Group #20287353 — subgroups

Subgroups — #20287353

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	Q₈×D₂₁₂	C₂	1	1	yes	maximal, characteristic
2	#16810973	C₂	1	3	yes	maximal, complemented
3	#16810908	C₂	1	3	yes	maximal, complemented
4	#16810825	C₂	1	3	yes	maximal
5	Q₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#11719924	C₂	1	3	yes	maximal
7	C₅₃×#557	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₅₃	—	2	1	no	—
9	C₂×#11719924	C₂²	1	3	yes	—
10	#13133217	C₂²	1	3	yes	—
11	#13133197	—	2	3	no	—
12	C₄×D₂₁₂	C₂²	1	3	yes	—
13	#13133148	C₂²	1	3	yes	—
14	C₄⋊#88322	C₂²	1	3	yes	—
15	#13133141	C₂²	1	6	yes	—
16	#13133141	C₂²	1	3	yes	—
17	#13133137	C₂²	1	1	yes	characteristic
18	C₄×#88322	C₂²	1	3	yes	—
19	C₄⋊Q₈×C₅₃	C₂²	1	3	yes	—
20	#125×C₅₃	—	2	3	no	—
21	C₄×Q₈×C₅₃	—	2	1	no	—
22	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₄×Q₈×C₅₃	C₂²	1	3	yes	—
24	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
25	C₂×Q₈×C₅₃	C₂³	1	3	yes	—
26	Q₈×D₅₃	—	4	4	no	—
27	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
28	C₂×C₄×D₅₃	—	2	3	no	—
29	C₂×Dic₁₀₆	—	2	3	no	—
30	C₄⋊C₄×C₅₃	—	2	3	no	—
31	C₄⋊C₄×C₅₃	—	2	3	no	—
32	C₄⋊C₄×C₅₃	—	2	3	no	—
33	C₄⋊C₄×C₅₃	D₄	1	3	yes	—
34	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
35	C₄²×C₅₃	—	2	3	no	—
36	C₄²×C₅₃	C₂³	1	3	yes	—
37	#11719924	—	2	6	no	—
38	#11719921	—	2	3	no	—
39	C₄⋊Dic₅₃	—	2	3	no	—
40	C₄⋊Dic₅₃	C₂³	1	3	yes	—
41	C₂×#88322	C₂³	1	1	yes	characteristic
42	C₂×#88322	C₂³	1	3	yes	—
43	C₂²×D₅₃	—	2	1	no	—
44	Q₈×C₅₃	—	2	3	no	—
45	Q₈×C₅₃	—	2	3	no	—
46	Q₈×C₅₃	SD₁₆	1	4	yes	complemented
47	C₂×C₄×C₅₃	—	2	3	no	—
48	C₂×C₄×C₅₃	—	2	1	no	—
49	C₂×C₄×C₅₃	C₂×D₄	1	3	yes	—
50	C₂×C₄×C₅₃	C₂×D₄	1	3	yes	—
51	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
52	C₂×Dic₅₃	—	2	3	no	—
53	D₂₁₂	—	2	1	no	—
54	D₂₁₂	#41	1	2	yes	—
55	C₄×D₅₃	—	4	6	no	—
56	Dic₁₀₆	—	4	6	no	—
57	#88322	—	2	3	no	—
58	#88322	—	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	—	4	3	no	—
65	C₄×C₅₃	—	4	1	no	—
66	C₄×C₅₃	C₂×SD₁₆	1	6	yes	—
67	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
68	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
