Group #20274191 — subgroups

Subgroups — #20274191

Order 6784.

198 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 55 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	#16810815	C₂	1	2	yes	maximal, complemented
2	#16810702	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810374	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810367	C₂	1	2	yes	maximal, complemented
5	C₅₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11719922	—	2	2	no	—
7	C₂×#11719912	—	2	1	no	—
8	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
9	C₄⋊#88322	C₂²	1	2	yes	—
10	C₄⋊#88322	C₂²	1	1	yes	characteristic
11	#13133138	—	2	2	no	—
12	#13133129	—	2	1	no	—
13	C₂×D₈×C₅₃	—	2	1	no	—
14	C₂×D₈×C₅₃	—	2	1	no	—
15	#127×C₅₃	C₂²	1	2	yes	—
16	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×D₄×C₅₃	—	2	2	no	—
18	C₂×D₄×C₅₃	—	2	2	no	—
19	C₂×D₄×C₅₃	—	2	2	no	—
20	C₂×D₂₁₂	D₄	1	1	yes	characteristic
21	C₂×C₄×D₅₃	—	2	1	no	—
22	D₈×C₅₃	—	4	2	no	—
23	D₈×C₅₃	—	4	2	no	—
24	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
25	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
26	C₄²×C₅₃	C₂³	1	1	yes	characteristic
27	#11719922	—	4	4	no	—
28	#11719921	—	2	1	no	—
29	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
30	C₂×#88322	—	2	1	no	—
31	C₂×#88322	—	2	2	no	—
32	#11719912	—	4	2	no	—
33	C₂³×C₅₃	—	4	2	no	—
34	C₂²×D₅₃	—	2	1	no	—
35	D₄×C₅₃	—	4	2	no	—
36	D₄×C₅₃	—	4	2	no	—
37	D₄×C₅₃	—	4	2	no	—
38	D₄×C₅₃	—	4	2	no	—
39	D₄×C₅₃	—	8	2	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	1	yes	characteristic
43	C₂×Dic₅₃	—	2	1	no	—
44	D₂₁₂	—	2	2	no	—
45	C₄×D₅₃	—	4	1	no	—
46	C₄×D₅₃	—	4	1	no	—
47	C₈×C₅₃	—	2	1	no	—
48	C₈×C₅₃	D₈	1	2	yes	complemented
49	#88322	—	4	1	no	—
50	#88322	—	4	2	no	—
51	C₂²×C₅₃	—	8	2	no	—
52	C₂²×C₅₃	—	8	2	no	—
53	C₂²×C₅₃	—	8	2	no	—
54	C₂²×C₅₃	#121	1	1	yes	characteristic
55	C₂×D₅₃	—	4	1	no	—
56	C₂×D₅₃	—	4	1	no	—
57	C₂×D₅₃	—	4	1	no	—
58	C₄×C₅₃	—	2	1	no	—
59	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
60	C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
61	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
62	Dic₅₃	—	4	1	no	—
63	#1687	—	53	1	no	maximal, Sylow
64	C₂×C₅₃	—	16	2	no	—
65	C₂×C₅₃	#459	1	1	yes	characteristic
66	C₂×C₅₃	#469	1	1	yes	characteristic
67	C₂×C₅₃	#472	1	1	yes	characteristic
68	D₅₃	—	8	1	no	—
69	#493	D₅₃	1	1	yes	characteristic, complemented
70	#459	—	53	2	no	—
71	#436	—	53	1	no	—
72	#334	—	53	1	no	—
73	#331	—	53	2	no	—
74	C₅₃	#1687	1	1	yes	characteristic, complemented, Sylow
75	C₂×D₈	—	2	1	no	—
76	C₂×D₈	—	2	1	no	—
77	C₂×D₈	—	106	2	no	—
78	C₂×M₁₆	—	106	1	no	—
79	#127	C₂×D₅₃	1	2	yes	—
80	C₄×D₄	—	53	1	no	—
81	C₄⋊C₈	—	53	1	no	—
82	C₄⋊C₈	—	53	2	no	—
83	#102	—	106	2	no	—
84	C₂²⋊C₈	—	106	1	no	—
85	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
86	C₂×D₄	—	2	2	no	—
87	C₂×D₄	—	2	2	no	—
88	C₂×D₄	—	2	2	no	—
89	C₂×D₄	—	53	1	no	—
90	C₂²×C₄	—	106	1	no	—
91	D₈	—	4	2	no	—
92	D₈	—	4	2	no	—
93	D₈	—	212	4	no	—
94	M₁₆	—	212	2	no	—
95	C₂×C₈	—	106	2	no	—
96	C₂×C₈	—	106	1	no	—
97	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
98	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
99	C₄⋊C₄	—	53	1	no	—
100	C₂²⋊C₄	—	106	1	no	—
101	C₄²	C₂²×D₅₃	1	1	yes	characteristic
102	C₂³	—	4	2	no	—
103	C₂³	—	106	1	no	—
104	D₄	—	4	2	no	—
105	D₄	—	4	2	no	—
106	D₄	—	4	2	no	—
107	D₄	—	4	2	no	—
108	D₄	—	8	2	no	—
109	D₄	—	106	2	no	—
110	C₂×C₄	—	106	1	no	—
111	C₂×C₄	—	212	1	no	—
112	C₂×C₄	—	212	1	no	—
113	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
114	C₂×C₄	#11719947	1	1	yes	characteristic
115	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
116	C₈	—	2	1	no	—
117	C₈	—	212	1	no	—
118	C₈	—	212	2	no	—
119	C₈	#11719922	1	2	yes	complemented
120	C₂²	—	8	2	no	—
121	C₂²	—	8	2	no	—
122	C₂²	—	8	2	no	—
123	C₂²	—	212	1	no	—
124	C₂²	—	212	1	no	—
125	C₂²	—	212	1	no	—
126	C₂²	#13133188	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	212	1	no	—
129	C₄	C₂×#11719922	1	1	yes	characteristic
130	C₄	#13133241	1	2	yes	—
131	C₄	#13133243	1	1	yes	characteristic
132	C₂	—	16	2	no	—
133	C₂	—	424	1	no	—
134	C₂	#16810815	1	1	yes	characteristic
135	C₂	#16810873	1	1	yes	characteristic
136	C₂	#16810890	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810815

