Group #20274762 — subgroups

Subgroups — #20274762

Order 6784.

252 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 89 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×#13133223	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133153	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810605	C₂	1	2	yes	maximal
4	#16810381	C₂	1	2	yes	maximal, complemented
5	C₂×#108×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11719945	C₂²	1	1	yes	characteristic
7	#13133223	—	2	2	no	—
8	#13133223	—	2	2	no	—
9	C₂×#11719917	C₂²	1	2	yes	—
10	C₂×#11719917	C₂²	1	1	yes	characteristic
11	C₂×#11719912	C₄	1	2	yes	—
12	C₂×C₈×D₅₃	C₄	1	2	yes	—
13	#13133153	—	2	2	no	—
14	#13133135	—	2	2	no	—
15	#13133133	—	2	2	no	—
16	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
17	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	#108×C₅₃	—	2	2	no	—
19	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
20	C₂×C₂²⋊D₅₃	Q₈	1	2	yes	—
21	#11719945	—	2	1	no	—
22	#11719945	—	2	1	no	—
23	#11719945	—	2	1	no	—
24	#11719945	—	2	1	no	—
25	C₂×D₂₁₂	D₄	1	1	yes	characteristic
26	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
27	C₂×Dic₁₀₆	D₄	1	1	yes	characteristic
28	M₁₆×C₅₃	—	2	2	no	—
29	M₁₆×C₅₃	—	2	2	no	—
30	C₂×C₈×C₅₃	—	2	2	no	—
31	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×C₅₃	D₄	1	2	yes	—
34	C₂×C₈×C₅₃	D₄	1	2	yes	—
35	#11719917	—	2	1	no	—
36	#11719917	—	2	1	no	—
37	#11719917	—	2	2	no	—
38	#11719917	—	2	2	no	—
39	C₂×#88322	—	2	2	no	—
40	C₂×#88322	C₂×C₄	1	2	yes	—
41	#11719912	—	2	4	no	—
42	C₈×D₅₃	—	2	4	no	—
43	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
44	C₂²×D₅₃	C₄⋊C₄	1	2	yes	—
45	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
50	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
51	C₂²⋊D₅₃	—	2	2	no	—
52	C₂²⋊D₅₃	—	2	2	no	—
53	C₂×Dic₅₃	C₄⋊C₄	1	2	yes	—
54	D₂₁₂	—	2	1	no	—
55	D₂₁₂	—	2	1	no	—
56	C₄×D₅₃	—	2	2	no	—
57	C₄×D₅₃	—	2	2	no	—
58	Dic₁₀₆	—	2	1	no	—
59	Dic₁₀₆	—	2	1	no	—
60	C₈×C₅₃	—	4	2	no	—
61	C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
62	C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
63	#88322	—	2	2	no	—
64	#88322	—	4	2	no	—
65	C₂²×C₅₃	—	4	1	no	—
66	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
68	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
69	C₂×D₅₃	—	2	2	no	—
70	C₂×D₅₃	—	4	2	no	—
71	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₄×C₅₃	#117	1	1	yes	characteristic
73	C₄×C₅₃	#121	1	1	yes	characteristic
74	C₄×C₅₃	#121	1	1	yes	characteristic
75	Dic₅₃	—	2	2	no	—
76	#1813	—	53	1	no	maximal, Sylow
77	C₂×C₅₃	—	2	1	no	—