69	Dic₅₃	—	4	3	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	D₅₃	1	1	yes	characteristic, complemented
76	D₄×Q₈	—	53	1	no	—
77	#487	—	53	3	no	—
78	#475	—	53	3	no	—
79	#461	—	53	3	no	—
80	C₈×Q₈	—	53	1	no	—
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	C₂×D₅₃	1	3	yes	—
87	#125	—	2	3	no	—
88	C₂²⋊Q₈	—	106	3	no	—
89	C₄×Q₈	—	2	1	no	—
90	C₄×Q₈	C₂×D₅₃	1	3	yes	—
91	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
92	C₄×D₄	—	53	3	no	—
93	#106	—	53	3	no	—
94	C₄⋊C₈	—	53	3	no	—
95	#103	—	53	3	no	—
96	#103	—	53	6	no	—
97	#102	—	53	1	no	—
98	C₄×C₈	—	53	3	no	—
99	C₂×Q₈	—	106	3	no	—
100	C₂×Q₈	—	212	4	no	—
101	C₂×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
102	C₂×Q₈	C₂²×D₅₃	1	3	yes	—
103	C₂×D₄	—	53	1	no	—
104	C₂²×C₄	—	106	3	no	—
105	SD₁₆	—	106	6	no	—
106	C₂×C₈	—	53	1	no	—
107	C₂×C₈	—	53	3	no	—
108	C₄⋊C₄	—	2	3	no	—
109	C₄⋊C₄	—	2	3	no	—
110	C₄⋊C₄	—	2	3	no	—
111	C₄⋊C₄	—	53	3	no	—
112	C₄⋊C₄	—	106	3	no	—
113	C₄⋊C₄	C₂²⋊D₅₃	1	3	yes	—
114	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
115	C₂²⋊C₄	—	106	3	no	—
116	C₄²	—	2	3	no	—
117	C₄²	C₂²×D₅₃	1	3	yes	—
118	C₂³	—	106	1	no	—
119	Q₈	—	2	3	no	—
120	Q₈	—	2	3	no	—
121	Q₈	—	212	6	no	—
122	Q₈	#11719924	1	4	yes	complemented
123	D₄	—	53	2	no	—
124	D₄	—	106	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	3	no	—
127	C₂×C₄	—	106	3	no	—
128	C₂×C₄	—	212	6	no	—
129	C₂×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
130	C₂×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
131	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
132	C₈	—	106	3	no	—
133	C₈	—	106	1	no	—
134	C₂²	—	106	2	no	—
135	C₂²	—	212	1	no	—
136	C₂²	—	212	1	no	—
137	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
138	C₄	—	2	3	no	—
139	C₄	—	4	3	no	—
140	C₄	—	4	1	no	—
141	C₄	—	212	3	no	—
142	C₄	C₂×#11719924	1	6	yes	—
143	C₄	#41×D₅₃	1	1	yes	characteristic
144	C₄	#13133264	1	1	yes	characteristic
145	C₂	—	212	2	no	—
146	C₂	#16811280	1	1	yes	characteristic
147	C₂	C₂²×#11719924	1	1	yes	characteristic
148	C₂	#16811438	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×D₂₁₂

