Group #20274774 — subgroups

Subgroups — #20274774

Order 6784.

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

normal subgroups only

This group has 264 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×#13133153	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13133149	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810624	C₂	1	2	yes	maximal
4	#16810391	C₂	1	2	yes	maximal, complemented
5	C₂×#131×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
8	#13133153	—	2	2	no	—
9	#13133149	—	2	1	no	—
10	#13133149	—	2	1	no	—
11	#13133141	—	2	2	no	—
12	#13133138	—	2	2	no	—
13	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
14	#131×C₅₃	—	2	2	no	—
15	#131×C₅₃	C₄	1	4	yes	complemented
16	C₂×M₁₆×C₅₃	—	2	1	no	—
17	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₅₃	—	2	1	no	—
19	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	#41×C₅₃	—	2	1	no	—
21	#41×C₅₃	—	2	1	no	—
22	#41×C₅₃	C₂×C₄	1	2	yes	—
23	#41×C₅₃	Q₈	1	2	yes	—
24	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
25	C₂×D₄×C₅₃	—	2	1	no	—
26	C₂×D₄×C₅₃	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₅₃	—	2	1	no	—
28	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
29	C₂²×Dic₅₃	—	2	2	no	—
30	M₁₆×C₅₃	—	2	1	no	—
31	M₁₆×C₅₃	—	2	4	no	—
32	M₁₆×C₅₃	C₂×C₄	1	2	yes	—
33	C₂×C₈×C₅₃	—	2	1	no	—
34	C₂×C₈×C₅₃	—	2	4	no	—
35	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
36	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
37	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
38	C₂×C₈×C₅₃	D₄	1	2	yes	—
39	C₄⋊Dic₅₃	—	2	2	no	—
40	C₄⋊Dic₅₃	—	2	2	no	—
41	#11719917	—	2	2	no	—
42	#11719917	—	2	2	no	—
43	C₂×#88322	—	2	2	no	—
44	C₂³×C₅₃	—	2	1	no	—
45	C₂³×C₅₃	#41	1	1	yes	characteristic
46	Q₈×C₅₃	—	2	1	no	—
47	Q₈×C₅₃	C₄⋊C₄	1	2	yes	—
48	D₄×C₅₃	—	2	2	no	—
49	D₄×C₅₃	—	2	2	no	—
50	D₄×C₅₃	—	2	1	no	—
51	D₄×C₅₃	C₄⋊C₄	1	2	yes	—
52	C₂×C₄×C₅₃	—	2	1	no	—
53	C₂×C₄×C₅₃	—	2	2	no	—
54	C₂×C₄×C₅₃	—	2	2	no	—
55	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
61	C₂×Dic₅₃	—	4	2	no	—
62	C₂×Dic₅₃	—	8	2	no	—
63	C₈×C₅₃	—	2	2	no	—
64	C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
65	C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
66	#88322	—	4	2	no	—
67	C₂²×C₅₃	—	2	2	no	—
68	C₂²×C₅₃	—	2	2	no	—
69	C₂²×C₅₃	—	4	2	no	—
70	C₂²×C₅₃	#117	1	1	yes	characteristic
71	C₂²×C₅₃	#121	1	1	yes	characteristic
72	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
73	C₄×C₅₃	—	2	2	no	—
74	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₄×C₅₃	#121	1	1	yes	characteristic
77	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
78	Dic₅₃	—	8	2	no	—
79	#1815	—	53	1	no	maximal, Sylow
80	C₂×C₅₃	—	2	1	no	—
81	C₂×C₅₃	—	2	1	no	—
82	C₂×C₅₃	—	4	2	no	—
83	C₂×C₅₃	#380	1	1	yes	characteristic
