Group #20275576 — subgroups

Subgroups — #20275576

Order 6784.

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

normal subgroups only

This group has 312 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	C₂²×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133151	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133142	C₂	1	2	yes	maximal
4	#16810529	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₀₆	C₂²	1	2	yes	—
7	C₂×Dic₂₁₂	—	2	2	no	—
8	C₂×Dic₂₁₂	—	2	2	no	—
9	C₂×Dic₂₁₂	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
11	#13133151	—	2	2	no	—
12	#13133142	—	2	4	no	—
13	#13133124	—	2	4	no	—
14	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₄×C₈×C₅₃	—	2	2	no	—
18	C₂²×C₄×C₅₃	D₄	1	2	yes	—
19	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
20	C₂²×Dic₅₃	—	2	2	no	—
21	C₂²×Dic₅₃	—	2	2	no	—
22	C₂×Dic₁₀₆	—	2	2	no	—
23	C₂×Dic₁₀₆	—	2	2	no	—
24	C₂×Dic₁₀₆	—	2	2	no	—
25	C₂×Dic₁₀₆	—	2	2	no	—
26	C₂×Dic₁₀₆	—	2	2	no	—
27	C₂×Dic₁₀₆	C₂×C₄	1	4	yes	—
28	C₂×C₈×C₅₃	—	2	1	no	—
29	C₂×C₈×C₅₃	—	2	1	no	—
30	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₅₃	D₄	1	2	yes	—
32	C₂×C₈×C₅₃	D₄	1	2	yes	—
33	C₂×C₈×C₅₃	D₄	1	2	yes	—
34	C₄²×C₅₃	—	2	1	no	—
35	C₄²×C₅₃	—	2	1	no	—
36	C₄⋊Dic₅₃	—	2	2	no	—
37	C₄⋊Dic₅₃	—	2	2	no	—
38	Dic₂₁₂	—	2	8	no	—
39	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₅₃	—	2	2	no	—
41	C₂×C₄×C₅₃	—	2	2	no	—
42	C₂×C₄×C₅₃	Q₁₆	1	4	yes	complemented
43	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
48	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
49	C₂×Dic₅₃	—	2	4	no	—
50	C₂×Dic₅₃	—	4	2	no	—
51	C₂×Dic₅₃	—	4	2	no	—
52	C₂×Dic₅₃	—	4	2	no	—
53	C₂×Dic₅₃	—	4	4	no	—
54	Dic₁₀₆	—	2	4	no	—
55	Dic₁₀₆	—	2	4	no	—
56	Dic₁₀₆	—	4	2	no	—
57	Dic₁₀₆	—	4	2	no	—
58	C₈×C₅₃	—	2	2	no	—
59	C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
60	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
61	C₂²×C₅₃	#121	1	1	yes	characteristic
62	C₂²×C₅₃	#127	1	1	yes	characteristic
63	C₂²×C₅₃	C₂×Q₁₆	1	2	yes	—
64	C₂²×C₅₃	#135	1	2	yes	—
65	C₄×C₅₃	—	2	4	no	—
66	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
68	C₄×C₅₃	#121	1	1	yes	characteristic
69	C₄×C₅₃	#124	1	1	yes	characteristic
70	Dic₅₃	—	4	4	no	—
71	Dic₅₃	—	8	2	no	—
72	#1969	—	53	1	no	maximal, Sylow
73	C₂×C₅₃	#390	1	1	yes	characteristic
74	C₂×C₅₃	C₄×Q₁₆	1	2	yes	—
75	C₂×C₅₃	#467	1	2	yes	—
76	C₂×C₅₃	#494	1	1	yes	characteristic
77	C₂×C₅₃	#495	1	1	yes	characteristic
78	C₂²×Q₁₆	—	53	1	no	—
79	C₂×#107	—	53	1	no	—
80	C₂×#103	—	53	2	no	—
81	C₂×C₄×C₈	D₅₃	1	1	yes	characteristic
82	#391	—	53	2	no	—
83	C₅₃	#1969	1	1	yes	characteristic, complemented, Sylow
84	C₂²×Q₈	—	53	2	no	—
85	C₂×Q₁₆	—	53	4	no	—
86	C₂×Q₁₆	—	106	2	no	—
87	C₂×Q₁₆	—	106	2	no	—
88	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
89	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
90	C₂×C₄⋊C₄	—	53	2	no	—
91	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
92	#107	—	106	2	no	—
93	#103	—	106	4	no	—
94	C₄×C₈	—	2	2	no	—
95	#95	—	106	4	no	—
96	C₂×Q₈	—	53	4	no	—
97	C₂×Q₈	—	106	2	no	—
98	C₂×Q₈	—	106	2	no	—
99	C₂×Q₈	—	106	2	no	—
100	C₂×Q₈	—	106	2	no	—
101	C₂×Q₈	—	106	2	no	—
102	C₂²×C₄	—	106	2	no	—
103	C₂²×C₄	—	106	2	no	—
104	C₂²×C₄	D₂₁₂	1	2	yes	—
105	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
106	Q₁₆	—	106	8	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	C₄×D₅₃	1	2	yes	—
110	C₂×C₈	D₂₁₂	1	2	yes	—
111	C₂×C₈	D₂₁₂	1	2	yes	—
112	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
113	C₄⋊C₄	—	106	2	no	—
114	C₄⋊C₄	—	106	2	no	—
115	C₄²	—	2	1	no	—
116	C₄²	—	2	1	no	—
117	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
118	Q₈	—	106	4	no	—
119	Q₈	—	106	4	no	—
120	Q₈	—	212	2	no	—
121	Q₈	—	212	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	106	4	no	—
125	C₂×C₄	—	212	2	no	—
126	C₂×C₄	—	212	4	no	—
127	C₂×C₄	—	212	2	no	—
128	C₂×C₄	—	212	2	no	—
129	C₂×C₄	Dic₂₁₂	1	4	yes	complemented
130	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
131	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
132	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
133	C₂×C₄	#11719945	1	1	yes	characteristic
134	C₂×C₄	#11719945	1	1	yes	characteristic
135	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
136	C₈	—	2	2	no	—
137	C₈	#11719921	1	4	yes	—
138	C₂²	C₄×D₂₁₂	1	1	yes	characteristic
139	C₂²	#13133191	1	1	yes	characteristic
140	C₂²	#13133214	1	1	yes	characteristic
141	C₂²	C₂×Dic₂₁₂	1	2	yes	—
142	C₂²	#13133239	1	2	yes	—
143	C₄	—	2	4	no	—
144	C₄	—	424	2	no	—
145	C₄	—	212	4	no	—
146	C₄	C₂×#11719921	1	1	yes	characteristic
147	C₄	C₄×D₂₁₂	1	1	yes	characteristic
148	C₄	#13133191	1	1	yes	characteristic
149	C₄	#13133205	1	1	yes	characteristic
150	C₂	#16810525	1	1	yes	characteristic
151	C₂	C₄×Dic₂₁₂	1	2	yes	—
152	C₂	#16810864	1	2	yes	—
153	C₂	#16811003	1	1	yes	characteristic
154	C₂	#16811007	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₂₁₂

