Group #20275521 — subgroups

Subgroups — #20275521

Order 6784.

436 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 141 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×#13133214	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133139	C₂	1	4	yes	maximal, complemented
3	#16810493	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₂₁₂	—	2	2	no	—
6	C₂²×D₂₁₂	C₂²	1	2	yes	—
7	#13133214	—	2	2	no	—
8	#13133214	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₅₃	—	2	2	no	—
10	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
11	#13133139	—	2	8	no	—
12	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₅₃	—	2	2	no	—
15	C₂³×D₅₃	—	2	2	no	—
16	C₂²×C₄×C₅₃	D₄	1	2	yes	—
17	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
18	C₂²×Dic₅₃	—	2	2	no	—
19	C₂×D₂₁₂	—	2	4	no	—
20	C₂×D₂₁₂	—	2	4	no	—
21	C₂×D₂₁₂	—	2	8	no	—
22	C₂×D₂₁₂	—	2	2	no	—
23	C₂×D₂₁₂	—	4	4	no	—
24	C₂×D₂₁₂	—	4	4	no	—
25	C₂×D₂₁₂	C₂×C₄	1	4	yes	—
26	C₂×C₈×C₅₃	—	2	2	no	—
27	C₂×C₈×C₅₃	—	2	2	no	—
28	C₂×C₈×C₅₃	D₄	1	4	yes	complemented
29	C₄²×C₅₃	C₂×C₄	1	2	yes	—
30	C₄²×C₅₃	D₄	1	2	yes	—
31	C₄⋊Dic₅₃	—	2	4	no	—
32	C₄⋊Dic₅₃	—	4	4	no	—
33	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
34	C₂²×D₅₃	—	2	4	no	—
35	C₂²×D₅₃	—	4	2	no	—
36	C₂²×D₅₃	—	4	2	no	—
37	C₂²×D₅₃	—	4	4	no	—
38	C₂²×D₅₃	—	4	4	no	—
39	C₂×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
40	C₂×C₄×C₅₃	D₈	1	4	yes	—
41	C₂×C₄×C₅₃	SD₁₆	1	4	yes	—
42	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	#41	1	2	yes	—
46	C₂×Dic₅₃	—	4	2	no	—
47	C₂×Dic₅₃	—	4	4	no	—
48	D₂₁₂	—	2	8	no	—
49	D₂₁₂	—	4	16	no	—
50	D₂₁₂	—	4	4	no	—
51	C₈×C₅₃	—	2	4	no	—
52	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₅₃	#124	1	1	yes	characteristic
54	C₂²×C₅₃	#127	1	1	yes	characteristic
55	C₂²×C₅₃	C₂×D₈	1	2	yes	—
56	C₂²×C₅₃	C₂×SD₁₆	1	2	yes	—
57	C₂×D₅₃	—	4	4	no	—
58	C₂×D₅₃	—	4	8	no	—
59	C₂×D₅₃	—	8	4	no	—
60	C₂×D₅₃	—	8	2	no	—
61	C₂×D₅₃	—	8	2	no	—
62	C₄×C₅₃	#102	1	8	yes	complemented
63	C₄×C₅₃	C₄×D₄	1	2	yes	—
64	C₄×C₅₃	#121	1	2	yes	—
65	Dic₅₃	—	8	2	no	—
66	#1957	—	53	1	no	maximal, Sylow
67	C₂×C₅₃	#390	1	1	yes	characteristic
68	C₂×C₅₃	C₂×#102	1	2	yes	—
69	C₂×C₅₃	#486	1	2	yes	—
