Group #20290953 — subgroups

Subgroups — #20290953

Order 6784.

220 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 78 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	C₂×#13133291	C₂	1	2	yes	maximal, complemented
2	#16811520	C₂	1	4	yes	maximal, complemented
3	C₂×#13133225	C₂	1	1	yes	maximal, characteristic
4	#13133291	—	2	2	no	—
5	#13133291	C₂²	1	4	yes	—
6	C₂×#11719910	—	2	2	no	—
7	C₂×#11719945	C₄	1	1	yes	characteristic
8	#13133225	—	2	2	no	—
9	#13133225	—	2	2	no	—
10	C₂×#11719912	C₂²	1	2	yes	—
11	C₂²×#88322	C₂²	1	1	yes	characteristic
12	C₂×C₈×D₅₃	C₄	1	2	yes	—
13	C₂×M₁₆×C₅₃	C₄	1	1	yes	characteristic
14	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂×C₂²⋊D₅₃	C₈	1	2	yes	—
16	#11719945	—	2	2	no	—
17	#11719945	—	2	2	no	—
18	C₂×D₂₁₂	C₈	1	1	yes	characteristic
19	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
20	C₂×Dic₁₀₆	C₈	1	1	yes	characteristic
21	M₁₆×C₅₃	—	2	2	no	—
22	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
23	C₂×#88322	C₂×C₄	1	1	yes	characteristic
24	C₂×#88322	D₄	1	2	yes	—
25	C₂×#88322	D₄	1	2	yes	—
26	C₂×#88322	C₂³	1	1	yes	characteristic
27	#11719912	—	2	4	no	—
28	C₈×D₅₃	—	2	2	no	—
29	C₈×D₅₃	—	2	2	no	—
30	#11719910	—	2	4	no	—
31	C₂³×C₅₃	C₂×C₈	1	1	yes	characteristic
32	C₂²×D₅₃	C₂×C₈	1	2	yes	—
33	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
34	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄×C₅₃	M₁₆	1	2	yes	—
36	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₂²⋊D₅₃	—	2	4	no	—
38	C₂×Dic₅₃	C₂×C₈	1	2	yes	—
39	D₂₁₂	—	2	2	no	—
40	C₄×D₅₃	—	2	2	no	—
41	C₄×D₅₃	—	2	2	no	—
42	Dic₁₀₆	—	2	2	no	—
43	C₈×C₅₃	—	2	2	no	—
44	#88322	C₂²⋊C₄	1	1	yes	characteristic
45	#88322	C₂²⋊C₄	1	1	yes	characteristic
46	#88322	C₂×D₄	1	2	yes	—
47	C₂²×C₅₃	—	4	1	no	—
48	C₂²×C₅₃	C₂²⋊C₈	1	2	yes	—
49	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
50	C₂×D₅₃	—	2	2	no	—
51	C₂×D₅₃	—	4	2	no	—
52	C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
53	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
55	Dic₅₃	—	2	2	no	—
56	#2115	—	53	1	no	maximal, Sylow
57	C₂×C₅₃	—	2	1	no	—
58	C₂×C₅₃	—	2	2	no	—
59	C₂×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
60	D₅₃	—	4	2	no	—
61	C₂×#131	—	53	1	no	—
62	C₂×M₃₂	—	53	2	no	—
63	#349	—	53	4	no	—
64	C₅₃	#2115	1	1	yes	characteristic, complemented, Sylow
65	C₂×#41	—	53	1	no	—
66	#131	—	106	2	no	—
67	#131	—	106	2	no	—
68	C₂×M₁₆	—	53	2	no	—
69	C₂×M₁₆	#6157	1	1	yes	characteristic
70	C₂²×C₈	—	53	1	no	—
71	C₂²×C₈	—	53	2	no	—
72	M₃₂	—	53	4	no	—
73	M₃₂	—	106	2	no	—
74	C₂×C₁₆	—	106	2	no	—
75	#41	—	106	2	no	—
76	#41	—	106	2	no	—
77	C₂×Q₈	—	53	1	no	—
78	C₂×D₄	—	53	2	no	—
79	C₂×D₄	—	53	1	no	—
80	C₂²×C₄	—	53	2	no	—
81	C₂²×C₄	C₂×#6157	1	1	yes	characteristic
82	M₁₆	—	2	2	no	—
83	M₁₆	—	106	4	no	—
84	C₂×C₈	—	53	2	no	—
85	C₂×C₈	—	53	2	no	—
86	C₂×C₈	—	53	1	no	—
87	C₂×C₈	—	53	1	no	—
88	C₂×C₈	—	106	2	no	—
89	C₂×C₈	—	106	2	no	—
90	C₂×C₈	C₂×#6157	1	2	yes	—
91	C₁₆	—	106	4	no	—
92	C₂³	—	53	2	no	—
93	C₂³	D₅₃⋊C₈	1	1	yes	characteristic
94	Q₈	—	106	2	no	—
95	D₄	—	106	4	no	—
96	D₄	—	106	2	no	—
97	C₂×C₄	—	53	2	no	—
98	C₂×C₄	—	106	2	no	—
99	C₂×C₄	—	106	2	no	—
100	C₂×C₄	D₅₃⋊C₈	1	1	yes	characteristic
101	C₂×C₄	#11719936	1	2	yes	—
102	C₂×C₄	C₂²⋊#6157	1	2	yes	—
103	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
104	C₈	—	2	2	no	—
105	C₈	—	53	1	no	—
106	C₈	—	53	1	no	—
107	C₈	—	53	2	no	—
108	C₂²	—	4	1	no	—
109	C₂²	—	106	2	no	—
110	C₂²	—	212	2	no	—
111	C₂²	#13133274	1	2	yes	—
112	C₂²	C₂×D₅₃⋊C₈	1	1	yes	characteristic
113	C₄	—	106	2	no	—
114	C₄	#13133274	1	2	yes	—
115	C₄	C₂×C₂²⋊#6157	1	1	yes	characteristic
116	C₄	C₂×#11719936	1	1	yes	characteristic
117	C₂	—	2	1	no	—
118	C₂	—	2	2	no	—
119	C₂	—	212	2	no	—
120	C₂	C₂×#13133274	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133291

