	212	2	no	—
143	C₂×C₄	#11719921	1	2	yes	—
144	C₂×C₄	#11719921	1	2	yes	—
145	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
146	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
147	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
148	C₂×C₄	#11719947	1	1	yes	characteristic
149	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
150	C₂×C₄	#11719949	1	1	yes	characteristic
151	C₈	—	212	2	no	—
152	C₂²	C₂×#11719921	1	1	yes	characteristic
153	C₂²	#13133188	1	1	yes	characteristic
154	C₂²	#13133193	1	1	yes	characteristic
155	C₂²	#13133237	1	2	yes	—
156	C₂²	#13133247	1	2	yes	—
157	C₄	—	2	2	no	—
158	C₄	—	2	2	no	—
159	C₄	—	8	2	no	—
160	C₄	—	106	4	no	—
161	C₄	—	212	2	no	—
162	C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
163	C₄	#13133175	1	1	yes	characteristic
164	C₄	#13133192	1	1	yes	characteristic
165	C₄	#13133197	1	1	yes	characteristic
166	C₂	#16810480	1	1	yes	characteristic
167	C₂	#16810617	1	1	yes	characteristic
168	C₂	#16810631	1	1	yes	characteristic
169	C₂	#16810842	1	2	yes	—
170	C₂	#16810927	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₀₆

C₂

yes

maximal, characteristic

C₂×C₄⋊#88322

C₂

yes

maximal, characteristic

C₂×#13133140

C₂

yes

maximal

#16810385

C₂

yes

maximal

C₅₃×#382

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

—

C₄⋊#88322

—

#13133140

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#95×C₅₃

—

#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×Dic₁₀₆

C₂×C₄

yes

—

C₂×Dic₁₀₆

D₄

yes

—

C₂×Dic₁₀₆

Q₈

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×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

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

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

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

C₄⋊C₄

yes

—

#88322

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

#121

yes

characteristic

C₂²×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₅₃

#135

yes

—

C₂²×C₅₃

#137

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄⋊C₄

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#479

yes

—

C₂×C₄×Q₈

—

C₂×C₄⋊C₈

—

C₂×#103

—

#382

D₅₃

yes

characteristic

#340

—

C₅₃

#1805

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

—

106

—

100

C₂×C₄²

C₂×D₅₃

yes

characteristic

101

C₄⋊C₈

—

106

—

102

#103

—

106

—

103

#95

—

104

#95

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

106

—

109

C₂×Q₈

—

106

—

110

C₂×Q₈

—

106

—

111

C₂²×C₄

—

112

C₂²×C₄

—

106

—

113

C₂²×C₄

—

106

—

114

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

115

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

116

C₂²×C₄

C₂²×D₅₃

yes

characteristic

117

C₂×C₈

—

106

—

118

C₂×C₈

—

106

—

119

C₂×C₈

—

106

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

106

—

124

C₄⋊C₄

—

106

—

125

C₄²

—

106

—

126

C₄²

C₄×D₅₃

yes

—

127

C₄²

D₂₁₂

yes

—

128

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

129

Q₈

—

130

Q₈

—

106

—

131

Q₈

—

106

—

132

Q₈

—

106

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

106

—

139

C₂×C₄

—

106

—

140

C₂×C₄

—

106

—

141

C₂×C₄

—

106

—

142

C₂×C₄

—

212

—

143

C₂×C₄

#11719921

yes

—

144

C₂×C₄

#11719921

yes

—

145

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

146

C₂×C₄

C₂×D₂₁₂

yes

characteristic

147

C₂×C₄

D₄×D₅₃

yes

characteristic

148

C₂×C₄

#11719947

yes

characteristic

149

C₂×C₄

Q₈×D₅₃

yes

characteristic

150

C₂×C₄

#11719949

yes

characteristic

151

C₈

—

212

—

152

C₂²

C₂×#11719921

yes

characteristic

153

C₂²

#13133188

yes

characteristic

154

C₂²

#13133193

yes

characteristic

155

C₂²

#13133237

yes

—

156

C₂²

#13133247

yes

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

106

—

161

C₄

—

212

—

162

C₄

C₄⋊C₄×D₅₃

yes

characteristic

163

C₄

#13133175

yes

characteristic

164

C₄

#13133192

yes

characteristic

165

C₄

#13133197

yes

characteristic

166

C₂

#16810480

yes

characteristic

167

C₂

#16810617

yes

characteristic

168

C₂

#16810631

yes

characteristic

169

C₂

#16810842

yes

—

170

C₂

#16810927

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#382	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	#382	D₅₃
hypocenter	C₅₃	#1805
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#382

C₂

socle

C₂³×C₅₃

C₂×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

N	H ≅ G/N	type
9. C₄×Dic₁₀₆	C₄	semidirect
91. C₅₃	#1805	semidirect

H ≅ G/N

type

9. C₄×Dic₁₀₆

C₄

semidirect

91. C₅₃

#1805

semidirect

Subgroups — #20274727

Conjugacy classes of subgroups

Complements