Group #20289780 — subgroups

Subgroups — #20289780

Order 6784.

136 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 52 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×#13133287	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133285	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133156	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719936	—	2	1	no	—
5	C₂×C₄⋊#6157	—	2	1	no	—
6	#13133287	—	2	1	no	—
7	#13133287	C₄	1	2	yes	complemented
8	#13133285	—	2	1	no	—
9	#13133285	C₄	1	2	yes	—
10	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
11	#13133156	—	2	1	no	—
12	#13133156	—	2	1	no	—
13	C₂×#11719908	C₄	1	1	yes	characteristic, complemented
14	C₂×C₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
15	C₂²×#6157	—	4	1	no	—
16	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
17	C₂×#88324	—	4	1	no	—
18	C₄⋊#6157	—	2	2	no	—
19	C₄⋊#6157	—	4	1	no	—
20	#11719936	—	2	2	no	—
21	#11719936	—	4	1	no	—
22	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×#88322	Q₈	1	1	yes	characteristic
24	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
25	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
26	C₈×D₅₃	D₄	1	1	yes	characteristic
27	C₈×D₅₃	D₄	1	1	yes	characteristic
28	C₁₆×C₅₃	—	2	1	no	—
29	#11719908	—	2	1	no	—
30	C₂²×D₅₃	SD₁₆	1	1	yes	characteristic
31	C₂×#6157	—	4	2	no	—
32	C₂×#6157	—	8	1	no	—
33	C₂×#6157	—	8	1	no	—
34	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×Dic₅₃	SD₁₆	1	1	yes	characteristic
36	C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₄×D₅₃	D₈	1	1	yes	characteristic
39	C₄×D₅₃	Q₁₆	1	1	yes	characteristic
40	#88324	—	4	2	no	—
41	C₈×C₅₃	C₄²	1	1	yes	characteristic
42	C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
43	#88322	C₄⋊C₄	1	1	yes	characteristic
44	#88322	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₅₃	#106	1	1	yes	characteristic
46	C₂×D₅₃	—	2	1	no	—
47	C₂×D₅₃	—	2	1	no	—
48	C₂×D₅₃	#102	1	1	yes	characteristic
49	C₂×D₅₃	#103	1	1	yes	characteristic
50	#6157	—	8	2	no	—
51	C₄×C₅₃	#95	1	1	yes	characteristic
52	C₄×C₅₃	#107	1	1	yes	characteristic
53	Dic₅₃	#102	1	1	yes	characteristic
54	Dic₅₃	#103	1	1	yes	characteristic
55	#1383	—	53	1	no	maximal, Sylow
56	C₂×C₅₃	#340	1	1	yes	characteristic
57	C₂×C₅₃	#365	1	1	yes	characteristic
58	C₂×C₅₃	#365	1	1	yes	characteristic
59	D₅₃	—	2	1	no	—
60	D₅₃	—	2	1	no	—
61	C₂×M₃₂	—	53	1	no	—
62	C₂×#108	—	53	1	no	—
63	C₂×#106	—	53	1	no	—
64	C₅₃	#1383	1	1	yes	characteristic, complemented, Sylow
65	C₂×M₁₆	—	106	1	no	—
66	C₂²×C₈	—	53	1	no	—
67	C₂×C₄⋊C₄	—	106	1	no	—
68	M₃₂	—	106	1	no	—
69	M₃₂	—	106	1	no	—
70	C₂×C₁₆	—	53	1	no	—
71	C₂×C₁₆	#6157	1	1	yes	characteristic, complemented
72	#108	—	53	2	no	—
73	#108	—	106	1	no	—
74	#106	—	53	2	no	—
75	#106	—	106	1	no	—
76	C₂²×C₄	—	53	1	no	—
77	C₂²×C₄	—	212	1	no	—
78	M₁₆	—	106	2	no	—
79	M₁₆	—	212	1	no	—
80	C₂×C₈	—	53	1	no	—
81	C₂×C₈	—	53	1	no	—
82	C₂×C₈	—	53	1	no	—
83	C₂×C₈	—	53	1	no	—
84	C₂×C₈	—	53	1	no	—
85	C₂×C₈	—	212	1	no	—
86	C₂×C₈	C₂×#6157	1	1	yes	characteristic
87	C₄⋊C₄	—	106	2	no	—
88	C₄⋊C₄	—	212	1	no	—
89	C₁₆	—	2	1	no	—
90	C₁₆	—	106	1	no	—
91	C₂³	—	53	1	no	—
92	C₂×C₄	—	53	1	no	—
93	C₂×C₄	—	53	1	no	—
94	C₂×C₄	—	53	1	no	—
95	C₂×C₄	—	53	1	no	—
96	C₂×C₄	—	53	1	no	—
97	C₂×C₄	—	424	1	no	—
98	C₂×C₄	—	424	1	no	—
99	C₂×C₄	—	212	2	no	—
100	C₂×C₄	C₄⋊#6157	1	1	yes	characteristic
101	C₈	—	53	1	no	—
102	C₈	—	53	1	no	—
103	C₈	—	212	2	no	—
104	C₈	C₄×#6157	1	1	yes	characteristic
105	C₈	C₂²⋊#6157	1	1	yes	characteristic
106	C₂²	—	53	1	no	—
107	C₂²	—	53	1	no	—
108	C₂²	—	106	1	no	—
109	C₂²	—	106	1	no	—
110	C₂²	#13133285	1	1	yes	characteristic
111	C₄	—	53	1	no	—
112	C₄	—	53	1	no	—
113	C₄	—	424	2	no	—
114	C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
115	C₄	#13133286	1	1	yes	characteristic
116	C₂	—	106	1	no	—
117	C₂	—	106	1	no	—
118	C₂	#16811497	1	1	yes	characteristic
119	C₂	#16811547	1	1	yes	characteristic
120	C₂	#16811548	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133287

