Group #20276128 — subgroups

Subgroups — #20276128

Order 6784.

258 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 81 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×#13133138	C₂	1	2	yes	maximal
2	#16810489	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810385	C₂	1	2	yes	maximal, 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	C₂²	1	2	yes	—
8	#13133138	—	2	4	no	—
9	#13133124	—	2	2	no	—
10	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
13	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
14	C₂⁴×C₅₃	—	2	1	no	—
15	C₂×D₄×C₅₃	—	2	2	no	—
16	C₂×D₄×C₅₃	—	2	2	no	—
17	C₂×D₄×C₅₃	—	2	1	no	—
18	C₂×D₄×C₅₃	D₄	1	2	yes	—
19	C₂²×C₄×C₅₃	—	2	1	no	—
20	C₂²×C₄×C₅₃	D₄	1	2	yes	—
21	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
22	C₂²×Dic₅₃	—	2	2	no	—
23	C₄⋊C₄×C₅₃	—	2	2	no	—
24	C₂²⋊C₄×C₅₃	—	4	4	no	—
25	C₄²×C₅₃	—	2	2	no	—
26	C₄⋊Dic₅₃	—	2	2	no	—
27	C₄⋊Dic₅₃	—	2	2	no	—
28	C₂×#88322	—	2	2	no	—
29	C₂×#88322	—	2	2	no	—
30	C₂×#88322	—	2	2	no	—
31	C₂³×C₅₃	—	2	2	no	—
32	C₂³×C₅₃	—	4	1	no	—
33	C₂³×C₅₃	—	4	1	no	—
34	C₂³×C₅₃	—	4	2	no	—
35	C₂³×C₅₃	—	4	2	no	—
36	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
37	D₄×C₅₃	—	2	4	no	—
38	D₄×C₅₃	—	4	2	no	—
39	C₂×C₄×C₅₃	—	2	2	no	—
40	C₂×C₄×C₅₃	—	2	2	no	—
41	C₂×C₄×C₅₃	—	2	2	no	—
42	C₂×C₄×C₅₃	—	4	1	no	—
43	C₂×C₄×C₅₃	—	4	2	no	—
44	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
46	C₂×C₄×C₅₃	#41	1	2	yes	—
47	C₂×C₄×C₅₃	#41	1	2	yes	—
48	C₂×Dic₅₃	—	4	2	no	—
49	C₂×Dic₅₃	—	4	2	no	—
50	C₂×Dic₅₃	—	4	2	no	—
51	#88322	—	4	2	no	—
52	C₂²×C₅₃	—	4	2	no	—
53	C₂²×C₅₃	—	4	4	no	—
54	C₂²×C₅₃	—	8	2	no	—
55	C₂²×C₅₃	—	8	1	no	—
56	C₂²×C₅₃	—	8	1	no	—
57	C₂²×C₅₃	#121	1	1	yes	characteristic
58	C₂²×C₅₃	#123	1	1	yes	characteristic
59	C₂²×C₅₃	#124	1	1	yes	characteristic
60	C₂²×C₅₃	#135	1	2	yes	—
61	C₂²×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
62	C₄×C₅₃	—	4	2	no	—
63	C₄×C₅₃	—	8	1	no	—
64	C₄×C₅₃	#123	1	2	yes	—
65	C₄×C₅₃	#126	1	2	yes	—
66	Dic₅₃	—	8	2	no	—
67	#2073	—	53	1	no	maximal, Sylow
68	C₂×C₅₃	—	8	2	no	—
69	C₂×C₅₃	#397	1	1	yes	characteristic
70	C₂×C₅₃	#463	1	2	yes	—
71	C₂×C₅₃	#482	1	2	yes	—
72	C₂×C₅₃	#488	1	1	yes	characteristic
73	C₂×C₅₃	#490	1	1	yes	characteristic
74	C₂×#102	—	53	2	no	—
75	#390	D₅₃	1	1	yes	characteristic
76	#382	—	53	1	no	—
77	#340	—	53	2	no	—
78	#336	—	53	1	no	—
79	C₅₃	#2073	1	1	yes	characteristic, complemented, Sylow
80	C₂²×D₄	C₂×D₅₃	1	1	yes	characteristic
81	C₂²×C₈	—	53	2	no	—
82	C₂×C₄⋊C₄	—	53	2	no	—
83	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
84	C₂×C₂²⋊C₄	—	2	2	no	—
85	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
86	#102	—	106	4	no	—
87	#95	—	106	2	no	—
88	C₂⁴	—	2	1	no	—
89	C₂×D₄	—	2	1	no	—
90	C₂×D₄	—	2	2	no	—
91	C₂×D₄	—	2	2	no	—
92	C₂×D₄	D₂₁₂	1	2	yes	—
93	C₂²×C₄	—	2	1	no	—
94	C₂²×C₄	—	106	2	no	—
95	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
96	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
97	C₂×C₈	—	106	2	no	—
98	C₂×C₈	—	106	2	no	—
99	C₂×C₈	—	106	2	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	106	2	no	—
102	C₄⋊C₄	—	106	2	no	—
103	C₂²⋊C₄	—	4	4	no	—
104	C₄²	—	2	2	no	—
105	C₂³	—	2	2	no	—
106	C₂³	—	4	1	no	—
107	C₂³	—	4	1	no	—
108	C₂³	—	4	2	no	—
109	C₂³	—	4	2	no	—
110	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
111	D₄	—	2	4	no	—
112	D₄	—	4	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	212	2	no	—
119	C₂×C₄	—	212	2	no	—
120	C₂×C₄	—	212	2	no	—
121	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
122	C₂×C₄	#11719945	1	1	yes	characteristic
123	C₂×C₄	#11719947	1	2	yes	—
124	C₂×C₄	#11719947	1	2	yes	—
125	C₈	—	212	2	no	—
126	C₂²	—	4	2	no	—
127	C₂²	—	4	4	no	—
128	C₂²	—	8	2	no	—
129	C₂²	—	8	1	no	—
130	C₂²	—	8	1	no	—
131	C₂²	#13133191	1	1	yes	characteristic
132	C₂²	#13133198	1	1	yes	characteristic
133	C₂²	#13133204	1	1	yes	characteristic
134	C₂²	#13133237	1	2	yes	—
135	C₂²	#13133243	1	2	yes	—
136	C₄	—	4	2	no	—
137	C₄	—	8	1	no	—
138	C₄	—	424	2	no	—
139	C₄	#13133201	1	2	yes	—
140	C₄	#13133210	1	2	yes	—
141	C₂	—	8	2	no	—
142	C₂	#16810548	1	1	yes	characteristic
143	C₂	#16810838	1	2	yes	—
144	C₂	#16810944	1	2	yes	—
145	C₂	#16810977	1	1	yes	characteristic
146	C₂	#16810987	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133138