C₂

yes

maximal, complemented

#16810702

C₂

yes

maximal, characteristic, complemented

#16810374

C₂

yes

maximal, characteristic, complemented

#16810367

C₂

yes

maximal, complemented

C₅₃×#493

C₂

yes

maximal, characteristic, complemented

C₂×#11719922

—

C₂×#11719912

—

C₄×D₂₁₂

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

—

C₄⋊#88322

C₂²

yes

characteristic

#13133138

—

#13133129

—

C₂×D₈×C₅₃

—

C₂×D₈×C₅₃

—

#127×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₂₁₂

D₄

yes

characteristic

C₂×C₄×D₅₃

—

D₈×C₅₃

—

D₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

#11719922

—

#11719921

—

C₄⋊Dic₅₃

D₄

yes

—

—

—

C₂³×C₅₃

—

C₂²×D₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

C₈×C₅₃

D₈

yes

complemented

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#121

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×D₈

yes

characteristic

C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₄×C₅₃

AGL(1,ℤ/8)

yes

characteristic

Dic₅₃

—

#1687

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

#493

D₅₃

yes

characteristic, complemented

—

—

—

—

C₅₃

#1687

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

106

—

C₂×M₁₆

—

106

—

#127

C₂×D₅₃

yes

—

C₄×D₄

—

C₄⋊C₈

—

C₄⋊C₈

—

#102

—

106

—

C₂²⋊C₈

—

106

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

106

—

D₈

—

D₈

—

D₈

—

212

—

M₁₆

—

212

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₄⋊C₄

—

100

C₂²⋊C₄

—

106

—

101

C₄²

C₂²×D₅₃

yes

characteristic

102

C₂³

—

103

C₂³

—

106

—

104

D₄

—

105

D₄

—

106

D₄

—

107

D₄

—

108

D₄

—

109

D₄

—

106

—

110

C₂×C₄

—

106

—

111

C₂×C₄

—

212

—

112

C₂×C₄

—

212

—

113

C₂×C₄

D₄×D₅₃

yes

characteristic

114

C₂×C₄

#11719947

yes

characteristic

115

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

116

C₈

—

117

C₈

—

212

—

118

C₈

—

212

—

119

C₈

#11719922

yes

complemented

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

212

—

124

C₂²

—

212

—

125

C₂²

—

212

—

126

C₂²

#13133188

yes

characteristic

127

C₄

—

128

C₄

—

212

—

129

C₄

C₂×#11719922

yes

characteristic

130

C₄

#13133241

yes

—

131

C₄

#13133243

yes

characteristic

132

C₂

—

133

C₂

—

424

—

134

C₂

#16810815

yes

characteristic

135

C₂

#16810873

yes

characteristic

136

C₂

#16810890

yes

characteristic

	subgroup	G/H
center	C₂²	#13133188
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#493	C₂
socle	C₂²×C₅₃	#121
hypercenter	#493	D₅₃
hypocenter	C₅₃	#1687
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#13133188

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#493

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #16810815	C₂	semidirect
2. #16810702	C₂	semidirect
3. #16810374	C₂	semidirect
4. #16810367	C₂	semidirect
5. C₅₃×#493	C₂	semidirect
48. C₈×C₅₃	D₈	semidirect
69. #493	D₅₃	semidirect
74. C₅₃	#1687	semidirect
119. C₈	#11719922	semidirect

H ≅ G/N

type

1. #16810815

C₂

semidirect

2. #16810702

C₂

semidirect

3. #16810374

C₂

semidirect

4. #16810367

C₂

semidirect

5. C₅₃×#493

C₂

semidirect

48. C₈×C₅₃

D₈

semidirect

69. #493

D₅₃

semidirect

74. C₅₃

#1687

semidirect

119. C₈

#11719922

semidirect

Subgroups — #20274191

Conjugacy classes of subgroups

Complements