Group #20285460 — subgroups

Subgroups — #20285460

Order 6784.

358 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 133 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	#16811180	C₂	1	2	yes	maximal
2	#16811021	C₂	1	1	yes	maximal, characteristic, complemented
3	#16811003	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810974	C₂	1	2	yes	maximal, complemented
5	#16810881	C₂	1	4	yes	maximal
6	#16810731	C₂	1	4	yes	maximal, complemented
7	C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₁₂	C₂²	1	4	yes	—
9	#13133217	C₂²	1	2	yes	—
10	#13133217	C₂²	1	2	yes	—
11	#13133209	—	2	2	no	—
12	#13133196	—	2	2	no	—
13	#13133196	C₂²	1	4	yes	—
14	C₄×Dic₁₀₆	—	2	2	no	—
15	C₄×Dic₁₀₆	C₂²	1	4	yes	—
16	#13133176	—	2	2	no	—
17	#13133149	C₂²	1	4	yes	—
18	#13133142	C₂²	1	8	yes	—
19	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
20	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
21	#97×C₅₃	C₂²	1	2	yes	—
22	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	D₄	1	2	yes	—
25	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
26	C₂×Dic₁₀₆	—	2	2	no	—
27	C₂×Dic₁₀₆	C₂³	1	4	yes	—
28	M₁₆×C₅₃	—	2	4	no	—
29	C₂×C₈×C₅₃	C₂³	1	2	yes	—
30	C₂×C₈×C₅₃	C₂³	1	2	yes	—
31	C₄²×C₅₃	D₄	1	2	yes	—
32	C₄²×C₅₃	C₂³	1	1	yes	characteristic
33	C₄²×C₅₃	C₂³	1	1	yes	characteristic
34	C₂²⋊Dic₅₃	—	2	4	no	—
35	C₄⋊Dic₅₃	—	2	2	no	—
36	C₄⋊Dic₅₃	—	2	2	no	—
37	C₄⋊Dic₅₃	—	2	2	no	—
38	C₄⋊Dic₅₃	C₂³	1	4	yes	—
39	#11719919	—	2	4	no	—
40	#11719919	—	2	4	no	—
41	C₄×Dic₅₃	—	2	4	no	—
42	Dic₂₁₂	—	2	8	no	—
43	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	—	2	4	no	—
45	C₂×C₄×C₅₃	—	2	2	no	—
46	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₅₃	—	2	4	no	—
51	C₂×Dic₅₃	—	2	4	no	—
52	Dic₁₀₆	—	2	8	no	—
53	Dic₁₀₆	—	4	4	no	—
54	C₈×C₅₃	—	2	2	no	—
55	C₈×C₅₃	#41	1	4	yes	—
56	C₂²×C₅₃	—	2	1	no	—
57	C₂²×C₅₃	—	2	2	no	—
58	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
59	C₄×C₅₃	—	2	1	no	—
60	C₄×C₅₃	—	2	2	no	—
61	C₄×C₅₃	#137	1	4	yes	—
62	C₄×C₅₃	C₂×#41	1	2	yes	—
63	Dic₅₃	—	4	4	no	—
64	Dic₅₃	—	4	4	no	—
65	#3185	—	53	1	no	maximal, Sylow
66	C₂×C₅₃	—	4	1	no	—
67	C₂×C₅₃	#532	1	1	yes	characteristic
68	C₂×C₅₃	C₂×#137	1	2	yes	—
69	#533	—	53	2	no	—
70	#498	—	53	1	no	—
71	#494	—	53	1	no	—
72	#487	—	53	2	no	—
73	#470	—	53	4	no	—
74	#441	—	53	4	no	—
75	C₄×M₁₆	D₅₃	1	1	yes	characteristic
76	C₅₃	#3185	1	1	yes	characteristic, complemented, Sylow
77	C₂×Q₁₆	—	53	4	no	—
78	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
79	C₄⋊Q₈	—	53	2	no	—
80	C₄⋊Q₈	—	53	2	no	—
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	C₂×D₅₃	1	1	yes	characteristic
88	#106	—	53	4	no	—
89	#103	—	53	8	no	—
90	#97	C₂×D₅₃	1	2	yes	—
91	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
92	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
93	C₂×Q₈	—	53	4	no	—
94	C₂×Q₈	—	106	2	no	—
95	C₂²×C₄	D₂₁₂	1	2	yes	—
96	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
97	Q₁₆	—	106	8	no	—
98	M₁₆	—	2	4	no	—
99	C₂×C₈	C₂²×D₅₃	1	2	yes	—
100	C₂×C₈	C₂²×D₅₃	1	2	yes	—
101	C₄⋊C₄	—	53	4	no	—
102	C₄⋊C₄	—	106	2	no	—
103	C₄⋊C₄	—	106	2	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	—	106	4	no	—
109	C₄²	D₂₁₂	1	2	yes	—
110	C₄²	C₂²×D₅₃	1	1	yes	characteristic
111	C₄²	C₂²×D₅₃	1	1	yes	characteristic
112	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
113	Q₈	—	106	8	no	—
114	Q₈	—	212	4	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	106	4	no	—
118	C₂×C₄	—	106	4	no	—
119	C₂×C₄	C₂×D₂₁₂	1	2	yes	—
120	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
121	C₂×C₄	C₂×D₂₁₂	1	2	yes	—
122	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
123	C₈	—	2	2	no	—
124	C₈	#11719945	1	4	yes	—
125	C₂²	—	2	2	no	—
126	C₂²	—	2	1	no	—
127	C₂²	C₂²×D₂₁₂	1	1	yes	characteristic
128	C₄	—	2	2	no	—
129	C₄	—	2	1	no	—
130	C₄	—	212	4	no	—
131	C₄	—	212	4	no	—
132	C₄	#13133250	1	4	yes	—
133	C₄	C₂×#11719945	1	2	yes	—
134	C₂	—	4	1	no	—
135	C₂	#16811173	1	1	yes	characteristic
136	C₂	C₂×#13133250	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16811180

