Group #20280836 — subgroups

Subgroups — #20280836

Order 6784.

448 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 165 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×#13133217	C₂	1	1	yes	maximal, characteristic
2	#16810553	C₂	1	4	yes	maximal
3	#16810543	C₂	1	4	yes	maximal
4	#16810492	C₂	1	2	yes	maximal
5	#16810488	C₂	1	2	yes	maximal
6	#16810469	C₂	1	1	yes	maximal, characteristic
7	C₂×#125×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₀₆	C₂²	1	2	yes	—
9	#13133217	—	2	4	no	—
10	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
12	C₂×#11719919	C₂²	1	8	yes	—
13	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
14	#13133124	C₂²	1	2	yes	—
15	#13133124	C₂²	1	8	yes	—
16	#13133124	C₂²	1	2	yes	—
17	#125×C₅₃	—	2	4	no	—
18	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
24	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
25	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
27	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
28	C₂²×Dic₅₃	C₂³	1	4	yes	—
29	C₂²×Dic₅₃	C₂³	1	4	yes	—
30	C₂×Dic₁₀₆	—	2	4	no	—
31	C₂×Dic₁₀₆	—	4	4	no	—
32	C₄⋊C₄×C₅₃	—	2	4	no	—
33	C₄⋊C₄×C₅₃	—	2	4	no	—
34	C₄⋊C₄×C₅₃	—	2	2	no	—
35	C₄⋊C₄×C₅₃	—	2	2	no	—
36	C₄²×C₅₃	D₄	1	4	yes	—
37	C₄⋊Dic₅₃	—	2	4	no	—
38	C₄⋊Dic₅₃	—	2	4	no	—
39	#11719919	—	4	8	no	—
40	C₄×Dic₅₃	Q₈	1	8	yes	complemented
41	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
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	—	2	4	no	—
47	C₂×C₄×C₅₃	—	2	4	no	—
48	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₅₃	C₂×Q₈	1	4	yes	—
52	C₂×Dic₅₃	—	2	4	no	—
53	C₂×Dic₅₃	—	2	4	no	—
54	C₂×Dic₅₃	—	2	4	no	—
55	C₂×Dic₅₃	—	2	8	no	—
56	C₂×Dic₅₃	C₂×Q₈	1	8	yes	—
57	Dic₁₀₆	—	4	8	no	—
58	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
59	C₂²×C₅₃	C₂²×Q₈	1	2	yes	—
60	C₂²×C₅₃	H(5,𝔽₂)	1	2	yes	—
61	C₂²×C₅₃	2₋¹⁺⁴	1	2	yes	—
62	C₄×C₅₃	—	2	2	no	—
63	C₄×C₅₃	—	2	2	no	—
64	C₄×C₅₃	—	4	2	no	—
65	C₄×C₅₃	—	4	2	no	—
66	C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
67	Dic₅₃	—	2	8	no	—
68	Dic₅₃	—	4	4	no	—
69	#2660	—	53	1	no	maximal, Sylow
70	C₂×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
71	C₂×C₅₃	#535	1	1	yes	characteristic
72	C₂×C₅₃	#536	1	1	yes	characteristic
73	C₂×C₅₃	#544	1	4	yes	—
74	C₂×C₄⋊Q₈	—	53	1	no	—
75	C₂×#125	D₅₃	1	1	yes	characteristic
76	#398	—	53	4	no	—
77	#395	—	53	4	no	—
78	#384	—	53	2	no	—
79	#382	—	53	2	no	—
80	#376	—	53	1	no	—
81	C₅₃	#2660	1	1	yes	characteristic, complemented, Sylow
82	C₂²×Q₈	—	53	2	no	—
83	C₄⋊Q₈	—	106	4	no	—
84	#125	—	2	4	no	—
85	C₂×C₄⋊C₄	—	53	2	no	—
86	C₂×C₄⋊C₄	—	53	2	no	—
87	C₂×C₄⋊C₄	—	53	8	no	—
88	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
89	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
90	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
91	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
92	C₂×C₄²	—	53	2	no	—
93	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
94	#95	—	53	2	no	—
95	#95	—	53	2	no	—
96	#95	—	53	8	no	—
97	C₂×Q₈	—	106	4	no	—
98	C₂×Q₈	—	212	4	no	—
99	C₂²×C₄	—	53	4	no	—
100	C₂²×C₄	—	53	4	no	—
101	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
102	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
103	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
104	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
105	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
106	C₄⋊C₄	—	2	4	no	—
107	C₄⋊C₄	—	2	4	no	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	106	4	no	—
111	C₄⋊C₄	—	106	4	no	—
112	C₄⋊C₄	—	212	8	no	—
113	C₄²	—	53	8	no	—
114	C₄²	C₂²⋊D₅₃	1	4	yes	—
115	C₂³	C₂³×D₅₃	1	1	yes	characteristic
116	Q₈	—	212	8	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	53	8	no	—
124	C₂×C₄	—	106	4	no	—
125	C₂×C₄	—	106	4	no	—
126	C₂×C₄	—	106	4	no	—
127	C₂×C₄	—	106	8	no	—
128	C₂×C₄	Q₈×D₅₃	1	4	yes	—
129	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
130	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
131	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
132	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
133	C₂²	C₂×Q₈×D₅₃	1	2	yes	—
134	C₂²	#13133263	1	2	yes	—
135	C₂²	#13133264	1	2	yes	—
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	4	2	no	—
139	C₄	—	4	2	no	—
140	C₄	—	106	8	no	—
141	C₄	—	212	4	no	—
142	C₄	#13133215	1	4	yes	—
143	C₂	C₂×#13133215	1	1	yes	characteristic
144	C₂	#16811187	1	1	yes	characteristic
145	C₂	#16811194	1	1	yes	characteristic
146	C₂	#16811253	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133217

