Group #20276139 — subgroups

Subgroups — #20276139

Order 6784.

256 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 81 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×#13133129	C₂	1	1	yes	maximal, characteristic
2	#16810480	C₂	1	2	yes	maximal
3	#16810386	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810385	C₂	1	2	yes	maximal, complemented
5	C₂×#107×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
7	C₂²×#88322	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
9	C₂×#11719921	—	2	2	no	—
10	#13133129	—	2	2	no	—
11	#13133124	—	2	2	no	—
12	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
14	#107×C₅₃	—	2	2	no	—
15	C₂³×D₅₃	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₅₃	—	2	2	no	—
17	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
18	C₂²×Dic₅₃	—	2	2	no	—
19	C₂²×Dic₅₃	D₄	1	1	yes	characteristic
20	C₂×C₄×D₅₃	—	2	1	no	—
21	C₂×C₄×D₅₃	—	2	2	no	—
22	C₂×C₄×D₅₃	—	2	2	no	—
23	C₂×C₄×D₅₃	—	2	1	no	—
24	C₂×C₈×C₅₃	—	2	2	no	—
25	C₂×C₈×C₅₃	D₄	1	2	yes	—
26	C₄⋊C₄×C₅₃	—	2	2	no	—
27	C₄⋊C₄×C₅₃	—	2	2	no	—
28	#11719921	—	4	4	no	—
29	C₄⋊Dic₅₃	—	2	2	no	—
30	C₄⋊Dic₅₃	—	2	2	no	—
31	C₂×#88322	—	2	1	no	—
32	C₂×#88322	—	2	2	no	—
33	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
34	C₂²×D₅₃	—	2	1	no	—
35	C₂²×D₅₃	—	2	2	no	—
36	C₂²×D₅₃	—	2	1	no	—
37	C₂²×D₅₃	—	2	2	no	—
38	C₂²×D₅₃	D₈	1	2	yes	—
39	C₂×C₄×C₅₃	—	4	2	no	—
40	C₂×C₄×C₅₃	—	4	2	no	—
41	C₂×C₄×C₅₃	—	4	2	no	—
42	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₅₃	#41	1	2	yes	—
44	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
45	C₂×C₄×C₅₃	#41	1	2	yes	—
46	C₂×Dic₅₃	—	2	1	no	—
47	C₂×Dic₅₃	—	2	2	no	—
48	C₂×Dic₅₃	—	4	2	no	—
49	C₂×Dic₅₃	—	4	2	no	—
50	C₂×Dic₅₃	—	4	2	no	—
51	C₄×D₅₃	—	4	1	no	—
52	C₄×D₅₃	—	4	1	no	—
53	C₄×D₅₃	—	4	2	no	—
54	C₈×C₅₃	—	2	2	no	—
55	#88322	—	4	1	no	—
56	C₂²×C₅₃	#121	1	1	yes	characteristic
57	C₂²×C₅₃	#123	1	2	yes	—
58	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
59	C₂²×C₅₃	#135	1	1	yes	characteristic
60	C₂²×C₅₃	#137	1	2	yes	—
61	C₂×D₅₃	—	2	2	no	—
62	C₂×D₅₃	—	2	4	no	—
63	C₂×D₅₃	—	4	1	no	—
64	C₂×D₅₃	—	4	1	no	—
65	C₂×D₅₃	—	4	2	no	—
66	C₄×C₅₃	—	8	2	no	—
67	C₄×C₅₃	#123	1	1	yes	characteristic
68	C₄×C₅₃	#124	1	1	yes	characteristic
69	C₄×C₅₃	#126	1	2	yes	—
70	Dic₅₃	—	4	1	no	—
71	Dic₅₃	—	8	2	no	—
72	#2075	—	53	1	no	maximal, Sylow
73	C₂×C₅₃	#397	1	1	yes	characteristic
74	C₂×C₅₃	#466	1	1	yes	characteristic
75	C₂×C₅₃	#470	1	1	yes	characteristic
76	C₂×C₅₃	#480	1	2	yes	—
77	C₂×C₅₃	#485	1	2	yes	—
78	D₅₃	—	4	2	no	—
79	C₂×#107	D₅₃	1	1	yes	characteristic, complemented
80	C₂×C₂²⋊C₈	—	53	1	no	—
81	#380	—	53	2	no	—
82	#340	—	53	1	no	—
83	#340	—	53	2	no	—
84	C₅₃	#2075	1	1	yes	characteristic, complemented, Sylow
85	C₂³×C₄	—	53	1	no	—
86	C₂²×C₈	—	53	1	no	—
87	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
88	C₂×C₄⋊C₄	—	53	2	no	—
89	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
90	C₂×C₂²⋊C₄	—	106	2	no	—
91	#107	—	2	2	no	—
92	C₂²⋊C₈	—	106	2	no	—
93	#95	—	106	2	no	—
94	C₂⁴	—	53	1	no	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	—	53	1	no	—
97	C₂²×C₄	—	106	2	no	—
98	C₂²×C₄	—	106	1	no	—
99	C₂²×C₄	—	106	2	no	—
100	C₂²×C₄	—	106	1	no	—
101	C₂²×C₄	—	106	2	no	—
102	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	106	2	no	—
105	C₂×C₈	—	106	1	no	—
106	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	106	2	no	—
110	C₄⋊C₄	—	106	2	no	—
111	C₂²⋊C₄	—	212	4	no	—
112	C₂³	—	53	2	no	—
113	C₂³	—	106	1	no	—
114	C₂³	—	106	2	no	—
115	C₂³	—	106	1	no	—
116	C₂³	—	106	2	no	—
117	C₂³	D₄×D₅₃	1	1	yes	characteristic
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	106	1	no	—
122	C₂×C₄	—	106	2	no	—
123	C₂×C₄	—	212	2	no	—
124	C₂×C₄	—	212	2	no	—
125	C₂×C₄	—	212	2	no	—
126	C₂×C₄	—	212	1	no	—
127	C₂×C₄	—	212	1	no	—
128	C₂×C₄	—	212	2	no	—
129	C₂×C₄	#11719945	1	2	yes	—
130	C₂×C₄	#11719947	1	1	yes	characteristic
131	C₂×C₄	#11719949	1	2	yes	—
132	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	—	212	1	no	—
135	C₂²	—	106	2	no	—
136	C₂²	—	106	4	no	—
137	C₂²	—	212	1	no	—
138	C₂²	—	212	1	no	—
139	C₂²	—	212	2	no	—
140	C₂²	#13133188	1	1	yes	characteristic
141	C₂²	#13133202	1	2	yes	—
142	C₂²	D₈×D₅₃	1	1	yes	characteristic
143	C₂²	#13133236	1	1	yes	characteristic
144	C₂²	#13133249	1	2	yes	—
145	C₄	—	8	2	no	—
146	C₄	—	424	2	no	—
147	C₄	—	212	1	no	—
148	C₄	#13133200	1	1	yes	characteristic
149	C₄	#13133206	1	1	yes	characteristic
150	C₄	#13133212	1	2	yes	—
151	C₂	—	212	2	no	—
152	C₂	#16810549	1	1	yes	characteristic
153	C₂	#16810856	1	1	yes	characteristic
154	C₂	#16810878	1	1	yes	characteristic
155	C₂	#16810938	1	2	yes	—
156	C₂	#16810966	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133129