C₂

yes

maximal, complemented

#16811520

C₂

yes

maximal, complemented

C₂×#13133225

C₂

yes

maximal, characteristic

#13133291

—

#13133291

C₂²

yes

—

C₂×#11719910

—

C₂×#11719945

C₄

yes

—

—

C₂²

yes

—

C₂²×#88322

C₂²

yes

characteristic

C₂×C₈×D₅₃

C₄

yes

—

C₂×M₁₆×C₅₃

C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂×C₂²⋊D₅₃

C₈

yes

—

#11719945

—

#11719945

—

C₂×D₂₁₂

C₈

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×Dic₁₀₆

C₈

yes

characteristic

M₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×#88322

C₂×C₄

yes

characteristic

C₂×#88322

D₄

yes

—

C₂×#88322

D₄

yes

—

C₂×#88322

C₂³

yes

characteristic

#11719912

—

C₈×D₅₃

—

C₈×D₅₃

—

#11719910

—

C₂³×C₅₃

C₂×C₈

yes

characteristic

C₂²×D₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

M₁₆

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²⋊D₅₃

—

C₂×Dic₅₃

C₂×C₈

yes

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

#88322

C₂²⋊C₄

yes

characteristic

#88322

C₂²⋊C₄

yes

characteristic

#88322

C₂×D₄

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²⋊C₈

yes

—

C₂²×C₅₃

C₂²×C₈

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

C₂²⋊C₈

yes

—

C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

C₂×M₁₆

yes

characteristic

Dic₅₃

—

#2115

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₂×C₂²⋊C₈

yes

characteristic

D₅₃

—

C₂×#131

—

C₂×M₃₂

—

#349

—

C₅₃

#2115

yes

characteristic, complemented, Sylow

C₂×#41

—

#131

—

106

—

#131

—

106

—

C₂×M₁₆

—

C₂×M₁₆

#6157

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

M₃₂

—

M₃₂

—

106

—

C₂×C₁₆

—

106

—

#41

—

106

—

#41

—

106

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#6157

yes

characteristic

M₁₆

—

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

C₂×#6157

yes

—

C₁₆

—

106

—

C₂³

—

C₂³

D₅₃⋊C₈

yes

characteristic

Q₈

—

106

—

D₄

—

106

—

D₄

—

106

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

100

C₂×C₄

D₅₃⋊C₈

yes

characteristic

101

C₂×C₄

#11719936

yes

—

102

C₂×C₄

C₂²⋊#6157

yes

—

103

C₂×C₄

C₂²×#6157

yes

characteristic

104

C₈

—

105

C₈

—

106

C₈

—

107

C₈

—

108

C₂²

—

109

C₂²

—

106

—

110

C₂²

—

212

—

111

C₂²

#13133274

yes

—

112

C₂²

C₂×D₅₃⋊C₈

yes

characteristic

113

C₄

—

106

—

114

C₄

#13133274

yes

—

115

C₄

C₂×C₂²⋊#6157

yes

characteristic

116

C₄

C₂×#11719936

yes

characteristic

117

C₂

—

118

C₂

—

119

C₂

—

212

—

120

C₂

C₂×#13133274

yes

characteristic

	subgroup	G/H
center	C₄	C₂×C₂²⋊#6157
commutator	C₂²×C₅₃	C₂²×C₈
Frattini	C₂×C₄	C₂²×#6157
Fitting	C₂×M₁₆×C₅₃	C₄
socle	C₂×C₅₃	C₂×C₂²⋊C₈
hypercenter	C₂×M₁₆	#6157
hypocenter	C₅₃	#2115
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

subgroup

G/H

center

C₄

C₂×C₂²⋊#6157

commutator

C₂²×C₅₃

C₂²×C₈

Frattini

C₂×C₄

C₂²×#6157

Fitting

C₂×M₁₆×C₅₃

C₄

socle

C₂×C₅₃

C₂×C₂²⋊C₈

C₂×M₁₆

C₅₃

G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
1. C₂×#13133291	C₂	semidirect
2. #16811520	C₂	semidirect
64. C₅₃	#2115	semidirect

H ≅ G/N

type

1. C₂×#13133291

C₂

semidirect

2. #16811520

C₂

semidirect

64. C₅₃

#2115

semidirect

Subgroups — #20290953

Conjugacy classes of subgroups

Complements