78	C₂×C₅₃	—	2	1	no	—
79	C₂×C₅₃	—	2	1	no	—
80	C₂×C₅₃	#380	1	1	yes	characteristic
81	D₅₃	—	4	2	no	—
82	C₂×#131	—	53	1	no	—
83	C₂×#108	—	53	1	no	—
84	C₂×#108	D₅₃	1	1	yes	characteristic, complemented
85	#413	—	53	2	no	—
86	#338	—	53	2	no	—
87	C₅₃	#1813	1	1	yes	characteristic, complemented, Sylow
88	C₂×#41	—	53	1	no	—
89	#131	—	106	2	no	—
90	#131	—	106	2	no	—
91	C₂×M₁₆	—	53	1	no	—
92	C₂×M₁₆	—	53	2	no	—
93	C₂×M₁₆	—	53	2	no	—
94	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
95	C₂²×C₈	—	53	2	no	—
96	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
97	#108	—	2	2	no	—
98	#108	—	106	2	no	—
99	#101	—	106	2	no	—
100	#100	—	106	2	no	—
101	#41	—	106	1	no	—
102	#41	—	106	1	no	—
103	#41	—	106	1	no	—
104	#41	—	106	1	no	—
105	C₂×Q₈	—	53	1	no	—
106	C₂×D₄	—	53	2	no	—
107	C₂×D₄	—	53	1	no	—
108	C₂²×C₄	—	53	2	no	—
109	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
110	M₁₆	—	2	2	no	—
111	M₁₆	—	2	2	no	—
112	M₁₆	—	106	1	no	—
113	M₁₆	—	106	1	no	—
114	M₁₆	—	106	4	no	—
115	M₁₆	—	106	2	no	—
116	M₁₆	—	106	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	53	2	no	—
119	C₂×C₈	—	106	4	no	—
120	C₂×C₈	—	106	2	no	—
121	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
122	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
123	C₂×C₈	D₂₁₂	1	2	yes	—
124	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
125	C₂³	—	53	2	no	—
126	C₂³	Q₈×D₅₃	1	1	yes	characteristic
127	Q₈	—	106	1	no	—
128	Q₈	—	106	1	no	—
129	D₄	—	106	2	no	—
130	D₄	—	106	2	no	—
131	D₄	—	106	1	no	—
132	D₄	—	106	1	no	—
133	C₂×C₄	—	53	2	no	—
134	C₂×C₄	—	106	2	no	—
135	C₂×C₄	—	106	2	no	—
136	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
137	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
138	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
139	C₂×C₄	#11719947	1	1	yes	characteristic
140	C₂×C₄	#11719949	1	1	yes	characteristic
141	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
142	C₈	—	4	2	no	—
143	C₈	—	106	2	no	—
144	C₈	—	212	2	no	—
145	C₈	#11719921	1	2	yes	—
146	C₈	#11719921	1	2	yes	—
147	C₂²	—	4	1	no	—
148	C₂²	—	106	2	no	—
149	C₂²	—	212	2	no	—
150	C₂²	C₄⋊C₄×D₅₃	1	1	yes	characteristic
151	C₂²	#13133193	1	1	yes	characteristic
152	C₂²	#13133197	1	1	yes	characteristic
153	C₄	—	106	2	no	—
154	C₄	C₂×#11719921	1	1	yes	characteristic
155	C₄	#13133175	1	1	yes	characteristic
156	C₄	#13133188	1	1	yes	characteristic
157	C₄	#13133192	1	1	yes	characteristic
158	C₂	—	2	1	no	—
159	C₂	—	2	1	no	—
160	C₂	—	2	1	no	—
161	C₂	—	212	2	no	—
162	C₂	#16810480	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133223