C₂

yes

maximal, characteristic

#16810973

C₂

yes

maximal, complemented

#16810908

C₂

yes

maximal, complemented

#16810825

C₂

yes

maximal

Q₈×#88322

C₂

yes

maximal, characteristic, complemented

C₄×#11719924

C₂

yes

maximal

C₅₃×#557

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₅₃

—

C₂×#11719924

C₂²

yes

—

#13133217

C₂²

yes

—

#13133197

—

C₄×D₂₁₂

C₂²

yes

—

#13133148

C₂²

yes

—

C₄⋊#88322

C₂²

yes

—

#13133141

C₂²

yes

—

#13133141

C₂²

yes

—

#13133137

C₂²

yes

characteristic

C₄×#88322

C₂²

yes

—

C₄⋊Q₈×C₅₃

C₂²

yes

—

#125×C₅₃

—

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₄×Q₈×C₅₃

C₂²

yes

—

C₂×Q₈×C₅₃

D₄

yes

characteristic

C₂×Q₈×C₅₃

C₂³

yes

—

Q₈×D₅₃

—

C₂×D₂₁₂

C₂³

yes

characteristic

C₂×C₄×D₅₃

—

C₂×Dic₁₀₆

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₄

yes

—

C₄⋊C₄×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

—

C₄²×C₅₃

C₂³

yes

—

#11719924

—

#11719921

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

C₂×#88322

C₂³

yes

characteristic

C₂×#88322

C₂³

yes

—

C₂²×D₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

SD₁₆

yes

complemented

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

#41

yes

—

C₄×D₅₃

—

Dic₁₀₆

—

#88322

—

#88322

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×SD₁₆

yes

—

C₄×C₅₃

C₂×#41

yes

characteristic

C₄×C₅₃

2₋¹⁺⁴

yes

characteristic

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

C₂²×SD₁₆

yes

characteristic

C₂×C₅₃

#578

yes

characteristic

D₅₃

—

#557

D₅₃

yes

characteristic, complemented

D₄×Q₈

—

#487

—

#475

—

#461

—

C₈×Q₈

—

C₄×SD₁₆

—

C₅₃

#3359

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

106

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅₃

yes

—

#125

—

C₂²⋊Q₈

—

106

—

C₄×Q₈

—

C₄×Q₈

C₂×D₅₃

yes

—

C₄×Q₈

C₂×D₅₃

yes

characteristic

C₄×D₄

—

#106

—

C₄⋊C₈

—

#103

—

#103

—

#102

—

C₄×C₈

—

C₂×Q₈

—

106

—

100

C₂×Q₈

—

212

—

101

C₂×Q₈

C₂²⋊D₅₃

yes

characteristic

102

C₂×Q₈

C₂²×D₅₃

yes

—

103

C₂×D₄

—

104

C₂²×C₄

—

106

—

105

SD₁₆

—

106

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

106

—

113

C₄⋊C₄

C₂²⋊D₅₃

yes

—

114

C₄⋊C₄

C₂²×D₅₃

yes

characteristic

115

C₂²⋊C₄

—

106

—

116

C₄²

—

117

C₄²

C₂²×D₅₃

yes

—

118

C₂³

—

106

—

119

Q₈

—

120

Q₈

—

121

Q₈

—

212

—

122

Q₈

#11719924

yes

complemented

123

D₄

—

124

D₄

—

106

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

106

—

128

C₂×C₄

—

212

—

129

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

130

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

131

C₂×C₄

C₂³×D₅₃

yes

characteristic

132

C₈

—

106

—

133

C₈

—

106

—

134

C₂²

—

106

—

135

C₂²

—

212

—

136

C₂²

—

212

—

137

C₂²

C₂²×C₂²⋊D₅₃

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

212

—

142

C₄

C₂×#11719924

yes

—

143

C₄

#41×D₅₃

yes

characteristic

144

C₄

#13133264

yes

characteristic

145

C₂

—

212

—

146

C₂

#16811280

yes

characteristic

147

C₂

C₂²×#11719924

yes

characteristic

148

C₂

#16811438

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#557	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#557	D₅₃
hypocenter	C₅₃	#3359
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #557

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#557

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #557

N	H ≅ G/N	type
2. #16810973	C₂	semidirect
3. #16810908	C₂	semidirect
5. Q₈×#88322	C₂	semidirect
7. C₅₃×#557	C₂	semidirect
46. Q₈×C₅₃	SD₁₆	semidirect
75. #557	D₅₃	semidirect
82. C₅₃	#3359	semidirect
122. Q₈	#11719924	semidirect

H ≅ G/N

type

2. #16810973

C₂

semidirect

3. #16810908

C₂

semidirect

5. Q₈×#88322

C₂

semidirect

7. C₅₃×#557

C₂

semidirect

46. Q₈×C₅₃

SD₁₆

semidirect

75. #557

D₅₃

semidirect

82. C₅₃

#3359

semidirect

122. Q₈

#11719924

semidirect

Subgroups — #20287353

Conjugacy classes of subgroups

Complements