Group #20275449 — subgroups

Subgroups — #20275449

Order 6784.

248 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₂×#13133151	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#16810522	C₂	1	2	yes	maximal
4	#16810385	C₂	1	2	yes	maximal
5	C₂×#107×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#88322	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₅₃	—	2	2	no	—
8	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
9	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
10	#13133151	—	2	2	no	—
11	C₈×Dic₅₃	—	2	2	no	—
12	#13133124	—	2	2	no	—
13	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
15	#107×C₅₃	C₄	1	4	yes	complemented
16	C₂²×C₄×C₅₃	—	2	2	no	—
17	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
18	C₂²×Dic₅₃	—	2	2	no	—
19	C₂²×Dic₅₃	D₄	1	1	yes	characteristic
20	C₂²×Dic₅₃	D₄	1	1	yes	characteristic
21	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
22	C₂×C₈×C₅₃	D₄	1	2	yes	—
23	C₂×C₈×C₅₃	Q₈	1	2	yes	—
24	C₄⋊C₄×C₅₃	—	2	2	no	—
25	C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
26	C₄⋊Dic₅₃	—	2	2	no	—
27	C₄⋊Dic₅₃	—	2	2	no	—
28	C₄⋊Dic₅₃	—	4	4	no	—
29	C₄×Dic₅₃	—	2	1	no	—
30	C₄×Dic₅₃	—	2	1	no	—
31	C₂×#88322	—	2	1	no	—
32	C₂×#88322	—	2	1	no	—
33	C₂×#88322	Q₈	1	2	yes	—
34	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₅₃	—	2	4	no	—
36	C₂×C₄×C₅₃	—	4	2	no	—
37	C₂×C₄×C₅₃	—	4	2	no	—
38	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
43	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
44	C₂×Dic₅₃	—	2	1	no	—
45	C₂×Dic₅₃	—	2	1	no	—
46	C₂×Dic₅₃	—	2	1	no	—
47	C₂×Dic₅₃	—	2	1	no	—
48	C₂×Dic₅₃	—	4	2	no	—
49	C₂×Dic₅₃	—	4	2	no	—
50	C₂×Dic₅₃	—	4	2	no	—
51	C₂×Dic₅₃	D₈	1	2	yes	—
52	C₂×Dic₅₃	Q₁₆	1	2	yes	—
53	C₈×C₅₃	C₄⋊C₄	1	4	yes	complemented
54	#88322	—	2	2	no	—
55	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
56	C₂²×C₅₃	#121	1	1	yes	characteristic
57	C₂²×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
58	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
59	C₂²×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
60	C₂²×C₅₃	#135	1	1	yes	characteristic
61	C₂²×C₅₃	#135	1	1	yes	characteristic
62	C₄×C₅₃	—	4	4	no	—
63	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
65	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
66	C₄×C₅₃	#125	1	1	yes	characteristic
67	Dic₅₃	—	2	2	no	—
68	Dic₅₃	—	2	2	no	—
69	Dic₅₃	—	8	2	no	—
70	#1942	—	53	1	no	maximal, Sylow
71	C₂×C₅₃	#389	1	1	yes	characteristic
72	C₂×C₅₃	C₄×D₈	1	1	yes	characteristic
73	C₂×C₅₃	C₄×Q₁₆	1	1	yes	characteristic
74	C₂×C₅₃	#466	1	1	yes	characteristic
75	C₂×C₅₃	#467	1	1	yes	characteristic
76	C₂×C₅₃	#499	1	1	yes	characteristic
77	C₂×C₅₃	#500	1	1	yes	characteristic
78	C₂×#107	—	53	1	no	—
79	C₂×#107	D₅₃	1	1	yes	characteristic
80	C₂×C₄×C₈	—	53	1	no	—
81	#389	—	53	2	no	—
82	#340	—	53	2	no	—
83	C₅₃	#1942	1	1	yes	characteristic, complemented, Sylow
84	C₂²×C₈	—	53	1	no	—
85	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
86	C₂×C₄⋊C₄	—	53	2	no	—
87	C₂×C₄⋊C₄	—	106	2	no	—
88	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
89	C₂×C₄²	—	53	1	no	—
90	#107	—	106	2	no	—
91	#107	Dic₅₃	1	4	yes	complemented
92	C₄×C₈	—	106	2	no	—
93	#95	—	106	2	no	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	—	53	1	no	—
96	C₂²×C₄	—	53	1	no	—
97	C₂²×C₄	—	106	2	no	—
98	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
99	C₂×C₈	—	53	2	no	—
100	C₂×C₈	—	106	1	no	—
101	C₂×C₈	—	106	1	no	—
102	C₂×C₈	Dic₁₀₆	1	2	yes	—
103	C₂×C₈	D₂₁₂	1	2	yes	—
104	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	—	106	2	no	—
107	C₄⋊C₄	—	106	2	no	—
108	C₄⋊C₄	—	212	4	no	—
109	C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
110	C₄²	—	106	1	no	—
111	C₄²	—	106	1	no	—
112	C₂³	D₄×D₅₃	1	1	yes	characteristic
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	53	2	no	—
117	C₂×C₄	—	53	2	no	—
118	C₂×C₄	—	106	1	no	—
119	C₂×C₄	—	106	1	no	—
120	C₂×C₄	—	106	1	no	—
121	C₂×C₄	—	106	1	no	—
122	C₂×C₄	—	212	2	no	—
123	C₂×C₄	—	212	2	no	—
124	C₂×C₄	—	212	2	no	—
125	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
126	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
127	C₂×C₄	#11719947	1	1	yes	characteristic
128	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
129	C₂×C₄	#11719949	1	1	yes	characteristic
130	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
131	C₈	—	106	2	no	—
132	C₈	C₄⋊Dic₅₃	1	4	yes	complemented
133	C₂²	D₄×Dic₅₃	1	1	yes	characteristic
134	C₂²	#13133192	1	1	yes	characteristic
135	C₂²	#13133216	1	1	yes	characteristic
136	C₂²	D₈×D₅₃	1	1	yes	characteristic
137	C₂²	Q₁₆×D₅₃	1	1	yes	characteristic
138	C₂²	#13133236	1	1	yes	characteristic
139	C₂²	#13133240	1	1	yes	characteristic
140	C₄	—	4	4	no	—
141	C₄	—	106	2	no	—
142	C₄	—	106	2	no	—
143	C₄	—	424	2	no	—
144	C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
145	C₄	Q₈×Dic₅₃	1	1	yes	characteristic
146	C₄	#13133197	1	1	yes	characteristic
147	C₄	#13133208	1	1	yes	characteristic
148	C₂	#16810522	1	1	yes	characteristic
149	C₂	D₈×Dic₅₃	1	1	yes	characteristic
150	C₂	Q₁₆×Dic₅₃	1	1	yes	characteristic
151	C₂	#16810860	1	1	yes	characteristic
152	C₂	#16810865	1	1	yes	characteristic
153	C₂	#16811024	1	1	yes	characteristic
154	C₂	#16811027	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133151