C₂

yes

maximal, characteristic

C₂×#13133153

C₂

yes

maximal, characteristic, complemented

#16810605

C₂

yes

maximal

#16810381

C₂

yes

maximal, complemented

C₂×#108×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂×#11719945

C₂²

yes

—

—

C₂²

yes

—

C₂×#11719917

C₂²

yes

characteristic

C₂×#11719912

C₄

yes

—

C₂×C₈×D₅₃

C₄

yes

—

#13133153

—

#13133135

—

#13133133

—

C₂×M₁₆×C₅₃

C₂²

yes

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

#108×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₂²⋊D₅₃

Q₈

yes

—

—

—

—

—

C₂×D₂₁₂

D₄

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×Dic₁₀₆

D₄

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

D₄

yes

—

—

—

—

—

—

C₂×C₄

yes

—

#11719912

—

C₈×D₅₃

—

C₂³×C₅₃

C₂×Q₈

yes

characteristic

C₂²×D₅₃

C₄⋊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×D₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂²⋊D₅₃

—

C₂²⋊D₅₃

—

C₂×Dic₅₃

C₄⋊C₄

yes

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

C₂²⋊C₄

yes

—

C₈×C₅₃

C₂²⋊C₄

yes

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

C₂×C₂²⋊C₄

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

#1813

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#380

yes

characteristic

D₅₃

—

C₂×#131

—

C₂×#108

—

C₂×#108

D₅₃

yes

characteristic, complemented

#413

—

#338

—

C₅₃

#1813

yes

characteristic, complemented, Sylow

C₂×#41

—

#131

—

106

—

#131

—

106

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

#108

—

#108

—

106

—

#101

—

106

—

100

#100

—

106

—

101

#41

—

106

—

102

#41

—

106

—

103

#41

—

106

—

104

#41

—

106

—

105

C₂×Q₈

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂²×C₄

—

109

C₂²×C₄

C₂²×D₅₃

yes

characteristic

110

M₁₆

—

111

M₁₆

—

112

M₁₆

—

106

—

113

M₁₆

—

106

—

114

M₁₆

—

106

—

115

M₁₆

—

106

—

116

M₁₆

—

106

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

106

—

120

C₂×C₈

—

106

—

121

C₂×C₈

C₄×D₅₃

yes

characteristic

122

C₂×C₈

C₄×D₅₃

yes

characteristic

123

C₂×C₈

D₂₁₂

yes

—

124

C₂×C₈

C₂²⋊D₅₃

yes

—

125

C₂³

—

126

C₂³

Q₈×D₅₃

yes

characteristic

127

Q₈

—

106

—

128

Q₈

—

106

—

129

D₄

—

106

—

130

D₄

—

106

—

131

D₄

—

106

—

132

D₄

—

106

—

133

C₂×C₄

—

134

C₂×C₄

—

106

—

135

C₂×C₄

—

106

—

136

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

137

C₂×C₄

C₂×D₂₁₂

yes

characteristic

138

C₂×C₄

D₄×D₅₃

yes

characteristic

139

C₂×C₄

#11719947

yes

characteristic

140

C₂×C₄

#11719949

yes

characteristic

141

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

142

C₈

—

143

C₈

—

106

—

144

C₈

—

212

—

145

C₈

#11719921

yes

—

146

C₈

#11719921

yes

—

147

C₂²

—

148

C₂²

—

106

—

149

C₂²

—

212

—

150

C₂²

C₄⋊C₄×D₅₃

yes

characteristic

151

C₂²

#13133193

yes

characteristic

152

C₂²

#13133197

yes

characteristic

153

C₄

—

106

—

154

C₄

C₂×#11719921

yes

characteristic

155

C₄

#13133175

yes

characteristic

156

C₄

#13133188

yes

characteristic

157

C₄

#13133192

yes

characteristic

158

C₂

—

159

C₂

—

160

C₂

—

161

C₂

—

212

—

162

C₂

#16810480

yes

characteristic

	subgroup	G/H
center	C₄	#13133175
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×#108×C₅₃	C₂
socle	C₂×C₅₃	#380
hypercenter	C₂×#108	D₅₃
hypocenter	C₅₃	#1813
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₄

#13133175

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×#108×C₅₃

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
2. C₂×#13133153	C₂	semidirect
4. #16810381	C₂	semidirect
5. C₂×#108×C₅₃	C₂	semidirect
84. C₂×#108	D₅₃	semidirect
87. C₅₃	#1813	semidirect

H ≅ G/N

type

2. C₂×#13133153

C₂

semidirect

4. #16810381

C₂

semidirect

5. C₂×#108×C₅₃

C₂

semidirect

84. C₂×#108

D₅₃

semidirect

87. C₅₃

#1813

semidirect

Subgroups — #20274762

Conjugacy classes of subgroups

Complements