Group #20283820 — subgroups

Subgroups — #20283820

Order 6784.

314 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 157 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	#16810752	C₂	1	4	yes	maximal, complemented
3	#16810751	C₂	1	4	yes	maximal, complemented
4	#16810681	C₂	1	4	yes	maximal
5	#16810680	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#432	C₂	1	1	yes	maximal, characteristic
7	#13133217	C₄	1	1	yes	characteristic
8	#13133217	C₄	1	1	yes	characteristic
9	#13133209	C₄	1	2	yes	—
10	#13133196	C₄	1	2	yes	—
11	#13133196	C₄	1	2	yes	—
12	C₄×Dic₁₀₆	C₂²	1	2	yes	—
13	C₄×Dic₁₀₆	C₂²	1	2	yes	—
14	#13133176	C₂²	1	2	yes	—
15	C₄⋊#88322	C₂²	1	2	yes	—
16	#13133146	C₂²	1	8	yes	—
17	C₂²⋊#88322	C₂²	1	2	yes	—
18	#13133128	C₂²	1	2	yes	—
19	#13133127	C₂²	1	4	yes	—
20	C₈×Dic₅₃	C₂²	1	4	yes	—
21	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
23	C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
24	#97×C₅₃	C₂²	1	2	yes	—
25	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
28	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
29	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₅₃	C₂³	1	4	yes	—
31	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₄²×C₅₃	C₂³	1	1	yes	characteristic
34	C₄²×C₅₃	C₂³	1	1	yes	characteristic
35	C₂²⋊Dic₅₃	C₂×C₄	1	4	yes	—
36	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
37	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
38	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
39	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
40	#11719919	C₂×C₄	1	4	yes	—
41	#11719919	C₂×C₄	1	4	yes	—
42	C₄×Dic₅₃	C₂³	1	4	yes	—
43	C₂×#88322	C₂³	1	4	yes	—
44	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	—	2	2	no	—
46	C₂×C₄×C₅₃	—	4	1	no	—
47	C₂×C₄×C₅₃	—	4	1	no	—
48	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
54	C₂×Dic₅₃	C₂²×C₄	1	4	yes	—
55	C₂×Dic₅₃	C₂²×C₄	1	4	yes	—
56	Dic₁₀₆	—	4	2	no	—
57	Dic₁₀₆	—	4	2	no	—
58	C₈×C₅₃	—	2	4	no	—
59	#88322	—	2	4	no	—
60	C₂²×C₅₃	—	2	1	no	—
61	C₂²×C₅₃	—	2	1	no	—
62	C₂²×C₅₃	—	2	1	no	—
63	C₂²×C₅₃	C₂³×C₄	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	—	2	1	no	—
69	C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
70	Dic₅₃	—	2	4	no	—
71	Dic₅₃	—	2	4	no	—
72	#2987	—	53	1	no	maximal, Sylow
73	C₂×C₅₃	—	4	1	no	—
74	C₂×C₅₃	#519	1	1	yes	characteristic
75	C₂×C₅₃	#568	1	1	yes	characteristic
76	C₂×C₅₃	#568	1	1	yes	characteristic
77	#533	—	53	1	no	—
78	#446	—	53	4	no	—
79	#446	—	53	4	no	—
80	#433	—	53	4	no	—
81	#432	—	53	1	no	—
82	#432	D₅₃	1	1	yes	characteristic
83	C₅₃	#2987	1	1	yes	characteristic, complemented, Sylow
84	C₄⋊Q₈	—	53	1	no	—
85	C₄⋊Q₈	—	53	1	no	—
86	#125	—	53	2	no	—
87	C₂²⋊Q₈	—	53	2	no	—
88	C₂²⋊Q₈	—	53	2	no	—
89	C₄×Q₈	—	53	2	no	—
90	C₄×Q₈	—	53	2	no	—
91	#117	—	53	2	no	—
92	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
93	C₄⋊C₈	—	53	8	no	—
94	C₄⋊C₈	—	53	2	no	—
95	C₄⋊C₈	C₂×D₅₃	1	2	yes	—
96	C₂²⋊C₈	—	53	2	no	—
97	C₂²⋊C₈	C₂×D₅₃	1	2	yes	—
98	#97	—	53	4	no	—
99	#97	—	53	2	no	—
100	#97	C₂×D₅₃	1	2	yes	—
101	C₄×C₈	—	53	4	no	—
102	C₂×Q₈	—	53	2	no	—
103	C₂×Q₈	—	53	2	no	—
104	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
105	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
107	C₂×C₈	—	53	4	no	—
108	C₂×C₈	C₂²×D₅₃	1	4	yes	—
109	C₄⋊C₄	—	53	2	no	—
110	C₄⋊C₄	—	53	2	no	—
111	C₄⋊C₄	—	53	4	no	—
112	C₄⋊C₄	—	53	4	no	—
113	C₄⋊C₄	—	53	2	no	—
114	C₄⋊C₄	—	53	2	no	—
115	C₂²⋊C₄	—	53	4	no	—
116	C₄²	—	53	4	no	—
117	C₄²	C₄×D₅₃	1	1	yes	characteristic
118	C₄²	C₄×D₅₃	1	1	yes	characteristic
119	C₄²	C₂²×D₅₃	1	1	yes	characteristic
120	C₄²	C₂²×D₅₃	1	1	yes	characteristic
121	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
122	Q₈	—	212	2	no	—
123	Q₈	—	212	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	53	4	no	—
128	C₂×C₄	—	53	4	no	—
129	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
130	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
131	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
132	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
133	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
134	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
135	C₈	—	2	4	no	—
136	C₈	—	106	4	no	—
137	C₂²	—	2	1	no	—
138	C₂²	—	2	1	no	—
139	C₂²	—	2	1	no	—
140	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	—	2	1	no	—
146	C₄	—	106	4	no	—
147	C₄	—	106	4	no	—
148	C₄	#13133264	1	2	yes	—
149	C₂	—	4	1	no	—
150	C₂	#16811100	1	1	yes	characteristic
151	C₂	#16811389	1	1	yes	characteristic
152	C₂	#16811389	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811180