84	C₂×C₅₃	#471	1	2	yes	—
85	C₂×#131	D₅₃	1	1	yes	characteristic
86	C₂×#108	—	53	1	no	—
87	C₂×#106	—	53	1	no	—
88	#417	—	53	2	no	—
89	#343	—	53	2	no	—
90	C₅₃	#1815	1	1	yes	characteristic, complemented, Sylow
91	C₂×#41	C₂×D₅₃	1	1	yes	characteristic
92	#131	—	2	2	no	—
93	#131	Dic₅₃	1	4	yes	complemented
94	C₂×M₁₆	—	2	1	no	—
95	C₂×M₁₆	—	53	2	no	—
96	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
97	C₂²×C₈	—	2	1	no	—
98	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
99	C₂×C₄⋊C₄	—	53	2	no	—
100	#108	—	106	2	no	—
101	#106	—	106	1	no	—
102	#106	—	106	1	no	—
103	#103	—	106	2	no	—
104	#102	—	106	2	no	—
105	#41	—	2	1	no	—
106	#41	—	2	1	no	—
107	#41	Dic₁₀₆	1	2	yes	—
108	#41	C₂×Dic₅₃	1	2	yes	—
109	C₂×Q₈	D₂₁₂	1	1	yes	characteristic
110	C₂×D₄	—	2	1	no	—
111	C₂×D₄	D₂₁₂	1	1	yes	characteristic
112	C₂²×C₄	—	2	1	no	—
113	C₂²×C₄	—	106	2	no	—
114	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
115	M₁₆	—	2	1	no	—
116	M₁₆	—	2	4	no	—
117	M₁₆	—	106	2	no	—
118	M₁₆	—	106	2	no	—
119	M₁₆	C₂×Dic₅₃	1	2	yes	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	2	4	no	—
122	C₂×C₈	—	106	2	no	—
123	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
124	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
125	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
126	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
127	C₄⋊C₄	—	106	2	no	—
128	C₄⋊C₄	—	106	2	no	—
129	C₂³	—	2	1	no	—
130	C₂³	#11719945	1	1	yes	characteristic
131	Q₈	—	2	1	no	—
132	Q₈	C₄⋊Dic₅₃	1	2	yes	—
133	D₄	—	2	2	no	—
134	D₄	—	2	2	no	—
135	D₄	—	2	1	no	—
136	D₄	C₄⋊Dic₅₃	1	2	yes	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	424	2	no	—
141	C₂×C₄	—	212	2	no	—
142	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
143	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
144	C₂×C₄	#11719945	1	1	yes	characteristic
145	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
146	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
147	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
148	C₈	—	2	2	no	—
149	C₈	—	212	2	no	—
150	C₈	C₂²⋊Dic₅₃	1	2	yes	—
151	C₈	C₂²⋊Dic₅₃	1	2	yes	—
152	C₂²	—	2	2	no	—
153	C₂²	—	2	2	no	—
154	C₂²	—	4	2	no	—
155	C₂²	#13133176	1	1	yes	characteristic
156	C₂²	#13133191	1	1	yes	characteristic
157	C₂²	#13133196	1	1	yes	characteristic
158	C₄	—	2	2	no	—
159	C₄	—	424	2	no	—
160	C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
161	C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
162	C₄	#13133191	1	1	yes	characteristic
163	C₄	#13133196	1	1	yes	characteristic
164	C₂	—	2	1	no	—
165	C₂	—	2	1	no	—
166	C₂	—	4	2	no	—
167	C₂	#16810483	1	1	yes	characteristic
168	C₂	#16810886	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133153

