Group #20285211 — subgroups

Subgroups — #498×D₅₃ #20285211

Order 6784.

424 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 165 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	2	yes	maximal
2	#16811021	C₂	1	1	yes	maximal, characteristic, complemented
3	#16811020	C₂	1	4	yes	maximal, complemented
4	#16811019	C₂	1	2	yes	maximal, complemented
5	#106×D₅₃	C₂	1	4	yes	maximal
6	C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#498	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₅₃	—	2	2	no	—
9	C₂×C₈×D₅₃	C₂²	1	2	yes	—
10	#13133217	C₂²	1	2	yes	—
11	#13133216	—	2	2	no	—
12	#13133216	C₂²	1	4	yes	—
13	#13133215	—	2	2	no	—
14	C₄⋊C₄×D₅₃	—	2	2	no	—
15	C₄⋊C₄×D₅₃	C₂²	1	4	yes	—
16	C₄²×D₅₃	C₂²	1	1	yes	characteristic
17	#13133149	C₂²	1	4	yes	—
18	#13133148	C₂²	1	8	yes	—
19	C₄×#88322	C₂²	1	1	yes	characteristic
20	C₈×Dic₅₃	C₂²	1	2	yes	—
21	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
22	#106×C₅₃	C₂²	1	4	yes	—
23	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×Q₈×C₅₃	—	2	2	no	—
25	Q₈×D₅₃	—	4	8	no	—
26	C₂×C₄×D₅₃	—	2	4	no	—
27	C₂×C₄×D₅₃	D₄	1	2	yes	—
28	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₀₆	—	2	2	no	—
30	C₂×Dic₁₀₆	—	2	4	no	—
31	C₂×C₈×C₅₃	C₂³	1	2	yes	—
32	C₄⋊C₄×C₅₃	—	2	2	no	—
33	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
34	C₄²×C₅₃	C₂³	1	1	yes	characteristic
35	C₄⋊Dic₅₃	—	2	2	no	—
36	C₄⋊Dic₅₃	C₂³	1	4	yes	—
37	#11719919	—	2	4	no	—
38	#11719919	—	2	4	no	—
39	C₄×Dic₅₃	D₄	1	2	yes	—
40	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
41	C₂×#88322	C₂³	1	2	yes	—
42	C₈×D₅₃	Q₈	1	8	yes	complemented
43	C₂²×D₅₃	C₂×D₄	1	1	yes	characteristic
44	Q₈×C₅₃	—	4	4	no	—
45	C₂×C₄×C₅₃	—	2	4	no	—
46	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
48	C₂×Dic₅₃	—	2	4	no	—
49	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
51	C₄×D₅₃	—	4	8	no	—
52	C₄×D₅₃	SD₁₆	1	8	yes	—
53	C₄×D₅₃	C₂×Q₈	1	4	yes	—
54	Dic₁₀₆	—	4	4	no	—
55	Dic₁₀₆	—	4	8	no	—
56	C₈×C₅₃	C₂×Q₈	1	4	yes	complemented
57	#88322	C₂×Q₈	1	4	yes	complemented
58	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
59	C₂×D₅₃	C₄⋊Q₈	1	2	yes	—
60	C₂×D₅₃	C₂×SD₁₆	1	4	yes	—
61	C₄×C₅₃	—	4	4	no	—
62	C₄×C₅₃	C₂×SD₁₆	1	4	yes	—
63	C₄×C₅₃	C₂²×Q₈	1	2	yes	—
