Group #20274336 — subgroups

Subgroups — #20274336

Order 6784.

198 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 61 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	#16811026	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811021	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810995	C₂	1	1	yes	maximal, characteristic
4	#16810995	C₂	1	1	yes	maximal, characteristic
5	#16810973	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810345	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#486	C₂	1	1	yes	maximal, characteristic
8	C₂×#11719923	—	2	2	no	—
9	C₂×#11719923	—	2	2	no	—
10	#13133217	C₂²	1	1	yes	characteristic
11	#13133217	C₂²	1	1	yes	characteristic
12	#13133150	—	2	2	no	—
13	#13133149	—	2	2	no	—
14	#13133141	—	2	2	no	—
15	C₄×#88322	C₂²	1	1	yes	characteristic
16	C₄×#88322	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
18	#127×C₅₃	C₂²	1	1	yes	characteristic
19	#102×C₅₃	—	2	2	no	—
20	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₅₃	—	2	1	no	—
22	C₂×D₄×C₅₃	—	2	1	no	—
23	C₂×D₄×C₅₃	—	2	1	no	—
24	C₂×D₄×C₅₃	—	2	1	no	—
25	C₂×Dic₁₀₆	—	2	1	no	—
26	C₂×Dic₁₀₆	—	2	1	no	—
27	C₂×C₈×C₅₃	D₄	1	2	yes	—
28	C₄⋊C₄×C₅₃	—	2	1	no	—
29	C₄⋊C₄×C₅₃	—	2	1	no	—
30	C₄²×C₅₃	C₂³	1	1	yes	characteristic
31	#11719923	—	4	4	no	—
32	#11719923	—	4	4	no	—
33	C₄⋊Dic₅₃	—	2	1	no	—
34	C₄⋊Dic₅₃	—	2	1	no	—
35	C₄⋊Dic₅₃	—	2	1	no	—
36	C₄⋊Dic₅₃	—	2	1	no	—
37	C₂×#88322	D₄	1	2	yes	—
38	C₂×#88322	D₄	1	2	yes	—
39	C₂³×C₅₃	—	4	1	no	—
40	Q₈×C₅₃	—	4	2	no	—
41	D₄×C₅₃	—	4	2	no	—
42	D₄×C₅₃	—	4	2	no	—
43	D₄×C₅₃	—	4	2	no	—
44	C₂×C₄×C₅₃	—	4	1	no	—
45	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂×Dic₅₃	—	4	1	no	—
49	C₂×Dic₅₃	—	4	1	no	—
50	Dic₁₀₆	—	4	2	no	—
51	Dic₁₀₆	—	4	2	no	—
52	C₈×C₅₃	—	2	2	no	—
53	#88322	—	2	2	no	—
54	#88322	SD₁₆	1	4	yes	complemented
55	C₂²×C₅₃	—	8	1	no	—
56	C₂²×C₅₃	—	8	1	no	—
57	C₂²×C₅₃	—	8	1	no	—
58	C₂²×C₅₃	#127	1	1	yes	characteristic
59	C₄×C₅₃	—	8	1	no	—
60	C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
61	C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
62	C₄×C₅₃	#137	1	2	yes	—
63	Dic₅₃	—	8	1	no	—
64	Dic₅₃	—	8	1	no	—
65	#1714	—	53	1	no	maximal, Sylow
66	C₂×C₅₃	—	16	1	no	—
67	C₂×C₅₃	#492	1	1	yes	characteristic
68	C₂×C₅₃	#496	1	1	yes	characteristic
69	C₂×C₅₃	#497	1	1	yes	characteristic
70	#500	—	53	1	no	—
71	#498	—	53	1	no	—
72	#492	—	53	1	no	—
73	#492	—	53	1	no	—
74	#487	—	53	1	no	—
75	#486	D₅₃	1	1	yes	characteristic
76	#322	—	53	1	no	—
77	C₅₃	#1714	1	1	yes	characteristic, complemented, Sylow
78	C₂×SD₁₆	—	106	2	no	—
79	C₂×SD₁₆	—	106	2	no	—
80	C₄⋊Q₈	—	53	1	no	—
81	C₄⋊Q₈	—	53	1	no	—
82	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
83	#127	C₂×D₅₃	1	1	yes	characteristic
84	#107	—	106	2	no	—
85	#106	—	106	2	no	—
86	#103	—	106	2	no	—
87	#102	—	2	2	no	—
88	C₄×C₈	—	53	1	no	—
89	C₄×C₈	—	53	1	no	—
90	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	106	1	no	—
93	C₂×Q₈	—	106	1	no	—
94	C₂×D₄	—	2	1	no	—
95	C₂×D₄	—	2	1	no	—
96	C₂×D₄	—	2	1	no	—
97	SD₁₆	—	212	4	no	—
98	SD₁₆	—	212	4	no	—
99	C₂×C₈	—	53	2	no	—
100	C₂×C₈	—	53	2	no	—
101	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	106	1	no	—
105	C₄⋊C₄	—	106	1	no	—
106	C₄⋊C₄	—	106	1	no	—
107	C₄⋊C₄	—	106	1	no	—
108	C₄²	C₂²×D₅₃	1	1	yes	characteristic
109	C₂³	—	4	1	no	—
110	Q₈	—	4	2	no	—
111	Q₈	—	212	2	no	—
112	Q₈	—	212	2	no	—
113	D₄	—	4	2	no	—
114	D₄	—	4	2	no	—
115	D₄	—	4	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	212	1	no	—
118	C₂×C₄	—	212	1	no	—
119	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
120	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
122	C₈	—	2	2	no	—
123	C₈	—	53	4	no	—
124	C₈	—	106	2	no	—
125	C₂²	—	8	1	no	—
126	C₂²	—	8	1	no	—
127	C₂²	—	8	1	no	—
128	C₂²	#13133213	1	1	yes	characteristic
129	C₄	—	8	1	no	—
130	C₄	—	424	1	no	—
131	C₄	—	424	1	no	—
132	C₄	SD₁₆×D₅₃	1	2	yes	—
133	C₄	#13133241	1	2	yes	—
134	C₄	#13133247	1	2	yes	—
135	C₂	—	16	1	no	—
136	C₂	#16810994	1	1	yes	characteristic
137	C₂	#16811011	1	1	yes	characteristic
138	C₂	#16811018	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811026

