Group #20274289 — subgroups

Subgroups — #20274289

Order 6784.

154 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 49 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	#16810973	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810914	C₂	1	1	yes	maximal, characteristic
3	#16810900	C₂	1	1	yes	maximal, characteristic
4	#16810700	C₂	1	1	yes	maximal, characteristic
5	#16810373	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810371	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#332	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11719912	—	2	1	no	—
9	#13133217	C₂²	1	1	yes	characteristic
10	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
11	#13133150	—	2	1	no	—
12	#13133149	—	2	1	no	—
13	C₄⋊#88322	C₂²	1	1	yes	characteristic
14	#13133141	—	2	1	no	—
15	#13133141	—	2	1	no	—
16	#13133139	—	2	1	no	—
17	#13133137	—	2	1	no	—
18	#13133129	—	2	1	no	—
19	C₄×#88322	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₅₃	—	2	1	no	—
24	C₂×D₂₁₂	D₄	1	1	yes	characteristic
25	C₂×C₄×D₅₃	—	2	1	no	—
26	C₂×Dic₁₀₆	—	2	1	no	—
27	C₂×C₈×C₅₃	—	2	1	no	—
28	C₂×C₈×C₅₃	—	2	1	no	—
29	C₄⋊C₄×C₅₃	—	2	1	no	—
30	C₄⋊C₄×C₅₃	—	2	1	no	—
31	C₄²×C₅₃	C₂³	1	1	yes	characteristic
32	#11719921	—	2	1	no	—
33	C₄⋊Dic₅₃	—	2	1	no	—
34	C₄⋊Dic₅₃	—	2	1	no	—
35	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
36	C₂×#88322	—	2	1	no	—
37	C₂×#88322	D₄	1	1	yes	characteristic
38	C₂×#88322	D₄	1	1	yes	characteristic
39	#11719912	—	4	2	no	—
40	C₂²×D₅₃	—	2	1	no	—
41	Q₈×C₅₃	—	4	1	no	—
42	Q₈×C₅₃	—	4	1	no	—
43	C₂×C₄×C₅₃	—	4	1	no	—
44	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
47	C₂×Dic₅₃	—	2	1	no	—
48	C₂×Dic₅₃	—	4	1	no	—
49	D₂₁₂	—	2	1	no	—
50	D₂₁₂	—	2	1	no	—
51	C₄×D₅₃	—	4	1	no	—
52	C₄×D₅₃	—	4	1	no	—
53	Dic₁₀₆	—	8	1	no	—
54	C₈×C₅₃	—	4	1	no	—
55	C₈×C₅₃	—	4	1	no	—
56	#88322	—	2	1	no	—
57	#88322	—	2	1	no	—
58	#88322	—	4	1	no	—
59	C₂²×C₅₃	#121	1	1	yes	characteristic
60	C₂×D₅₃	—	4	1	no	—
61	C₂×D₅₃	—	4	1	no	—
62	C₂×D₅₃	—	4	1	no	—
63	C₄×C₅₃	—	2	1	no	—
64	C₄×C₅₃	—	8	1	no	—
65	C₄×C₅₃	#135	1	1	yes	characteristic
66	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
67	C₄×C₅₃	#137	1	1	yes	characteristic
68	C₄×C₅₃	#137	1	1	yes	characteristic
69	Dic₅₃	—	4	1	no	—
70	Dic₅₃	—	8	1	no	—
71	#1703	—	53	1	no	maximal, Sylow
72	C₂×C₅₃	#463	1	1	yes	characteristic
73	C₂×C₅₃	#470	1	1	yes	characteristic
74	C₂×C₅₃	#471	1	1	yes	characteristic
75	D₅₃	—	8	1	no	—
76	#487	—	53	1	no	—
77	#476	—	53	1	no	—
78	#474	—	53	1	no	—
79	#436	—	53	1	no	—
80	#334	—	53	1	no	—
81	#333	—	53	1	no	—
82	#332	D₅₃	1	1	yes	characteristic, complemented
83	C₅₃	#1703	1	1	yes	characteristic, complemented, Sylow
84	C₂×M₁₆	—	106	1	no	—
85	C₄⋊Q₈	—	53	1	no	—
86	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
87	C₄×D₄	—	53	1	no	—
88	#107	—	106	1	no	—
89	#106	—	106	1	no	—
90	C₄⋊C₈	—	53	1	no	—
91	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
92	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
93	#103	—	106	1	no	—
94	#103	—	106	1	no	—
95	#102	—	106	1	no	—
96	#102	—	106	1	no	—
97	C₂²⋊C₈	—	106	1	no	—
98	C₄×C₈	—	53	1	no	—
99	C₂×Q₈	—	2	1	no	—
100	C₂×Q₈	—	106	1	no	—
101	C₂×D₄	—	53	1	no	—
102	C₂²×C₄	—	106	1	no	—
103	M₁₆	—	212	2	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	53	1	no	—
107	C₂×C₈	—	53	1	no	—
108	C₂×C₈	—	106	1	no	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	53	1	no	—
112	C₄⋊C₄	—	106	1	no	—
113	C₄⋊C₄	—	106	1	no	—
114	C₂²⋊C₄	—	106	1	no	—
115	C₄²	C₂²×D₅₃	1	1	yes	characteristic
116	C₂³	—	106	1	no	—
117	Q₈	—	4	1	no	—
118	Q₈	—	4	1	no	—
119	Q₈	—	424	1	no	—
120	D₄	—	106	1	no	—
121	D₄	—	106	1	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	106	1	no	—
124	C₂×C₄	—	212	1	no	—
125	C₂×C₄	—	212	1	no	—
126	C₂×C₄	—	212	1	no	—
127	C₂×C₄	#11719945	1	1	yes	characteristic
128	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
129	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
130	C₈	—	4	1	no	—
131	C₈	—	4	1	no	—
132	C₈	—	106	1	no	—
133	C₈	—	106	1	no	—
134	C₈	—	212	1	no	—
135	C₂²	—	212	1	no	—
136	C₂²	—	212	1	no	—
137	C₂²	—	212	1	no	—
138	C₂²	#13133190	1	1	yes	characteristic
139	C₄	—	2	1	no	—
140	C₄	—	8	1	no	—
141	C₄	—	424	1	no	—
142	C₄	—	212	1	no	—
143	C₄	#13133238	1	1	yes	characteristic
144	C₄	#13133241	1	1	yes	characteristic
145	C₄	#13133246	1	1	yes	characteristic
146	C₄	#13133249	1	1	yes	characteristic
147	C₂	—	424	1	no	—
148	C₂	#16810837	1	1	yes	characteristic
149	C₂	#16810880	1	1	yes	characteristic
150	C₂	#16810889	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810973

