Group #20280690 — subgroups

Subgroups — #20280690

Order 6784.

360 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 149 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	#16810565	C₂	1	4	yes	maximal
2	#16810557	C₂	1	4	yes	maximal
3	#16810521	C₂	1	2	yes	maximal
4	#16810521	C₂	1	2	yes	maximal
5	#16810493	C₂	1	1	yes	maximal, characteristic
6	#16810489	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#376	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₅₃	C₂²	1	4	yes	—
11	C₂×#11719919	C₂²	1	4	yes	—
12	C₂×#11719919	C₂²	1	4	yes	—
13	#13133124	C₂²	1	4	yes	—
14	#13133124	C₂²	1	4	yes	—
15	#13133124	C₂²	1	4	yes	—
16	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
19	#95×C₅₃	C₂²	1	2	yes	—
20	#95×C₅₃	C₂²	1	2	yes	—
21	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
22	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
23	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
25	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
26	C₂²×Dic₅₃	C₂³	1	4	yes	—
27	C₂²×Dic₅₃	C₂³	1	4	yes	—
28	C₄²×C₅₃	—	2	2	no	—
29	C₄²×C₅₃	—	2	2	no	—
30	C₄²×C₅₃	Q₈	1	4	yes	—
31	C₄⋊Dic₅₃	—	2	4	no	—
32	C₄⋊Dic₅₃	—	2	4	no	—
33	C₄⋊Dic₅₃	—	4	4	no	—
34	#11719919	—	4	4	no	—
35	#11719919	—	4	4	no	—
36	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
37	C₂×C₄×C₅₃	—	2	2	no	—
38	C₂×C₄×C₅₃	—	2	2	no	—
39	C₂×C₄×C₅₃	—	2	1	no	—
40	C₂×C₄×C₅₃	—	2	1	no	—
41	C₂×C₄×C₅₃	—	2	2	no	—
42	C₂×C₄×C₅₃	—	2	2	no	—
43	C₂×C₄×C₅₃	—	2	2	no	—
44	C₂×C₄×C₅₃	—	2	2	no	—
45	C₂×C₄×C₅₃	—	2	2	no	—
46	C₂×C₄×C₅₃	C₂×Q₈	1	4	yes	—
47	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
48	C₂×C₄×C₅₃	#41	1	2	yes	—
49	C₂×C₄×C₅₃	#41	1	2	yes	—
50	C₂×Dic₅₃	—	2	4	no	—
51	C₂×Dic₅₃	—	2	4	no	—
52	C₂×Dic₅₃	—	2	4	no	—
53	C₂×Dic₅₃	—	2	4	no	—
54	C₂×Dic₅₃	—	2	4	no	—
55	C₂×Dic₅₃	—	2	4	no	—
56	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
57	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
58	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
59	C₂²×C₅₃	H(5,𝔽₂)	1	2	yes	—
60	C₂²×C₅₃	2₋¹⁺⁴	1	2	yes	—
61	C₄×C₅₃	—	2	4	no	—
62	C₄×C₅₃	—	4	1	no	—
63	C₄×C₅₃	—	4	1	no	—
64	C₄×C₅₃	—	4	2	no	—
65	C₄×C₅₃	#125	1	4	yes	—
66	Dic₅₃	—	4	4	no	—
67	Dic₅₃	—	4	4	no	—
68	#2647	—	53	1	no	maximal, Sylow
69	C₂×C₅₃	C₂×#125	1	1	yes	characteristic
70	C₂×C₅₃	#540	1	1	yes	characteristic
71	C₂×C₅₃	#541	1	1	yes	characteristic
72	C₂×C₅₃	#542	1	2	yes	—
73	C₂×C₅₃	#544	1	2	yes	—
74	#400	—	53	4	no	—
75	#398	—	53	4	no	—
76	#389	—	53	2	no	—
77	#389	—	53	2	no	—
78	#384	—	53	1	no	—
79	#382	—	53	1	no	—
80	#376	D₅₃	1	1	yes	characteristic
81	C₅₃	#2647	1	1	yes	characteristic, complemented, Sylow
82	C₂×C₄⋊C₄	—	53	2	no	—
83	C₂×C₄⋊C₄	—	53	2	no	—
84	C₂×C₄⋊C₄	—	53	4	no	—
85	C₂×C₄⋊C₄	—	53	4	no	—
86	C₂×C₄⋊C₄	—	53	4	no	—
87	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
88	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
89	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
90	#95	—	53	4	no	—
91	#95	—	53	4	no	—
92	#95	—	53	4	no	—
93	#95	C₂×D₅₃	1	2	yes	—
94	#95	C₂×D₅₃	1	2	yes	—
95	C₂²×C₄	—	53	4	no	—
96	C₂²×C₄	—	53	4	no	—
97	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
99	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
100	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
101	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
102	C₄⋊C₄	—	106	4	no	—
103	C₄⋊C₄	—	106	4	no	—
104	C₄⋊C₄	—	212	4	no	—
105	C₄⋊C₄	—	212	4	no	—
106	C₄⋊C₄	—	212	4	no	—
107	C₄²	—	2	2	no	—
108	C₄²	—	2	2	no	—
109	C₄²	Dic₁₀₆	1	4	yes	—
110	C₂³	C₂³×D₅₃	1	1	yes	characteristic
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	106	4	no	—
121	C₂×C₄	—	106	4	no	—
122	C₂×C₄	—	106	4	no	—
123	C₂×C₄	—	106	4	no	—
124	C₂×C₄	—	106	4	no	—
125	C₂×C₄	—	106	4	no	—
126	C₂×C₄	C₂×Dic₁₀₆	1	4	yes	—
127	C₂×C₄	C₂×Dic₁₀₆	1	2	yes	—
128	C₂×C₄	#11719945	1	2	yes	—
129	C₂×C₄	#11719945	1	2	yes	—
130	C₂²	C₂²×Dic₁₀₆	1	1	yes	characteristic
131	C₂²	C₂×#11719945	1	1	yes	characteristic
132	C₂²	C₂×#11719945	1	1	yes	characteristic
133	C₂²	#13133263	1	2	yes	—
134	C₂²	#13133264	1	2	yes	—
135	C₄	—	2	4	no	—
136	C₄	—	4	1	no	—
137	C₄	—	4	1	no	—
138	C₄	—	4	2	no	—
139	C₄	—	212	4	no	—
140	C₄	—	212	4	no	—
141	C₄	#13133209	1	4	yes	—
142	C₂	C₂×#13133209	1	1	yes	characteristic
143	C₂	#16811228	1	1	yes	characteristic
144	C₂	#16811235	1	1	yes	characteristic
145	C₂	#16811244	1	2	yes	—
146	C₂	#16811251	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16810565