70	C₂×C₅₃	#492	1	1	yes	characteristic
71	C₂×C₅₃	#493	1	1	yes	characteristic
72	D₅₃	—	8	4	no	—
73	C₂×#127	—	53	1	no	—
74	C₂×#102	—	53	4	no	—
75	C₂×C₄×C₈	D₅₃	1	1	yes	characteristic, complemented
76	#384	—	53	1	no	—
77	C₅₃	#1957	1	1	yes	characteristic, complemented, Sylow
78	C₂²×D₄	—	53	2	no	—
79	C₂²×D₄	—	106	2	no	—
80	C₂²×C₈	C₂×D₅₃	1	2	yes	—
81	#127	—	53	4	no	—
82	#127	—	106	2	no	—
83	C₂×C₄⋊C₄	—	53	2	no	—
84	C₂×C₄⋊C₄	—	106	2	no	—
85	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
86	#102	—	106	8	no	—
87	C₄×C₈	—	2	2	no	—
88	C₂⁴	—	106	2	no	—
89	C₂×D₄	—	53	4	no	—
90	C₂×D₄	—	106	4	no	—
91	C₂×D₄	—	106	4	no	—
92	C₂×D₄	—	106	2	no	—
93	C₂×D₄	—	106	8	no	—
94	C₂×D₄	—	212	4	no	—
95	C₂×D₄	—	212	4	no	—
96	C₂²×C₄	—	106	2	no	—
97	C₂²×C₄	D₂₁₂	1	2	yes	—
98	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	D₂₁₂	1	4	yes	complemented
102	C₄⋊C₄	—	106	4	no	—
103	C₄⋊C₄	—	212	4	no	—
104	C₄²	C₄×D₅₃	1	2	yes	—
105	C₄²	C₂²⋊D₅₃	1	2	yes	—
106	C₂³	—	106	4	no	—
107	C₂³	—	212	2	no	—
108	C₂³	—	212	2	no	—
109	C₂³	—	212	4	no	—
110	C₂³	—	212	4	no	—
111	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
112	D₄	—	106	8	no	—
113	D₄	—	212	16	no	—
114	D₄	—	212	4	no	—
115	C₂×C₄	—	212	2	no	—
116	C₂×C₄	—	212	4	no	—
117	C₂×C₄	#11719913	1	4	yes	—
118	C₂×C₄	D₄₂₄	1	4	yes	—
119	C₂×C₄	#11719921	1	4	yes	—
120	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
121	C₂×C₄	C₂×D₂₁₂	1	2	yes	—
122	C₂×C₄	#11719945	1	2	yes	—
123	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
124	C₈	—	2	4	no	—
125	C₂²	—	424	4	no	—
126	C₂²	—	424	2	no	—
127	C₂²	—	424	2	no	—
128	C₂²	—	212	4	no	—
129	C₂²	—	212	8	no	—
130	C₂²	C₂×#11719921	1	1	yes	characteristic
131	C₂²	#13133205	1	1	yes	characteristic
132	C₂²	#13133214	1	1	yes	characteristic
133	C₂²	C₂×D₄₂₄	1	2	yes	—
134	C₂²	C₂×#11719913	1	2	yes	—
135	C₄	—	424	2	no	—
136	C₄	#13133139	1	8	yes	complemented
137	C₄	C₄×D₂₁₂	1	2	yes	—
138	C₄	#13133191	1	2	yes	—
139	C₂	—	424	4	no	—
140	C₂	#16810525	1	1	yes	characteristic
141	C₂	C₂×#13133139	1	2	yes	—
142	C₂	#16810970	1	2	yes	—
143	C₂	#16810996	1	1	yes	characteristic
144	C₂	#16811000	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133214

