Group #20276981 — subgroups

Subgroups — #20276981

Order 6784.

198 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 61 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×#13133162	C₂	1	2	yes	maximal, complemented
2	C₂×D₈₄₈	C₂	1	2	yes	maximal, complemented
3	#16811007	C₂	1	1	yes	maximal, characteristic, complemented
4	#16811000	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₂₁₂	C₂²	1	1	yes	characteristic
7	C₂×#11719913	—	2	1	no	—
8	C₂×D₄₂₄	—	2	1	no	—
9	C₂×D₄₂₄	C₂²	1	1	yes	characteristic
10	C₂×D₄₂₄	C₂²	1	2	yes	—
11	#13133214	—	2	1	no	—
12	#13133205	—	2	1	no	—
13	#13133162	—	2	4	no	—
14	D₈₄₈	—	2	4	no	—
15	C₂×C₁₆×C₅₃	C₂²	1	2	yes	—
16	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×D₂₁₂	—	2	1	no	—
18	C₂×D₂₁₂	—	2	2	no	—
19	C₂×D₂₁₂	—	4	2	no	—
20	C₂×Dic₁₀₆	—	2	1	no	—
21	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
22	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
23	C₄²×C₅₃	D₄	1	1	yes	characteristic
24	#11719921	—	4	2	no	—
25	Dic₂₁₂	—	2	2	no	—
26	D₄₂₄	—	2	2	no	—
27	D₄₂₄	—	2	2	no	—
28	D₄₂₄	—	2	2	no	—
29	D₄₂₄	—	4	2	no	—
30	#11719913	—	4	2	no	—
31	C₁₆×C₅₃	D₄	1	4	yes	complemented
32	C₂²×D₅₃	—	4	2	no	—
33	C₂²×D₅₃	—	4	1	no	—
34	C₂×C₄×C₅₃	D₈	1	2	yes	—
35	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₂×Dic₅₃	—	4	1	no	—
37	D₂₁₂	—	4	1	no	—
38	D₂₁₂	—	4	2	no	—
39	D₂₁₂	—	4	1	no	—
40	D₂₁₂	—	4	2	no	—
41	D₂₁₂	—	8	4	no	—
42	Dic₁₀₆	—	4	1	no	—
43	Dic₁₀₆	—	4	1	no	—
44	C₈×C₅₃	D₈	1	2	yes	—
45	C₈×C₅₃	C₂×D₄	1	2	yes	—
46	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
47	C₂×D₅₃	—	8	1	no	—
48	C₂×D₅₃	—	8	2	no	—
49	C₂×D₅₃	—	8	4	no	—
50	C₂×D₅₃	—	8	2	no	—
51	C₄×C₅₃	—	2	2	no	—
52	C₄×C₅₃	#127	1	1	yes	characteristic
53	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
54	Dic₅₃	—	8	1	no	—
55	#2250	—	53	1	no	maximal, Sylow
56	C₂×C₅₃	#493	1	1	yes	characteristic
57	C₂×C₅₃	#509	1	2	yes	—
58	D₅₃	—	16	2	no	—
59	D₅₃	—	16	1	no	—
60	C₂×SD₃₂	—	53	2	no	—
61	C₂×D₁₆	—	53	2	no	—
62	#495	—	53	1	no	—
63	#493	—	53	1	no	—
64	#346	D₅₃	1	1	yes	characteristic, complemented
65	C₅₃	#2250	1	1	yes	characteristic, complemented, Sylow
66	C₂×Q₁₆	—	53	1	no	—
67	C₂×SD₁₆	—	106	1	no	—
68	C₂×D₈	—	53	2	no	—
69	C₂×D₈	—	53	1	no	—
70	C₂×D₈	—	106	1	no	—
71	#127	—	106	1	no	—
72	#124	—	106	1	no	—
73	SD₃₂	—	106	4	no	—
74	D₁₆	—	106	4	no	—
75	C₂×C₁₆	C₂×D₅₃	1	2	yes	—
76	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
77	C₂×Q₈	—	106	1	no	—
78	C₂×D₄	—	106	1	no	—
79	C₂×D₄	—	106	2	no	—
80	C₂×D₄	—	212	2	no	—
81	Q₁₆	—	106	2	no	—
82	SD₁₆	—	212	2	no	—
83	D₈	—	106	2	no	—
84	D₈	—	106	2	no	—
85	D₈	—	106	2	no	—
86	D₈	—	212	2	no	—
87	C₂×C₈	D₂₁₂	1	1	yes	characteristic
88	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
89	C₂²⋊C₄	—	212	2	no	—
90	C₄²	D₂₁₂	1	1	yes	characteristic
91	C₁₆	D₂₁₂	1	4	yes	complemented
92	C₂³	—	212	1	no	—
93	C₂³	—	212	2	no	—
94	Q₈	—	212	1	no	—
95	Q₈	—	212	1	no	—
96	D₄	—	424	4	no	—
97	D₄	—	212	1	no	—
98	D₄	—	212	2	no	—
99	D₄	—	212	2	no	—
100	D₄	—	212	1	no	—
101	C₂×C₄	—	212	1	no	—
102	C₂×C₄	D₄₂₄	1	2	yes	—
103	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
104	C₈	D₄₂₄	1	2	yes	—
105	C₈	C₂×D₂₁₂	1	2	yes	—
106	C₂²	—	424	4	no	—
107	C₂²	—	424	2	no	—
108	C₂²	—	424	2	no	—
109	C₂²	—	424	1	no	—
110	C₂²	C₂×D₄₂₄	1	1	yes	characteristic
111	C₄	—	2	2	no	—
112	C₄	—	424	1	no	—
113	C₄	#13133214	1	1	yes	characteristic
114	C₄	C₂×D₄₂₄	1	1	yes	characteristic
115	C₂	—	848	2	no	—
116	C₂	—	848	1	no	—
117	C₂	#16811000	1	1	yes	characteristic
118	C₂	#16811061	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133162