64	Dic₅₃	—	4	4	no	—
65	Dic₅₃	C₄⋊Q₈	1	2	yes	—
66	Dic₅₃	C₂×SD₁₆	1	4	yes	—
67	C₂×#498	—	53	1	no	maximal, Sylow
68	C₂×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₅₃	C₂²×SD₁₆	1	2	yes	—
70	D₅₃	#498	1	4	yes	direct product factor
71	C₂×C₄⋊Q₈	—	53	2	no	—
72	#498	—	53	2	no	—
73	#498	—	53	4	no	—
74	#498	—	53	1	no	—
75	#498	D₅₃	1	1	yes	characteristic, direct product factor
76	C₂×#106	—	53	4	no	—
77	C₂×C₄×C₈	—	53	1	no	—
78	C₅₃	C₂×#498	1	1	yes	characteristic, complemented, Sylow
79	C₂²×Q₈	—	106	2	no	—
80	C₂²×C₈	—	53	2	no	—
81	C₄⋊Q₈	—	53	4	no	—
82	C₄⋊Q₈	—	53	2	no	—
83	C₄⋊Q₈	—	106	2	no	—
84	C₄⋊Q₈	—	106	2	no	—
85	C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
86	C₂×C₄⋊C₄	—	53	4	no	—
87	C₂×C₄⋊C₄	—	106	2	no	—
88	C₂×C₄²	—	53	1	no	—
89	#106	—	53	4	no	—
90	#106	—	53	8	no	—
91	#106	C₂×D₅₃	1	4	yes	—
92	C₄×C₈	—	53	2	no	—
93	C₄×C₈	—	53	1	no	—
94	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
95	C₂×Q₈	—	2	2	no	—
96	C₂×Q₈	—	106	4	no	—
97	C₂×Q₈	—	106	2	no	—
98	C₂×Q₈	—	212	8	no	—
99	C₂²×C₄	—	53	2	no	—
100	C₂²×C₄	—	53	1	no	—
101	C₂²×C₄	—	106	4	no	—
102	C₂×C₈	—	53	2	no	—
103	C₂×C₈	—	53	8	no	—
104	C₂×C₈	C₂²×D₅₃	1	2	yes	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	—	53	4	no	—
107	C₄⋊C₄	—	106	4	no	—
108	C₄⋊C₄	—	106	4	no	—
109	C₄⋊C₄	—	106	2	no	—
110	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
111	C₄²	—	53	2	no	—
112	C₄²	—	53	1	no	—
113	C₄²	C₂²×D₅₃	1	1	yes	characteristic
114	C₂³	—	53	1	no	—
115	Q₈	—	4	4	no	—
116	Q₈	—	212	8	no	—
117	Q₈	—	212	4	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	53	1	no	—
120	C₂×C₄	—	53	2	no	—
121	C₂×C₄	—	53	8	no	—
122	C₂×C₄	—	53	4	no	—
123	C₂×C₄	—	106	4	no	—
124	C₂×C₄	—	212	8	no	—
125	C₂×C₄	D₄×D₅₃	1	2	yes	—
126	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
127	C₈	—	53	4	no	—
128	C₈	Q₈×D₅₃	1	4	yes	complemented
129	C₂²	—	53	4	no	—
130	C₂²	—	53	2	no	—
131	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
132	C₄	—	4	4	no	—
133	C₄	—	53	4	no	—
134	C₄	—	53	2	no	—
135	C₄	—	212	4	no	—
136	C₄	SD₁₆×D₅₃	1	4	yes	—
137	C₄	C₂×Q₈×D₅₃	1	2	yes	—
138	C₂	—	53	4	no	—
139	C₂	C₄⋊Q₈×D₅₃	1	1	yes	characteristic
140	C₂	C₂×SD₁₆×D₅₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊Q₈×D₅₃