C₂

yes

maximal, characteristic

#16810480

C₂

yes

maximal

#16810386

C₂

yes

maximal, characteristic, complemented

#16810385

C₂

yes

maximal, complemented

C₂×#107×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₅₃

C₂²

yes

characteristic

C₂²×#88322

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×#11719921

—

#13133129

—

#13133124

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

#107×C₅₃

—

C₂³×D₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

D₄

yes

characteristic

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

#11719921

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

D₈

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

#121

yes

characteristic

C₂²×C₅₃

#123

yes

—

C₂²×C₅₃

C₂×D₈

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#485

yes

—

D₅₃

—

C₂×#107

D₅₃

yes

characteristic, complemented

C₂×C₂²⋊C₈

—

#380

—

#340

—

#340

—

C₅₃

#2075

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₂²⋊C₄

—

106

—

#107

—

C₂²⋊C₈

—

106

—

#95

—

106

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

—

106

—

100

C₂²×C₄

—

106

—

101

C₂²×C₄

—

106

—

102

C₂²×C₄

C₂²×D₅₃

yes

characteristic

103

C₂×C₈

—

104

C₂×C₈

—

106

—

105

C₂×C₈

—

106

—

106

C₂×C₈

C₂²⋊D₅₃

yes

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

106

—

110

C₄⋊C₄

—

106

—

111

C₂²⋊C₄

—

212

—

112

C₂³

—

113

C₂³

—

106

—

114

C₂³

—

106

—

115

C₂³

—

106

—

116

C₂³

—

106

—

117

C₂³

D₄×D₅₃

yes

characteristic

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

106

—

122

C₂×C₄

—

106

—

123

C₂×C₄

—

212

—

124

C₂×C₄

—

212

—

125

C₂×C₄

—

212

—

126

C₂×C₄

—

212

—

127

C₂×C₄

—

212

—

128

C₂×C₄

—

212

—

129

C₂×C₄

#11719945

yes

—

130

C₂×C₄

#11719947

yes

characteristic

131

C₂×C₄

#11719949

yes

—

132

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

133

C₈

—

134

C₈

—

212

—

135

C₂²

—

106

—

136

C₂²

—

106

—

137

C₂²

—

212

—

138

C₂²

—

212

—

139

C₂²

—

212

—

140

C₂²

#13133188

yes

characteristic

141

C₂²

#13133202

yes

—

142

C₂²

D₈×D₅₃

yes

characteristic

143

C₂²

#13133236

yes

characteristic

144

C₂²

#13133249

yes

—

145

C₄

—

146

C₄

—

424

—

147

C₄

—

212

—

148

C₄

#13133200

yes

characteristic

149

C₄

#13133206

yes

characteristic

150

C₄

#13133212

yes

—

151

C₂

—

212

—

152

C₂

#16810549

yes

characteristic

153

C₂

#16810856

yes

characteristic

154

C₂

#16810878

yes

characteristic

155

C₂

#16810938

yes

—

156

C₂

#16810966

yes

—

	subgroup	G/H
center	C₂³	D₄×D₅₃
commutator	C₂²×C₄×C₅₃	C₂³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×#107×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×#107	D₅₃
hypocenter	C₅₃	#2075
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₅₃

commutator

C₂²×C₄×C₅₃

C₂³

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×#107×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
3. #16810386	C₂	semidirect
4. #16810385	C₂	semidirect
5. C₂×#107×C₅₃	C₂	semidirect
79. C₂×#107	D₅₃	semidirect
84. C₅₃	#2075	semidirect

H ≅ G/N

type

3. #16810386

C₂

semidirect

4. #16810385

C₂

semidirect

5. C₂×#107×C₅₃

C₂

semidirect

79. C₂×#107

D₅₃

semidirect

84. C₅₃

#2075

semidirect

Subgroups — #20276139

Conjugacy classes of subgroups

Complements