C₂

yes

maximal, complemented

C₂×D₈₄₈

C₂

yes

maximal, complemented

#16811007

C₂

yes

maximal, characteristic, complemented

#16811000

C₂

yes

maximal, characteristic, complemented

C₅₃×#346

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₁₂

C₂²

yes

characteristic

C₂×#11719913

—

C₂×D₄₂₄

—

C₂×D₄₂₄

C₂²

yes

characteristic

C₂×D₄₂₄

C₂²

yes

—

#13133214

—

#13133205

—

#13133162

—

D₈₄₈

—

C₂×C₁₆×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

#11719921

—

Dic₂₁₂

—

D₄₂₄

—

D₄₂₄

—

D₄₂₄

—

D₄₂₄

—

#11719913

—

C₁₆×C₅₃

D₄

yes

complemented

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

D₈

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

D₈

yes

—

C₈×C₅₃

C₂×D₄

yes

—

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#127

yes

characteristic

C₄×C₅₃

C₂×D₈

yes

characteristic

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

—

D₅₃

—

C₂×SD₃₂

—

C₂×D₁₆

—

#495

—

#493

—

#346

D₅₃

yes

characteristic, complemented

C₅₃

#2250

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×SD₁₆

—

106

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

106

—

#127

—

106

—

#124

—

106

—

SD₃₂

—

106

—

D₁₆

—

106

—

C₂×C₁₆

C₂×D₅₃

yes

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×Q₈

—

106

—

C₂×D₄

—

106

—

C₂×D₄

—

106

—

C₂×D₄

—

212

—

Q₁₆

—

106

—

SD₁₆

—

212

—

D₈

—

106

—

D₈

—

106

—

D₈

—

106

—

D₈

—

212

—

C₂×C₈

D₂₁₂

yes

characteristic

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₂²⋊C₄

—

212

—

C₄²

D₂₁₂

yes

characteristic

C₁₆

D₂₁₂

yes

complemented

C₂³

—

212

—

C₂³

—

212

—

Q₈

—

212

—

Q₈

—

212

—

D₄

—

424

—

D₄

—

212

—

D₄

—

212

—

D₄

—

212

—

100

D₄

—

212

—

101

C₂×C₄

—

212

—

102

C₂×C₄

D₄₂₄

yes

—

103

C₂×C₄

C₂×D₂₁₂

yes

characteristic

104

C₈

D₄₂₄

yes

—

105

C₈

C₂×D₂₁₂

yes

—

106

C₂²

—

424

—

107

C₂²

—

424

—

108

C₂²

—

424

—

109

C₂²

—

424

—

110

C₂²

C₂×D₄₂₄

yes

characteristic

111

C₄

—

112

C₄

—

424

—

113

C₄

#13133214

yes

characteristic

114

C₄

C₂×D₄₂₄

yes

characteristic

115

C₂

—

848

—

116

C₂

—

848

—

117

C₂

#16811000

yes

characteristic

118

C₂

#16811061

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄₂₄
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#346	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	#346	D₅₃
hypocenter	C₅₃	#2250
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

subgroup

G/H

center

C₂²

C₂×D₄₂₄

commutator

C₂×C₈×C₅₃

C₂³

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₅₃×#346

C₂

socle

C₂²×C₅₃

C₂×D₈

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

N	H ≅ G/N	type
1. C₂×#13133162	C₂	semidirect
2. C₂×D₈₄₈	C₂	semidirect
3. #16811007	C₂	semidirect
4. #16811000	C₂	semidirect
5. C₅₃×#346	C₂	semidirect
31. C₁₆×C₅₃	D₄	semidirect
64. #346	D₅₃	semidirect
65. C₅₃	#2250	semidirect
91. C₁₆	D₂₁₂	semidirect

H ≅ G/N

type

1. C₂×#13133162

C₂

semidirect

2. C₂×D₈₄₈

C₂

semidirect

3. #16811007

C₂

semidirect

4. #16811000

C₂

semidirect

5. C₅₃×#346

C₂

semidirect

31. C₁₆×C₅₃

D₄

semidirect

64. #346

D₅₃

semidirect

65. C₅₃

#2250

semidirect

91. C₁₆

D₂₁₂

semidirect

Subgroups — #20276981

Conjugacy classes of subgroups

Complements