Group #20275459 — subgroups

Subgroups — #20275459

Order 6784.

240 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 113 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×#13133151	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133127	C₂	1	1	yes	maximal, characteristic
3	#16810522	C₂	1	2	yes	maximal
4	#16810385	C₂	1	2	yes	maximal
5	C₂×#106×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#88322	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₅₃	—	2	2	no	—
8	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
9	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
10	#13133151	—	2	2	no	—
11	#13133127	—	2	2	no	—
12	#13133124	—	2	2	no	—
13	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
15	#106×C₅₃	C₄	1	4	yes	complemented
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	C₂²×Dic₅₃	D₄	1	1	yes	characteristic
21	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
22	C₂×C₈×C₅₃	D₄	1	2	yes	—
23	C₂×C₈×C₅₃	Q₈	1	2	yes	—
24	C₄⋊C₄×C₅₃	—	2	2	no	—
25	C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
26	C₄⋊Dic₅₃	—	2	2	no	—
27	C₄⋊Dic₅₃	—	2	2	no	—
28	C₄⋊Dic₅₃	—	4	4	no	—
29	C₄×Dic₅₃	—	2	1	no	—
30	C₄×Dic₅₃	—	2	1	no	—
31	C₂×#88322	—	2	1	no	—
32	C₂×#88322	—	2	1	no	—
33	C₂×#88322	Q₈	1	2	yes	—
34	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₅₃	—	2	4	no	—
36	C₂×C₄×C₅₃	—	4	2	no	—
37	C₂×C₄×C₅₃	—	4	2	no	—
38	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
43	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
44	C₂×Dic₅₃	—	2	1	no	—
45	C₂×Dic₅₃	—	2	1	no	—
46	C₂×Dic₅₃	—	2	2	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	C₈×C₅₃	C₄⋊C₄	1	4	yes	complemented
52	#88322	—	2	2	no	—
53	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
54	C₂²×C₅₃	#121	1	1	yes	characteristic
55	C₂²×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
56	C₂²×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
57	C₂²×C₅₃	#137	1	2	yes	—
58	C₄×C₅₃	—	4	4	no	—
59	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
61	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
62	C₄×C₅₃	#125	1	1	yes	characteristic
63	Dic₅₃	—	4	1	no	—
64	Dic₅₃	—	4	1	no	—
65	Dic₅₃	—	8	2	no	—
66	#1944	—	53	1	no	maximal, Sylow
67	C₂×C₅₃	#389	1	1	yes	characteristic
68	C₂×C₅₃	#440	1	2	yes	—
69	C₂×C₅₃	#469	1	1	yes	characteristic
70	C₂×C₅₃	#470	1	1	yes	characteristic
71	C₂×C₅₃	#501	1	2	yes	—
72	C₂×#107	—	53	1	no	—
73	C₂×#106	D₅₃	1	1	yes	characteristic
74	C₂×#97	—	53	1	no	—
75	#389	—	53	2	no	—
76	#340	—	53	2	no	—
77	C₅₃	#1944	1	1	yes	characteristic, complemented, Sylow
78	C₂²×C₈	—	53	1	no	—
79	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
80	C₂×C₄⋊C₄	—	53	2	no	—
81	C₂×C₄⋊C₄	—	106	2	no	—
82	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
83	C₂×C₄²	—	53	1	no	—
84	#107	—	106	2	no	—
85	#106	Dic₅₃	1	4	yes	complemented
86	#97	—	106	2	no	—
87	#95	—	106	2	no	—
88	C₂²×C₄	—	2	2	no	—
89	C₂²×C₄	—	53	1	no	—
90	C₂²×C₄	—	53	1	no	—
91	C₂²×C₄	—	106	2	no	—
92	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
93	C₂×C₈	—	53	2	no	—
94	C₂×C₈	—	106	1	no	—
95	C₂×C₈	—	106	1	no	—
96	C₂×C₈	Dic₁₀₆	1	2	yes	—
97	C₂×C₈	D₂₁₂	1	2	yes	—
98	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	106	2	no	—
101	C₄⋊C₄	—	106	2	no	—
102	C₄⋊C₄	—	212	4	no	—
103	C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
104	C₄²	—	106	1	no	—
105	C₄²	—	106	1	no	—
106	C₂³	D₄×D₅₃	1	1	yes	characteristic
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	106	1	no	—
111	C₂×C₄	—	106	1	no	—
112	C₂×C₄	—	106	2	no	—
113	C₂×C₄	—	106	2	no	—
114	C₂×C₄	—	212	2	no	—
115	C₂×C₄	—	212	2	no	—
116	C₂×C₄	—	212	2	no	—
117	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
118	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
119	C₂×C₄	#11719947	1	1	yes	characteristic
120	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
121	C₂×C₄	#11719949	1	1	yes	characteristic
122	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
123	C₈	—	106	2	no	—
124	C₈	C₄⋊Dic₅₃	1	4	yes	complemented
125	C₂²	D₄×Dic₅₃	1	1	yes	characteristic
126	C₂²	#13133192	1	1	yes	characteristic
127	C₂²	#13133216	1	1	yes	characteristic
128	C₂²	#13133244	1	2	yes	—
129	C₂²	#13133246	1	2	yes	—
130	C₄	—	4	4	no	—
131	C₄	—	424	2	no	—
132	C₄	—	212	1	no	—
133	C₄	—	212	1	no	—
134	C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
135	C₄	Q₈×Dic₅₃	1	1	yes	characteristic
136	C₄	#13133197	1	1	yes	characteristic
137	C₄	#13133208	1	1	yes	characteristic
138	C₂	#16810522	1	1	yes	characteristic
139	C₂	#16810724	1	2	yes	—
140	C₂	#16810877	1	1	yes	characteristic
141	C₂	#16810882	1	1	yes	characteristic
142	C₂	#16811030	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133151

