Group #20287358 — subgroups

Subgroups — #20287358

Order 6784.

370 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 139 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	#16811352	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×D₂₁₂	C₂	1	1	yes	maximal, characteristic
3	#16810974	C₂	1	3	yes	maximal, complemented
4	#16810909	C₂	1	3	yes	maximal, complemented
5	#16810829	C₂	1	3	yes	maximal
6	C₄×#11719913	C₂	1	3	yes	maximal
7	C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₅₃	—	2	1	no	—
9	C₂×#11719913	C₂²	1	3	yes	—
10	#13133217	C₂²	1	3	yes	—
11	#13133217	C₂²	1	3	yes	—
12	#13133208	—	2	3	no	—
13	#13133197	—	2	3	no	—
14	Q₈×Dic₅₃	—	2	1	no	—
15	C₄×Dic₁₀₆	C₂²	1	3	yes	—
16	C₄×D₂₁₂	C₂²	1	3	yes	—
17	#13133149	C₂²	1	3	yes	—
18	#13133142	C₂²	1	6	yes	—
19	#13133142	C₂²	1	3	yes	—
20	#13133139	C₂²	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
25	Q₈×D₅₃	—	4	4	no	—
26	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
27	C₂×C₄×D₅₃	—	2	3	no	—
28	C₂×Dic₁₀₆	—	2	3	no	—
29	C₂×Dic₁₀₆	C₂³	1	3	yes	—
30	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₅₃	C₂³	1	3	yes	—
32	C₄⋊C₄×C₅₃	D₄	1	3	yes	—
33	C₄²×C₅₃	C₂³	1	3	yes	—
34	#11719921	—	2	3	no	—
35	C₄⋊Dic₅₃	—	2	3	no	—
36	C₄⋊Dic₅₃	—	2	3	no	—
37	C₄⋊Dic₅₃	—	2	3	no	—
38	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
39	C₄⋊Dic₅₃	C₂³	1	3	yes	—
40	#11719919	—	2	3	no	—
41	C₄×Dic₅₃	—	2	3	no	—
42	#11719913	—	2	6	no	—
43	C₂²×D₅₃	—	2	1	no	—
44	Q₈×C₅₃	SD₁₆	1	4	yes	complemented
45	C₂×C₄×C₅₃	C₂×D₄	1	3	yes	—
46	C₂×C₄×C₅₃	C₂×D₄	1	3	yes	—
47	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
48	C₂×Dic₅₃	—	2	3	no	—
49	C₂×Dic₅₃	—	2	3	no	—
50	C₂×Dic₅₃	—	2	1	no	—
51	D₂₁₂	—	2	1	no	—
52	D₂₁₂	#41	1	2	yes	—
53	C₄×D₅₃	—	4	6	no	—
54	Dic₁₀₆	—	2	3	no	—
55	Dic₁₀₆	—	2	3	no	—
56	Dic₁₀₆	—	4	6	no	—
57	C₈×C₅₃	—	2	3	no	—
58	C₈×C₅₃	—	2	1	no	—
59	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
60	C₂×D₅₃	—	2	2	no	—
61	C₂×D₅₃	—	4	1	no	—
62	C₂×D₅₃	—	4	1	no	—
63	C₄×C₅₃	—	2	3	no	—
64	C₄×C₅₃	C₂×SD₁₆	1	6	yes	—
65	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
66	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
67	Dic₅₃	—	4	3	no	—
68	Dic₅₃	—	4	3	no	—
69	Dic₅₃	—	4	1	no	—
