Group #20285458 — subgroups

Subgroups — #20285458

Order 6784.

358 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 125 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	#16811178	C₂	1	2	yes	maximal
2	#16811021	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810973	C₂	1	2	yes	maximal, complemented
4	#16810878	C₂	1	4	yes	maximal
5	#16810730	C₂	1	4	yes	maximal, complemented
6	C₄×#11719912	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11719912	C₂²	1	2	yes	—
9	#13133217	C₂²	1	2	yes	—
10	#13133208	—	2	2	no	—
11	#13133197	—	2	2	no	—
12	#13133193	C₂²	1	4	yes	—
13	Q₈×Dic₅₃	C₂²	1	4	yes	—
14	#13133182	—	2	2	no	—
15	#13133175	—	2	2	no	—
16	C₄²×D₅₃	C₂²	1	1	yes	characteristic
17	#13133149	C₂²	1	4	yes	—
18	#13133141	C₂²	1	8	yes	—
19	#13133127	C₂²	1	2	yes	—
20	C₄×#88322	C₂²	1	1	yes	characteristic
21	C₂×Q₁₆×C₅₃	C₂²	1	4	yes	—
22	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
23	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×Q₈×C₅₃	C₂³	1	4	yes	—
25	C₂×C₄×D₅₃	D₄	1	2	yes	—
26	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
27	C₂×Dic₁₀₆	—	2	2	no	—
28	Q₁₆×C₅₃	—	2	8	no	—
29	C₂×C₈×C₅₃	C₂³	1	2	yes	—
30	C₄⋊C₄×C₅₃	—	2	2	no	—
31	C₄⋊C₄×C₅₃	—	2	2	no	—
32	C₄²×C₅₃	C₂³	1	1	yes	characteristic
33	#11719921	—	2	4	no	—
34	C₄⋊Dic₅₃	—	2	2	no	—
35	C₄⋊Dic₅₃	—	2	4	no	—
36	C₄⋊Dic₅₃	C₂³	1	4	yes	—
37	#11719919	—	2	4	no	—
38	C₄×Dic₅₃	—	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	#11719912	—	2	4	no	—
43	C₂²×D₅₃	C₂×D₄	1	1	yes	characteristic
44	Q₈×C₅₃	—	2	8	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	2	yes	—
50	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
51	C₄×D₅₃	—	2	4	no	—
52	C₄×D₅₃	—	2	2	no	—
53	Dic₁₀₆	—	4	4	no	—
54	C₈×C₅₃	#41	1	4	yes	—
55	#88322	—	2	2	no	—
56	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
57	C₂×D₅₃	—	2	1	no	—
58	C₂×D₅₃	—	2	2	no	—
59	C₄×C₅₃	—	4	4	no	—
60	C₄×C₅₃	#137	1	4	yes	—
61	C₄×C₅₃	C₂×#41	1	2	yes	—
62	Dic₅₃	—	2	1	no	—
63	Dic₅₃	—	2	2	no	—
64	Dic₅₃	—	4	4	no	—
65	#3185	—	53	1	no	maximal, Sylow
66	C₂×C₅₃	#532	1	1	yes	characteristic
67	C₂×C₅₃	C₂×#137	1	2	yes	—
68	D₅₃	—	4	1	no	—
69	#533	—	53	2	no	—
70	#498	—	53	1	no	—
71	#494	D₅₃	1	1	yes	characteristic, complemented
72	#487	—	53	2	no	—
73	#470	—	53	4	no	—
74	#441	—	53	4	no	—
75	C₄×M₁₆	—	53	1	no	—
76	C₅₃	#3185	1	1	yes	characteristic, complemented, Sylow
77	C₂×Q₁₆	C₂×D₅₃	1	4	yes	—
78	C₂×M₁₆	—	53	2	no	—
79	C₄⋊Q₈	—	53	2	no	—
80	C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
81	#125	—	106	2	no	—
82	C₂²⋊Q₈	—	53	4	no	—
83	C₂²⋊Q₈	—	106	2	no	—
84	C₄×Q₈	—	53	4	no	—
85	C₄×Q₈	—	106	2	no	—
86	#117	—	106	2	no	—
87	C₂×C₄²	—	53	1	no	—
88	#106	—	53	4	no	—
89	#103	—	53	8	no	—
90	#97	—	53	2	no	—
91	C₄×C₈	—	53	1	no	—
92	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
93	C₂×Q₈	—	106	2	no	—
94	C₂×Q₈	C₂²×D₅₃	1	4	yes	—
95	C₂²×C₄	—	53	2	no	—
96	C₂²×C₄	—	53	1	no	—
97	Q₁₆	—	2	8	no	—
98	M₁₆	—	106	4	no	—
99	C₂×C₈	—	53	2	no	—
100	C₂×C₈	C₂²×D₅₃	1	2	yes	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	—	53	4	no	—
104	C₄⋊C₄	—	106	4	no	—
105	C₄⋊C₄	—	106	4	no	—
106	C₄⋊C₄	—	106	2	no	—
107	C₂²⋊C₄	—	106	4	no	—
108	C₄²	—	53	2	no	—
109	C₄²	—	53	1	no	—
110	C₄²	—	106	4	no	—
111	C₄²	C₂²×D₅₃	1	1	yes	characteristic
112	C₂³	—	53	1	no	—
113	Q₈	—	2	8	no	—
114	Q₈	—	212	4	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	53	2	no	—
117	C₂×C₄	—	53	1	no	—
118	C₂×C₄	—	106	4	no	—
119	C₂×C₄	—	106	2	no	—
120	C₂×C₄	—	106	4	no	—
121	C₂×C₄	D₄×D₅₃	1	2	yes	—
122	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
123	C₈	—	106	2	no	—
124	C₈	#11719947	1	4	yes	—
125	C₂²	—	106	2	no	—
126	C₂²	—	106	1	no	—
127	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
128	C₄	—	4	4	no	—
129	C₄	—	106	2	no	—
130	C₄	—	106	1	no	—
131	C₄	—	212	4	no	—
132	C₄	#13133249	1	4	yes	—
133	C₄	C₂×#11719947	1	2	yes	—
134	C₂	—	212	1	no	—
135	C₂	#16811171	1	1	yes	characteristic
136	C₂	C₂×#13133249	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16811178

