Group #20283794 — subgroups

Subgroups — #20283794

Order 6784.

396 conjugacy classes of subgroups; only the 217 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 217 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	C₄×Q₈×D₅₃	C₂	1	1	yes	maximal, characteristic
2	Q₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×Dic₁₀₆	C₂	1	3	yes	maximal, complemented
4	#16810747	C₂	1	3	yes	maximal, complemented
5	#16810675	C₂	1	3	yes	maximal
6	#16810674	C₂	1	3	yes	maximal
7	C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₅₃	C₄	1	1	yes	characteristic
9	Q₈×Dic₅₃	C₄	1	1	yes	characteristic
10	C₄×Dic₁₀₆	C₂²	1	3	yes	—
11	#13133182	C₄	1	3	yes	—
12	C₄⋊C₄×D₅₃	C₄	1	3	yes	—
13	C₄²×D₅₃	C₂²	1	3	yes	—
14	C₄⋊#88322	C₂²	1	3	yes	—
15	#13133146	C₂²	1	3	yes	—
16	#13133146	C₂²	1	3	yes	—
17	#13133129	C₂²	1	1	yes	characteristic
18	#13133129	C₂²	1	3	yes	—
19	C₄×#88322	C₂²	1	3	yes	—
20	C₈×Dic₅₃	C₂²	1	6	yes	—
21	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₅₃	C₂²	1	3	yes	—
23	C₄×C₈×C₅₃	C₂²	1	3	yes	—
24	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
25	Q₈×D₅₃	C₈	1	8	yes	complemented
26	C₂×C₄×D₅₃	C₂×C₄	1	3	yes	—
27	C₂×C₄×D₅₃	C₂×C₄	1	3	yes	—
28	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₀₆	C₂×C₄	1	3	yes	—
30	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₅₃	C₂³	1	3	yes	—
32	C₄⋊C₄×C₅₃	C₂×C₄	1	3	yes	—
33	C₄²×C₅₃	C₂³	1	3	yes	—
34	C₄⋊Dic₅₃	C₂×C₄	1	3	yes	—
35	#11719919	C₂×C₄	1	6	yes	—
36	C₄×Dic₅₃	C₂×C₄	1	3	yes	—
37	C₄×Dic₅₃	C₂×C₄	1	3	yes	—
38	C₄×Dic₅₃	C₂³	1	3	yes	—
39	C₂×#88322	C₂³	1	3	yes	—
40	C₂×#88322	C₂³	1	1	yes	characteristic
41	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
42	Q₈×C₅₃	C₂×C₈	1	4	yes	—
43	C₂×C₄×C₅₃	C₂²×C₄	1	3	yes	—
44	C₂×C₄×C₅₃	C₂²×C₄	1	3	yes	—
45	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
46	C₂×Dic₅₃	C₂²×C₄	1	3	yes	—
47	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×Dic₅₃	C₂²×C₄	1	3	yes	—
49	C₄×D₅₃	—	2	1	no	—
50	C₄×D₅₃	—	2	1	no	—
51	C₄×D₅₃	—	4	3	no	—
52	C₄×D₅₃	C₂×C₈	1	12	yes	—
53	Dic₁₀₆	C₂×C₈	1	12	yes	—
54	C₈×C₅₃	—	2	3	no	—
55	C₈×C₅₃	—	2	1	no	—
56	#88322	—	2	1	no	—
57	#88322	—	2	3	no	—
58	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
59	C₂×D₅₃	—	2	1	no	—
60	C₂×D₅₃	—	2	1	no	—
61	C₂×D₅₃	C₂²×C₈	1	2	yes	—
62	C₄×C₅₃	—	2	3	no	—
63	C₄×C₅₃	C₂²×C₈	1	6	yes	—
64	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
