Group #20274743 — subgroups

Subgroups — #20274743

Order 6784.

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

normal subgroups only

This group has 336 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	C₂²×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133150	C₂	1	2	yes	maximal, complemented
3	#16810480	C₂	1	2	yes	maximal
4	#16810386	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#340	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×C₈×D₅₃	—	2	4	no	—
9	C₂×C₈×D₅₃	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
11	C₂×#11719921	—	2	2	no	—
12	#13133150	—	2	4	no	—
13	#13133124	—	2	2	no	—
14	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
16	C₂³×D₅₃	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₅₃	—	2	2	no	—
18	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
19	C₂²×Dic₅₃	—	2	2	no	—
20	C₂²×Dic₅₃	D₄	1	1	yes	characteristic
21	C₂×C₄×D₅₃	—	2	2	no	—
22	C₂×C₄×D₅₃	—	2	2	no	—
23	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
24	C₂×C₄×D₅₃	Q₈	1	2	yes	—
25	C₂×C₈×C₅₃	—	2	2	no	—
26	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
27	C₄⋊C₄×C₅₃	—	2	2	no	—
28	C₄⋊C₄×C₅₃	—	2	2	no	—
29	#11719921	—	4	4	no	—
30	C₄⋊Dic₅₃	—	2	2	no	—
31	C₄⋊Dic₅₃	—	2	2	no	—
32	C₂×#88322	C₂×C₄	1	2	yes	—
33	C₂×#88322	D₄	1	4	yes	—
34	C₈×D₅₃	—	2	8	no	—
35	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₂²×D₅₃	—	2	2	no	—
37	C₂²×D₅₃	—	2	2	no	—
38	C₂²×D₅₃	C₄⋊C₄	1	2	yes	—
39	C₂²×D₅₃	D₈	1	2	yes	—
40	C₂²×D₅₃	Q₁₆	1	2	yes	—
41	C₂×C₄×C₅₃	—	4	2	no	—
42	C₂×C₄×C₅₃	—	4	2	no	—
43	C₂×C₄×C₅₃	—	4	2	no	—
44	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₅₃	#41	1	2	yes	—
48	C₂×Dic₅₃	—	2	2	no	—
49	C₂×Dic₅₃	—	4	2	no	—
50	C₂×Dic₅₃	—	4	2	no	—
51	C₂×Dic₅₃	—	4	2	no	—
52	C₂×Dic₅₃	C₄⋊C₄	1	2	yes	—
53	C₄×D₅₃	—	2	2	no	—
54	C₄×D₅₃	—	2	2	no	—
55	C₄×D₅₃	—	2	4	no	—
56	C₈×C₅₃	—	2	2	no	—
57	#88322	C₂²⋊C₄	1	4	yes	complemented
58	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₅₃	#121	1	2	yes	—
60	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
61	C₂²×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
62	C₂²×C₅₃	#135	1	2	yes	—
63	C₂×D₅₃	—	2	2	no	—
64	C₂×D₅₃	—	2	2	no	—
65	C₂×D₅₃	—	2	4	no	—
66	C₂×D₅₃	—	2	4	no	—
67	C₂×D₅₃	#107	1	4	yes	complemented
68	C₄×C₅₃	—	8	2	no	—
69	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₄×C₅₃	#117	1	1	yes	characteristic
71	C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
72	Dic₅₃	—	2	2	no	—
73	Dic₅₃	—	8	2	no	—
