Group #20274195 — subgroups

Subgroups — #20274195

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	#16811000	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810819	C₂	1	2	yes	maximal, complemented
3	#16810702	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810366	C₂	1	2	yes	maximal, complemented
5	C₅₃×#334	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄₂₄	—	2	1	no	—
7	C₂×D₄₂₄	—	2	1	no	—
8	C₂×D₄₂₄	—	2	2	no	—
9	C₂×#11719912	—	2	1	no	—
10	#13133214	C₂²	1	2	yes	—
11	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
12	C₄⋊#88322	C₂²	1	1	yes	characteristic
13	#13133139	—	2	2	no	—
14	#13133129	—	2	1	no	—
15	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
16	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×D₂₁₂	—	2	2	no	—
18	C₂×D₂₁₂	—	2	2	no	—
19	C₂×D₂₁₂	—	2	2	no	—
20	C₂×D₂₁₂	D₄	1	1	yes	characteristic
21	C₂×C₄×D₅₃	—	2	1	no	—
22	C₂×C₈×C₅₃	—	2	2	no	—
23	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
25	C₄²×C₅₃	C₂³	1	1	yes	characteristic
26	#11719921	—	2	1	no	—
27	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
28	C₂×#88322	—	2	1	no	—
29	D₄₂₄	—	4	2	no	—
30	D₄₂₄	—	4	4	no	—
31	D₄₂₄	—	4	2	no	—
32	#11719912	—	4	2	no	—
33	C₂²×D₅₃	—	2	1	no	—
34	C₂²×D₅₃	—	4	2	no	—
35	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
38	C₂×Dic₅₃	—	2	1	no	—
39	D₂₁₂	—	2	2	no	—
40	D₂₁₂	—	4	2	no	—
41	D₂₁₂	—	4	2	no	—
42	D₂₁₂	—	4	2	no	—
43	D₂₁₂	—	4	2	no	—
44	D₂₁₂	—	8	2	no	—
45	C₄×D₅₃	—	4	1	no	—
46	C₄×D₅₃	—	4	1	no	—
47	C₈×C₅₃	—	2	1	no	—
48	C₈×C₅₃	—	4	2	no	—
49	C₈×C₅₃	D₈	1	2	yes	complemented
50	#88322	—	4	1	no	—
51	C₂²×C₅₃	#121	1	1	yes	characteristic
52	C₂×D₅₃	—	4	1	no	—
53	C₂×D₅₃	—	4	1	no	—
54	C₂×D₅₃	—	4	1	no	—
55	C₂×D₅₃	—	8	2	no	—
56	C₂×D₅₃	—	8	2	no	—
57	C₂×D₅₃	—	8	2	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	#459	1	1	yes	characteristic
65	C₂×C₅₃	#469	1	1	yes	characteristic
66	C₂×C₅₃	#472	1	1	yes	characteristic
67	D₅₃	—	8	1	no	—
68	D₅₃	—	16	2	no	—
69	#493	—	53	1	no	—
70	#459	—	53	2	no	—
71	#436	—	53	1	no	—
72	#334	D₅₃	1	1	yes	characteristic, complemented
73	#331	—	53	2	no	—
74	C₅₃	#1687	1	1	yes	characteristic, complemented, Sylow
75	C₂×D₈	—	106	2	no	—
76	C₂×D₈	—	106	1	no	—
77	C₂×D₈	—	106	1	no	—
78	C₂×M₁₆	—	106	1	no	—
79	#127	—	53	2	no	—
80	C₄×D₄	—	53	1	no	—
81	C₄⋊C₈	—	53	1	no	—
82	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
83	#102	—	106	2	no	—
84	C₂²⋊C₈	—	106	1	no	—
85	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
86	C₂×D₄	—	53	1	no	—
87	C₂×D₄	—	106	2	no	—
88	C₂×D₄	—	106	2	no	—
89	C₂×D₄	—	106	2	no	—
90	C₂²×C₄	—	106	1	no	—
91	D₈	—	212	2	no	—
92	D₈	—	212	4	no	—
93	D₈	—	212	2	no	—
94	M₁₆	—	212	2	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	106	1	no	—
97	C₂×C₈	D₂₁₂	1	1	yes	characteristic
98	C₂×C₈	D₂₁₂	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	—	106	1	no	—
103	C₂³	—	212	2	no	—
104	D₄	—	106	2	no	—
105	D₄	—	424	2	no	—
106	D₄	—	212	2	no	—
107	D₄	—	212	2	no	—
108	D₄	—	212	2	no	—
109	D₄	—	212	2	no	—
110	C₂×C₄	—	106	1	no	—
111	C₂×C₄	—	212	1	no	—
112	C₂×C₄	—	212	1	no	—
113	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
114	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
115	C₂×C₄	#11719949	1	1	yes	characteristic
116	C₈	—	2	1	no	—
117	C₈	—	4	2	no	—
118	C₈	—	212	1	no	—
119	C₈	D₄₂₄	1	2	yes	complemented
120	C₂²	—	424	2	no	—
121	C₂²	—	424	2	no	—
122	C₂²	—	424	2	no	—
123	C₂²	—	212	1	no	—
124	C₂²	—	212	1	no	—
125	C₂²	—	212	1	no	—
126	C₂²	#13133192	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	212	1	no	—
129	C₄	C₂×D₄₂₄	1	1	yes	characteristic
130	C₄	#13133244	1	2	yes	—
131	C₄	#13133245	1	1	yes	characteristic
132	C₂	—	848	2	no	—
133	C₂	—	424	1	no	—
134	C₂	#16810819	1	1	yes	characteristic
135	C₂	#16810877	1	1	yes	characteristic
136	C₂	#16810894	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811000