C₂

yes

maximal, characteristic, complemented

#16811021

C₂

yes

maximal, characteristic, complemented

#16810995

C₂

yes

maximal, characteristic

#16810995

C₂

yes

maximal, characteristic

#16810973

C₂

yes

maximal, characteristic, complemented

#16810345

C₂

yes

maximal, characteristic, complemented

C₅₃×#486

C₂

yes

maximal, characteristic

C₂×#11719923

—

C₂×#11719923

—

#13133217

C₂²

yes

characteristic

#13133217

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₄×#88322

C₂²

yes

characteristic

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

#127×C₅₃

C₂²

yes

characteristic

#102×C₅₃

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

D₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

#11719923

—

#11719923

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

D₄

yes

—

C₂×#88322

D₄

yes

—

C₂³×C₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

#88322

—

#88322

SD₁₆

yes

complemented

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#127

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₂×SD₁₆

yes

—

C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₄×C₅₃

#137

yes

—

Dic₅₃

—

Dic₅₃

—

#1714

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

—

—

—

D₅₃

yes

characteristic

#322

—

C₅₃

#1714

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

106

—

C₂×SD₁₆

—

106

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

#127

C₂×D₅₃

yes

characteristic

#107

—

106

—

#106

—

106

—

#103

—

106

—

#102

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

212

—

SD₁₆

—

212

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

C₂²⋊D₅₃

yes

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

106

—

105

C₄⋊C₄

—

106

—

106

C₄⋊C₄

—

106

—

107

C₄⋊C₄

—

106

—

108

C₄²

C₂²×D₅₃

yes

characteristic

109

C₂³

—

110

Q₈

—

111

Q₈

—

212

—

112

Q₈

—

212

—

113

D₄

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

212

—

118

C₂×C₄

—

212

—

119

C₂×C₄

D₄×D₅₃

yes

characteristic

120

C₂×C₄

D₄×D₅₃

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

122

C₈

—

123

C₈

—

124

C₈

—

106

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

#13133213

yes

characteristic

129

C₄

—

130

C₄

—

424

—

131

C₄

—

424

—

132

C₄

SD₁₆×D₅₃

yes

—

133

C₄

#13133241

yes

—

134

C₄

#13133247

yes

—

135

C₂

—

136

C₂

#16810994

yes

characteristic

137

C₂

#16811011

yes

characteristic

138

C₂

#16811018

yes

characteristic

	subgroup	G/H
center	C₂²	#13133213
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#486	C₂
socle	C₂²×C₅₃	#127
hypercenter	#486	D₅₃
hypocenter	C₅₃	#1714
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #486

subgroup

G/H

center

C₂²

#13133213

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#486

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #486

N	H ≅ G/N	type
1. #16811026	C₂	semidirect
2. #16811021	C₂	semidirect
5. #16810973	C₂	semidirect
6. #16810345	C₂	semidirect
54. #88322	SD₁₆	semidirect
77. C₅₃	#1714	semidirect

H ≅ G/N

type

1. #16811026

C₂

semidirect

2. #16811021

C₂

semidirect

5. #16810973

C₂

semidirect

6. #16810345

C₂

semidirect

54. #88322

SD₁₆

semidirect

77. C₅₃

#1714

semidirect

Subgroups — #20274336

Conjugacy classes of subgroups

Complements