C₂

yes

maximal, characteristic, complemented

#16810914

C₂

yes

maximal, characteristic

#16810900

C₂

yes

maximal, characteristic

#16810700

C₂

yes

maximal, characteristic

#16810373

C₂

yes

maximal, characteristic, complemented

#16810371

C₂

yes

maximal, characteristic, complemented

C₅₃×#332

C₂

yes

maximal, characteristic, complemented

C₂×#11719912

—

#13133217

C₂²

yes

characteristic

C₄×D₂₁₂

C₂²

yes

—

—

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×D₂₁₂

D₄

yes

characteristic

C₂×C₄×D₅₃

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

#11719921

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

characteristic

C₂×#88322

—

C₂×#88322

D₄

yes

characteristic

C₂×#88322

D₄

yes

characteristic

#11719912

—

C₂²×D₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

—

#88322

—

C₂²×C₅₃

#121

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

—

—

—

—

—

—

D₅₃

yes

characteristic, complemented

C₅₃

#1703

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

106

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

C₄×D₄

—

#107

—

106

—

#106

—

106

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₅₃

yes

characteristic

C₄⋊C₈

C₂×D₅₃

yes

characteristic

#103

—

106

—

#103

—

106

—

#102

—

106

—

#102

—

106

—

C₂²⋊C₈

—

106

—

C₄×C₈

—

C₂×Q₈

—

100

C₂×Q₈

—

106

—

101

C₂×D₄

—

102

C₂²×C₄

—

106

—

103

M₁₆

—

212

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

106

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

106

—

113

C₄⋊C₄

—

106

—

114

C₂²⋊C₄

—

106

—

115

C₄²

C₂²×D₅₃

yes

characteristic

116

C₂³

—

106

—

117

Q₈

—

118

Q₈

—

119

Q₈

—

424

—

120

D₄

—

106

—

121

D₄

—

106

—

122

C₂×C₄

—

123

C₂×C₄

—

106

—

124

C₂×C₄

—

212

—

125

C₂×C₄

—

212

—

126

C₂×C₄

—

212

—

127

C₂×C₄

#11719945

yes

characteristic

128

C₂×C₄

D₄×D₅₃

yes

characteristic

129

C₂×C₄

D₄×D₅₃

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

—

106

—

133

C₈

—

106

—

134

C₈

—

212

—

135

C₂²

—

212

—

136

C₂²

—

212

—

137

C₂²

—

212

—

138

C₂²

#13133190

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

424

—

142

C₄

—

212

—

143

C₄

#13133238

yes

characteristic

144

C₄

#13133241

yes

characteristic

145

C₄

#13133246

yes

characteristic

146

C₄

#13133249

yes

characteristic

147

C₂

—

424

—

148

C₂

#16810837

yes

characteristic

149

C₂

#16810880

yes

characteristic

150

C₂

#16810889

yes

characteristic

	subgroup	G/H
center	C₂²	#13133190
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#332	C₂
socle	C₂²×C₅₃	#121
hypercenter	#332	D₅₃
hypocenter	C₅₃	#1703
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

#13133190

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#332

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
1. #16810973	C₂	semidirect
5. #16810373	C₂	semidirect
6. #16810371	C₂	semidirect
7. C₅₃×#332	C₂	semidirect
82. #332	D₅₃	semidirect
83. C₅₃	#1703	semidirect

H ≅ G/N

type

1. #16810973

C₂

semidirect

5. #16810373

C₂

semidirect

6. #16810371

C₂

semidirect

7. C₅₃×#332

C₂

semidirect

82. #332

D₅₃

semidirect

83. C₅₃

#1703

semidirect

Subgroups — #20274289

Conjugacy classes of subgroups

Complements