C₂

yes

maximal, characteristic

C₂×#13133127

C₂

yes

maximal, characteristic

#16810522

C₂

yes

maximal

#16810385

C₂

yes

maximal

C₂×#106×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

—

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

#106×C₅₃

C₄

yes

complemented

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

D₄

yes

characteristic

C₂²×Dic₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

Q₈

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

Q₈

yes

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

C₄⋊C₄

yes

complemented

#88322

—

C₂²×C₅₃

C₄×D₄

yes

characteristic

C₂²×C₅₃

#121

yes

characteristic

C₂²×C₅₃

C₄⋊Q₈

yes

characteristic

C₂²×C₅₃

AGL(1,ℤ/8)

yes

—

C₂²×C₅₃

#137

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

C₄×Q₈

yes

characteristic

C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₄×C₅₃

#125

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×#107

—

C₂×#106

D₅₃

yes

characteristic

C₂×#97

—

#389

—

#340

—

C₅₃

#1944

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

—

#107

—

106

—

#106

Dic₅₃

yes

complemented

#97

—

106

—

#95

—

106

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

Dic₁₀₆

yes

—

C₂×C₈

D₂₁₂

yes

—

C₂×C₈

C₂×Dic₅₃

yes

—

C₄⋊C₄

—

100

C₄⋊C₄

—

106

—

101

C₄⋊C₄

—

106

—

102

C₄⋊C₄

—

212

—

103

C₄⋊C₄

C₂×Dic₅₃

yes

—

104

C₄²

—

106

—

105

C₄²

—

106

—

106

C₂³

D₄×D₅₃

yes

characteristic

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

106

—

111

C₂×C₄

—

106

—

112

C₂×C₄

—

106

—

113

C₂×C₄

—

106

—

114

C₂×C₄

—

212

—

115

C₂×C₄

—

212

—

116

C₂×C₄

—

212

—

117

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

118

C₂×C₄

C₂×D₂₁₂

yes

characteristic

119

C₂×C₄

#11719947

yes

characteristic

120

C₂×C₄

Q₈×D₅₃

yes

characteristic

121

C₂×C₄

#11719949

yes

characteristic

122

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

123

C₈

—

106

—

124

C₈

C₄⋊Dic₅₃

yes

complemented

125

C₂²

D₄×Dic₅₃

yes

characteristic

126

C₂²

#13133192

yes

characteristic

127

C₂²

#13133216

yes

characteristic

128

C₂²

#13133244

yes

—

129

C₂²

#13133246

yes

—

130

C₄

—

131

C₄

—

424

—

132

C₄

—

212

—

133

C₄

—

212

—

134

C₄

C₂×C₄⋊Dic₅₃

yes

characteristic

135

C₄

Q₈×Dic₅₃

yes

characteristic

136

C₄

#13133197

yes

characteristic

137

C₄

#13133208

yes

characteristic

138

C₂

#16810522

yes

characteristic

139

C₂

#16810724

yes

—

140

C₂

#16810877

yes

characteristic

141

C₂

#16810882

yes

characteristic

142

C₂

#16811030

yes

—

	subgroup	G/H
center	C₂³	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×#106×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×#106	D₅₃
hypocenter	C₅₃	#1944
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×#106×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
15. #106×C₅₃	C₄	semidirect
51. C₈×C₅₃	C₄⋊C₄	semidirect
77. C₅₃	#1944	semidirect
85. #106	Dic₅₃	semidirect
124. C₈	C₄⋊Dic₅₃	semidirect

H ≅ G/N

type

15. #106×C₅₃

C₄

semidirect

51. C₈×C₅₃

C₄⋊C₄

semidirect

77. C₅₃

#1944

semidirect

85. #106

Dic₅₃

semidirect

124. C₈

C₄⋊Dic₅₃

semidirect

Subgroups — #20275459

Conjugacy classes of subgroups

Complements