C₂

yes

maximal, characteristic

#16810553

C₂

yes

maximal

#16810543

C₂

yes

maximal

#16810492

C₂

yes

maximal

#16810488

C₂

yes

maximal

#16810469

C₂

yes

maximal, characteristic

C₂×#125×C₅₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₂²

yes

—

#13133217

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×#11719919

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

#13133124

C₂²

yes

—

#13133124

C₂²

yes

—

#13133124

C₂²

yes

—

#125×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

D₄

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

C₄×Dic₅₃

Q₈

yes

complemented

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×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×Q₈

yes

—

Dic₁₀₆

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂²×C₅₃

C₂²×Q₈

yes

—

C₂²×C₅₃

H(5,𝔽₂)

yes

—

C₂²×C₅₃

2₋¹⁺⁴

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄⋊Q₈

yes

—

Dic₅₃

—

Dic₅₃

—

#2660

—

maximal, Sylow

C₂×C₅₃

C₂×C₄⋊Q₈

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₄⋊Q₈

—

C₂×#125

D₅₃

yes

—

—

—

—

—

C₅₃

#2660

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₄⋊Q₈

—

106

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

characteristic

C₂×C₄⋊C₄

C₂×D₅₃

yes

characteristic

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₃

yes

—

—

—

C₂×Q₈

—

106

—

C₂×Q₈

—

212

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

C₂²×D₅₃

yes

—

102

C₂²×C₄

C₂²×D₅₃

yes

—

103

C₂²×C₄

C₂²×D₅₃

yes

characteristic

104

C₂²×C₄

C₂²×D₅₃

yes

characteristic

105

C₂²×C₄

C₂²×D₅₃

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

106

—

111

C₄⋊C₄

—

106

—

112

C₄⋊C₄

—

212

—

113

C₄²

—

114

C₄²

C₂²⋊D₅₃

yes

—

115

C₂³

C₂³×D₅₃

yes

characteristic

116

Q₈

—

212

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

106

—

125

C₂×C₄

—

106

—

126

C₂×C₄

—

106

—

127

C₂×C₄

—

106

—

128

C₂×C₄

Q₈×D₅₃

yes

—

129

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

130

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

131

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

132

C₂²

C₂²×C₂²⋊D₅₃

yes

characteristic

133

C₂²

C₂×Q₈×D₅₃

yes

—

134

C₂²

#13133263

yes

—

135

C₂²

#13133264

yes

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

106

—

141

C₄

—

212

—

142

C₄

#13133215

yes

—

143

C₂

C₂×#13133215

yes

characteristic

144

C₂

#16811187

yes

characteristic

145

C₂

#16811194

yes

characteristic

146

C₂

#16811253

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂⁴
Fitting	C₂×#125×C₅₃	C₂
hypercenter	C₂×#125	D₅₃
hypocenter	C₅₃	#2660
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₃

commutator, socle

C₂³×C₅₃

C₂⁴

Fitting

C₂×#125×C₅₃

C₂

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
40. C₄×Dic₅₃	Q₈	semidirect
81. C₅₃	#2660	semidirect

H ≅ G/N

type

40. C₄×Dic₅₃

Q₈

semidirect

81. C₅₃

#2660

semidirect

Subgroups — #20280836

Conjugacy classes of subgroups

Complements