C₂

yes

maximal

#16810557

C₂

yes

maximal

#16810521

C₂

yes

maximal

#16810521

C₂

yes

maximal

#16810493

C₂

yes

maximal, characteristic

#16810489

C₂

yes

maximal, characteristic

C₅₃×#376

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×#11719919

C₂²

yes

—

C₂×#11719919

C₂²

yes

—

#13133124

C₂²

yes

—

#13133124

C₂²

yes

—

#13133124

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#95×C₅₃

C₂²

yes

—

#95×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

Q₈

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

#11719919

—

C₂³×C₅₃

C₂⁴

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂²×C₅₃

C₂²×Q₈

yes

characteristic

C₂²×C₅₃

C₂×#41

yes

characteristic

C₂²×C₅₃

C₂×#41

yes

characteristic

C₂²×C₅₃

H(5,𝔽₂)

yes

—

C₂²×C₅₃

2₋¹⁺⁴

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#125

yes

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#544

yes

—

—

—

—

—

—

—

D₅₃

yes

characteristic

C₅₃

#2647

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₂×C₄²

C₂×D₅₃

yes

—

—

—

C₂×D₅₃

yes

—

#95

C₂×D₅₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

100

C₂²×C₄

C₂²×D₅₃

yes

—

101

C₂²×C₄

C₂²×D₅₃

yes

characteristic

102

C₄⋊C₄

—

106

—

103

C₄⋊C₄

—

106

—

104

C₄⋊C₄

—

212

—

105

C₄⋊C₄

—

212

—

106

C₄⋊C₄

—

212

—

107

C₄²

—

108

C₄²

—

109

C₄²

Dic₁₀₆

yes

—

110

C₂³

C₂³×D₅₃

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

106

—

121

C₂×C₄

—

106

—

122

C₂×C₄

—

106

—

123

C₂×C₄

—

106

—

124

C₂×C₄

—

106

—

125

C₂×C₄

—

106

—

126

C₂×C₄

C₂×Dic₁₀₆

yes

—

127

C₂×C₄

C₂×Dic₁₀₆

yes

—

128

C₂×C₄

#11719945

yes

—

129

C₂×C₄

#11719945

yes

—

130

C₂²

C₂²×Dic₁₀₆

yes

characteristic

131

C₂²

C₂×#11719945

yes

characteristic

132

C₂²

C₂×#11719945

yes

characteristic

133

C₂²

#13133263

yes

—

134

C₂²

#13133264

yes

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

212

—

140

C₄

—

212

—

141

C₄

#13133209

yes

—

142

C₂

C₂×#13133209

yes

characteristic

143

C₂

#16811228

yes

characteristic

144

C₂

#16811235

yes

characteristic

145

C₂

#16811244

yes

—

146

C₂

#16811251

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂⁴
Fitting	C₅₃×#376	C₂
hypercenter	#376	D₅₃
hypocenter	C₅₃	#2647
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #376

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₃

commutator, socle

C₂³×C₅₃

C₂⁴

Fitting

C₅₃×#376

C₂

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ #376

N	H ≅ G/N	type
81. C₅₃	#2647	semidirect

H ≅ G/N

type

81. C₅₃

#2647

semidirect

Subgroups — #20280690

Conjugacy classes of subgroups

Complements