70	#3359	—	53	1	no	maximal, Sylow
71	C₂×C₅₃	#548	1	1	yes	characteristic
72	C₂×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
73	C₂×C₅₃	#578	1	1	yes	characteristic
74	D₅₃	—	4	2	no	—
75	#557	—	53	1	no	—
76	D₄×Q₈	—	53	1	no	—
77	#487	—	53	3	no	—
78	#475	—	53	3	no	—
79	#461	—	53	3	no	—
80	C₈×Q₈	D₅₃	1	1	yes	characteristic, complemented
81	C₄×SD₁₆	—	53	3	no	—
82	C₅₃	#3359	1	1	yes	characteristic, complemented, Sylow
83	C₂²×Q₈	—	106	1	no	—
84	C₂×SD₁₆	—	53	3	no	—
85	C₄⋊Q₈	—	53	3	no	—
86	C₄⋊Q₈	—	53	3	no	—
87	#125	—	106	3	no	—
88	C₂²⋊Q₈	—	106	3	no	—
89	C₄×Q₈	—	53	3	no	—
90	C₄×Q₈	—	106	1	no	—
91	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
92	C₄×D₄	—	53	3	no	—
93	#106	—	53	3	no	—
94	C₄⋊C₈	C₂×D₅₃	1	3	yes	—
95	#103	—	53	3	no	—
96	#103	—	53	6	no	—
97	#102	—	53	1	no	—
98	C₄×C₈	C₂×D₅₃	1	3	yes	—
99	C₂×Q₈	—	53	3	no	—
100	C₂×Q₈	—	106	3	no	—
101	C₂×Q₈	—	212	4	no	—
102	C₂×Q₈	D₂₁₂	1	1	yes	characteristic
103	C₂×D₄	—	53	1	no	—
104	C₂²×C₄	—	106	3	no	—
105	SD₁₆	—	106	6	no	—
106	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
107	C₂×C₈	C₂²×D₅₃	1	3	yes	—
108	C₄⋊C₄	—	53	1	no	—
109	C₄⋊C₄	—	53	3	no	—
110	C₄⋊C₄	—	106	3	no	—
111	C₄⋊C₄	—	106	3	no	—
112	C₄⋊C₄	—	106	3	no	—
113	C₄⋊C₄	—	106	3	no	—
114	C₄⋊C₄	D₂₁₂	1	3	yes	—
115	C₂²⋊C₄	—	106	3	no	—
116	C₄²	—	106	3	no	—
117	C₄²	C₂²×D₅₃	1	3	yes	—
118	C₂³	—	106	1	no	—
119	Q₈	—	106	3	no	—
120	Q₈	—	106	3	no	—
121	Q₈	—	212	6	no	—
122	Q₈	#11719913	1	4	yes	complemented
123	D₄	—	53	2	no	—
124	D₄	—	106	1	no	—
125	C₂×C₄	—	106	1	no	—
126	C₂×C₄	—	106	3	no	—
127	C₂×C₄	—	106	3	no	—
128	C₂×C₄	—	212	6	no	—
129	C₂×C₄	C₂×D₂₁₂	1	3	yes	—
130	C₂×C₄	C₂×D₂₁₂	1	3	yes	—
131	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
132	C₈	—	2	3	no	—
133	C₈	—	2	1	no	—
134	C₂²	—	106	2	no	—
135	C₂²	—	212	1	no	—
136	C₂²	—	212	1	no	—
137	C₂²	C₂²×D₂₁₂	1	1	yes	characteristic
138	C₄	—	2	3	no	—
139	C₄	—	212	3	no	—
140	C₄	—	212	3	no	—
141	C₄	—	212	1	no	—
142	C₄	C₂×#11719913	1	6	yes	—
143	C₄	#41×D₅₃	1	1	yes	characteristic
144	C₄	#13133265	1	1	yes	characteristic
145	C₂	—	212	2	no	—
146	C₂	#16811285	1	1	yes	characteristic
147	C₂	C₂²×#11719913	1	1	yes	characteristic
148	C₂	#16811440	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811352

