Group #20276160 — subgroups

Subgroups — #20276160

Order 6784.

198 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 65 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×#13133149	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810624	C₂	1	2	yes	maximal
3	#16810391	C₂	1	2	yes	maximal, complemented
4	#16810388	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#408	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
8	#13133149	—	2	2	no	—
9	#13133141	—	2	2	no	—
10	#13133138	—	2	2	no	—
11	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²⋊C₈×C₅₃	—	2	2	no	—
15	#41×C₅₃	—	2	2	no	—
16	#41×C₅₃	—	2	2	no	—
17	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
18	C₂×D₄×C₅₃	—	2	1	no	—
19	C₂×D₄×C₅₃	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₅₃	—	2	1	no	—
21	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
22	C₂²×Dic₅₃	—	2	2	no	—
23	M₁₆×C₅₃	—	2	2	no	—
24	C₂×C₈×C₅₃	—	2	2	no	—
25	C₂×C₈×C₅₃	—	2	1	no	—
26	C₂×C₈×C₅₃	D₄	1	2	yes	—
27	C₄⋊Dic₅₃	—	2	2	no	—
28	C₄⋊Dic₅₃	—	2	2	no	—
29	#11719917	—	2	2	no	—
30	#11719917	—	2	2	no	—
31	C₂×#88322	—	2	2	no	—
32	C₂³×C₅₃	—	2	1	no	—
33	C₂³×C₅₃	#41	1	1	yes	characteristic
34	Q₈×C₅₃	—	2	2	no	—
35	D₄×C₅₃	—	2	2	no	—
36	D₄×C₅₃	—	4	2	no	—
37	C₂×C₄×C₅₃	—	2	1	no	—
38	C₂×C₄×C₅₃	—	4	1	no	—
39	C₂×C₄×C₅₃	—	4	1	no	—
40	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₅₃	#41	1	2	yes	—
43	C₂×C₄×C₅₃	#41	1	2	yes	—
44	C₂×Dic₅₃	—	4	2	no	—
45	C₂×Dic₅₃	—	8	2	no	—
46	C₈×C₅₃	—	2	2	no	—
47	C₈×C₅₃	—	4	1	no	—
48	#88322	—	4	2	no	—
49	C₂²×C₅₃	—	2	2	no	—
50	C₂²×C₅₃	—	4	1	no	—
51	C₂²×C₅₃	—	4	2	no	—
52	C₂²×C₅₃	#121	1	1	yes	characteristic
53	C₂²×C₅₃	#126	1	2	yes	—
54	C₄×C₅₃	—	4	1	no	—
55	C₄×C₅₃	#123	1	1	yes	characteristic
56	C₄×C₅₃	#123	1	2	yes	—
57	C₄×C₅₃	#124	1	1	yes	characteristic
58	Dic₅₃	—	8	2	no	—
59	#2079	—	53	1	no	maximal, Sylow
60	C₂×C₅₃	—	2	2	no	—
61	C₂×C₅₃	—	8	1	no	—
62	C₂×C₅₃	#397	1	1	yes	characteristic
63	C₂×C₅₃	#471	1	2	yes	—
64	C₂×#106	—	53	1	no	—
65	#417	—	53	2	no	—
66	#408	D₅₃	1	1	yes	characteristic
67	#343	—	53	2	no	—
68	#341	—	53	1	no	—
69	C₅₃	#2079	1	1	yes	characteristic, complemented, Sylow
70	C₂×#41	C₂×D₅₃	1	1	yes	characteristic
71	C₂×M₁₆	—	53	2	no	—
72	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
73	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
74	C₂×C₄⋊C₄	—	53	2	no	—
75	#106	—	106	2	no	—
76	#103	—	106	2	no	—
77	#102	—	106	2	no	—
78	C₂²⋊C₈	—	2	2	no	—
79	#41	—	2	2	no	—
80	#41	—	2	2	no	—
81	C₂×Q₈	D₂₁₂	1	1	yes	characteristic
82	C₂×D₄	—	2	1	no	—
83	C₂×D₄	D₂₁₂	1	1	yes	characteristic
84	C₂²×C₄	—	2	1	no	—
85	C₂²×C₄	—	106	2	no	—
86	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
87	M₁₆	—	2	2	no	—
88	M₁₆	—	106	2	no	—
89	M₁₆	—	106	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	106	2	no	—
93	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
94	C₄⋊C₄	—	106	2	no	—
95	C₄⋊C₄	—	106	2	no	—
96	C₂³	—	2	1	no	—
97	C₂³	#11719945	1	1	yes	characteristic
98	Q₈	—	2	2	no	—
99	D₄	—	2	2	no	—
100	D₄	—	4	2	no	—
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	4	1	no	—
103	C₂×C₄	—	4	1	no	—
104	C₂×C₄	—	424	2	no	—
105	C₂×C₄	—	212	2	no	—
106	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
107	C₂×C₄	#11719947	1	2	yes	—
108	C₂×C₄	#11719947	1	2	yes	—
109	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
110	C₈	—	2	2	no	—
111	C₈	—	4	1	no	—
112	C₈	—	212	2	no	—
113	C₂²	—	2	2	no	—
114	C₂²	—	4	1	no	—
115	C₂²	—	4	2	no	—
116	C₂²	#13133191	1	1	yes	characteristic
117	C₂²	#13133210	1	2	yes	—
118	C₄	—	4	1	no	—
119	C₄	—	424	2	no	—
120	C₄	#13133198	1	1	yes	characteristic
121	C₄	#13133201	1	2	yes	—
122	C₄	#13133204	1	1	yes	characteristic
123	C₂	—	2	2	no	—
124	C₂	—	8	1	no	—
125	C₂	#16810548	1	1	yes	characteristic
126	C₂	#16810886	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133149

