Group #20286060 — subgroups

Subgroups — #20286060

Order 6784.

348 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 137 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	#16810919	C₂	1	4	yes	maximal, complemented
3	#16810864	C₂	1	4	yes	maximal
4	#16810834	C₂	1	4	yes	maximal, complemented
5	C₂×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₁₂	C₂²	1	4	yes	—
7	#13133217	C₂²	1	2	yes	—
8	#13133217	C₂²	1	2	yes	—
9	#13133209	—	2	2	no	—
10	#13133196	—	2	2	no	—
11	#13133196	C₂²	1	4	yes	—
12	C₄×Dic₁₀₆	—	2	2	no	—
13	C₄×Dic₁₀₆	C₂²	1	4	yes	—
14	#13133176	—	2	2	no	—
15	#13133151	C₂²	1	4	yes	—
16	#13133142	C₂²	1	8	yes	—
17	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₅₃	C₂²	1	4	yes	—
20	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
23	C₂×Dic₁₀₆	—	2	2	no	—
24	C₂×Dic₁₀₆	C₂³	1	4	yes	—
25	C₂×C₈×C₅₃	—	4	2	no	—
26	C₂×C₈×C₅₃	C₂³	1	4	yes	—
27	C₄²×C₅₃	D₄	1	2	yes	—
28	C₄²×C₅₃	C₂³	1	2	yes	—
29	C₂²⋊Dic₅₃	—	2	4	no	—
30	C₄⋊Dic₅₃	—	2	2	no	—
31	C₄⋊Dic₅₃	—	2	2	no	—
32	C₄⋊Dic₅₃	—	2	2	no	—
33	C₄⋊Dic₅₃	C₂³	1	4	yes	—
34	#11719919	—	2	4	no	—
35	#11719919	—	2	4	no	—
36	C₄×Dic₅₃	—	2	4	no	—
37	Dic₂₁₂	—	4	4	no	—
38	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₅₃	—	2	1	no	—
40	C₂×C₄×C₅₃	—	2	1	no	—
41	C₂×C₄×C₅₃	—	4	1	no	—
42	C₂×C₄×C₅₃	Q₁₆	1	4	yes	complemented
43	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
47	C₂×Dic₅₃	—	2	4	no	—
48	C₂×Dic₅₃	—	2	4	no	—
49	Dic₁₀₆	—	2	4	no	—
50	Dic₁₀₆	—	2	4	no	—
51	Dic₁₀₆	—	4	4	no	—
52	C₈×C₅₃	—	2	4	no	—
53	C₂²×C₅₃	—	2	1	no	—
54	C₂²×C₅₃	—	2	1	no	—
55	C₂²×C₅₃	C₂×Q₁₆	1	2	yes	—
56	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
57	C₄×C₅₃	—	2	1	no	—
58	C₄×C₅₃	—	2	2	no	—
59	C₄×C₅₃	C₂×Q₁₆	1	4	yes	—
60	C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
61	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
62	Dic₅₃	—	4	4	no	—
63	Dic₅₃	—	4	4	no	—
64	#3249	—	53	1	no	maximal, Sylow
65	C₂×C₅₃	—	2	2	no	—
66	C₂×C₅₃	#537	1	1	yes	characteristic
67	C₂×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
68	C₂×C₅₃	#575	1	1	yes	characteristic
69	#533	—	53	2	no	—
70	#477	—	53	4	no	—
71	#467	—	53	4	no	—
72	#462	—	53	4	no	—
73	C₂×C₄⋊C₈	D₅₃	1	1	yes	characteristic
74	C₅₃	#3249	1	1	yes	characteristic, complemented, Sylow
75	C₂×Q₁₆	—	53	4	no	—
76	C₂²×C₈	C₂×D₅₃	1	2	yes	—
77	C₄⋊Q₈	—	53	2	no	—
78	C₄⋊Q₈	—	53	2	no	—
79	#125	—	106	2	no	—
80	C₂²⋊Q₈	—	53	4	no	—
81	C₂²⋊Q₈	—	106	2	no	—
82	C₄×Q₈	—	53	4	no	—
83	C₄×Q₈	—	106	2	no	—
84	#117	—	106	2	no	—
85	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
86	#107	—	53	4	no	—
87	C₄⋊C₈	C₂×D₅₃	1	4	yes	—
88	#103	—	53	8	no	—
89	C₂×Q₈	—	53	4	no	—
90	C₂×Q₈	—	106	2	no	—
91	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
92	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
93	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
94	Q₁₆	—	212	4	no	—
95	C₂×C₈	—	4	2	no	—
96	C₂×C₈	C₂²×D₅₃	1	4	yes	—
97	C₄⋊C₄	—	53	4	no	—
98	C₄⋊C₄	—	106	2	no	—
99	C₄⋊C₄	—	106	2	no	—
100	C₄⋊C₄	—	106	4	no	—
101	C₄⋊C₄	—	106	4	no	—
102	C₄⋊C₄	—	106	2	no	—
103	C₂²⋊C₄	—	106	4	no	—
104	C₄²	—	106	4	no	—
105	C₄²	D₂₁₂	1	2	yes	—
106	C₄²	C₂²×D₅₃	1	2	yes	—
107	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
108	Q₈	—	106	4	no	—
109	Q₈	—	106	4	no	—
110	Q₈	—	212	4	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	4	1	no	—
114	C₂×C₄	—	106	4	no	—
115	C₂×C₄	—	106	4	no	—
116	C₂×C₄	Dic₂₁₂	1	4	yes	complemented
117	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
118	C₂×C₄	C₂×D₂₁₂	1	2	yes	—
119	C₂×C₄	C₂×D₂₁₂	1	2	yes	—
120	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
121	C₈	—	2	4	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	C₂×Dic₂₁₂	1	2	yes	—
125	C₂²	C₂²×D₂₁₂	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	2	1	no	—
128	C₄	—	212	4	no	—
129	C₄	—	212	4	no	—
130	C₄	C₂×Dic₂₁₂	1	4	yes	—
131	C₄	#13133262	1	1	yes	characteristic
132	C₄	#13133265	1	1	yes	characteristic
133	C₂	—	2	2	no	—
134	C₂	#16811204	1	1	yes	characteristic
135	C₂	C₂²×Dic₂₁₂	1	1	yes	characteristic
136	C₂	#16811423	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811180