C₂

yes

maximal, characteristic, complemented

Q₈×D₂₁₂

C₂

yes

maximal, characteristic

#16810974

C₂

yes

maximal, complemented

#16810909

C₂

yes

maximal, complemented

#16810829

C₂

yes

maximal

C₄×#11719913

C₂

yes

maximal

C₈×Q₈×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₅₃

—

C₂×#11719913

C₂²

yes

—

#13133217

C₂²

yes

—

#13133217

C₂²

yes

—

#13133208

—

#13133197

—

Q₈×Dic₅₃

—

C₄×Dic₁₀₆

C₂²

yes

—

C₄×D₂₁₂

C₂²

yes

—

#13133149

C₂²

yes

—

#13133142

C₂²

yes

—

#13133142

C₂²

yes

—

#13133139

C₂²

yes

characteristic

C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

—

C₂×Q₈×C₅₃

D₄

yes

characteristic

Q₈×D₅₃

—

C₂×D₂₁₂

C₂³

yes

characteristic

C₂×C₄×D₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂³

yes

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

—

#11719921

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

characteristic

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

C₄×Dic₅₃

—

#11719913

—

C₂²×D₅₃

—

Q₈×C₅₃

SD₁₆

yes

complemented

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

#41

yes

—

C₄×D₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×SD₁₆

yes

—

C₄×C₅₃

C₂×#41

yes

characteristic

C₄×C₅₃

2₋¹⁺⁴

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

C₂²×SD₁₆

yes

characteristic

C₂×C₅₃

#578

yes

characteristic

D₅₃

—

#557

—

D₄×Q₈

—

#487

—

#475

—

#461

—

C₈×Q₈

D₅₃

yes

characteristic, complemented

C₄×SD₁₆

—

C₅₃

#3359

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

106

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

106

—

C₂²⋊Q₈

—

106

—

C₄×Q₈

—

C₄×Q₈

—

106

—

C₄×Q₈

C₂×D₅₃

yes

characteristic

C₄×D₄

—

#106

—

C₄⋊C₈

C₂×D₅₃

yes

—

#103

—

#103

—

#102

—

C₄×C₈

C₂×D₅₃

yes

—

C₂×Q₈

—

100

C₂×Q₈

—

106

—

101

C₂×Q₈

—

212

—

102

C₂×Q₈

D₂₁₂

yes

characteristic

103

C₂×D₄

—

104

C₂²×C₄

—

106

—

105

SD₁₆

—

106

—

106

C₂×C₈

C₂²×D₅₃

yes

characteristic

107

C₂×C₈

C₂²×D₅₃

yes

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

106

—

111

C₄⋊C₄

—

106

—

112

C₄⋊C₄

—

106

—

113

C₄⋊C₄

—

106

—

114

C₄⋊C₄

D₂₁₂

yes

—

115

C₂²⋊C₄

—

106

—

116

C₄²

—

106

—

117

C₄²

C₂²×D₅₃

yes

—

118

C₂³

—

106

—

119

Q₈

—

106

—

120

Q₈

—

106

—

121

Q₈

—

212

—

122

Q₈

#11719913

yes

complemented

123

D₄

—

124

D₄

—

106

—

125

C₂×C₄

—

106

—

126

C₂×C₄

—

106

—

127

C₂×C₄

—

106

—

128

C₂×C₄

—

212

—

129

C₂×C₄

C₂×D₂₁₂

yes

—

130

C₂×C₄

C₂×D₂₁₂

yes

—

131

C₂×C₄

C₂³×D₅₃

yes

characteristic

132

C₈

—

133

C₈

—

134

C₂²

—

106

—

135

C₂²

—

212

—

136

C₂²

—

212

—

137

C₂²

C₂²×D₂₁₂

yes

characteristic

138

C₄

—

139

C₄

—

212

—

140

C₄

—

212

—

141

C₄

—

212

—

142

C₄

C₂×#11719913

yes

—

143

C₄

#41×D₅₃

yes

characteristic

144

C₄

#13133265

yes

characteristic

145

C₂

—

212

—

146

C₂

#16811285

yes

characteristic

147

C₂

C₂²×#11719913

yes

characteristic

148

C₂

#16811440

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₈×Q₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	C₈×Q₈	D₅₃
hypocenter	C₅₃	#3359
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×D₂₁₂

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₈×Q₈×C₅₃

C₂

socle

C₂²×C₅₃

C₂²×D₄

hypercenter

C₈×Q₈

D₅₃

hypocenter

C₅₃

#3359

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
1. #16811352	C₂	semidirect
3. #16810974	C₂	semidirect
4. #16810909	C₂	semidirect
7. C₈×Q₈×C₅₃	C₂	semidirect
44. Q₈×C₅₃	SD₁₆	semidirect
80. C₈×Q₈	D₅₃	semidirect
82. C₅₃	#3359	semidirect
122. Q₈	#11719913	semidirect

H ≅ G/N

type

1. #16811352

C₂

semidirect

3. #16810974

C₂

semidirect

4. #16810909

C₂

semidirect

7. C₈×Q₈×C₅₃

C₂

semidirect

44. Q₈×C₅₃

SD₁₆

semidirect

80. C₈×Q₈

D₅₃

semidirect

82. C₅₃

#3359

semidirect

122. Q₈

#11719913

semidirect

Subgroups — #20287358

Conjugacy classes of subgroups

Complements