Group #20276243 — subgroups

Subgroups — #20276243

Order 6784.

290 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 93 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×#13133150	C₂	1	2	yes	maximal, complemented
2	C₂×#13133138	C₂	1	2	yes	maximal
3	#16810493	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810378	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#390	C₂	1	1	yes	maximal, characteristic
6	C₂²×#88322	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₅₃	—	2	2	no	—
8	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
9	#13133150	—	2	4	no	—
10	#13133138	—	2	4	no	—
11	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
14	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
15	C₂⁴×C₅₃	—	2	1	no	—
16	C₂×D₄×C₅₃	—	2	2	no	—
17	C₂×D₄×C₅₃	—	2	2	no	—
18	C₂×D₄×C₅₃	—	2	1	no	—
19	C₂×D₄×C₅₃	Q₈	1	2	yes	—
20	C₂²×C₄×C₅₃	—	2	1	no	—
21	C₂²×C₄×C₅₃	D₄	1	2	yes	—
22	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
23	C₂²×Dic₅₃	—	2	2	no	—
24	C₄⋊C₄×C₅₃	—	2	2	no	—
25	C₂²⋊C₄×C₅₃	—	4	4	no	—
26	C₄²×C₅₃	—	2	2	no	—
27	C₄⋊Dic₅₃	—	2	2	no	—
28	C₄⋊Dic₅₃	—	2	2	no	—
29	C₄⋊Dic₅₃	—	4	4	no	—
30	C₂×#88322	—	2	2	no	—
31	C₂×#88322	—	2	2	no	—
32	C₂×#88322	D₄	1	4	yes	—
33	C₂³×C₅₃	—	2	2	no	—
34	C₂³×C₅₃	—	4	1	no	—
35	C₂³×C₅₃	—	4	1	no	—
36	C₂³×C₅₃	—	4	2	no	—
37	C₂³×C₅₃	—	4	2	no	—
38	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
39	D₄×C₅₃	—	2	4	no	—
40	D₄×C₅₃	—	4	2	no	—
41	C₂×C₄×C₅₃	—	2	2	no	—
42	C₂×C₄×C₅₃	—	2	2	no	—
43	C₂×C₄×C₅₃	—	4	1	no	—
44	C₂×C₄×C₅₃	—	4	2	no	—
45	C₂×C₄×C₅₃	D₈	1	4	yes	—
46	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
49	C₂×C₄×C₅₃	#41	1	2	yes	—
50	C₂×Dic₅₃	—	4	2	no	—
51	C₂×Dic₅₃	—	4	4	no	—
52	#88322	—	2	4	no	—
53	C₂²×C₅₃	—	4	2	no	—
54	C₂²×C₅₃	—	4	4	no	—
55	C₂²×C₅₃	—	8	2	no	—
56	C₂²×C₅₃	—	8	1	no	—
57	C₂²×C₅₃	—	8	1	no	—
58	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
59	C₂²×C₅₃	#123	1	1	yes	characteristic
60	C₂²×C₅₃	#127	1	1	yes	characteristic
61	C₂²×C₅₃	C₂×D₈	1	2	yes	—
62	C₂²×C₅₃	#137	1	2	yes	—
63	C₄×C₅₃	—	2	4	no	—
64	C₄×C₅₃	—	8	1	no	—
65	C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
66	C₄×C₅₃	#123	1	2	yes	—
67	Dic₅₃	—	8	2	no	—
68	#2093	—	53	1	no	maximal, Sylow
69	C₂×C₅₃	—	8	2	no	—
70	C₂×C₅₃	#399	1	1	yes	characteristic
71	C₂×C₅₃	#474	1	2	yes	—
72	C₂×C₅₃	#480	1	2	yes	—
73	C₂×C₅₃	#493	1	1	yes	characteristic
74	C₂×C₅₃	#497	1	1	yes	characteristic
75	C₂×#107	—	53	2	no	—
76	C₂×#102	—	53	2	no	—
77	#390	D₅₃	1	1	yes	characteristic
78	#384	—	53	1	no	—
79	#336	—	53	1	no	—
80	C₅₃	#2093	1	1	yes	characteristic, complemented, Sylow
81	C₂²×D₄	C₂×D₅₃	1	1	yes	characteristic
82	C₂²×C₈	—	53	2	no	—
83	C₂×C₄⋊C₄	—	53	2	no	—
84	C₂×C₄⋊C₄	—	106	2	no	—
85	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
86	C₂×C₂²⋊C₄	—	2	2	no	—
87	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
88	#107	—	106	4	no	—
89	#102	—	106	4	no	—
90	C₂⁴	—	2	1	no	—
91	C₂×D₄	—	2	1	no	—
92	C₂×D₄	—	2	2	no	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	Dic₁₀₆	1	2	yes	—
95	C₂²×C₄	—	2	1	no	—
96	C₂²×C₄	—	106	2	no	—
97	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
98	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
99	C₂×C₈	—	53	4	no	—
100	C₂×C₈	—	106	2	no	—
101	C₂×C₈	—	106	2	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	—	106	2	no	—
104	C₄⋊C₄	—	106	2	no	—
105	C₄⋊C₄	—	212	4	no	—
106	C₂²⋊C₄	—	4	4	no	—
107	C₄²	—	2	2	no	—
108	C₂³	—	2	2	no	—
109	C₂³	—	4	1	no	—
110	C₂³	—	4	1	no	—
111	C₂³	—	4	2	no	—
112	C₂³	—	4	2	no	—
113	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
114	D₄	—	2	4	no	—
115	D₄	—	4	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	212	2	no	—
121	C₂×C₄	—	212	4	no	—
122	C₂×C₄	#11719922	1	4	yes	—
123	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
124	C₂×C₄	#11719945	1	1	yes	characteristic
125	C₂×C₄	D₄×D₅₃	1	2	yes	—
126	C₂×C₄	#11719947	1	2	yes	—
127	C₈	—	106	4	no	—
128	C₂²	—	4	2	no	—
129	C₂²	—	4	4	no	—
130	C₂²	—	8	1	no	—
131	C₂²	—	8	2	no	—
132	C₂²	—	8	1	no	—
133	C₂²	#13133196	1	1	yes	characteristic
134	C₂²	#13133198	1	1	yes	characteristic
135	C₂²	#13133213	1	1	yes	characteristic
136	C₂²	C₂×#11719922	1	2	yes	—
137	C₂²	#13133247	1	2	yes	—
138	C₄	—	2	4	no	—
139	C₄	—	8	1	no	—
140	C₄	—	424	2	no	—
141	C₄	#13133194	1	2	yes	—
142	C₄	#13133199	1	2	yes	—
143	C₂	—	8	2	no	—
144	C₂	#16810559	1	1	yes	characteristic
145	C₂	#16810903	1	2	yes	—
146	C₂	#16810934	1	2	yes	—
147	C₂	#16810999	1	1	yes	characteristic
148	C₂	#16811018	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133150