74	#1808	—	53	1	no	maximal, Sylow
75	C₂×C₅₃	#380	1	1	yes	characteristic
76	C₂×C₅₃	C₂×#107	1	1	yes	characteristic
77	C₂×C₅₃	#427	1	1	yes	characteristic
78	C₂×C₅₃	#466	1	2	yes	—
79	C₂×C₅₃	#467	1	2	yes	—
80	D₅₃	—	2	4	no	—
81	C₂³×C₈	—	53	1	no	—
82	C₂×#107	—	53	2	no	—
83	#380	—	53	2	no	—
84	#340	—	53	1	no	—
85	#340	D₅₃	1	1	yes	characteristic, complemented
86	C₅₃	#1808	1	1	yes	characteristic, complemented, Sylow
87	C₂³×C₄	—	53	1	no	—
88	C₂²×C₈	—	53	1	no	—
89	C₂²×C₈	—	53	4	no	—
90	C₂²×C₈	—	106	4	no	—
91	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
92	C₂×C₄⋊C₄	—	53	2	no	—
93	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
94	C₂×C₂²⋊C₄	—	106	2	no	—
95	#107	—	106	4	no	—
96	#95	—	106	2	no	—
97	C₂⁴	—	53	1	no	—
98	C₂²×C₄	—	2	2	no	—
99	C₂²×C₄	—	53	1	no	—
100	C₂²×C₄	—	53	2	no	—
101	C₂²×C₄	—	53	2	no	—
102	C₂²×C₄	—	106	2	no	—
103	C₂²×C₄	—	106	2	no	—
104	C₂²×C₄	—	106	2	no	—
105	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	53	2	no	—
108	C₂×C₈	—	53	4	no	—
109	C₂×C₈	—	106	8	no	—
110	C₂×C₈	C₄×D₅₃	1	2	yes	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	106	2	no	—
114	C₄⋊C₄	—	106	2	no	—
115	C₂²⋊C₄	—	212	4	no	—
116	C₂³	—	53	2	no	—
117	C₂³	—	53	2	no	—
118	C₂³	—	53	2	no	—
119	C₂³	—	106	2	no	—
120	C₂³	—	106	2	no	—
121	C₂³	D₄×D₅₃	1	1	yes	characteristic
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	53	2	no	—
126	C₂×C₄	—	106	2	no	—
127	C₂×C₄	—	106	2	no	—
128	C₂×C₄	—	106	2	no	—
129	C₂×C₄	—	106	4	no	—
130	C₂×C₄	—	212	2	no	—
131	C₂×C₄	—	212	2	no	—
132	C₂×C₄	—	212	2	no	—
133	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
134	C₂×C₄	#11719945	1	2	yes	—
135	C₂×C₄	D₄×D₅₃	1	2	yes	—
136	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	—	53	4	no	—
139	C₂²	—	53	4	no	—
140	C₂²	—	106	4	no	—
141	C₂²	—	106	4	no	—
142	C₂²	—	106	2	no	—
143	C₂²	—	106	2	no	—
144	C₂²	C₄⋊C₄×D₅₃	1	1	yes	characteristic
145	C₂²	#13133190	1	2	yes	—
146	C₂²	D₈×D₅₃	1	1	yes	characteristic
147	C₂²	Q₁₆×D₅₃	1	1	yes	characteristic
148	C₂²	#13133238	1	2	yes	—
149	C₄	—	8	2	no	—
150	C₄	—	106	2	no	—
151	C₄	—	424	2	no	—
152	C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
153	C₄	#13133174	1	1	yes	characteristic
154	C₄	#13133195	1	2	yes	—
155	C₂	—	106	4	no	—
156	C₂	#16810482	1	1	yes	characteristic
157	C₂	#107×D₅₃	1	1	yes	characteristic
158	C₂	#16810658	1	1	yes	characteristic
159	C₂	#16810858	1	2	yes	—
160	C₂	#16810863	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₈×D₅₃

