Group #20274113 — subgroups

Subgroups — #20274113

Order 6784.

160 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 53 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	#16810914	C₂	1	1	yes	maximal, characteristic
3	#16810910	C₂	1	1	yes	maximal, characteristic
4	#16810688	C₂	1	1	yes	maximal, characteristic
5	#16810375	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810371	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#333	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×D₅₃	—	2	1	no	—
9	#13133217	C₂²	1	1	yes	characteristic
10	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
11	#13133150	—	2	1	no	—
12	#13133150	—	2	1	no	—
13	#13133149	—	2	1	no	—
14	#13133148	—	2	1	no	—
15	C₄⋊#88322	C₂²	1	1	yes	characteristic
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	C₈×D₅₃	—	4	2	no	—
40	C₂²×D₅₃	—	2	1	no	—
41	Q₈×C₅₃	—	8	1	no	—
42	C₂×C₄×C₅₃	—	4	1	no	—
43	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
46	C₂×Dic₅₃	—	2	1	no	—
47	C₂×Dic₅₃	—	4	1	no	—
48	D₂₁₂	—	2	1	no	—
49	D₂₁₂	Q₁₆	1	2	yes	—
50	C₄×D₅₃	—	4	1	no	—
51	C₄×D₅₃	—	4	1	no	—
52	Dic₁₀₆	—	8	1	no	—
53	C₈×C₅₃	—	4	1	no	—
54	C₈×C₅₃	—	4	1	no	—
55	#88322	—	2	1	no	—
56	#88322	—	4	1	no	—
57	#88322	SD₁₆	1	2	yes	—
58	C₂²×C₅₃	#121	1	1	yes	characteristic
59	C₂×D₅₃	—	2	2	no	—
60	C₂×D₅₃	—	4	1	no	—
61	C₂×D₅₃	—	4	1	no	—
62	C₄×C₅₃	—	2	1	no	—
63	C₄×C₅₃	—	8	1	no	—
64	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
65	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
66	C₄×C₅₃	#135	1	1	yes	characteristic
67	C₄×C₅₃	#137	1	1	yes	characteristic
68	Dic₅₃	—	4	1	no	—
69	Dic₅₃	—	8	1	no	—
70	#1675	—	53	1	no	maximal, Sylow
71	C₂×C₅₃	#461	1	1	yes	characteristic
72	C₂×C₅₃	#467	1	1	yes	characteristic
73	C₂×C₅₃	#472	1	1	yes	characteristic
74	D₅₃	—	4	2	no	—
75	#500	—	53	1	no	—
76	#476	—	53	1	no	—
77	#476	—	53	1	no	—
78	C₈×D₄	—	53	1	no	—
79	#335	—	53	1	no	—
80	#333	—	53	1	no	—
81	#333	D₅₃	1	1	yes	characteristic, complemented
82	C₅₃	#1675	1	1	yes	characteristic, complemented, Sylow
83	C₂²×C₈	—	106	1	no	—
84	C₄⋊Q₈	—	53	1	no	—
85	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
86	C₄×D₄	—	53	1	no	—
87	#107	—	106	1	no	—
88	#107	—	106	1	no	—
89	#106	—	106	1	no	—
90	#106	—	106	1	no	—
91	C₄⋊C₈	—	53	1	no	—
92	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
93	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
94	#102	—	106	1	no	—
95	#102	—	106	1	no	—
96	C₂²⋊C₈	—	106	1	no	—
97	C₄×C₈	—	53	1	no	—
98	C₂×Q₈	—	2	1	no	—
99	C₂×Q₈	—	106	1	no	—
100	C₂×D₄	—	53	1	no	—
101	C₂²×C₄	—	106	1	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	53	1	no	—
105	C₂×C₈	—	53	1	no	—
106	C₂×C₈	—	106	1	no	—
107	C₂×C₈	—	212	2	no	—
108	C₄⋊C₄	—	2	1	no	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	53	1	no	—
111	C₄⋊C₄	—	106	1	no	—
112	C₄⋊C₄	—	106	1	no	—
113	C₂²⋊C₄	—	106	1	no	—
114	C₄²	C₂²×D₅₃	1	1	yes	characteristic
115	C₂³	—	106	1	no	—
116	Q₈	—	8	1	no	—
117	Q₈	—	424	1	no	—
118	D₄	—	53	2	no	—
119	D₄	—	106	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	106	1	no	—
122	C₂×C₄	—	212	1	no	—
123	C₂×C₄	—	212	1	no	—
124	C₂×C₄	—	212	1	no	—
125	C₂×C₄	#11719945	1	1	yes	characteristic
126	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
127	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
128	C₈	—	4	1	no	—
129	C₈	—	4	1	no	—
130	C₈	—	53	2	no	—
131	C₈	—	106	1	no	—
132	C₈	—	212	1	no	—
133	C₂²	—	106	2	no	—
134	C₂²	—	212	1	no	—
135	C₂²	—	212	1	no	—
136	C₂²	#13133190	1	1	yes	characteristic
137	C₄	—	2	1	no	—
138	C₄	—	8	1	no	—
139	C₄	—	424	1	no	—
140	C₄	—	212	1	no	—
141	C₄	SD₁₆×D₅₃	1	1	yes	characteristic
142	C₄	Q₁₆×D₅₃	1	1	yes	characteristic
143	C₄	#13133238	1	1	yes	characteristic
144	C₄	#13133249	1	1	yes	characteristic
145	C₂	—	212	2	no	—
146	C₂	#16810827	1	1	yes	characteristic
147	C₂	#16810863	1	1	yes	characteristic
148	C₂	#16810892	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811026