C₂

yes

maximal, characteristic

C₂×#13133151

C₂

yes

maximal, characteristic

C₂×#13133142

C₂

yes

maximal

#16810529

C₂

yes

maximal

C₂×C₄×C₈×C₅₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₂²

yes

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

C₄

yes

complemented

C₂×C₄⋊Dic₅₃

C₂²

yes

—

#13133151

—

#13133142

—

#13133124

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂×C₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

D₄

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

Dic₂₁₂

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

Q₁₆

yes

complemented

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₅₃

C₄×D₄

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

C₂×Q₁₆

yes

—

C₂²×C₅₃

#135

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

C₄×D₄

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

C₄×Q₁₆

yes

—

C₂×C₅₃

#467

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂²×Q₁₆

—

C₂×#107

—

C₂×#103

—

C₂×C₄×C₈

D₅₃

yes

characteristic

#391

—

C₅₃

#1969

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

106

—

C₂×Q₁₆

—

106

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#107

—

106

—

#103

—

106

—

C₄×C₈

—

#95

—

106

—

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

100

C₂×Q₈

—

106

—

101

C₂×Q₈

—

106

—

102

C₂²×C₄

—

106

—

103

C₂²×C₄

—

106

—

104

C₂²×C₄

D₂₁₂

yes

—

105

C₂²×C₄

C₂²×D₅₃

yes

characteristic

106

Q₁₆

—

106

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

C₄×D₅₃

yes

—

110

C₂×C₈

D₂₁₂

yes

—

111

C₂×C₈

D₂₁₂

yes

—

112

C₂×C₈

C₂²⋊D₅₃

yes

—

113

C₄⋊C₄

—

106

—

114

C₄⋊C₄

—

106

—

115

C₄²

—

116

C₄²

—

117

C₂³

C₂×D₂₁₂

yes

characteristic

118

Q₈

—

106

—

119

Q₈

—

106

—

120

Q₈

—

212

—

121

Q₈

—

212

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

106

—

125

C₂×C₄

—

212

—

126

C₂×C₄

—

212

—

127

C₂×C₄

—

212

—

128

C₂×C₄

—

212

—

129

C₂×C₄

Dic₂₁₂

yes

complemented

130

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

131

C₂×C₄

C₂×D₂₁₂

yes

characteristic

132

C₂×C₄

C₂×D₂₁₂

yes

characteristic

133

C₂×C₄

#11719945

yes

characteristic

134

C₂×C₄

#11719945

yes

characteristic

135

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

136

C₈

—

137

C₈

#11719921

yes

—

138

C₂²

C₄×D₂₁₂

yes

characteristic

139

C₂²

#13133191

yes

characteristic

140

C₂²

#13133214

yes

characteristic

141

C₂²

C₂×Dic₂₁₂

yes

—

142

C₂²

#13133239

yes

—

143

C₄

—

144

C₄

—

424

—

145

C₄

—

212

—

146

C₄

C₂×#11719921

yes

characteristic

147

C₄

C₄×D₂₁₂

yes

characteristic

148

C₄

#13133191

yes

characteristic

149

C₄

#13133205

yes

characteristic

150

C₂

#16810525

yes

characteristic

151

C₂

C₄×Dic₂₁₂

yes

—

152

C₂

#16810864

yes

—

153

C₂

#16811003

yes

characteristic

154

C₂

#16811007

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×C₄×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×C₄×C₈	D₅₃
hypocenter	C₅₃	#1969
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₂₁₂

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×C₄×C₈×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

hypercenter

C₂×C₄×C₈

D₅₃

hypocenter

C₅₃

#1969

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
9. C₂×Dic₂₁₂	C₄	semidirect
42. C₂×C₄×C₅₃	Q₁₆	semidirect
83. C₅₃	#1969	semidirect
129. C₂×C₄	Dic₂₁₂	semidirect

H ≅ G/N

type

9. C₂×Dic₂₁₂

C₄

semidirect

42. C₂×C₄×C₅₃

Q₁₆

semidirect

83. C₅₃

#1969

semidirect

129. C₂×C₄

Dic₂₁₂

semidirect

Subgroups — #20275576

Conjugacy classes of subgroups

Complements