Group #20276134 — subgroups

Subgroups — #20276134

Order 6784.

222 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 79 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×#13133140	C₂	1	2	yes	maximal
2	#16810529	C₂	1	1	yes	maximal, characteristic
3	#16810385	C₂	1	2	yes	maximal
4	#16810378	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#382	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₀₆	C₂²	1	1	yes	characteristic
7	C₂²×#88322	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
9	#13133140	—	2	4	no	—
10	#13133124	—	2	2	no	—
11	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
12	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
13	#95×C₅₃	—	2	2	no	—
14	C₂²×C₄×C₅₃	—	2	2	no	—
15	C₂²×C₄×C₅₃	D₄	1	2	yes	—
16	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
17	C₂²×Dic₅₃	—	2	1	no	—
18	C₂²×Dic₅₃	—	2	1	no	—
19	C₂×Dic₁₀₆	—	2	2	no	—
20	C₂×Dic₁₀₆	—	2	2	no	—
21	C₂×Dic₁₀₆	—	2	1	no	—
22	C₂×Dic₁₀₆	D₄	1	2	yes	—
23	C₄⋊C₄×C₅₃	—	2	2	no	—
24	C₄⋊C₄×C₅₃	—	2	2	no	—
25	C₄²×C₅₃	—	2	2	no	—
26	C₄⋊Dic₅₃	—	2	2	no	—
27	C₂×#88322	—	2	2	no	—
28	C₂×#88322	—	2	2	no	—
29	C₂×#88322	—	2	2	no	—
30	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₅₃	—	2	2	no	—
32	C₂×C₄×C₅₃	—	2	2	no	—
33	C₂×C₄×C₅₃	—	2	2	no	—
34	C₂×C₄×C₅₃	—	4	2	no	—
35	C₂×C₄×C₅₃	—	4	2	no	—
36	C₂×C₄×C₅₃	—	4	2	no	—
37	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₅₃	#41	1	2	yes	—
39	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
40	C₂×C₄×C₅₃	#41	1	2	yes	—
41	C₂×Dic₅₃	—	2	2	no	—
42	C₂×Dic₅₃	—	4	1	no	—
43	C₂×Dic₅₃	—	4	1	no	—
44	C₂×Dic₅₃	—	4	1	no	—
45	C₂×Dic₅₃	—	4	2	no	—
46	Dic₁₀₆	—	2	4	no	—
47	Dic₁₀₆	—	4	2	no	—
48	#88322	—	4	2	no	—
49	C₂²×C₅₃	#121	1	1	yes	characteristic
50	C₂²×C₅₃	#123	1	1	yes	characteristic
51	C₂²×C₅₃	#124	1	1	yes	characteristic
52	C₂²×C₅₃	#135	1	2	yes	—
53	C₂²×C₅₃	#137	1	2	yes	—
54	C₄×C₅₃	—	4	2	no	—
55	C₄×C₅₃	—	8	2	no	—
56	C₄×C₅₃	#123	1	2	yes	—
57	C₄×C₅₃	#126	1	2	yes	—
58	Dic₅₃	—	4	2	no	—
59	Dic₅₃	—	8	1	no	—
60	#2074	—	53	1	no	maximal, Sylow
61	C₂×C₅₃	#397	1	1	yes	characteristic
62	C₂×C₅₃	#464	1	2	yes	—
63	C₂×C₅₃	#485	1	2	yes	—
64	C₂×C₅₃	#488	1	1	yes	characteristic
65	C₂×C₅₃	#491	1	1	yes	characteristic
66	C₂×#103	—	53	2	no	—
67	#391	—	53	1	no	—
68	#382	D₅₃	1	1	yes	characteristic
69	#340	—	53	2	no	—
70	#336	—	53	1	no	—
71	C₅₃	#2074	1	1	yes	characteristic, complemented, Sylow
72	C₂²×Q₈	—	53	1	no	—
73	C₂²×C₈	—	53	2	no	—
74	C₂×C₄⋊C₄	—	53	1	no	—
75	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
76	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
77	#103	—	106	4	no	—
78	#95	—	2	2	no	—
79	#95	—	106	2	no	—
80	C₂×Q₈	—	53	2	no	—
81	C₂×Q₈	—	106	1	no	—
82	C₂×Q₈	—	106	2	no	—
83	C₂×Q₈	—	106	2	no	—
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	—	106	1	no	—
86	C₂²×C₄	—	106	1	no	—
87	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
88	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
89	C₂×C₈	—	106	2	no	—
90	C₂×C₈	—	106	2	no	—
91	C₂×C₈	—	106	2	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	106	2	no	—
95	C₄²	—	2	2	no	—
96	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
97	Q₈	—	106	4	no	—
98	Q₈	—	212	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	106	2	no	—
106	C₂×C₄	—	212	1	no	—
107	C₂×C₄	—	212	2	no	—
108	C₂×C₄	—	212	1	no	—
109	C₂×C₄	—	212	1	no	—
110	C₂×C₄	#11719945	1	2	yes	—
111	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
112	C₂×C₄	#11719947	1	1	yes	characteristic
113	C₂×C₄	#11719949	1	2	yes	—
114	C₈	—	212	2	no	—
115	C₂²	#13133188	1	1	yes	characteristic
116	C₂²	#13133200	1	1	yes	characteristic
117	C₂²	#13133206	1	1	yes	characteristic
118	C₂²	#13133237	1	2	yes	—
119	C₂²	#13133247	1	2	yes	—
120	C₄	—	4	2	no	—
121	C₄	—	8	2	no	—
122	C₄	—	424	1	no	—
123	C₄	—	212	2	no	—
124	C₄	#13133202	1	2	yes	—
125	C₄	#13133212	1	2	yes	—
126	C₂	#16810549	1	1	yes	characteristic
127	C₂	#16810842	1	2	yes	—
128	C₂	#16810964	1	2	yes	—
129	C₂	#16810980	1	1	yes	characteristic
130	C₂	#16810993	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133140