C₂

yes

maximal

#16811021

C₂

yes

maximal, characteristic, complemented

#16811020

C₂

yes

maximal, complemented

#16811019

C₂

yes

maximal, complemented

#106×D₅₃

C₂

yes

maximal

C₄×C₈×D₅₃

C₂

yes

maximal, characteristic

C₅₃×#498

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₅₃

—

C₂×C₈×D₅₃

C₂²

yes

—

#13133217

C₂²

yes

—

#13133216

—

#13133216

C₂²

yes

—

#13133215

—

C₄⋊C₄×D₅₃

—

C₄⋊C₄×D₅₃

C₂²

yes

—

C₄²×D₅₃

C₂²

yes

characteristic

#13133149

C₂²

yes

—

#13133148

C₂²

yes

—

C₄×#88322

C₂²

yes

characteristic

C₈×Dic₅₃

C₂²

yes

—

C₄⋊Q₈×C₅₃

C₂²

yes

—

#106×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

Q₈×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

D₄

yes

—

C₂×C₄×D₅₃

C₂³

yes

characteristic

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

#11719919

—

C₄×Dic₅₃

D₄

yes

—

C₄×Dic₅₃

C₂³

yes

characteristic

C₂×#88322

C₂³

yes

—

C₈×D₅₃

Q₈

yes

complemented

C₂²×D₅₃

C₂×D₄

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

C₂×D₄

yes

—

C₄×D₅₃

—

C₄×D₅₃

SD₁₆

yes

—

C₄×D₅₃

C₂×Q₈

yes

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

C₂×Q₈

yes

complemented

#88322

C₂×Q₈

yes

complemented

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂×D₅₃

C₄⋊Q₈

yes

—

C₂×D₅₃

C₂×SD₁₆

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×SD₁₆

yes

—

C₄×C₅₃

C₂²×Q₈

yes

—

Dic₅₃

—

Dic₅₃

C₄⋊Q₈

yes

—

Dic₅₃

C₂×SD₁₆

yes

—

C₂×#498

—

maximal, Sylow

C₂×C₅₃

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₅₃

C₂²×SD₁₆

yes

—

D₅₃

#498

yes

direct product factor

C₂×C₄⋊Q₈

—

—

—

—

D₅₃

yes

characteristic, direct product factor

C₂×#106

—

C₂×C₄×C₈

—

C₅₃

C₂×#498

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

106

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

106

—

C₄⋊Q₈

—

106

—

C₄⋊Q₈

C₂×D₅₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

—

#106

—

#106

—

#106

C₂×D₅₃

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

C₂×Q₈

—

212

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

106

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

C₂²×D₅₃

yes

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

106

—

108

C₄⋊C₄

—

106

—

109

C₄⋊C₄

—

106

—

110

C₄⋊C₄

C₂²×D₅₃

yes

—

111

C₄²

—

112

C₄²

—

113

C₄²

C₂²×D₅₃

yes

characteristic

114

C₂³

—

115

Q₈

—

116

Q₈

—

212

—

117

Q₈

—

212

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

106

—

124

C₂×C₄

—

212

—

125

C₂×C₄

D₄×D₅₃

yes

—

126

C₂×C₄

C₂³×D₅₃

yes

characteristic

127

C₈

—

128

C₈

Q₈×D₅₃

yes

complemented

129

C₂²

—

130

C₂²

—

131

C₂²

C₂×D₄×D₅₃

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

212

—

136

C₄

SD₁₆×D₅₃

yes

—

137

C₄

C₂×Q₈×D₅₃

yes

—

138

C₂

—

139

C₂

C₄⋊Q₈×D₅₃

yes

characteristic

140

C₂

C₂×SD₁₆×D₅₃

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#498	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#498	D₅₃
hypocenter	C₅₃	C₂×#498
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

C₂×D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#498

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
2. #16811021	C₂	semidirect
3. #16811020	C₂	semidirect
4. #16811019	C₂	semidirect
7. C₅₃×#498	C₂	semidirect
42. C₈×D₅₃	Q₈	semidirect
56. C₈×C₅₃	C₂×Q₈	semidirect
57. #88322	C₂×Q₈	semidirect
70. D₅₃	#498	direct (N × H)
75. #498	D₅₃	direct (N × H)
78. C₅₃	C₂×#498	semidirect
128. C₈	Q₈×D₅₃	semidirect

H ≅ G/N

type

2. #16811021

C₂

semidirect

3. #16811020

C₂

semidirect

4. #16811019

C₂

semidirect

7. C₅₃×#498

C₂

semidirect

42. C₈×D₅₃

Q₈

semidirect

56. C₈×C₅₃

C₂×Q₈

semidirect

57. #88322

C₂×Q₈

semidirect

70. D₅₃

#498

direct (N × H)

75. #498

D₅₃

direct (N × H)

78. C₅₃

C₂×#498

semidirect

128. C₈

Q₈×D₅₃

semidirect

Subgroups — #498×D₅₃ #20285211

Conjugacy classes of subgroups

Complements

Subgroups — #498×D53 #20285211

Conjugacy classes of subgroups

Complements

Subgroups — #498×D₅₃ #20285211