C₂

yes

maximal

#16810919

C₂

yes

maximal, complemented

#16810864

C₂

yes

maximal

#16810834

C₂

yes

maximal, complemented

C₂×C₄⋊C₈×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

—

#13133151

C₂²

yes

—

#13133142

C₂²

yes

—

C₂²×C₈×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂³

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

—

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

—

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×C₄×C₅₃

Q₁₆

yes

complemented

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

—

Dic₁₀₆

—

C₈×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×Q₁₆

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×Q₁₆

yes

—

C₄×C₅₃

H(5,𝔽₂)

yes

characteristic

C₄×C₅₃

2₋¹⁺⁴

yes

characteristic

Dic₅₃

—

Dic₅₃

—

#3249

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#537

yes

characteristic

C₂×C₅₃

C₂²×Q₁₆

yes

C₂×C₅₃

yes

—

—

—

—

C₂×C₄⋊C₈

D₅₃

yes

characteristic

C₅₃

#3249

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂²×C₈

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

characteristic

#107

—

C₄⋊C₈

C₂×D₅₃

yes

—

#103

—

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂²×C₄

D₂₁₂

yes

characteristic

C₂²×C₄

D₂₁₂

yes

characteristic

C₂²×C₄

C₂²×D₅₃

yes

characteristic

Q₁₆

—

212

—

C₂×C₈

—

C₂×C₈

C₂²×D₅₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

100

C₄⋊C₄

—

106

—

101

C₄⋊C₄

—

106

—

102

C₄⋊C₄

—

106

—

103

C₂²⋊C₄

—

106

—

104

C₄²

—

106

—

105

C₄²

D₂₁₂

yes

—

106

C₄²

C₂²×D₅₃

yes

—

107

C₂³

C₂×D₂₁₂

yes

characteristic

108

Q₈

—

106

—

109

Q₈

—

106

—

110

Q₈

—

212

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

106

—

115

C₂×C₄

—

106

—

116

C₂×C₄

Dic₂₁₂

yes

complemented

117

C₂×C₄

C₂×D₂₁₂

yes

characteristic

118

C₂×C₄

C₂×D₂₁₂

yes

—

119

C₂×C₄

C₂×D₂₁₂

yes

—

120

C₂×C₄

C₂³×D₅₃

yes

characteristic

121

C₈

—

122

C₂²

—

123

C₂²

—

124

C₂²

C₂×Dic₂₁₂

yes

—

125

C₂²

C₂²×D₂₁₂

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

212

—

129

C₄

—

212

—

130

C₄

C₂×Dic₂₁₂

yes

—

131

C₄

#13133262

yes

characteristic

132

C₄

#13133265

yes

characteristic

133

C₂

—

134

C₂

#16811204

yes

characteristic

135

C₂

C₂²×Dic₂₁₂

yes

characteristic

136

C₂

#16811423

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂×C₄⋊C₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	C₂×C₄⋊C₈	D₅₃
hypocenter	C₅₃	#3249
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂²

C₂²×D₂₁₂

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₂×C₄⋊C₈×C₅₃

C₂

socle

C₂²×C₅₃

C₂²×D₄

hypercenter

C₂×C₄⋊C₈

D₅₃

hypocenter

C₅₃

#3249

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
2. #16810919	C₂	semidirect
4. #16810834	C₂	semidirect
42. C₂×C₄×C₅₃	Q₁₆	semidirect
74. C₅₃	#3249	semidirect
116. C₂×C₄	Dic₂₁₂	semidirect

H ≅ G/N

type

2. #16810919

C₂

semidirect

4. #16810834

C₂

semidirect

42. C₂×C₄×C₅₃

Q₁₆

semidirect

74. C₅₃

#3249

semidirect

116. C₂×C₄

Dic₂₁₂

semidirect

Subgroups — #20286060

Conjugacy classes of subgroups

Complements