C₂

yes

maximal, characteristic, complemented

C₂×#13133149

C₂

yes

maximal, characteristic, complemented

#16810624

C₂

yes

maximal

#16810391

C₂

yes

maximal, complemented

C₂×#131×C₅₃

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

—

—

—

—

—

C₂×#41×C₅₃

C₂²

yes

characteristic

#131×C₅₃

—

#131×C₅₃

C₄

yes

complemented

C₂×M₁₆×C₅₃

—

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

#41×C₅₃

—

#41×C₅₃

—

#41×C₅₃

C₂×C₄

yes

—

#41×C₅₃

Q₈

yes

—

C₂×Q₈×C₅₃

D₄

yes

characteristic

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719917

—

#11719917

—

C₂×#88322

—

C₂³×C₅₃

—

C₂³×C₅₃

#41

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

C₄⋊C₄

yes

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

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

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₂²⋊C₄

yes

—

C₈×C₅₃

C₂²⋊C₄

yes

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

C₂²⋊Q₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

#121

yes

characteristic

C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

Dic₅₃

—

#1815

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#380

yes

characteristic

C₂×C₅₃

#471

yes

—

C₂×#131

D₅₃

yes

—

—

—

—

C₅₃

#1815

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₅₃

yes

characteristic

#131

—

#131

Dic₅₃

yes

complemented

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂×C₄⋊C₄

—

100

#108

—

106

—

101

#106

—

106

—

102

#106

—

106

—

103

#103

—

106

—

104

#102

—

106

—

105

#41

—

106

#41

—

107

#41

Dic₁₀₆

yes

—

108

#41

C₂×Dic₅₃

yes

—

109

C₂×Q₈

D₂₁₂

yes

characteristic

110

C₂×D₄

—

111

C₂×D₄

D₂₁₂

yes

characteristic

112

C₂²×C₄

—

113

C₂²×C₄

—

106

—

114

C₂²×C₄

C₂²×D₅₃

yes

characteristic

115

M₁₆

—

116

M₁₆

—

117

M₁₆

—

106

—

118

M₁₆

—

106

—

119

M₁₆

C₂×Dic₅₃

yes

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

106

—

123

C₂×C₈

C₂×Dic₅₃

yes

—

124

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

125

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

126

C₂×C₈

C₂²⋊D₅₃

yes

—

127

C₄⋊C₄

—

106

—

128

C₄⋊C₄

—

106

—

129

C₂³

—

130

C₂³

#11719945

yes

characteristic

131

Q₈

—

132

Q₈

C₄⋊Dic₅₃

yes

—

133

D₄

—

134

D₄

—

135

D₄

—

136

D₄

C₄⋊Dic₅₃

yes

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

424

—

141

C₂×C₄

—

212

—

142

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

143

C₂×C₄

C₂×D₂₁₂

yes

characteristic

144

C₂×C₄

#11719945

yes

characteristic

145

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

146

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

147

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

148

C₈

—

149

C₈

—

212

—

150

C₈

C₂²⋊Dic₅₃

yes

—

151

C₈

C₂²⋊Dic₅₃

yes

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

#13133176

yes

characteristic

156

C₂²

#13133191

yes

characteristic

157

C₂²

#13133196

yes

characteristic

158

C₄

—

159

C₄

—

424

—

160

C₄

C₂×C₂²⋊Dic₅₃

yes

characteristic

161

C₄

C₂×C₄⋊Dic₅₃

yes

characteristic

162

C₄

#13133191

yes

characteristic

163

C₄

#13133196

yes

characteristic

164

C₂

—

165

C₂

—

166

C₂

—

167

C₂

#16810483

yes

characteristic

168

C₂

#16810886

yes

—

	subgroup	G/H
center	C₂²	#13133191
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×#131×C₅₃	C₂
socle	C₂²×C₅₃	#121
hypercenter	C₂×#131	D₅₃
hypocenter	C₅₃	#1815
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131

subgroup

G/H

center

C₂²

#13133191

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×#131×C₅₃

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131

N	H ≅ G/N	type
1. C₂×#13133153	C₂	semidirect
2. C₂×#13133149	C₂	semidirect
4. #16810391	C₂	semidirect
15. #131×C₅₃	C₄	semidirect
90. C₅₃	#1815	semidirect
93. #131	Dic₅₃	semidirect

H ≅ G/N

type

1. C₂×#13133153

C₂

semidirect

2. C₂×#13133149

C₂

semidirect

4. #16810391

C₂

semidirect

15. #131×C₅₃

C₄

semidirect

90. C₅₃

#1815

semidirect

93. #131

Dic₅₃

semidirect

Subgroups — #20274774

Conjugacy classes of subgroups

Complements