C₂

yes

maximal, characteristic, complemented

#16810624

C₂

yes

maximal

#16810391

C₂

yes

maximal, complemented

#16810388

C₂

yes

maximal, characteristic, complemented

C₅₃×#408

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

#13133149

—

#13133141

—

#13133138

—

C₂×#41×C₅₃

C₂²

yes

characteristic

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂²⋊C₈×C₅₃

—

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

D₄

yes

characteristic

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719917

—

#11719917

—

C₂×#88322

—

C₂³×C₅₃

—

C₂³×C₅₃

#41

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#121

yes

characteristic

C₂²×C₅₃

#126

yes

—

C₄×C₅₃

—

C₄×C₅₃

#123

yes

characteristic

C₄×C₅₃

#123

yes

—

C₄×C₅₃

#124

yes

characteristic

Dic₅₃

—

#2079

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#397

yes

characteristic

C₂×C₅₃

#471

yes

—

C₂×#106

—

#417

—

#408

D₅₃

yes

characteristic

#343

—

#341

—

C₅₃

#2079

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₅₃

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

characteristic

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂×C₄⋊C₄

—

#106

—

106

—

#103

—

106

—

#102

—

106

—

C₂²⋊C₈

—

#41

—

#41

—

C₂×Q₈

D₂₁₂

yes

characteristic

C₂×D₄

—

C₂×D₄

D₂₁₂

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

106

—

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

C₂²⋊D₅₃

yes

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₂³

—

C₂³

#11719945

yes

characteristic

Q₈

—

D₄

—

100

D₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

424

—

105

C₂×C₄

—

212

—

106

C₂×C₄

C₂×D₂₁₂

yes

characteristic

107

C₂×C₄

#11719947

yes

—

108

C₂×C₄

#11719947

yes

—

109

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

110

C₈

—

111

C₈

—

112

C₈

—

212

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

#13133191

yes

characteristic

117

C₂²

#13133210

yes

—

118

C₄

—

119

C₄

—

424

—

120

C₄

#13133198

yes

characteristic

121

C₄

#13133201

yes

—

122

C₄

#13133204

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

#16810548

yes

characteristic

126

C₂

#16810886

yes

—

	subgroup	G/H
center	C₂²	#13133191
commutator	C₂²×C₄×C₅₃	C₂³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#408	C₂
socle	C₂²×C₅₃	#121
hypercenter	#408	D₅₃
hypocenter	C₅₃	#2079
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #408

subgroup

G/H

center

C₂²

#13133191

commutator

C₂²×C₄×C₅₃

C₂³

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#408

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #408

N	H ≅ G/N	type
1. C₂×#13133149	C₂	semidirect
3. #16810391	C₂	semidirect
4. #16810388	C₂	semidirect
69. C₅₃	#2079	semidirect

H ≅ G/N

type

1. C₂×#13133149

C₂

semidirect

3. #16810391

C₂

semidirect

4. #16810388

C₂

semidirect

69. C₅₃

#2079

semidirect

Subgroups — #20276160

Conjugacy classes of subgroups

Complements