C₂

yes

maximal

#16811021

C₂

yes

maximal, characteristic, complemented

#16810973

C₂

yes

maximal, complemented

#16810878

C₂

yes

maximal

#16810730

C₂

yes

maximal, complemented

C₄×#11719912

C₂

yes

maximal, characteristic

C₅₃×#494

C₂

yes

maximal, characteristic, complemented

C₂×#11719912

C₂²

yes

—

#13133217

C₂²

yes

—

#13133208

—

#13133197

—

#13133193

C₂²

yes

—

Q₈×Dic₅₃

C₂²

yes

—

#13133182

—

#13133175

—

C₄²×D₅₃

C₂²

yes

characteristic

#13133149

C₂²

yes

—

#13133141

C₂²

yes

—

#13133127

C₂²

yes

—

C₄×#88322

C₂²

yes

characteristic

C₂×Q₁₆×C₅₃

C₂²

yes

—

C₄⋊Q₈×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

C₂³

yes

—

C₂×C₄×D₅₃

D₄

yes

—

C₂×C₄×D₅₃

C₂³

yes

characteristic

C₂×Dic₁₀₆

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

#11719921

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

D₄

yes

—

C₄×Dic₅₃

C₂³

yes

characteristic

C₂×#88322

C₂³

yes

—

#11719912

—

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

—

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

C₈×C₅₃

#41

yes

—

#88322

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#137

yes

—

C₄×C₅₃

C₂×#41

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

—

#533

—

#498

—

#494

D₅₃

yes

characteristic, complemented

#487

—

#470

—

#441

—

C₄×M₁₆

—

C₅₃

#3185

yes

characteristic, complemented, Sylow

C₂×Q₁₆

C₂×D₅₃

yes

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅₃

yes

—

#125

—

106

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

106

—

C₄×Q₈

—

C₄×Q₈

—

106

—

#117

—

106

—

C₂×C₄²

—

#106

—

#103

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×Q₈

—

106

—

C₂×Q₈

C₂²×D₅₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

Q₁₆

—

M₁₆

—

106

—

C₂×C₈

—

100

C₂×C₈

C₂²×D₅₃

yes

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

106

—

105

C₄⋊C₄

—

106

—

106

C₄⋊C₄

—

106

—

107

C₂²⋊C₄

—

106

—

108

C₄²

—

109

C₄²

—

110

C₄²

—

106

—

111

C₄²

C₂²×D₅₃

yes

characteristic

112

C₂³

—

113

Q₈

—

114

Q₈

—

212

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

106

—

119

C₂×C₄

—

106

—

120

C₂×C₄

—

106

—

121

C₂×C₄

D₄×D₅₃

yes

—

122

C₂×C₄

C₂³×D₅₃

yes

characteristic

123

C₈

—

106

—

124

C₈

#11719947

yes

—

125

C₂²

—

106

—

126

C₂²

—

106

—

127

C₂²

C₂×D₄×D₅₃

yes

characteristic

128

C₄

—

129

C₄

—

106

—

130

C₄

—

106

—

131

C₄

—

212

—

132

C₄

#13133249

yes

—

133

C₄

C₂×#11719947

yes

—

134

C₂

—

212

—

135

C₂

#16811171

yes

characteristic

136

C₂

C₂×#13133249

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#494	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#494	D₅₃
hypocenter	C₅₃	#3185
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

C₂×D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#494

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
2. #16811021	C₂	semidirect
3. #16810973	C₂	semidirect
5. #16810730	C₂	semidirect
7. C₅₃×#494	C₂	semidirect
71. #494	D₅₃	semidirect
76. C₅₃	#3185	semidirect

H ≅ G/N

type

2. #16811021

C₂

semidirect

3. #16810973

C₂

semidirect

5. #16810730

C₂

semidirect

7. C₅₃×#494

C₂

semidirect

71. #494

D₅₃

semidirect

76. C₅₃

#3185

semidirect

Subgroups — #20285458

Conjugacy classes of subgroups

Complements