Group #20273278 — subgroups

Subgroups — #20273278

Order 6784.

220 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 85 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	#16811180	C₂	1	1	yes	maximal, characteristic
2	#16810676	C₂	1	1	yes	maximal, characteristic
3	#16810359	C₂	1	4	yes	maximal
4	C₅₃×#382	C₂	1	1	yes	maximal, characteristic
5	#13133217	C₄	1	1	yes	characteristic, complemented
6	#13133217	C₄	1	1	yes	characteristic, complemented
7	#13133209	—	2	1	no	—
8	#13133196	—	2	1	no	—
9	#13133196	C₂²	1	2	yes	—
10	C₄×Dic₁₀₆	—	2	1	no	—
11	C₄×Dic₁₀₆	C₄	1	2	yes	complemented
12	#13133176	—	2	1	no	—
13	C₄⋊#88322	—	2	1	no	—
14	C₂²⋊#88322	C₂²	1	2	yes	—
15	C₄×#88322	—	2	1	no	—
16	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
17	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
18	#95×C₅₃	—	2	1	no	—
19	#95×C₅₃	C₂²	1	2	yes	—
20	C₂²×C₄×C₅₃	—	2	2	no	—
21	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
24	C₂×Dic₁₀₆	—	2	1	no	—
25	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
26	C₄⋊C₄×C₅₃	—	4	2	no	—
27	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₅₃	D₄	1	1	yes	characteristic
30	C₄²×C₅₃	D₄	1	1	yes	characteristic
31	C₂²⋊Dic₅₃	—	2	2	no	—
32	C₄⋊Dic₅₃	—	2	1	no	—
33	C₄⋊Dic₅₃	—	2	1	no	—
34	C₄⋊Dic₅₃	—	2	1	no	—
35	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
36	#11719919	—	2	2	no	—
37	#11719919	—	2	2	no	—
38	C₄×Dic₅₃	—	2	2	no	—
39	C₂×#88322	—	2	2	no	—
40	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₅₃	—	4	1	no	—
42	C₂×C₄×C₅₃	—	4	1	no	—
43	C₂×C₄×C₅₃	—	4	2	no	—
44	C₂×C₄×C₅₃	—	4	2	no	—
45	C₂×C₄×C₅₃	—	4	2	no	—
46	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₅₃	SD₁₆	1	2	yes	—
51	C₂×C₄×C₅₃	Q₁₆	1	2	yes	—
52	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂×Dic₅₃	—	2	2	no	—
55	C₂×Dic₅₃	—	2	2	no	—
56	Dic₁₀₆	—	4	2	no	—
57	Dic₁₀₆	—	4	2	no	—
58	#88322	—	4	2	no	—
59	C₂²×C₅₃	—	2	1	no	—
60	C₂²×C₅₃	—	2	1	no	—
61	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
63	C₂²×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
64	C₄×C₅₃	—	2	1	no	—
65	C₄×C₅₃	—	2	1	no	—
66	C₄×C₅₃	—	2	1	no	—
67	C₄×C₅₃	—	2	1	no	—
68	C₄×C₅₃	—	8	2	no	—
69	C₄×C₅₃	#103	1	2	yes	—
70	C₄×C₅₃	#103	1	2	yes	—
71	Dic₅₃	—	4	2	no	—
72	Dic₅₃	—	4	2	no	—
73	#1518	—	53	1	no	maximal, Sylow
74	C₂×C₅₃	—	2	1	no	—
75	C₂×C₅₃	—	2	1	no	—
76	C₂×C₅₃	#410	1	1	yes	characteristic
77	C₂×C₅₃	C₂×#103	1	1	yes	characteristic
78	C₂×C₅₃	#421	1	1	yes	characteristic
79	#533	—	53	1	no	—
80	#431	—	53	1	no	—
81	#382	D₅₃	1	1	yes	characteristic
82	#328	—	53	4	no	—
83	C₅₃	#1518	1	1	yes	characteristic, complemented, Sylow
84	C₄⋊Q₈	—	53	1	no	—
85	C₄⋊Q₈	—	53	1	no	—
86	#125	—	106	1	no	—
87	C₂²⋊Q₈	—	53	2	no	—
88	C₂²⋊Q₈	—	106	1	no	—
89	C₄×Q₈	—	53	2	no	—
90	C₄×Q₈	—	106	1	no	—
91	#117	—	106	1	no	—
92	C₂×C₄⋊C₄	—	2	1	no	—
93	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
94	C₄⋊C₈	—	106	1	no	—
95	C₂²⋊C₈	—	53	2	no	—
96	C₄×C₈	—	106	1	no	—
97	#95	—	2	1	no	—
98	#95	C₂×D₅₃	1	2	yes	—
99	C₂×Q₈	—	53	2	no	—
100	C₂×Q₈	—	106	1	no	—
101	C₂²×C₄	—	2	2	no	—
102	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
103	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
104	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
105	C₂×C₈	—	106	2	no	—
106	C₄⋊C₄	—	4	2	no	—
107	C₄⋊C₄	—	53	2	no	—
108	C₄⋊C₄	—	106	2	no	—
109	C₄⋊C₄	—	106	1	no	—
110	C₄⋊C₄	—	106	1	no	—
111	C₄⋊C₄	—	106	2	no	—
112	C₄⋊C₄	—	106	1	no	—
113	C₂²⋊C₄	—	106	2	no	—
114	C₄²	—	106	2	no	—
115	C₄²	C₄×D₅₃	1	1	yes	characteristic
116	C₄²	C₄×D₅₃	1	1	yes	characteristic
117	C₄²	D₂₁₂	1	1	yes	characteristic
118	C₄²	D₂₁₂	1	1	yes	characteristic
119	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
120	Q₈	—	212	2	no	—
121	Q₈	—	212	2	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	—	106	2	no	—
128	C₂×C₄	—	106	2	no	—
129	C₂×C₄	#11719921	1	1	yes	characteristic
130	C₂×C₄	#11719921	1	1	yes	characteristic
131	C₂×C₄	#11719921	1	1	yes	characteristic
132	C₂×C₄	#11719921	1	1	yes	characteristic
133	C₂×C₄	#11719923	1	2	yes	—
134	C₂×C₄	#11719925	1	2	yes	—
135	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
136	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
137	C₈	—	212	2	no	—
138	C₂²	—	2	1	no	—
139	C₂²	—	2	1	no	—
140	C₂²	C₂×#11719921	1	1	yes	characteristic
141	C₂²	C₂×#11719923	1	1	yes	characteristic
142	C₂²	C₂×#11719925	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	8	2	no	—
148	C₄	—	212	2	no	—
149	C₄	—	212	2	no	—
150	C₄	#13133140	1	2	yes	—
151	C₄	#13133140	1	2	yes	—
152	C₂	—	2	1	no	—
153	C₂	—	2	1	no	—
154	C₂	#16810596	1	1	yes	characteristic
155	C₂	C₂×#13133140	1	1	yes	characteristic
156	C₂	#16810641	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811180