C₂

yes

maximal, characteristic, complemented

#16810819

C₂

yes

maximal, complemented

#16810702

C₂

yes

maximal, characteristic, complemented

#16810366

C₂

yes

maximal, complemented

C₅₃×#334

C₂

yes

maximal, characteristic, complemented

C₂×D₄₂₄

—

C₂×D₄₂₄

—

C₂×D₄₂₄

—

C₂×#11719912

—

#13133214

C₂²

yes

—

C₄×D₂₁₂

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

characteristic

#13133139

—

#13133129

—

C₄⋊C₈×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

D₄

yes

characteristic

C₂×C₄×D₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

#11719921

—

C₄⋊Dic₅₃

D₄

yes

characteristic

C₂×#88322

—

D₄₂₄

—

D₄₂₄

—

D₄₂₄

—

#11719912

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

D₈

yes

complemented

#88322

—

C₂²×C₅₃

#121

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

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

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

D₅₃

—

—

—

—

D₅₃

yes

characteristic, complemented

#331

—

C₅₃

#1687

yes

characteristic, complemented, Sylow

C₂×D₈

—

106

—

C₂×D₈

—

106

—

C₂×D₈

—

106

—

C₂×M₁₆

—

106

—

#127

—

C₄×D₄

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₅₃

yes

—

#102

—

106

—

C₂²⋊C₈

—

106

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×D₄

—

C₂×D₄

—

106

—

C₂×D₄

—

106

—

C₂×D₄

—

106

—

C₂²×C₄

—

106

—

D₈

—

212

—

D₈

—

212

—

D₈

—

212

—

M₁₆

—

212

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

D₂₁₂

yes

characteristic

C₂×C₈

D₂₁₂

yes

characteristic

C₄⋊C₄

—

100

C₂²⋊C₄

—

106

—

101

C₄²

C₂²×D₅₃

yes

characteristic

102

C₂³

—

106

—

103

C₂³

—

212

—

104

D₄

—

106

—

105

D₄

—

424

—

106

D₄

—

212

—

107

D₄

—

212

—

108

D₄

—

212

—

109

D₄

—

212

—

110

C₂×C₄

—

106

—

111

C₂×C₄

—

212

—

112

C₂×C₄

—

212

—

113

C₂×C₄

C₂×D₂₁₂

yes

characteristic

114

C₂×C₄

D₄×D₅₃

yes

characteristic

115

C₂×C₄

#11719949

yes

characteristic

116

C₈

—

117

C₈

—

118

C₈

—

212

—

119

C₈

D₄₂₄

yes

complemented

120

C₂²

—

424

—

121

C₂²

—

424

—

122

C₂²

—

424

—

123

C₂²

—

212

—

124

C₂²

—

212

—

125

C₂²

—

212

—

126

C₂²

#13133192

yes

characteristic

127

C₄

—

128

C₄

—

212

—

129

C₄

C₂×D₄₂₄

yes

characteristic

130

C₄

#13133244

yes

—

131

C₄

#13133245

yes

characteristic

132

C₂

—

848

—

133

C₂

—

424

—

134

C₂

#16810819

yes

characteristic

135

C₂

#16810877

yes

characteristic

136

C₂

#16810894

yes

characteristic

	subgroup	G/H
center	C₂²	#13133192
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#334	C₂
socle	C₂²×C₅₃	#121
hypercenter	#334	D₅₃
hypocenter	C₅₃	#1687
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #334

subgroup

G/H

center

C₂²

#13133192

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#334

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #334

N	H ≅ G/N	type
1. #16811000	C₂	semidirect
2. #16810819	C₂	semidirect
3. #16810702	C₂	semidirect
4. #16810366	C₂	semidirect
5. C₅₃×#334	C₂	semidirect
49. C₈×C₅₃	D₈	semidirect
72. #334	D₅₃	semidirect
74. C₅₃	#1687	semidirect
119. C₈	D₄₂₄	semidirect

H ≅ G/N

type

1. #16811000

C₂

semidirect

2. #16810819

C₂

semidirect

3. #16810702

C₂

semidirect

4. #16810366

C₂

semidirect

5. C₅₃×#334

C₂

semidirect

49. C₈×C₅₃

D₈

semidirect

72. #334

D₅₃

semidirect

74. C₅₃

#1687

semidirect

119. C₈

D₄₂₄

semidirect

Subgroups — #20274195

Conjugacy classes of subgroups

Complements