C₂

yes

maximal, characteristic

C₂×C₈×Dic₅₃

C₂

yes

maximal, characteristic

#16810522

C₂

yes

maximal

#16810385

C₂

yes

maximal

C₂×#107×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

#13133151

—

C₈×Dic₅₃

—

#13133124

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

#107×C₅₃

C₄

yes

complemented

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

D₄

yes

characteristic

C₂²×Dic₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

Q₈

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

Q₈

yes

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

D₈

yes

—

C₂×Dic₅₃

Q₁₆

yes

—

C₈×C₅₃

C₄⋊C₄

yes

complemented

#88322

—

C₂²×C₅₃

C₄×D₄

yes

characteristic

C₂²×C₅₃

#121

yes

characteristic

C₂²×C₅₃

C₄⋊Q₈

yes

characteristic

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₂²×C₅₃

C₂×Q₁₆

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

C₄×Q₈

yes

characteristic

C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₄×C₅₃

#125

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

C₄×D₈

yes

characteristic

C₂×C₅₃

C₄×Q₁₆

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

yes

characteristic

C₂×C₄×C₈

—

#389

—

#340

—

C₅₃

#1942

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

—

#107

—

106

—

#107

Dic₅₃

yes

complemented

C₄×C₈

—

106

—

#95

—

106

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

106

—

101

C₂×C₈

—

106

—

102

C₂×C₈

Dic₁₀₆

yes

—

103

C₂×C₈

D₂₁₂

yes

—

104

C₂×C₈

C₂×Dic₅₃

yes

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

106

—

107

C₄⋊C₄

—

106

—

108

C₄⋊C₄

—

212

—

109

C₄⋊C₄

C₂×Dic₅₃

yes

—

110

C₄²

—

106

—

111

C₄²

—

106

—

112

C₂³

D₄×D₅₃

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

106

—

119

C₂×C₄

—

106

—

120

C₂×C₄

—

106

—

121

C₂×C₄

—

106

—

122

C₂×C₄

—

212

—

123

C₂×C₄

—

212

—

124

C₂×C₄

—

212

—

125

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

126

C₂×C₄

C₂×D₂₁₂

yes

characteristic

127

C₂×C₄

#11719947

yes

characteristic

128

C₂×C₄

Q₈×D₅₃

yes

characteristic

129

C₂×C₄

#11719949

yes

characteristic

130

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

131

C₈

—

106

—

132

C₈

C₄⋊Dic₅₃

yes

complemented

133

C₂²

D₄×Dic₅₃

yes

characteristic

134

C₂²

#13133192

yes

characteristic

135

C₂²

#13133216

yes

characteristic

136

C₂²

D₈×D₅₃

yes

characteristic

137

C₂²

Q₁₆×D₅₃

yes

characteristic

138

C₂²

#13133236

yes

characteristic

139

C₂²

#13133240

yes

characteristic

140

C₄

—

141

C₄

—

106

—

142

C₄

—

106

—

143

C₄

—

424

—

144

C₄

C₂×C₄⋊Dic₅₃

yes

characteristic

145

C₄

Q₈×Dic₅₃

yes

characteristic

146

C₄

#13133197

yes

characteristic

147

C₄

#13133208

yes

characteristic

148

C₂

#16810522

yes

characteristic

149

C₂

D₈×Dic₅₃

yes

characteristic

150

C₂

Q₁₆×Dic₅₃

yes

characteristic

151

C₂

#16810860

yes

characteristic

152

C₂

#16810865

yes

characteristic

153

C₂

#16811024

yes

characteristic

154

C₂

#16811027

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×#107×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×#107	D₅₃
hypocenter	C₅₃	#1942
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×#107×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
15. #107×C₅₃	C₄	semidirect
53. C₈×C₅₃	C₄⋊C₄	semidirect
83. C₅₃	#1942	semidirect
91. #107	Dic₅₃	semidirect
132. C₈	C₄⋊Dic₅₃	semidirect

H ≅ G/N

type

15. #107×C₅₃

C₄

semidirect

53. C₈×C₅₃

C₄⋊C₄

semidirect

83. C₅₃

#1942

semidirect

91. #107

Dic₅₃

semidirect

132. C₈

C₄⋊Dic₅₃

semidirect

Subgroups — #20275449

Conjugacy classes of subgroups

Complements