C₂

yes

maximal, characteristic

#16810676

C₂

yes

maximal, characteristic

#16810359

C₂

yes

maximal

C₅₃×#382

C₂

yes

maximal, characteristic

#13133217

C₄

yes

characteristic, complemented

#13133217

C₄

yes

characteristic, complemented

#13133209

—

#13133196

—

#13133196

C₂²

yes

—

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

C₄

yes

—

—

C₂²

yes

—

C₄×#88322

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#95×C₅₃

—

#95×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂×C₄

yes

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

#11719919

—

#11719919

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

SD₁₆

yes

—

C₂×C₄×C₅₃

Q₁₆

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

C₂×SD₁₆

yes

characteristic

C₂²×C₅₃

C₂×Q₁₆

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#103

yes

—

C₄×C₅₃

#103

yes

—

Dic₅₃

—

Dic₅₃

—

#1518

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

D₅₃

yes

characteristic

#328

—

C₅₃

#1518

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

106

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

106

—

C₄×Q₈

—

C₄×Q₈

—

106

—

#117

—

106

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

C₄×C₈

—

106

—

#95

—

#95

C₂×D₅₃

yes

—

C₂×Q₈

—

100

C₂×Q₈

—

106

—

101

C₂²×C₄

—

102

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

103

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

104

C₂²×C₄

C₂²×D₅₃

yes

characteristic

105

C₂×C₈

—

106

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

106

—

109

C₄⋊C₄

—

106

—

110

C₄⋊C₄

—

106

—

111

C₄⋊C₄

—

106

—

112

C₄⋊C₄

—

106

—

113

C₂²⋊C₄

—

106

—

114

C₄²

—

106

—

115

C₄²

C₄×D₅₃

yes

characteristic

116

C₄²

C₄×D₅₃

yes

characteristic

117

C₄²

D₂₁₂

yes

characteristic

118

C₄²

D₂₁₂

yes

characteristic

119

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

120

Q₈

—

212

—

121

Q₈

—

212

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

106

—

128

C₂×C₄

—

106

—

129

C₂×C₄

#11719921

yes

characteristic

130

C₂×C₄

#11719921

yes

characteristic

131

C₂×C₄

#11719921

yes

characteristic

132

C₂×C₄

#11719921

yes

characteristic

133

C₂×C₄

#11719923

yes

—

134

C₂×C₄

#11719925

yes

—

135

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

136

C₂×C₄

C₂×D₂₁₂

yes

characteristic

137

C₈

—

212

—

138

C₂²

—

139

C₂²

—

140

C₂²

C₂×#11719921

yes

characteristic

141

C₂²

C₂×#11719923

yes

characteristic

142

C₂²

C₂×#11719925

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

212

—

149

C₄

—

212

—

150

C₄

#13133140

yes

—

151

C₄

#13133140

yes

—

152

C₂

—

153

C₂

—

154

C₂

#16810596

yes

characteristic

155

C₂

C₂×#13133140

yes

characteristic

156

C₂

#16810641

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11719921
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#382	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	#382	D₅₃
hypocenter	C₅₃	#1518
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂²

C₂×#11719921

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#382

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #382

N	H ≅ G/N	type
5. #13133217	C₄	semidirect
6. #13133217	C₄	semidirect
11. C₄×Dic₁₀₆	C₄	semidirect
83. C₅₃	#1518	semidirect

H ≅ G/N

type

5. #13133217

C₄

semidirect

6. #13133217

C₄

semidirect

11. C₄×Dic₁₀₆

C₄

semidirect

83. C₅₃

#1518

semidirect

Subgroups — #20273278

Conjugacy classes of subgroups

Complements