65	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
66	Dic₅₃	—	2	1	no	—
67	Dic₅₃	—	2	3	no	—
68	Dic₅₃	C₂²×C₈	1	6	yes	—
69	#2982	—	53	1	no	maximal, Sylow
70	C₂×C₅₃	#519	1	1	yes	characteristic
71	C₂×C₅₃	C₂³×C₈	1	1	yes	characteristic
72	C₂×C₅₃	#568	1	1	yes	characteristic
73	D₅₃	—	2	2	no	—
74	C₂×C₄×Q₈	—	53	1	no	—
75	C₈×Q₈	—	53	1	no	—
76	C₈×Q₈	—	53	3	no	—
77	C₈×Q₈	—	53	3	no	—
78	C₈×Q₈	D₅₃	1	1	yes	characteristic, complemented
79	#431	—	53	3	no	—
80	#431	—	53	3	no	—
81	C₅₃	#2982	1	1	yes	characteristic, complemented, Sylow
82	C₂²×Q₈	—	53	1	no	—
83	C₄×Q₈	—	53	1	no	—
84	C₄×Q₈	—	53	3	no	—
85	C₄×Q₈	—	53	3	no	—
86	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
87	C₂×C₄⋊C₄	—	53	3	no	—
88	C₂×C₄²	—	53	3	no	—
89	C₄⋊C₈	—	53	3	no	—
90	C₄⋊C₈	—	53	3	no	—
91	C₄⋊C₈	—	53	3	no	—
92	C₄⋊C₈	C₂×D₅₃	1	3	yes	—
93	C₂²⋊C₈	—	53	3	no	—
94	C₂²⋊C₈	—	53	1	no	—
95	C₄×C₈	—	53	6	no	—
96	C₄×C₈	—	53	3	no	—
97	C₄×C₈	C₂×D₅₃	1	3	yes	—
98	C₂×Q₈	—	53	8	no	—
99	C₂×Q₈	—	53	3	no	—
100	C₂×Q₈	C₄×D₅₃	1	1	yes	characteristic
101	C₂²×C₄	—	53	3	no	—
102	C₂²×C₄	—	53	1	no	—
103	C₂²×C₄	—	53	3	no	—
104	C₂×C₈	—	53	1	no	—
105	C₂×C₈	—	53	3	no	—
106	C₂×C₈	C₂²×D₅₃	1	3	yes	—
107	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
108	C₄⋊C₄	—	53	6	no	—
109	C₄⋊C₄	—	53	3	no	—
110	C₄⋊C₄	C₄×D₅₃	1	3	yes	—
111	C₄²	—	53	3	no	—
112	C₄²	—	53	3	no	—
113	C₄²	—	53	3	no	—
114	C₄²	C₂²×D₅₃	1	3	yes	—
115	C₂³	—	53	1	no	—
116	Q₈	—	53	12	no	—
117	Q₈	C₈×D₅₃	1	4	yes	—
118	C₂×C₄	—	53	3	no	—
119	C₂×C₄	—	53	12	no	—
120	C₂×C₄	—	53	1	no	—
121	C₂×C₄	—	53	3	no	—
122	C₂×C₄	—	106	1	no	—
123	C₂×C₄	—	106	1	no	—
124	C₂×C₄	—	212	3	no	—
125	C₂×C₄	C₂×C₄×D₅₃	1	3	yes	—
126	C₂×C₄	C₂×C₄×D₅₃	1	3	yes	—
127	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
128	C₈	—	2	3	no	—
129	C₈	—	2	1	no	—
130	C₈	—	106	1	no	—
131	C₈	—	106	3	no	—
132	C₂²	—	53	2	no	—
133	C₂²	—	106	1	no	—
134	C₂²	—	106	1	no	—
135	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
136	C₄	—	2	3	no	—
137	C₄	—	53	6	no	—
138	C₄	—	106	1	no	—
139	C₄	—	106	3	no	—
140	C₄	C₂×C₈×D₅₃	1	6	yes	—
141	C₄	#13133264	1	1	yes	characteristic
142	C₄	#13133265	1	1	yes	characteristic
143	C₂	—	106	2	no	—
144	C₂	#16811099	1	1	yes	characteristic
145	C₂	C₂²×C₈×D₅₃	1	1	yes	characteristic
146	C₂	#16811388	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×Q₈×D₅₃