C₂

yes

maximal, characteristic

C₂×#13133285

C₂

yes

maximal, characteristic

C₂×#13133156

C₂

yes

maximal, characteristic

C₂×#11719936

—

C₂×C₄⋊#6157

—

#13133287

—

#13133287

C₄

yes

complemented

#13133285

—

#13133285

C₄

yes

—

C₂×C₈×D₅₃

C₂²

yes

—

—

C₄

yes

characteristic, complemented

C₂×C₁₆×C₅₃

C₄

yes

characteristic, complemented

C₂²×#6157

—

C₂×C₄×D₅₃

D₄

yes

—

—

—

—

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×#88322

Q₈

yes

characteristic

C₈×D₅₃

C₂×C₄

yes

characteristic

C₈×D₅₃

C₂×C₄

yes

characteristic

C₈×D₅₃

D₄

yes

characteristic

C₈×D₅₃

D₄

yes

characteristic

C₁₆×C₅₃

—

#11719908

—

C₂²×D₅₃

SD₁₆

yes

—

—

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

SD₁₆

yes

characteristic

C₄×D₅₃

C₂²⋊C₄

yes

characteristic

C₄×D₅₃

C₂²⋊C₄

yes

characteristic

C₄×D₅₃

D₈

yes

characteristic

C₄×D₅₃

Q₁₆

yes

characteristic

#88324

—

C₈×C₅₃

C₄²

yes

characteristic

C₈×C₅₃

C₂²⋊C₄

yes

characteristic

#88322

C₄⋊C₄

yes

characteristic

#88322

C₄⋊C₄

yes

characteristic

C₂²×C₅₃

#106

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

yes

C₂×D₅₃

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

yes

Dic₅₃

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

D₅₃

—

C₂×M₃₂

—

C₂×#108

—

C₂×#106

—

C₅₃

#1383

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

106

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

106

—

M₃₂

—

106

—

M₃₂

—

106

—

C₂×C₁₆

—

C₂×C₁₆

#6157

yes

characteristic, complemented

#108

—

#108

—

106

—

#106

—

#106

—

106

—

C₂²×C₄

—

C₂²×C₄

—

212

—

M₁₆

—

106

—

M₁₆

—

212

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

212

—

C₂×C₈

C₂×#6157

yes

characteristic

C₄⋊C₄

—

106

—

C₄⋊C₄

—

212

—

C₁₆

—

C₁₆

—

106

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

424

—

C₂×C₄

—

424

—

C₂×C₄

—

212

—

100

C₂×C₄

C₄⋊#6157

yes

characteristic

101

C₈

—

102

C₈

—

103

C₈

—

212

—

104

C₈

C₄×#6157

yes

characteristic

105

C₈

C₂²⋊#6157

yes

characteristic

106

C₂²

—

107

C₂²

—

108

C₂²

—

106

—

109

C₂²

—

106

—

110

C₂²

#13133285

yes

characteristic

111

C₄

—

112

C₄

—

113

C₄

—

424

—

114

C₄

(C₂×#6157)⋊C₄

yes

characteristic

115

C₄

#13133286

yes

characteristic

116

C₂

—

106

—

117

C₂

—

106

—

118

C₂

#16811497

yes

characteristic

119

C₂

#16811547

yes

characteristic

120

C₂

#16811548

yes

characteristic

	subgroup	G/H
center	C₂²	#13133285
commutator	C₈×C₅₃	C₄²
Frattini	C₂×C₈	C₂×#6157
Fitting	C₂×C₁₆×C₅₃	C₄
socle	C₂²×C₅₃	#106
hypercenter	C₂×C₁₆	#6157
hypocenter	C₅₃	#1383
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

subgroup

G/H

center

C₂²

#13133285

commutator

C₈×C₅₃

C₄²

Frattini

C₂×C₈

C₂×#6157

Fitting

C₂×C₁₆×C₅₃

C₄

C₂²×C₅₃

C₂×C₁₆

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

N	H ≅ G/N	type
7. #13133287	C₄	semidirect
13. C₂×#11719908	C₄	semidirect
14. C₂×C₁₆×C₅₃	C₄	semidirect
64. C₅₃	#1383	semidirect
71. C₂×C₁₆	#6157	semidirect

H ≅ G/N

type

7. #13133287

C₄

semidirect

13. C₂×#11719908

C₄

semidirect

14. C₂×C₁₆×C₅₃

C₄

semidirect

64. C₅₃

#1383

semidirect

71. C₂×C₁₆

#6157

semidirect

Subgroups — #20289780

Conjugacy classes of subgroups

Complements