C₂

yes

maximal

#16810489

C₂

yes

maximal, characteristic, complemented

#16810385

C₂

yes

maximal, complemented

#16810378

C₂

yes

maximal, characteristic, complemented

C₅₃×#390

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

#13133138

—

#13133124

—

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

D₄

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

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

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

—

C₂×C₄×C₅₃

C₂×D₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

AGL(1,ℤ/8)

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#123

yes

—

C₄×C₅₃

#126

yes

—

Dic₅₃

—

#2073

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#397

yes

characteristic

C₂×C₅₃

#463

yes

—

C₂×C₅₃

#482

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

yes

—

—

—

C₅₃

#2073

yes

characteristic, complemented, Sylow

C₂²×D₄

C₂×D₅₃

yes

characteristic

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#102

—

106

—

#95

—

106

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

D₂₁₂

yes

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

106

—

102

C₄⋊C₄

—

106

—

103

C₂²⋊C₄

—

104

C₄²

—

105

C₂³

—

106

C₂³

—

107

C₂³

—

108

C₂³

—

109

C₂³

—

110

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

111

D₄

—

112

D₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

212

—

119

C₂×C₄

—

212

—

120

C₂×C₄

—

212

—

121

C₂×C₄

C₂×D₂₁₂

yes

characteristic

122

C₂×C₄

#11719945

yes

characteristic

123

C₂×C₄

#11719947

yes

—

124

C₂×C₄

#11719947

yes

—

125

C₈

—

212

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

#13133191

yes

characteristic

132

C₂²

#13133198

yes

characteristic

133

C₂²

#13133204

yes

characteristic

134

C₂²

#13133237

yes

—

135

C₂²

#13133243

yes

—

136

C₄

—

137

C₄

—

138

C₄

—

424

—

139

C₄

#13133201

yes

—

140

C₄

#13133210

yes

—

141

C₂

—

142

C₂

#16810548

yes

characteristic

143

C₂

#16810838

yes

—

144

C₂

#16810944

yes

—

145

C₂

#16810977

yes

characteristic

146

C₂

#16810987

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	#2073
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
2. #16810489	C₂	semidirect
3. #16810385	C₂	semidirect
4. #16810378	C₂	semidirect
79. C₅₃	#2073	semidirect

H ≅ G/N

type

2. #16810489

C₂

semidirect

3. #16810385

C₂

semidirect

4. #16810378

C₂

semidirect

79. C₅₃

#2073

semidirect

Subgroups — #20276128

Conjugacy classes of subgroups

Complements