C₂

yes

maximal, characteristic

C₂×#13133150

C₂

yes

maximal, complemented

#16810480

C₂

yes

maximal

#16810386

C₂

yes

maximal, characteristic, complemented

C₅₃×#340

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₅₃

C₂²

yes

characteristic

C₂²×#88322

C₂²

yes

characteristic

C₂×C₈×D₅₃

—

C₂×C₈×D₅₃

C₄

yes

complemented

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×#11719921

—

#13133150

—

#13133124

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

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

yes

—

C₂×C₄×D₅₃

Q₈

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

#11719921

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

C₂×C₄

yes

—

C₂×#88322

D₄

yes

—

C₈×D₅₃

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

C₄⋊C₄

yes

—

C₂²×D₅₃

D₈

yes

—

C₂²×D₅₃

Q₁₆

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₄⋊C₄

yes

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

#88322

C₂²⋊C₄

yes

complemented

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₃

#121

yes

—

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₂²×C₅₃

C₂×Q₁₆

yes

characteristic

C₂²×C₅₃

#135

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

#107

yes

complemented

C₄×C₅₃

—

C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

#117

yes

characteristic

C₄×C₅₃

C₂²⋊Q₈

yes

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#467

yes

—

D₅₃

—

C₂³×C₈

—

—

—

—

D₅₃

yes

characteristic, complemented

C₅₃

#1808

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

106

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₂²⋊C₄

—

106

—

#107

—

106

—

#95

—

106

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

106

—

103

C₂²×C₄

—

106

—

104

C₂²×C₄

—

106

—

105

C₂²×C₄

C₂²×D₅₃

yes

characteristic

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

106

—

110

C₂×C₈

C₄×D₅₃

yes

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

106

—

114

C₄⋊C₄

—

106

—

115

C₂²⋊C₄

—

212

—

116

C₂³

—

117

C₂³

—

118

C₂³

—

119

C₂³

—

106

—

120

C₂³

—

106

—

121

C₂³

D₄×D₅₃

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

106

—

127

C₂×C₄

—

106

—

128

C₂×C₄

—

106

—

129

C₂×C₄

—

106

—

130

C₂×C₄

—

212

—

131

C₂×C₄

—

212

—

132

C₂×C₄

—

212

—

133

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

134

C₂×C₄

#11719945

yes

—

135

C₂×C₄

D₄×D₅₃

yes

—

136

C₂×C₄

Q₈×D₅₃

yes

characteristic

137

C₈

—

138

C₈

—

139

C₂²

—

140

C₂²

—

106

—

141

C₂²

—

106

—

142

C₂²

—

106

—

143

C₂²

—

106

—

144

C₂²

C₄⋊C₄×D₅₃

yes

characteristic

145

C₂²

#13133190

yes

—

146

C₂²

D₈×D₅₃

yes

characteristic

147

C₂²

Q₁₆×D₅₃

yes

characteristic

148

C₂²

#13133238

yes

—

149

C₄

—

150

C₄

—

106

—

151

C₄

—

424

—

152

C₄

C₂²⋊C₄×D₅₃

yes

characteristic

153

C₄

#13133174

yes

characteristic

154

C₄

#13133195

yes

—

155

C₂

—

106

—

156

C₂

#16810482

yes

characteristic

157

C₂

#107×D₅₃

yes

characteristic

158

C₂

#16810658

yes

characteristic

159

C₂

#16810858

yes

—

160

C₂

#16810863

yes

—

	subgroup	G/H
center	C₂³	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#340	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	#340	D₅₃
hypocenter	C₅₃	#1808
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#340

C₂

socle

C₂³×C₅₃

C₂×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
2. C₂×#13133150	C₂	semidirect
4. #16810386	C₂	semidirect
5. C₅₃×#340	C₂	semidirect
9. C₂×C₈×D₅₃	C₄	semidirect
57. #88322	C₂²⋊C₄	semidirect
67. C₂×D₅₃	#107	semidirect
85. #340	D₅₃	semidirect
86. C₅₃	#1808	semidirect

H ≅ G/N

type

2. C₂×#13133150

C₂

semidirect

4. #16810386

C₂

semidirect

5. C₅₃×#340

C₂

semidirect

9. C₂×C₈×D₅₃

C₄

semidirect

57. #88322

C₂²⋊C₄

semidirect

67. C₂×D₅₃

#107

semidirect

85. #340

D₅₃

semidirect

86. C₅₃

#1808

semidirect

Subgroups — #20274743

Conjugacy classes of subgroups

Complements