C₂

yes

maximal, characteristic, complemented

#16810914

C₂

yes

maximal, characteristic

#16810910

C₂

yes

maximal, characteristic

#16810688

C₂

yes

maximal, characteristic

#16810375

C₂

yes

maximal, characteristic, complemented

#16810371

C₂

yes

maximal, characteristic, complemented

C₅₃×#333

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₅₃

—

#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

C₈×D₅₃

—

C₂²×D₅₃

—

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

Q₁₆

yes

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

—

#88322

SD₁₆

yes

—

C₂²×C₅₃

#121

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₄×C₅₃

C₂×Q₁₆

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

#500

—

#476

—

#476

—

C₈×D₄

—

#335

—

#333

—

#333

D₅₃

yes

characteristic, complemented

C₅₃

#1675

yes

characteristic, complemented, Sylow

C₂²×C₈

—

106

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

C₄×D₄

—

#107

—

106

—

#107

—

106

—

#106

—

106

—

#106

—

106

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₅₃

yes

characteristic

C₄⋊C₈

C₂×D₅₃

yes

characteristic

#102

—

106

—

#102

—

106

—

C₂²⋊C₈

—

106

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

106

—

100

C₂×D₄

—

101

C₂²×C₄

—

106

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

106

—

107

C₂×C₈

—

212

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

106

—

112

C₄⋊C₄

—

106

—

113

C₂²⋊C₄

—

106

—

114

C₄²

C₂²×D₅₃

yes

characteristic

115

C₂³

—

106

—

116

Q₈

—

117

Q₈

—

424

—

118

D₄

—

119

D₄

—

106

—

120

C₂×C₄

—

121

C₂×C₄

—

106

—

122

C₂×C₄

—

212

—

123

C₂×C₄

—

212

—

124

C₂×C₄

—

212

—

125

C₂×C₄

#11719945

yes

characteristic

126

C₂×C₄

D₄×D₅₃

yes

characteristic

127

C₂×C₄

D₄×D₅₃

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

—

106

—

132

C₈

—

212

—

133

C₂²

—

106

—

134

C₂²

—

212

—

135

C₂²

—

212

—

136

C₂²

#13133190

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

424

—

140

C₄

—

212

—

141

C₄

SD₁₆×D₅₃

yes

characteristic

142

C₄

Q₁₆×D₅₃

yes

characteristic

143

C₄

#13133238

yes

characteristic

144

C₄

#13133249

yes

characteristic

145

C₂

—

212

—

146

C₂

#16810827

yes

characteristic

147

C₂

#16810863

yes

characteristic

148

C₂

#16810892

yes

characteristic

	subgroup	G/H
center	C₂²	#13133190
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#333	C₂
socle	C₂²×C₅₃	#121
hypercenter	#333	D₅₃
hypocenter	C₅₃	#1675
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

#13133190

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#333

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
1. #16811026	C₂	semidirect
5. #16810375	C₂	semidirect
6. #16810371	C₂	semidirect
7. C₅₃×#333	C₂	semidirect
81. #333	D₅₃	semidirect
82. C₅₃	#1675	semidirect

H ≅ G/N

type

1. #16811026

C₂

semidirect

5. #16810375

C₂

semidirect

6. #16810371

C₂

semidirect

7. C₅₃×#333

C₂

semidirect

81. #333

D₅₃

semidirect

82. C₅₃

#1675

semidirect

Subgroups — #20274113

Conjugacy classes of subgroups

Complements