C₂

yes

maximal, characteristic

Q₈×#88322

C₂

yes

maximal, characteristic, complemented

C₈×Dic₁₀₆

C₂

yes

maximal, complemented

#16810747

C₂

yes

maximal, complemented

#16810675

C₂

yes

maximal

#16810674

C₂

yes

maximal

C₈×Q₈×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₅₃

C₄

yes

characteristic

Q₈×Dic₅₃

C₄

yes

characteristic

C₄×Dic₁₀₆

C₂²

yes

—

#13133182

C₄

yes

—

C₄⋊C₄×D₅₃

C₄

yes

—

C₄²×D₅₃

C₂²

yes

—

C₄⋊#88322

C₂²

yes

—

#13133146

C₂²

yes

—

#13133146

C₂²

yes

—

#13133129

C₂²

yes

characteristic

#13133129

C₂²

yes

—

C₄×#88322

C₂²

yes

—

C₈×Dic₅₃

C₂²

yes

—

C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

—

C₂×Q₈×C₅₃

C₂×C₄

yes

characteristic

Q₈×D₅₃

C₈

yes

complemented

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₂³

yes

characteristic

C₂×Dic₁₀₆

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂³

yes

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

#11719919

C₂×C₄

yes

—

C₄×Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

C₂³

yes

—

C₂×#88322

C₂³

yes

—

C₂×#88322

C₂³

yes

characteristic

C₂²×D₅₃

C₂²×C₄

yes

characteristic

Q₈×C₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

C₂²×C₄

yes

—

C₂×Dic₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

C₂²×C₄

yes

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

C₂×C₈

yes

—

Dic₁₀₆

C₂×C₈

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

C₂²×C₈

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂²×C₈

yes

—

C₄×C₅₃

2₋¹⁺⁴

yes

characteristic

C₄×C₅₃

2₋¹⁺⁴

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

C₂²×C₈

yes

—

—

C₂×C₅₃

yes

C₂×C₅₃

C₂³×C₈

yes

characteristic

C₂×C₅₃

#568

yes

characteristic

D₅₃

—

C₂×C₄×Q₈

—

C₈×Q₈

—

C₈×Q₈

—

C₈×Q₈

—

C₈×Q₈

D₅₃

yes

characteristic, complemented

#431

—

#431

—

C₅₃

#2982

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×D₅₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₅₃

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₅₃

yes

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

C₄×D₅₃

yes

characteristic

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

C₂²×D₅₃

yes

—

107

C₂×C₈

C₂²×D₅₃

yes

characteristic

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

C₄×D₅₃

yes

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₄²

C₂²×D₅₃

yes

—

115

C₂³

—

116

Q₈

—

117

Q₈

C₈×D₅₃

yes

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

106

—

123

C₂×C₄

—

106

—

124

C₂×C₄

—

212

—

125

C₂×C₄

C₂×C₄×D₅₃

yes

—

126

C₂×C₄

C₂×C₄×D₅₃

yes

—

127

C₂×C₄

C₂³×D₅₃

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

106

—

131

C₈

—

106

—

132

C₂²

—

133

C₂²

—

106

—

134

C₂²

—

106

—

135

C₂²

C₂²×C₄×D₅₃

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

106

—

139

C₄

—

106

—

140

C₄

C₂×C₈×D₅₃

yes

—

141

C₄

#13133264

yes

characteristic

142

C₄

#13133265

yes

characteristic

143

C₂

—

106

—

144

C₂

#16811099

yes

characteristic

145

C₂

C₂²×C₈×D₅₃

yes

characteristic

146

C₂

#16811388

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator	C₂×C₅₃	C₂³×C₈
Fitting	C₈×Q₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂³×C₄
hypercenter	C₈×Q₈	D₅₃
hypocenter	C₅₃	#2982
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₈×Q₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₅₃

commutator

C₂×C₅₃

C₂³×C₈

Fitting

C₈×Q₈×C₅₃

C₂

socle

C₂²×C₅₃

C₂³×C₄

hypercenter

C₈×Q₈

D₅₃

C₅₃

G ▷ C₂×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ C₈×Q₈

N	H ≅ G/N	type
2. Q₈×#88322	C₂	semidirect
3. C₈×Dic₁₀₆	C₂	semidirect
4. #16810747	C₂	semidirect
7. C₈×Q₈×C₅₃	C₂	semidirect
25. Q₈×D₅₃	C₈	semidirect
78. C₈×Q₈	D₅₃	semidirect
81. C₅₃	#2982	semidirect

H ≅ G/N

type

2. Q₈×#88322

C₂

semidirect

3. C₈×Dic₁₀₆

C₂

semidirect

4. #16810747

C₂

semidirect

7. C₈×Q₈×C₅₃

C₂

semidirect

25. Q₈×D₅₃

C₈

semidirect

78. C₈×Q₈

D₅₃

semidirect

81. C₅₃

#2982

semidirect

Subgroups — #20283794

Conjugacy classes of subgroups

Complements