C₂

yes

maximal, characteristic

#16810752

C₂

yes

maximal, complemented

#16810751

C₂

yes

maximal, complemented

#16810681

C₂

yes

maximal

#16810680

C₂

yes

maximal, characteristic

C₅₃×#432

C₂

yes

maximal, characteristic

#13133217

C₄

yes

characteristic

#13133217

C₄

yes

characteristic

#13133209

C₄

yes

—

#13133196

C₄

yes

—

#13133196

C₄

yes

—

C₄×Dic₁₀₆

C₂²

yes

—

C₄×Dic₁₀₆

C₂²

yes

—

#13133176

C₂²

yes

—

C₄⋊#88322

C₂²

yes

—

#13133146

C₂²

yes

—

C₂²⋊#88322

C₂²

yes

—

#13133128

C₂²

yes

—

#13133127

C₂²

yes

—

C₈×Dic₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

—

C₂²⋊C₈×C₅₃

C₂²

yes

—

#97×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×Dic₁₀₆

C₂×C₄

yes

—

C₂×Dic₁₀₆

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂³

yes

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₂²⋊Dic₅₃

C₂×C₄

yes

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

#11719919

C₂×C₄

yes

—

#11719919

C₂×C₄

yes

—

C₄×Dic₅₃

C₂³

yes

—

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

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

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

C₂²×C₄

yes

—

C₂×Dic₅₃

C₂²×C₄

yes

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

2₋¹⁺⁴

yes

—

Dic₅₃

—

Dic₅₃

—

#2987

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

—

—

—

D₅₃

yes

characteristic

C₅₃

#2987

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₅₃

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

C₂×D₅₃

yes

—

#97

—

#97

—

100

#97

C₂×D₅₃

yes

—

101

C₄×C₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂²×C₄

C₄×D₅₃

yes

characteristic

105

C₂²×C₄

C₄×D₅₃

yes

characteristic

106

C₂²×C₄

C₂²×D₅₃

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

C₂²×D₅₃

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₂²⋊C₄

—

116

C₄²

—

117

C₄²

C₄×D₅₃

yes

characteristic

118

C₄²

C₄×D₅₃

yes

characteristic

119

C₄²

C₂²×D₅₃

yes

characteristic

120

C₄²

C₂²×D₅₃

yes

characteristic

121

C₂³

C₂×C₄×D₅₃

yes

characteristic

122

Q₈

—

212

—

123

Q₈

—

212

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

130

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

131

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

132

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

133

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

134

C₂×C₄

C₂³×D₅₃

yes

characteristic

135

C₈

—

136

C₈

—

106

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

C₂²×C₄×D₅₃

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

106

—

147

C₄

—

106

—

148

C₄

#13133264

yes

—

149

C₂

—

150

C₂

#16811100

yes

characteristic

151

C₂

#16811389

yes

characteristic

152

C₂

#16811389

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₅₃×#432	C₂
hypercenter	#432	D₅₃
hypocenter	C₅₃	#2987
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #432

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₅₃

commutator, socle

C₂²×C₅₃

C₂³×C₄

Fitting

C₅₃×#432

C₂

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ #432

N	H ≅ G/N	type
2. #16810752	C₂	semidirect
3. #16810751	C₂	semidirect
83. C₅₃	#2987	semidirect

H ≅ G/N

type

2. #16810752

C₂

semidirect

3. #16810751

C₂

semidirect

83. C₅₃

#2987

semidirect

Subgroups — #20283820

Conjugacy classes of subgroups

Complements