C₂

yes

maximal, complemented

C₂×#13133138

C₂

yes

maximal

#16810493

C₂

yes

maximal, characteristic, complemented

#16810378

C₂

yes

maximal, characteristic, complemented

C₅₃×#390

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

#13133150

—

#13133138

—

C₂²×D₄×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂⁴×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

D₄

yes

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

D₈

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²⋊Q₈

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

C₂×D₈

yes

—

C₂²×C₅₃

#137

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²⋊Q₈

yes

—

C₄×C₅₃

#123

yes

—

Dic₅₃

—

#2093

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#399

yes

characteristic

C₂×C₅₃

#474

yes

—

C₂×C₅₃

#480

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

D₅₃

yes

characteristic

#384

—

#336

—

C₅₃

#2093

yes

characteristic, complemented, Sylow

C₂²×D₄

C₂×D₅₃

yes

characteristic

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#107

—

106

—

#102

—

106

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

Dic₁₀₆

yes

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

106

—

101

C₂×C₈

—

106

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

106

—

104

C₄⋊C₄

—

106

—

105

C₄⋊C₄

—

212

—

106

C₂²⋊C₄

—

107

C₄²

—

108

C₂³

—

109

C₂³

—

110

C₂³

—

111

C₂³

—

112

C₂³

—

113

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

212

—

121

C₂×C₄

—

212

—

122

C₂×C₄

#11719922

yes

—

123

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

124

C₂×C₄

#11719945

yes

characteristic

125

C₂×C₄

D₄×D₅₃

yes

—

126

C₂×C₄

#11719947

yes

—

127

C₈

—

106

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

#13133196

yes

characteristic

134

C₂²

#13133198

yes

characteristic

135

C₂²

#13133213

yes

characteristic

136

C₂²

C₂×#11719922

yes

—

137

C₂²

#13133247

yes

—

138

C₄

—

139

C₄

—

140

C₄

—

424

—

141

C₄

#13133194

yes

—

142

C₄

#13133199

yes

—

143

C₂

—

144

C₂

#16810559

yes

characteristic

145

C₂

#16810903

yes

—

146

C₂

#16810934

yes

—

147

C₂

#16810999

yes

characteristic

148

C₂

#16811018

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₃
commutator	C₂²×C₄×C₅₃	C₂³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#390	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	#390	D₅₃
hypocenter	C₅₃	#2093
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₃

commutator

C₂²×C₄×C₅₃

C₂³

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#390

C₂

socle

C₂³×C₅₃

C₂×D₄

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

N	H ≅ G/N	type
1. C₂×#13133150	C₂	semidirect
3. #16810493	C₂	semidirect
4. #16810378	C₂	semidirect
80. C₅₃	#2093	semidirect

H ≅ G/N

type

1. C₂×#13133150

C₂

semidirect

3. #16810493

C₂

semidirect

4. #16810378

C₂

semidirect

80. C₅₃

#2093

semidirect

Subgroups — #20276243

Conjugacy classes of subgroups

Complements