C₂

yes

maximal

#16810529

C₂

yes

maximal, characteristic

#16810385

C₂

yes

maximal

#16810378

C₂

yes

maximal, characteristic

C₅₃×#382

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₂²

yes

characteristic

C₂²×#88322

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

characteristic

#13133140

—

#13133124

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#95×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

D₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂×D₄

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

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

#88322

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#137

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#123

yes

—

C₄×C₅₃

#126

yes

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#485

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

D₅₃

yes

characteristic

#340

—

#336

—

C₅₃

#2074

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#103

—

106

—

#95

—

#95

—

106

—

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄²

—

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

Q₈

—

106

—

Q₈

—

212

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

—

106

C₂×C₄

—

212

—

107

C₂×C₄

—

212

—

108

C₂×C₄

—

212

—

109

C₂×C₄

—

212

—

110

C₂×C₄

#11719945

yes

—

111

C₂×C₄

D₄×D₅₃

yes

characteristic

112

C₂×C₄

#11719947

yes

characteristic

113

C₂×C₄

#11719949

yes

—

114

C₈

—

212

—

115

C₂²

#13133188

yes

characteristic

116

C₂²

#13133200

yes

characteristic

117

C₂²

#13133206

yes

characteristic

118

C₂²

#13133237

yes

—

119

C₂²

#13133247

yes

—

120

C₄

—

121

C₄

—

122

C₄

—

424

—

123

C₄

—

212

—

124

C₄

#13133202

yes

—

125

C₄

#13133212

yes

—

126

C₂

#16810549

yes

characteristic

127

C₂

#16810842

yes

—

128

C₂

#16810964

yes

—

129

C₂

#16810980

yes

characteristic

130

C₂

#16810993

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₃
commutator	C₂²×C₄×C₅₃	C₂³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#382	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	#382	D₅₃
hypocenter	C₅₃	#2074
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₃

commutator

C₂²×C₄×C₅₃

C₂³

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#382

C₂

socle

C₂³×C₅₃

C₂×D₄

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

N	H ≅ G/N	type
71. C₅₃	#2074	semidirect

H ≅ G/N

type

71. C₅₃

#2074

semidirect

Subgroups — #20276134

Conjugacy classes of subgroups

Complements