C₂

yes

maximal

#16811021

C₂

yes

maximal, characteristic, complemented

#16811003

C₂

yes

maximal, characteristic, complemented

#16810974

C₂

yes

maximal, complemented

#16810881

C₂

yes

maximal

#16810731

C₂

yes

maximal, complemented

C₄×M₁₆×C₅₃

C₂

yes

maximal, characteristic

C₂×Dic₂₁₂

C₂²

yes

—

#13133217

C₂²

yes

—

#13133217

C₂²

yes

—

#13133209

—

#13133196

—

#13133196

C₂²

yes

—

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

C₂²

yes

—

#13133176

—

#13133149

C₂²

yes

—

#13133142

C₂²

yes

—

C₂×M₁₆×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#97×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂³

yes

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

—

C₂×C₈×C₅₃

C₂³

yes

—

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

#11719919

—

C₄×Dic₅₃

—

Dic₂₁₂

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

#41

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#137

yes

—

C₄×C₅₃

C₂×#41

yes

—

Dic₅₃

—

Dic₅₃

—

#3185

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#532

yes

characteristic

C₂×C₅₃

C₂×#137

yes

—

—

—

—

—

—

—

C₄×M₁₆

D₅₃

yes

characteristic

C₅₃

#3185

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

106

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

106

—

C₄×Q₈

—

C₄×Q₈

—

106

—

#117

—

106

—

C₂×C₄²

C₂×D₅₃

yes

—

—

C₂×D₅₃

yes

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂²×C₄

D₂₁₂

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

Q₁₆

—

106

—

M₁₆

—

C₂×C₈

C₂²×D₅₃

yes

—

100

C₂×C₈

C₂²×D₅₃

yes

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

106

—

103

C₄⋊C₄

—

106

—

104

C₄⋊C₄

—

106

—

105

C₄⋊C₄

—

106

—

106

C₄⋊C₄

—

106

—

107

C₂²⋊C₄

—

106

—

108

C₄²

—

106

—

109

C₄²

D₂₁₂

yes

—

110

C₄²

C₂²×D₅₃

yes

characteristic

111

C₄²

C₂²×D₅₃

yes

characteristic

112

C₂³

C₂×D₂₁₂

yes

characteristic

113

Q₈

—

106

—

114

Q₈

—

212

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

106

—

118

C₂×C₄

—

106

—

119

C₂×C₄

C₂×D₂₁₂

yes

—

120

C₂×C₄

C₂×D₂₁₂

yes

characteristic

121

C₂×C₄

C₂×D₂₁₂

yes

—

122

C₂×C₄

C₂³×D₅₃

yes

characteristic

123

C₈

—

124

C₈

#11719945

yes

—

125

C₂²

—

126

C₂²

—

127

C₂²

C₂²×D₂₁₂

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

212

—

131

C₄

—

212

—

132

C₄

#13133250

yes

—

133

C₄

C₂×#11719945

yes

—

134

C₂

—

135

C₂

#16811173

yes

characteristic

136

C₂

C₂×#13133250

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₄×M₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	C₄×M₁₆	D₅₃
hypocenter	C₅₃	#3185
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂²

C₂²×D₂₁₂

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₄×M₁₆×C₅₃

C₂

socle

C₂²×C₅₃

C₂²×D₄

hypercenter

C₄×M₁₆

D₅₃

hypocenter

C₅₃

#3185

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
2. #16811021	C₂	semidirect
3. #16811003	C₂	semidirect
4. #16810974	C₂	semidirect
6. #16810731	C₂	semidirect
76. C₅₃	#3185	semidirect

H ≅ G/N

type

2. #16811021

C₂

semidirect

3. #16811003

C₂

semidirect

4. #16810974

C₂

semidirect

6. #16810731

C₂

semidirect

76. C₅₃

#3185

semidirect

Subgroups — #20285460

Conjugacy classes of subgroups

Complements