C₂

yes

maximal, characteristic

C₂×#13133139

C₂

yes

maximal, complemented

#16810493

C₂

yes

maximal, characteristic, complemented

C₂×C₄×C₈×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₁₂

—

C₂²×D₂₁₂

C₂²

yes

—

#13133214

—

#13133214

C₄

yes

complemented

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

#13133139

—

C₂²×C₈×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

—

C₂³×D₅₃

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

C₂×C₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

complemented

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

D₄

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

D₈

yes

—

C₂×C₄×C₅₃

SD₁₆

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₈×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

C₂×D₈

yes

—

C₂²×C₅₃

C₂×SD₁₆

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

#102

yes

complemented

C₄×C₅₃

C₄×D₄

yes

—

C₄×C₅₃

#121

yes

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#486

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

C₂×#127

—

C₂×#102

—

C₂×C₄×C₈

D₅₃

yes

characteristic, complemented

#384

—

C₅₃

#1957

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×D₄

—

106

—

C₂²×C₈

C₂×D₅₃

yes

—

#127

—

#127

—

106

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#102

—

106

—

C₄×C₈

—

C₂⁴

—

106

—

C₂×D₄

—

C₂×D₄

—

106

—

C₂×D₄

—

106

—

C₂×D₄

—

106

—

C₂×D₄

—

106

—

C₂×D₄

—

212

—

C₂×D₄

—

212

—

C₂²×C₄

—

106

—

C₂²×C₄

D₂₁₂

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

D₂₁₂

yes

complemented

102

C₄⋊C₄

—

106

—

103

C₄⋊C₄

—

212

—

104

C₄²

C₄×D₅₃

yes

—

105

C₄²

C₂²⋊D₅₃

yes

—

106

C₂³

—

106

—

107

C₂³

—

212

—

108

C₂³

—

212

—

109

C₂³

—

212

—

110

C₂³

—

212

—

111

C₂³

C₂×D₂₁₂

yes

characteristic

112

D₄

—

106

—

113

D₄

—

212

—

114

D₄

—

212

—

115

C₂×C₄

—

212

—

116

C₂×C₄

—

212

—

117

C₂×C₄

#11719913

yes

—

118

C₂×C₄

D₄₂₄

yes

—

119

C₂×C₄

#11719921

yes

—

120

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

121

C₂×C₄

C₂×D₂₁₂

yes

—

122

C₂×C₄

#11719945

yes

—

123

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

124

C₈

—

125

C₂²

—

424

—

126

C₂²

—

424

—

127

C₂²

—

424

—

128

C₂²

—

212

—

129

C₂²

—

212

—

130

C₂²

C₂×#11719921

yes

characteristic

131

C₂²

#13133205

yes

characteristic

132

C₂²

#13133214

yes

characteristic

133

C₂²

C₂×D₄₂₄

yes

—

134

C₂²

C₂×#11719913

yes

—

135

C₄

—

424

—

136

C₄

#13133139

yes

complemented

137

C₄

C₄×D₂₁₂

yes

—

138

C₄

#13133191

yes

—

139

C₂

—

424

—

140

C₂

#16810525

yes

characteristic

141

C₂

C₂×#13133139

yes

—

142

C₂

#16810970

yes

—

143

C₂

#16810996

yes

characteristic

144

C₂

#16811000

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	#1957
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

#1957

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. C₂×#13133139	C₂	semidirect
3. #16810493	C₂	semidirect
4. C₂×C₄×C₈×C₅₃	C₂	semidirect
8. #13133214	C₄	semidirect
28. C₂×C₈×C₅₃	D₄	semidirect
62. C₄×C₅₃	#102	semidirect
75. C₂×C₄×C₈	D₅₃	semidirect
77. C₅₃	#1957	semidirect
101. C₂×C₈	D₂₁₂	semidirect
136. C₄	#13133139	semidirect

H ≅ G/N

type

2. C₂×#13133139

C₂

semidirect

3. #16810493

C₂

semidirect

4. C₂×C₄×C₈×C₅₃

C₂

semidirect

8. #13133214

C₄

semidirect

28. C₂×C₈×C₅₃

D₄

semidirect

62. C₄×C₅₃

#102

semidirect

75. C₂×C₄×C₈

D₅₃

semidirect

77. C₅₃

#1957

semidirect

101. C₂×C₈

D₂₁₂

semidirect

136. C₄

#13133139

semidirect

Subgroups — #20275521

Conjugacy classes of subgroups

Complements